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https://www.princeton.edu/~achaney/tmve/wiki100k/docs/Exosphere.html	# Exosphere related topics {math, energy, light} {acid, form, water} The exosphere is the uppermost layer of the atmosphere. In the exosphere, an upward travelling molecule moving fast enough to attain escape velocity can escape to space with a low chance of collisions; if it is moving below escape velocity it will be prevented from escaping from the celestial body by gravity. In either case, such a molecule is unlikely to collide with another molecule due to the exosphere's low density. ## Contents ### Earth's exosphere The main gases within the Earth's exosphere are the lightest gases, mainly hydrogen, with some helium, carbon dioxide, and atomic oxygen near the exobase. The exosphere is the last layer before outer space. Since there is no clear boundary between outer space and the exosphere, the exosphere is sometimes considered a part of outer space. ### Lower boundary The altitude of its lower boundary, known as the thermopause and exobase, ranges from about 250 to 500 kilometres (160 to 310 mi) depending on solar activity.[citation needed] Its lower boundary at the edge of the thermosphere has sometimes been estimated to be 500 to 1,000 km (310 to 620 mi) above the Earth's surface.[citation needed] The exobase is also called the critical level, the lowest altitude of the exosphere, and is typically defined in one of two ways: If we define the exobase as the height at which upward traveling molecules experience one collision on average, then at this position the mean free path of a molecule is equal to one pressure scale height. This is shown in the following. Consider a volume of air, with horizontal area A and height equal to the mean free path l, at pressure p and temperature T. For an ideal gas, the number of molecules contained in it is: where R is the universal gas constant. From the requirement that each molecule traveling upward undergoes on average one collision, the pressure is: where mA is the mean molecular mass of the gas. Solving these two equations gives: which is the equation for the pressure scale height. As the pressure scale height is almost equal to the density scale height of the primary constituent, and since the Knudsen number is the ratio of mean free path and typical density fluctuation scale, this means that the exobase lies in the region where $\mathrm{Kn}(h_{EB}) \simeq 1$.	2017-11-24 18:58:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7332723736763, "perplexity": 787.1525050111455}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934808742.58/warc/CC-MAIN-20171124180349-20171124200349-00420.warc.gz"}
https://plainmath.net/19961/determine-whether-polynomial-degree-polynomial-polynomial-standard	Question # Determine whether g(x) = x^{3}/2 − x^{2} + 2 is a polynomial. If it is, state its degree. If not, say why it is not a polynomial. If it is a polynomial, write it in standard form. Polynomial arithmetic Determine whether $$\displaystyle{g{{\left({x}\right)}}}=\frac{{x}^{{{3}}}}{{2}}−{x}^{{{2}}}+{2}$$ is a polynomial. If it is, state its degree. If not, say why it is not a polynomial. If it is a polynomial, write it in standard form. Identify the leading term and the constant term. 2021-08-04 Step 1 Given: $$\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}^{{{3}}}}}{{{2}}}}-{x}^{{{2}}}+{2}$$ Step 2 $$\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}^{{{3}}}}}{{{2}}}}-{x}^{{{2}}}+{2}$$ Convert element to fraction: $$\displaystyle{x}^{{{2}}}={\frac{{{x}^{{{2}}}{2}}}{{{2}}}},{2}={\frac{{{2}\cdot{2}}}{{{2}}}}$$ $$\displaystyle={\frac{{{x}^{{{3}}}}}{{{2}}}}-{\frac{{{x}^{{{2}}}\cdot{2}}}{{{2}}}}+{\frac{{{2}\cdot{2}}}{{{2}}}}$$ Since the denominators are equal, combine the fractions: $$\displaystyle{\frac{{{a}}}{{{c}}}}\pm{\frac{{{b}}}{{{c}}}}={\frac{{{a}\pm{b}}}{{{c}}}}$$ $$\displaystyle={\frac{{{x}^{{{3}}}-{x}^{{{2}}}\cdot{2}+{2}\cdot{2}}}{{{2}}}}$$ Multiply the numbers: $$\displaystyle{2}\cdot{2}={4}$$ $$\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}^{{{3}}}-{2}{x}^{{{2}}}+{4}}}{{{2}}}}$$ Hence, it is a polynomial with its degree 3, Step 3 Now, convert 3 degree polynomial into standard form which is $$\displaystyle{a}{x}^{{{3}}}+{b}{x}^{{{3}}}+{c}{x}+{d}$$ Standard form $$\displaystyle\rightarrow{g{{\left({x}\right)}}}={\frac{{{x}^{{{3}}}}}{{{2}}}}=-{2}{x}^{{{2}}}+{0}{x}+{2}$$ Leading term $$\displaystyle\rightarrow{\frac{{{1}}}{{{2}}}}$$ Constant term $$\displaystyle\rightarrow{2}$$	2021-09-18 06:48:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7504417300224304, "perplexity": 1080.2340682027245}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056348.59/warc/CC-MAIN-20210918062845-20210918092845-00296.warc.gz"}
https://worldbuilding.stackexchange.com/questions/148924/what-could-be-the-physiological-mechanism-for-a-biological-geiger-counter	# What could be the physiological mechanism for a biological Geiger counter? Some humans have been genetically engineered to be able to detect ionizing radiation to better survive the harsh environments found outside of earth's protective magnetosphere. What could be the physiological mechanism for such a biological Geiger counter, and would they be able to tell what direction the radiation is coming from? Editted to take account of the fact that cosmic radiation is important for this question Important thing to note: what benefit do you expect to gain from your enhancements? In space, and on the surface of airless worlds, cosmic radiation is basically omnipresent. Neutron radiation is only really a risk around badly shielded nuclear reactors and engines (so don't hang around those). Alpha and beta radiation is mostly a risk from nuclear waste. Most people simply won't benefit from being able to detect radiation, and those people whose job involves radioactive hazards can just carry a regular geiger counter. Colonisation of space isn't a good justification. Recolonisation after a nuclear war might well be. You should also consider how your new senses interact with medical technology. If you need macroscopic metal deposits in your body, you might find yourself having problems if you ever need an MRI, and they'll obscure x-rays, too. If you're sensitive to x-rays, then medical x-ray imaging may become quite unpleasant and require sedation instead of being a simple and painless procedure. If you're sensitive to beta radiation, branchytherapy becomes more difficult... radiotherapy in general is may be problematic and unpleasant. Better hope you have other good cancer treatment options! With that out of the way: You can roughly divide ionising radiation into three categories: • minimally penetrating particles, such as alpha or beta radiation • highly penetrating particles, such as neutrons or cosmic radiation • short wavelength EM, such as x-rays or gamma rays You can't trivially detect all of them with the same sort of device... even "geiger counters" need special materials for their windows to allow alpha and beta particles in, and carefully chosen materials that will convert EM radiation to charged particle radiation (ionisation via the photoelectric effect) to make it easier to detect these wavelengths that don't interact readily with the gas inside the tube. Neutrons are also inconvenient and various strategies are needed to reliably detect them. Each of these probably needs a separate sensing mechanism; there's unlikely to be one single organ that will do it all. Alpha and beta The principle source of these is from nuclear reactions and radioactive decay. Outside of earth's magnetosphere, a good source of these is of course the sun, plus any nuclear powerplant (fission or fusion) you might have brought with you. Alpha particles won't penetrate the skin, and beta particles wont readily penetrate the skin unless the source is close and the flux is high. You can't easily have a sense for these that which uses sensory neurons in the skin, because the epidermis (outermost layer of the skin) will block too much of the radiation to be useful. Sensors for these, then, will have to be inside the body. I'm not totally certain how you'd make an alpha or beta sensitive nerve, unfortunately, but the particle do very readily give up their energy so artificially evolving or maybe even designing a suitable mechanism seems like it might be possible. L'Dutch's suggestion of an ion channel is part of a solution, but is unlikely to be a solution in itself In Ken MacLeod's book The Stone Canal, one of the characters has modified olfactory receptors in her nose which tickle when she inhales radioactive gas or dust. When the stimulus is strong enough she sneezes, which isn't a bad reflex to have if you've just snorted a noseful of radioactive materials. This same approach might be extended to the lining of the lungs and maybe the eye mucosa... exposure to radioactive materials might then cause hayfever-like symptoms which might be a good way to clear the material out of the lungs and breathing passages even if it isn't big enough or irritating enough to trigger the normal coughing and sneezing reflexes. Making the lining of the disgestive tract more irritable, resulting in vomiting and diarrhoea on exposure to radiation (but crucially, before that radiation has had much time to actually kill enough cells to cause sickness) will help speed the offending material out of the body before it can do too much damage. Shielding against these kinds of radiation is straightfoward and lightweight. Furthermore, there's not a lot of air in most places in space, so being sensitive to it isn't particularly useful, as your spacesuit or clothing will block external sources and a portable air supply will prevent you inhaling any sources. This is only really a useful trait if you work around badly made fission reactors or have to deal with fallout from nuclear weapons. Neutrons and cosmic rays For people on earth, the easy option here is to just not bother. If you're not in space and not hanging around things undergoing fission or fusion you just won't encounter them (neutron emission is rare except in very short lived isotopes being formed as part of an ongoing nuclear reaction). They're also very highly penetrating, which would make them very difficult to localise, but at least means you could have your sensory organs for them almost anywhere. One possible solution is to have some kind of moderating material that converts them into another kind of radiation that you are sensitive to... charged particles hitting metal will release bremmstrahlung radiation and if you're sensitive to x- and gamma rays you could feel that. Large natural deposits of metal in your body might be awkward to form, however, but carrying around a bit of metal sheet as an external moderator might work. If you're in a metal-hulled spacecraft without adequate particle shielding, some of the incoming charged particle radiation will have already generated x-rays via this mechanism. Other approaches exist to convert neutron radiation into more easily detectable forms. Natural internal deposits of suitable boron compounds might be practical, or again, you can carry around a piece of suitable material with you and wait til your spidey radiation senses near it start tingling. Trying to localise the source of cosmic radiation will be largely pointless. In space, it comes from everywhere, more or less. On a world without a decent atmosphere and magnetosphere, it will come from the sky. because it is so highly penetrating, your sensory bits will probably be triggered regardless of their orientation or meagre shielding; you'll simply be able to feel that you're exposed to a hostile environment and maybe you'll be able to tell how hostile it is. Localising sources of neutron radiation will be quite difficult, again because they are highly penetrating (moreso than charged particles, for the same energy levels) and so shielding your sensory gear so it can detect neutrons from a specific direction is probably impractical. You'd probably just get a sense of impending doom, and maybe a rough idea of the strength of the neutron flux. Gamma rays and x-rays Another kind of highly penetrating radiation. In space, a major source of these will be charged particle radiation hitting metal objects and releasing bremmstrahlung radiation. Your nuclear reactors will also be a good source of x- and gamma rays. Interestingly, there already exist some naturally (ish) evolved radiotrophic fungi which have been found in the Chernobyl sarcophagus and use a form of photosynthesis using melanin that reacts to short wavelength light like gamma radiation. Humans can't be photosynthetic, but they do have melanin so copying the same approach could potentially be used... specialist melanosomes could be formed in some skin sensory neurons that could provide whole-body sensitivity to short wavelength radiation. It might feel like someone is shining a heat lamp at you, for example. This would also improve UV sensitivity, so you'd be less likely to hang around in the sun and get sunburnt, though increased levels of melanin in the skin would make you less prone to sunburn anyway. Highly penetrating radiation might be difficult to sense with directionality (because it might trigger the sensory nerves when it enters and leaves the body, and if it passes through your arms and torso you'll get multiple entrace an exit points, etc) but if you bring a handy metal sheet as a partial shield and you still feel the radiation with no change in intensity, then it is probably coming from the unshielded direction. If you can get your body to naturally grow highly x-ray opaque materials around sensory clusters, that would probably help too. You might be able to modify retinal cells to respond to x-rays and gamma rays using the same sort of techniques, but because your eyes will not be able to focus or block much short wavelength radiation you won't be able to get a very good idea of what is emitting the xrays becuase your whole retina will be stimulated. The sensation may be fainter or absent for sources behind you where your skull is providing shielding. You might just end up with foggy vision or a sort of optical glare which might be irritating or extremely debilitating, and not what you want when you're in a highly radioactive environment! • Nice answer, but one quibble: to see a glowing object (instead of the generalized foggy glare you describe later), your eye needs to focus the radiation that the object is glowing with. Focusing x-rays is a tough problem. At best, you might be able to do some sort of bony pinhole-camera arrangement; even better if you can deposit some heavy-metal-ish elements in the bone to make them more radioopaque...? – jeffB Jun 15 '19 at 13:37 • @jeffB I've fixed that issue, thanks for pointing it out. I wonder about increasing the radio-opacity of the body though... there's a tradeoff to be made between improved innate radiosensing and being able to make use of medical imaging and radiotherapy techniques. Made a note about that, too. – Starfish Prime Jun 15 '19 at 14:58 • But, hey, you could X-ray yourself just by looking in a mirror! Except there aren't good X-ray mirrors, either. Rats. – jeffB Jun 17 '19 at 3:10 Actually the human body already has an organ that comes close to this; the eye. Astronauts since the 1960s have reported seeing flashes in their eyes, even when closed while in space. It turns out the flashes are the retina being impacted with cosmic rays outside the Earth's magnetosphere. In effect this means that astronauts that received more rapid flashes would be experiencing higher doses of cosmic rays than those with only an occasional flash. Directionality is difficult because these things go so fast and radiation would be in a similar vein. Besides, we know the source of cosmic rays in our solar system; it's the sun. Something similar would be happening in other harsh environments and to put it bluntly, people would likely just head in the direction that leads to a lower rate of flashes occurring. This is not a perfect system; these cosmic rays are doing damage to the eye and to the body every time they pass through you, so eventually you're going to be blind if this is a regular occurrence. Also, this is not a finely tuned mechanism for detecting different kinds of radiation, but then neither is a Geiger Counter. If all you're looking for is a reading on the radiation levels in a given location, then the best approach may well be to modify the eye to 'see' radiation a little better (after all, light is just radiation of a specific wavelength, and different colours are just minor differences in that wavelength) and toughen the eye a little more biologically to be able to withstand long term exposure to low levels of radiation over a longer period. Ultimately, nature is far more likely to adapt an existing system to a new purpose than it is to generate a new organ from scratch, and we already have organs that detect very specific wavelengths of radiation called light. It just seems far more likely that humans in the conditions you describe would have their eyes enhanced over time by evolution to detect more bands of radiation as visible light to meet the need you describe. • Just a note that the radiation we see as light (electromagnetic radiation) and the radiation a Geiger counter picks up (particle radiation) are not the same physical phenomenon – katatahito Jun 14 '19 at 5:02 • @katatahito the flashes this answer is describing arise from high energy charged particles interacting with a human retina. They do not arise from x-rays or gamma rays, and are absolutely the kind of radiation a geiger counters can and do detect. – Starfish Prime Jun 14 '19 at 9:21 • Evolving the ability to create a thin layer of zinc sulfide would increase the sensitivity many-fold. See en.wikipedia.org/wiki/Spinthariscope – DJohnM Jun 14 '19 at 23:11 • "Besides, we know the source of cosmic rays in our solar system; it's the sun". Not quite -- technically cosmic rays are defined as originating from outside our solar system. Our sun actually helps shield some against true cosmic rays (especially in the combined Earth-Sun system). Perhaps you are thinking energetic particles from solar flares? – madscientist159 Jun 14 '19 at 23:37 • @madscientist159 yes, you're right. In my rush I was confusing the two. I'll come back and re-work the answer when I have more time... – Tim B II Jun 15 '19 at 2:06 Ionizing radiation, as the name says, induce ionization in the matter with which it interacts. Cells are able to interact and control ionized charges (see for instance the $$Na^+$$/$$K^+$$ pump), so it is plausible that a mechanism able to detect charge formation can be obtained by the cells: as soon as a meaningful charge is balanced by the cells, this trigger a signal. In order to detect the direction of the impinging radiation, considering that the probability of interaction is proportional to the depth of penetration, one can imagine 3 orthogonal tubes as a mean for directionality discrimination. • Could this sensing be applied to the inner ear so as to save on creation of new structures? - As a secondary thought, thematically this would fit if the stimulus promoted a clicking in the ears, much like a Geiger counter. – XenoDwarf Jun 14 '19 at 5:39 • @XenoDwarf combining radiation sensitivity with your balance system sounds like it might have awkward but amusing results. "Oh, Bob's fallen over and vomited again, better get out the lead overcoats". – Starfish Prime Jun 14 '19 at 9:58 • @StarfishPrime well it’s not like just the ears are affected by the radiation and I think cochlea, not the semilunar canals, would be the right place as it’s the part that possesses the mechanical analog for distance and space sensing. – XenoDwarf Jun 14 '19 at 10:03 • @XenoDwarf I'd argue that the eyes might be better for that; radiation won't neatly be collected by the structures of the ear in the same way that sound is. – Starfish Prime Jun 14 '19 at 10:05 • @StarfishPrime I mentioned ears because they have orthogonal tubes, even if I did suggest the cochlea. – XenoDwarf Jun 14 '19 at 10:17 You don't want a Geiger counter: that just tells you how much radiation there is. To figure out where the radiation is coming from, you want a scintillation detector. A scintillator is a material that gives off visible light when struck by ionizing radiation. Engineer your modified humans to accumulate suitable compounds in the retina, and you've got someone who can see radiation. The classic scintillator is zinc sulfide, which is sensitive to X-rays and beta radiation, but there are a wide range of other organic and inorganic substances (usually crystals) that you can use, with different sensitivities. The vitreous humor of the eye, for example, appears to be sensitive to high-energy nuclei. Keep in mind that most radiation passes through the human body without trouble. Except for low-energy beta radiation, your modified humans won't be able to tell where the radiation is coming from without assistance. But give them radiation-blocking helmets with eyeholes to provide a pinhole lens effect, and they'll be able to look around for the source, rather than just seeing a general radiation glow. You're probably not going to be able to engineer someone to see alpha radiation: although there are substances that glow in response to the radiation, it's too easy to block. Suitable substances for alpha shielding include "clothes", "skin", and "air". • To be picky, GM tubes generally do give you an indication of where the radiation is coming from, at least for alpha and beta, because there's generally only one window into the tube that those particles can penetrate. – Starfish Prime Jun 15 '19 at 15:03 The body is able to detect radiation. When there is enough radiation in the enviroment to make you sick, the telltale sign is that you get sick. Just the same, when the level of radiation is lethal, the body signals that by dying. That is a characteristic shared with most animals. Humans, however, have evolved beyond that and gained a skill that allows us to detect radiation without suffering any ill effects. It's called taming. Send a dog into the cave first, and if he comes back alive after an hour, you're probably good. Usually small vertebrates are affected faster and more strongly than humans. In the absence of animals, you can use the skills of language and bullying to send others to test radiation levels for you. To find the direction of the source, have three sacrificial aninals or people stand in different places and triangulate based on how long they take to get sick. Make sure all test subjects belong to the same species and have approximately the same body mass and age. • This plan is problematic when dealing with levels of background radiation that can lead to seriously increased risks of cancer, birth defects or infertility but won't actually kill anyone even after months of exposure. "Set up a sacrificial settlement and wait for a generation or two" has a bit of a long latency. – Starfish Prime Jun 14 '19 at 9:10 • @StarfishPrime that is not in the scope of the question ;) – The Square-Cube Law Jun 14 '19 at 13:21 • I was going to answer similarly, albeit even more psychopathically: breed humans who are incredibly sensitive to radiation so they get sick and die at levels below max-safe-for-normals, and they do so quickly. We could call them "canaries." – Carl Witthoft Jun 14 '19 at 15:30 • @CarlWitthoft Except, Your approach would actually work and is, in fact, in use right now (e.g. when winegrowing, people plant more sensitive roses next to the vines). Just take other species and given the dire situation, there would hardly be a moral issue. – hajef Jun 15 '19 at 8:06 • 8000 Rads is an immediately lethal dose; nuclearweaponarchive.org/Nwfaq/Nfaq5.html. The average human body would make a perfect detector under those conditions because it would die. – Richard Jun 16 '19 at 16:15 I asked myself the same question (except for evolution instead of genetical engineering). The answer I found back then was more basic but if you go for full genetical engineering, here are my thoughts: Unfortunately, I could not find a reference but I know that there are some bacteria that benefit from radiation by using it for their energy conversion. They basically use the radiation energy to create some ATP. Based on that, we could engineer an organ that produces heat or fires pain signals only if this form of energy is available. For natural evolution this probably would be a single organ that changes its purpose (like a Lymph node that becomes a sense organ) but if we get to jump to the most desirable form I would spread this across the skin so the subject could detect the source of alpha and probably beta radiation (alpha radiation does not penetrate and beta radiation gets severely reduced). That way one would feel radiation as a form of heat/pain (under the skin/ only at certain locations/ unlike the not-modified companions). This way one might even be able to tell the types of radiation apart, considering how alpha radiation only affects one side and gamma radiation merely gets reduced at all. You could, for example, mix in plant DNA that encodes the structure of the lower epidermis of leaves modifying the stoma to heat up/ firing pain signals when there is enough radiation to use it for cellular respiration. It is worth mentioning though, that genetical engineering is neither our only nor our best option. Cyborgs are. For example, the colorblind Neil Harbisson got an artificial sensor implanted that allows him to sense colors (ranging from ultraviolet to infrared) as vibrations in his skull. It took time for his brain to adjust but now it can process the information and he has a natural feeling for it (he can even stream a camera's point of view allowing him to take "VR" tours through space and so on). In an interview, he mentioned that he considers adding more artificial senses and named radiation as one. Creating a cyborg takes way less time and its outcome is way easier to predict than genetical engineering. Given enough knowledge and simulation, you could engineer someone who has the character traits and physiology you need but building an artificial sensor and implanting it to someone with the required character traits is way easier and cheaper. Given the level of technology needed to make the genetical engineering work at all, we could easily build an [Edit: conveniently small] implant with little to no maintenance and little to no risk of rejection. Also, note that plot tools like blackmailing these people with implanted bombs or artificial needs that only you can cover or addressing moral questions are possible for both scenarios though I'd expect a genetically engineered person to be programmed to be loyal, fearless and so on. If this is what you want to play with you can use brain implants and/or hormone-based brainwashing for the cyborg but if you don't want obedient soldiers, going for genetical engineering would rise a minor plot whole since loyalty is much easier to genetically engineer than new organs. [Eidt: that is, as long as it's a rare and expensive procedure available only to the government. If this is available for civilians as well, they would make their children smart rather than oediant.] • +1 for the cybernetics being a much simpler solution, though I'd argue that simply carrying a geiger counter would be just fine for 99% of people, or more! – Starfish Prime Jun 14 '19 at 14:08 • @StarfishPrime That depends on the situation. If you suspect there might be dangerous radiation and got time to prepare, your implants won't be as accurate as the carried one is considering your hazard suit. But living everyday life with a gadget you have to carry along all the time is super inconvenient but if you can't rely on being able to equip yourself before encountering potential radiation sources demands that you can detect radiation at all time. A small implant should be able to warn you of dangerous but not yet lethal radiation without being inconvenient in almost any situation. – hajef Jun 14 '19 at 14:26 • If you're not prepared to deal with an everyday inconvenience like carrying a dosimeter, then life outside of earth's biosphere is not for you. More importantly, if you're not prepared to use safety equipment all the time, you become a liability to everyone else, as well. – Starfish Prime Jun 14 '19 at 14:49 • I am prepared to do both but if I have the choice, I'll go for the more convenient scenario. Also, constant vigilance is beyond human capabilities so you can't rely on being fully equipped all the time and the whole point of these modified humans would be that they are able to detect a kind of danger that normal humans are unaware of. If you rely on having Geiger counters available at any corner, you wouldn't need mutants or cyborgs. – hajef Jun 14 '19 at 16:24 • The convenient scenario is the one that doesn't require invasive surgery and ongoing maintenance ;-) Wearing a dosimeter is a pretty far cry from "constant vigilance", too. I feel that your standards may be a little low! – Starfish Prime Jun 14 '19 at 16:59 As said elsewhere, the body already detects radiation in the form of radiation sickness. The root cause of this is the death of cells, especially rapidly dividing ones. The cause of the cell death is DNA damage - radiation breaks the chemical structure of DNA. Because DNA damage can lead to mutations and cancer, many organisms actually terminate cells when it occurs. Eukaryotic DNA is linear, and the ends (telomeres) are capped with special structures and proteins. When there are uncapped free ends, these almost certainly come from some sort of DNA damage event, and can be recognized by proteins. The famous p53 protein is present in all human cells, and when it detects DNA damage it triggers a special mechanism (apoptosis) for the cell to kill itself. In the majority of cancers, p53 must first be impaired before the carcinogen can produce a cancer cell. You can imagine a specialized group of cells that have extremely fragile DNA (not all base sequences are equally strong, so this is quite easy). They arise from specialized stem cells, but then terminally differentiate (ie. cannot divide anymore). Their purpose is to be a radiation detector - whenever there is radiation, their DNA breaks first. Modified versions of proteins like p53 detect this, but instead of triggering apoptosis they effect some other signal. This could even be directly stimulating neurons for instant radiation sensation, or it could be slow release of some chemicals for delayed sensing. These cells are very differentiated, like red blood cells, and are wholly incapable of replicating, so despite their lack of anti-DNA damage machinery they are not likely to become cancerous (or maybe their stem cells are likely to become cancerous, and that is the drawback?). This is not feasible with today's biotechnology (we don't understand differentiation enough to create new organs and we don't understand DNA damage that well) but certainly plausible as a molecular mechanism. Creating a whole new category of sensation in the brain seems like a hopelessly complex task, so the neural feedback of such an organ would probably hijack an existing one - so radiation might feel "hot", "wet", "sweet", "fuzzy" depending on which existing sense you pick. Keep in mind that there are actually not just 5 senses as per the colloquialism - there's at least 12 or 13. If the organ is internal, it can detect only gamma rays, which penetrate many materials. Beta and alpha radiation cannot pass through the outer layer of dead skin. The exception is that beta-emitting radioactive particles might be inhaled and there get very close to cells, so perhaps your organ should be in the airways to detect beta-emitting airborne particles. However, you could also have small organs (the "rad eyes") on some part of the body that people scan over an object like you would with a Geiger counter. • After reading the first paragraph, I almost skipped the rest of this very interesting answer. We do not detect radiation by getting sick but exploring the reasons and mechanisms of radiation sickness like you did is a good idea. You might want to clarify that. – hajef Jun 15 '19 at 7:55 • @hajef I feel that's a subjective, semantic argument - so I'll leave it as it is. It is no different than saying a canary "detects" poison gas. Glad you enjoyed the answer. – Withadel Jun 19 '19 at 2:26 One of the ways radiation kills you is by dissolving the blood-brain barrier. When the cells start detecting radiation damage, they could send a signal to your brain and you would know if you are being exposed to an abnormal amount of radiation. • If you need a citation: ncbi.nlm.nih.gov/m/pubmed/2262385 – ThisRandomGuy Jun 14 '19 at 14:34 • Thank you for the update (though I'd have put the citation in your answer rather than in a comment). The issue remains that the blood-brain barrier is only a small part of the body, and lethal radiation exposures could conceivably not affect it in the slightest. – Starfish Prime Jun 14 '19 at 14:54 • Bacteria, viruses,etc. could get through and I'm pretty sure that have blood leaking into your brain is a bad thing (even though brain fluid has blood in it, I'm quite sure that if it's oversaturated with blood, bad things will happen). – ThisRandomGuy Jun 14 '19 at 15:22 • I'm not disputing that damage to the BBB is bad. I'm disputing that monitoring damage to it is a good way to measure radiation, especially given that it is small relative to the body and lethal doses of radiation may still be received without the BBB being affected in the slightest. – Starfish Prime Jun 14 '19 at 15:26 • If part of the BBBisn't there it's quite likely that it's possible to detect that – ThisRandomGuy Jun 14 '19 at 15:27	2020-10-24 03:07:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4603711664676666, "perplexity": 1280.274140097358}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107881640.29/warc/CC-MAIN-20201024022853-20201024052853-00677.warc.gz"}
https://physics.stackexchange.com/questions/534173/why-does-the-minimum-energy-principle-work	# Why does the minimum energy principle work? The principle of minimum energy states that in a thermodynamic system the equilibrium state corresponds to the minimum energy state among a set of states of constant entropy. I believe I understand the mathematical derivation of this, however, my immediate intuition is that this should not be the case. People sometimes handwave something like "Thermodynamics should agree with mechanics when entropy is constant" or similar. Other arguments imply some sort of "interaction" with the environment, which increases entropy when one reaches a minimum value of energy (I'm not sure where I have read this, I wish I had a source) but I would prefer to steer away from those kinds of arguments. It is clear to me from the mathematical derivation that this principle does not rely on dynamics, mechanics, or other auxiliary systems to be true, only on the fact that the entropy is a concave function of its variables and that its hessian is negative definite at equilibrium. My intuition, however, says that if a system has a bunch of states available to it, and all states have the same entropy, then it should not prefer one state over the other and all of them should be equally good "equilibrium states". This is for sure valid when the energy is constant; I know this has to be wrong when the states have different energies, I just don't see how. Edit for clarity: As an example of the application of the principle of maximum entropy, consider a system composed of two ideal gasses with fixed numbers of particles in different compartments. The total energy and volume of the system are held constant but the entropies and volumes of both gasses are allowed to change subject to constraints, so that $$U(S_1, S_2, V_1, V_2)$$ has to be a constant, $$V_1 + V_2 = V$$ has to be a constant but $$S_1$$ and $$S_2$$ can freely change. There are many possible states for this system, but the principle of maximum entropy says that the state that corresponds to thermodynamic equilibrium is the state with maximum entropy $$S_1 + S_2$$. The principle of minimum energy is analogous but the roles of $$S$$ and $$U$$ are reversed, and the energy is actually a minimum at thermodynamic equilibrium instead of a maximum. • – ACuriousMind Mar 2 '20 at 18:44 • It is definitely related but it is not asking the same thing, that question is asking about the derivation of the principle which I don't really have much problems with. I'm having trouble finding an intuition for the conclusion that doesn't relly on mechanics or extra stuff. – Ignacio Mar 2 '20 at 18:55 This is figure 17.3 from Thermodynamics, a complete undergraduate course by myself (Steane), published by OUP (2016). Here is what I hope is an intuitive argument. For a $$pV$$ system, consider the situation at given $$S$$ (the volume $$V$$ being also fixed). Let $$X$$ be an internal parameter. The states at various $$X$$ have different internal energies to one another. Of these states, the one with the least internal energy is the equilibrium state when the system has the given $$S$$. Proof: • This is exactly the same as the argument given in amazon.com/Thermodynamics-Herbert-B-Callen/dp/0471130354 – Ignacio Mar 2 '20 at 19:44 • My issue with it is the following: you say "suppose the system is in equilibrium, at a a given S and V, with an internal energy larger than the minimum" so this means you have fixed S and it can't change at all, then afterwards you say you change your parameter and get a system which has a larger S, and the same U, but I don't really care about this new state because I wanted to know what the equilibrium state is with the S you fixed before, right? – Ignacio Mar 2 '20 at 19:50 • Put another way: what possible physical reason does your system have to evolve to the actual "equilibrium state" if left alone in your first state which is "away from equilibrium", but constrained to move only on paths of constant entropy? when your system is evolving the entropy is not changing at all, so why go somewhere specific? – Ignacio Mar 2 '20 at 20:04 • Yes you are right, in that if the system starts out of equilibrium, and then is constrained to move only at constant entropy, then it will never reach equilibrium. It will oscillate too and fro. The argument is about the nature of the state space, in which we introduce another process (doing some work and then supplying heat) in order to show what must be true of the state space. – Andrew Steane Mar 2 '20 at 20:18 • @Ignacio If one wants to allow the system to reach equilibrium by relaxation without changing its own entropy, then one may allow it to do work on a vane immersed in a viscous fluid somewhere else. While that is happening, the system is not isolated of course. Equilibrium is now the maximum of $S_{tot} = S + S_d$ with $S$ constant. In this process the internal energy of the other system (d) grows as its entropy does, hence a max of $S_d$ gives a max of $U_d$ hence a min of $U$ (since $U+U_d$ is fixed). – Andrew Steane Mar 2 '20 at 20:22 The principle of minimum energy states that in a thermodynamic system the equilibrium state corresponds to the minimum energy state among a set of states of constant entropy. which is very close to the statement in the introductory part of wikipedia page you cited. However, this is not a consistent way to express the minimum energy principle in thermodynamics. The reason for inconsistency should become clear by looking at formulas. In the case a thermodynamic state is fixed by the value of entropy, volume, and number of particles, the fundamental function from which the whole thermodynamic behavior can be obtained is the internal energy $$U(S,V,N)$$. Now, it is clear that once the independent variables are fixed, a unique value for $$U$$ is possible. There is one thermodynamic state and it is not clear which should be the states "among which energy should be minimum". Actually, the correct statement of the minimum principle for energy is the following: in an equilibrium system at fixed entropy, volume and number of particles, and subject to internal constraints controlled by a set of parameters $$X_{\alpha}$$, the internal energy is a function $$U(S,V,N;\{X_{\alpha}\})$$ and the final equilibrium state, obtained after removal of the constraints, corresponds to the minimum of the energy among the all the possible values of the constraint variables $$X_{\alpha}$$ (see Callen's textbook on Thermodynamics for a reference). Starting from the correct statement of the minimum principle, a first observation is that it is more general than just the convexity property of the function $$U(S,V,N)$$. Indeed, from the minimum principle, one can derive convexity of $$U(S,V,N)$$. But there are cases where the minimum principle provides results which are not derivable from convexity. For example, if one can determine different functions of energy at fixed $$S,N$$, as a function of $$V$$, minimum energy allows to determine for each $$V$$ the equilibrium state. What about intuition? Frankly, I think that in the case of the minimum energy principle, is far from being intuitive. The main reason is that the underlying condition of constant entropy is difficult to manage both from the experimental and from the conceptual point of view. However, since from the minimum of energy $$U(S,V,N;\{X_{\alpha}\})$$ one can easily obtain similar minimum principles for the Legendre transforms of energy (Helmholtz free energy, Gibbs free energy), the difficult condition of fixed volume and entropy can be transformed into the conceptually and experimentally easier conditions of minimum at fixed temperature and volume or temperature and pressure. Edit after a few comments and the editing of the question. Notwithstanding the previous words of caution about the non-intuitive condition of constant entropy, an example with a fluid system could help to get a better understanding. Let me start recasting in a correct way the situation, if it should be analyzed in term of minimum energy principle. There is a composite system made by two compartments such that initially the first compartment contains a fluid (the same in both compartments for simplicity) described by the thermodynamic variables $$S_1,V_1,N_1$$, and the second by $$S_2,V_2,N_2$$. $$V_1,N_1$$ and $$V_2,N_2$$ remain always fixed. The energy of this composite system is the sum of the energies of the two subsystems and, being filled with the same fluid (for example both Neon gas), the same function $$U$$ of entropy, volume and number of particles describes both. By introducing the subscript $$tot$$ for the extensive quantities describing the composite system we have $$S_{tot}=S_1+S_2$$, $$V_{tot}=V_1+V_2$$ and $$N_{tot}=N_1+N_2$$. For a given partition of the total entropy into a value $$S_1$$ and $$S_2=S_{tot}-S_1$$ (this is the constraint on our composite system) we have $$U_{tot}(S_{tot},V_{tot},N_{tot};S_1)=U(S_1,V_1,N_1)+U(S_{tot}-S_1,V_2,N_2).$$ The minimum energy principle applied to the present case says that if we eliminate the constraint that system $$1$$ should have entropy $$S_1$$, but always keeping fixed $$S_{tot}$$, the final equilibrium state of the composite system will correspond to the value of $$S_1$$ which minimize $$U_{tot}$$. That there should be a minimum can be seen by noting that $$U(S,V,N)$$, at fixed $$V$$ and $$N$$ must be an increasing function of $$S$$ (let's recall that $$\left.\frac{\partial{U}}{\partial{S}}\right\|_{V,N}=T\gt 0$$). So, $$U_{tot}$$ is the sum of an increasing and a decreasing (convex) function in the interval $$0 and therefore there should have a minimum. It is possible to check everything explicitly in the case of a perfect gas in two equal volume containers with the same density. The total energy is $$U_{tot} \propto \left( e^{\frac{2S_1}{3N_1k_B}} + e^{\frac{2(S_{tot}-S_1)}{3N_1k_B}} \right),$$ which has a minimum at $$S_1=S_{tot}/2$$. In a less formal way, one could say that the reason for the minimum is directly connected to the constraint of keeping fixed the total entropy. Since entropy is proportional to the logarithm of the number of states, a fixed total entropy in our composite system is equivalent to keep fixed the product of the number of states of system $$1$$ and system $$2$$. The way the number of states varies with energy provides the mechanism on which the minimum principle is based. A final remark on microstates. Discussion of the minimum energy principle can be based, as in the previous paragraphs on a completely macroscopic thermodynamic description. Of course, thermodynamic variational principles can be translated into the language of statistical mechanics. However, statistical mechanics is more naturally expressed in the framework of entropy and its Legendre transforms. So, in the case of microscopic description it is easier (more intuitive) to work with maximum principles. • You start by saying that it is not clear which should be the states among which energy should be minimum. I think those are clear, for instance, given two ideal gases in a container with constant volume, the internal energy is U(S, V1, V2, N1, N2), say we fix V = V1 + V2, N1 and N2. There are many possible states for the two gasses still. I do think that if you have just ONE homogeneous system with U(S, V, N) then it doesn't make much sense to fix V, N and ask what are the possible values of U for fixed S, but if you read closely I never say that in my question at all. – Ignacio Mar 2 '20 at 23:08 • You say: " But there are cases where the minimum principle provides results which are not derivable from convexity. For example, if one can determine different functions of energy at fixed 𝑆,𝑁, as a function of 𝑉, minimum energy allows to determine for each 𝑉 the equilibrium state." Can you clarify? – Ignacio Mar 2 '20 at 23:11 • Also, I agree on everything you said about legendre transforms and statistical mechanics, but I think they are not really the point of the question. The issue for me is that there is a priori no reason I would expect for a system to evolve towards a state of minimum energy if constraints are released and but its entropy is held constant. I think I can conceptualize constant entropy mathematically, you just restrict your system to evolve along paths of constant entropy, right? – Ignacio Mar 2 '20 at 23:15 • I guess I should have said the energy is U(S1, S2, V1, V2, N1, N2) but the point still stands – Ignacio Mar 2 '20 at 23:18 • @Ignacio You did not write functions and their arguments. However, you wrote about states with the same entropy, which is at last ambiguous. On possible example could be $U(S,V,N;S_1)$, where $S_1$ is the entropy of a subsystem. – GiorgioP Mar 3 '20 at 0:20 I find this question very interesting, as it deals with crucial concepts, common misunderstandings and often-encountered unclear reasoning. Part of the reply by Andrew Steane points to an answer (in the legend of his Fig. 17.3). Yet, on the other hand, I don’t feel that the demonstration that follows is fully appropriate or that it correctly addresses the issue (for example, the maximum entropy principle does not apply to a system that is not isolated). An important thing to understand (often a source of misconception) is that each point of the curve of Fig. 17.3 represents the entropy of a system at equilibrium for different constrained values of some internal parameters. As a consequence, plotting a "trajectory" on such a curve doesn't actually define any specific process, it just represents "loci of equilibrium states" to borrow the words of Herbert B. Callen. To be concrete I will take the nice example in the legend of Fig. 17.3 from Andrew Steane: a cylinder filled with an internal piston and some gas in each compartment. Assume the cylinder to be of constant volume and with adiabatic walls. If the position of the piston is changed reversibly, the entropy of the system stays constant. Now, whether the piston itself is adiabatic or not won’t change the following reasoning, but for simplicity I will fist assume that the piston is adiabatic; I will come back to the diathermal case afterwards. Imagine that the piston is manipulated from the outside to fix it at various positions while keeping the overall entropy of the cylinder constant. This can be done for example by moving the piston very slowly in order to avoid that any turbulence builds up. During this process, work is either received by or extracted from the cylinder and the internal energy of the system changes. Now, there will be a position where the pressure applied on the piston on each side by the gas in each compartment will be the same. (As a side remark for later, note that in this scenario of an adiabatic piston, the temperatures of each gas in each compartment play no role and can be of any values, only their pressures are relevant.) If a new constrained state is to be reached from this initial state of balance pressure by reversibly changing the position of the piston to a new constrained position, the pressure in one of the compartment will increase whereas the pressure in the other compartment will decrease. (As a side remark for later, note that at the same time the entropy of each compartment will stay the same as no heat is transferred to any of the compartments.) Therefore, to reach this new state some energy has to be transferred to the system in the form of work to counteract the difference in pressure that builds-up and the internal energy of the cylinder will be increased. This shows that the state with equal pressures is the state of minimal energy. Now, from any initially constrained position of the piston, imagine that the constraint is released. From the moment the constraint is released, we consider the cylinder to be isolated. If the piston is initially at the position of equal pressures (that is, at the position of minimal internal energy), nothing happens: for a system of constant entropy, the state of minimal energy is stable. If the piston is not initially at the position of equal pressures the piston will be spontaneously displaced by the difference of pressures and the system will be spontaneously pushed towards the state of minimal internal energy: for a system of constant entropy, the states that are not of minimal internal energy are unstable whereas, again, the state of minimal internal energy is stable. This is where the equilibrium thermodynamics reasoning stops: stating which of the constrained equilibria is an overall equilibrium when some constraints are removed. If one were to compute what happens next and how the system would evolve, one would need to build a mechanical dynamic model of the piston moving inside the cylinder under the influence of the forces of pressure from the gases in each compartment. In the hypothesis of reversibility, this would give as solutions oscillating movements of the piston inside the cylinder around the position of minimal internal energy (thermodynamic equilibrium) –that is, around the position of minimal potential energy (because at thermodynamic equilibrium there is no macroscopic kinetic energy to be considered since the system is static). We see here the analogy of static equilibrium position between thermodynamics and mechanics. Case of a diathermal piston To be complete, let’s now assume that the piston is diathermal. This implies that for all constrained equilibrium states considered, the temperatures of the gases in each compartment are always equal to each other. If the piston, initially at a position of equal pressures in each compartment (and so here also of equal temperatures between the gases) is brought reversibly to a new constrained position, similarly as before the pressure in one compartment will increase whereas the pressure in the other compartment will decrease. The difference with the adiabatic case here is that at the same time heat will now also flow from one compartment to the other to keep both compartments at the same temperature. Note that the new temperature of the gases in the compartments may differ from the initial temperature, but the important point here is that they will stay equal to each other. This heat transfer corresponds to a flux of entropy between the two compartments, but the total entropy of the cylinder stays the same: since the temperatures of both compartments are always equal on reversible paths, one can write (hypothesis of reversibility): $$dS = dS_1 + dS_2 = \delta Q_1/T + \delta Q_2/T$$, which, with $$\delta Q_1 = - \delta Q_2$$, gives $$dS = 0$$. So, similarly as for the adiabatic case, the total entropy stays constant, however, compared to the adiabatic case, there is no build-up of temperature difference here, and the total difference in pressure reached might not be the same as previously. Yet, from here the reasoning on the stability of the different positions of the piston when the constraint on the position is removed stays the same, and one finds that for a system of constant entropy the equilibrium position is the position of minimal internal energy. Case of a diathermal cylinder in contact with a thermal bath The above reasoning can of course also be followed in the case where the cylinder has diathermal walls and is in contact with a thermal bath that maintains the system at a constant temperature $$T$$. The crucial point here is that now, during a reversible process that moves the piston, there is also a flux of entropy between the cylinder and the thermal bath ($$dS = \delta Q/T$$, with $$\delta Q$$ the heat received (algebraically) by the system from the thermal bath), so the cylinder is not anymore at a constant entropy. If one would want to reason with a constant entropy, one would need to consider the total internal energy of the overall system composed of the cylinder plus the thermal bath. If one would want to reason only on the cylinder, neither the principle of maximal entropy nor the principle of minimal energy applies to the cylinder alone. To reason on the cylinder only, which is maintained at constant temperature, one can only reason on the Helmoltz free energy $$F=U-TS$$ of the cylinder. In this case, the equilibrium position of the piston would be the one with minimal Helmoltz free energy for the cylinder, over all the constrained equilibrium positions of the piston at constant temperature (vs the minimum of internal energy at constant entropy). To explore those issues further, I suggest especially Problems 2.7-3 and 3.4-8 of Callen's Thermodynamics and an Introduction to Thermostatistics (note that 3.4-8 gives different results whether you consider a reversible or an irreversible process) and Problem 4.3-1. Equilibrium states are states that can be defined with just a few parameters like $$V,T,S,P,N \text{ and }E$$, that are related by an equation of state. So if you are defining an equilibrium state completely, there is only one such state. If any one of them are different, then they are two different equilibrium states. However, if you are referring to the many different internal states (microstates) that your system can be in that lead up to the correct macroscopic equilibrium state, then you are right about each such microstate being equally likely to be present. • I don't really understand what you mean. Suppose you have two ideal gasses that are enclosed inside an adiabatic, rigid container, these can be in many possible states, temperature and pressure of each one can be very different in many of those states. There is only one state, however, that is the equilibrium state of the sytem, the state where the temperatures and pressures are equal. That state is obtained by maximizing entropy. Now, lets think about minimizing energy: the system has many states available to it, all of them with the same entropy, why choose the one with minimum energy? – Ignacio Mar 2 '20 at 19:35 • Say, in my first example, the gasses had an extra variable, call it color or whatever. The color of the gasses can change but that doesn't affect the entropy, then any state which had gasses of different colours would be fine as an "equilibrium" state. If the system had some dynamics that allowed switching between these then it would probably jump between them stochastically and it would make sense to consider them "the same thing" somehow and call them a "microstate" that is fine, however, I don't think it answers my question. – Ignacio Mar 2 '20 at 19:41 Your wrote: "My intuition, however, says that if a system has a bunch of states available to it, and all states have the same entropy, then it should not prefer one state over the other and all of them should be equally good "equilibrium states"." If I understand you correctly you are echoing Pippard's view $$[1]$$ of the maximum entropy principle. Let me quote from his magnificent book: Now for any given set of constraints a thermodynamic system has only one true equilibrium state, and we may therefore formulate the entropy law in a slightly different way: It is not possible to vary the constraints of an isolated system in such a way as to decrease the entropy. When the gas is in equilibrium in the larger volume its density is very nearly uniform, but is subject to continual minute fluctuations. Very occasionally larger fluctuations will occur, and there is a continuous spectrum of possible fluctuations ranging, with decreasing probability, from the very small to the very large; so that it is a theoretical possibility (though it is overwhelmingly improbable of observation even on a cosmic time scale) that the gas may spontaneously collapse into the smaller volume from which it originally escaped at the piercing of the wall. It will subsequently expand again to fill the full volume at just the same rate as at the first escape. We may now inquire what happens to the entropy of the gas during this large-scale fluctuation, and to this question the only satisfactory answer is the perhaps surprising one-nothing.[...] and the punch line(s): Thus we see that the entropy (and of course other thermodynamic functions) must be regarded as a property of the system and of its constraints, and that once these are fixed the entropy also is fixed. Only in this sense can any meaning be attached to the statement that the entropy of an isolated mass of gas, confined to a given volume, is a function of its internal energy and volume, $$S=S(U, V)$$. It follows from this that when the gas is confined to the smaller volume it has one value of the entropy, when the wall is pierced it has another value, and that it is the act of piercing the wall and not the subsequent expansion that increases the entropy. In the same way when two bodies at different temperatures are placed in thermal contact by removal of an adiabatic wall, it is the act of removing the wall and not the subsequent flow of heat which increases the entropy. [1]: Pippard: ELEMENTS OF CLASSICAL THERMODYNAMICS, pp. 96-98 • This is a neat quote, I will look into this book. I think it is a bit of an artificial distinction to say that the entropy increase "comes from punching the hole but not from the gas going into the other container" though. If this was literally true then if you punched a hole and then quickly covered it with your hand you would increase the entropy of the system and then decrease it again by covering the hole right? I think entropy is only clearly definable if you allow the gas to visit all states after changing a constraint. – Ignacio Mar 4 '20 at 20:40 • Even though I really like your quotes and I will check out the book I'm not sure if I see the conection with the minimum energy principle. – Ignacio Mar 4 '20 at 20:41 • I think Prof. Steane answered your question regarding the "equivalence" of the minimum energy and maximum entropy principles. My intent with the Pippard quote was to help you with any more lingering doubts you may have had. – hyportnex Mar 4 '20 at 22:05 • Regarding your keen observation about quickly plugging the hole and its effect on entropy change I suggest that you read Pippard's discussion following the quoted paragraph above. Enjoy! – hyportnex Mar 4 '20 at 22:09 • Thanks, yes, I guess I'll end up accepting that answer, its not 100% convincing to me but maybe I just have not sat on it enough. – Ignacio Mar 4 '20 at 22:18	2021-06-20 00:09:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 62, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8032331466674805, "perplexity": 250.49486827857817}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487653461.74/warc/CC-MAIN-20210619233720-20210620023720-00299.warc.gz"}
https://www.statemath.com/2021/08/how-to-find-the-eigenvalues-of-a-matrix.html	We learn to you how to find the eigenvalues of a matrix as well as the associated eigenvectors. This notion is very useful because it makes it possible to simplify the matrices only by changing the base of the space. We can prove in some cases that some matrices are similar to diagonal matrices, so matrix power calculations become simple in such cases. We assume that you are familiarized with the matrix operations. ## The answer to how to find the eigenvalues of a matrix. Exercise: Determine the eigenvalues and the associated characteristic spaces of the following matrix: \begin{align} A=\begin{pmatrix} 2&-1&1\\ -1&2&-1\\ -1&1&0\end{pmatrix}\end{align}. Solution: If we denote by $\sigma(A)$ the set of all eigenvalues of $A$ ”called also the spectrum of $A$”, then by definition we have \begin{align} \sigma(A)&=\{\lambda \in\mathbb{C}: \lambda I_3-A\;\text{is not injective}\}\cr &= \{\lambda\in\mathbb{C}:\det( \lambda I_3-A)=0\}. \end{align} For any $\lambda\in\mathbb{C}$ we have \begin{align} \det( \lambda I_3-A)= \begin{vmatrix} \lambda-2&1&-1\\1&\lambda-2&1\\1&-1&\lambda\end{vmatrix}. \end{align} We know that the determinant is unchanged if we replace any line of the matrix with a combination of the other lines “also if we replace any column with a linear combination of the others columns”. We denote by $L_i$ for $i=1,2,3,$ the lines of any matrix of order $3$. By replacing the line $L_1$ by $L_1+L_3$, we obtain \begin{align} \det( \lambda I_3-A)&= \begin{vmatrix} \lambda-1&0&\lambda-1\\1&\lambda-2&1\\1&-1&\lambda\end{vmatrix}\cr &= (\lambda-1)\begin{vmatrix} 1&0&1\\1&\lambda-2&1\\1&-1&\lambda\end{vmatrix} \end{align} In the lest determinant we replace the line $L_2$ by $L_2-L_3,$ we obtain \begin{align} \det( \lambda I_3-A)&= (\lambda-1)\begin{vmatrix} 1&0&1\\0&\lambda-1&1-\lambda\\1&-1&\lambda\end{vmatrix}\cr &=(\lambda-1)^2\begin{vmatrix} 1&0&1\\0&1&-1\\1&-1&\lambda\end{vmatrix} \end{align} We replace the line $L_3$ by $L_3+L_2-L_1$, we obtain \begin{align} \det( \lambda I_3-A) &=(\lambda-1)^2\begin{vmatrix} 1&0&1\\0&1&-1\\0&0&\lambda-2\end{vmatrix}\cr &= (\lambda-1)^2(\lambda-2). \end{align} Then the matrix $A$ process two eigenvalues “$\lambda_1=1$ is a double eigenvalue and $\lambda_2=2$ is a simple eigenvalue”. Let us denote by $E_1$ the characteristic space associated to eigenvalue $\lambda_1=1$ and $E_2$ the characteristic space associated to eigenvalue $\lambda_2=2$. By definition we have \begin{align} E_1=\ker(I_3-A),\quad E_2=\ker(2I_3-A). \end{align} Then \begin{align} X=\left(\begin{smallmatrix}x\\y\\z\end{smallmatrix}\right)\in E_1&\;\Longleftrightarrow\; A X=X\cr &\;\Longleftrightarrow\;\begin{cases} x-y+z=0\\ -x+y_z=0\\ -x+y-z=0\end{cases} \cr &\;\Longleftrightarrow\; x-y+z=0 \cr &\;\Longleftrightarrow\; X=\left(\begin{smallmatrix}y-z\\y\\z\end{smallmatrix}\right) \cr &\;\Longleftrightarrow\; X=\left(\begin{smallmatrix}y\\y\\0\end{smallmatrix}\right)+\left(\begin{smallmatrix}-z\\0\\z\end{smallmatrix}\right)\cr &\;\Longleftrightarrow\; X=y\left(\begin{smallmatrix}1\\1\\0\end{smallmatrix}\right)+z\left(\begin{smallmatrix}-1\\0\\1\end{smallmatrix}\right)\cr &;\Longleftrightarrow\; X\in {\rm span}\left\{\left(\begin{smallmatrix}1\\1\\0\end{smallmatrix}\right),\left(\begin{smallmatrix}-1\\0\\1\end{smallmatrix}\right)\right\}. \end{align} This shows that \begin{align} E_1={\rm span}\left\{\left(\begin{smallmatrix}1\\1\\0\end{smallmatrix}\right),\left(\begin{smallmatrix}-1\\0\\1\end{smallmatrix}\right)\right\}. \end{align} Now we calculate $E_2$. Let $X=\left(\begin{smallmatrix}x\\y\\z\end{smallmatrix}\right)$. Then \begin{align} X\in E^2&\;\Longleftrightarrow\; AX=2X\cr &\;\Longleftrightarrow\;\begin{cases} -y+z=0\\ -x-z=0\\-x+y-2z=0\end{cases} \cr &\;\Longleftrightarrow\; y=z,\quad x=-z \cr &\;\Longleftrightarrow\; X=\left(\begin{smallmatrix}-z\\z\\z\end{smallmatrix}\right)\cr &\;\Longleftrightarrow\; X=z\left(\begin{smallmatrix}-1\\1\\1\end{smallmatrix}\right)\cr &\;\Longleftrightarrow\; X\in {\rm span}\left\{\left(\begin{smallmatrix}-1\\1\\1\end{smallmatrix}\right)\right\}. \end{align} Hence \begin{align} E_2={\rm span}\left\{\left(\begin{smallmatrix}-1\\1\\1\end{smallmatrix}\right)\right\}. \end{align} ## Eigenvalues of endomorphism Exercise: Determine eigenvalues and eigenvectors of the endomorphism \begin{align}f:\mathbb{C}&[X]\longrightarrow \mathbb{C}[X]\cr & P\longmapsto f(P)=P-(X-1)P’.\end{align} Solution: Let $\lambda\in\mathbb{C}$ be an eigenvalue of $f$. Then there exists a polynomial $P\in \mathbb{C}$ non-null such that $f(P)=\lambda P$. This means that \begin{align} (1-\lambda)P-(X-1)P’=0. \end{align}If $\lambda=1$ then we have $P’=0$. So $P$ is a non-null constant. Assume that $\lambda\neq 1$. Define the polynomial function $u(x)=P(x)$ for any $x\in (1,\infty)$. This implies that the function $u$ is the solution of the following differential equation \begin{align} (x-1)u’-(1-\lambda)u. \end{align} As $P\neq 0$ then $u$ is non null. Then \begin{align} \frac{u’}{u}=\frac{1-\lambda}{x-1}. \end{align} This is equivalent to \begin{align} u(x)=c(x-1)^{1-\lambda},\qquad c\in \mathbb{C}^\ast. \end{align} For $u$ to be a polynomial function it is necessary that $1-\lambda\in \mathbb{N}$. This means that $\lambda=1-n$ with $n\in\mathbb{N}^\ast$, because $\lambda\neq 1$. We then have $u(x)=c (x-1)^n$ for any $x\in (1,+\infty)$ and $n\in\mathbb{N}^\ast$. Then \begin{align} P=c(X-1)^n. \end{align} Conversely, if $n\in\mathbb{N}$ and $P=c(X-1)^n$ with $c\in \mathbb{C}^\ast,$ then $f(P)=(1-n)P$. Thus the eigenvalues of $f$ are $\{1-n:n\in\mathbb{N}\}$ and the associated eigenvectors are $P=c(X-1)^n$ with $c\in \mathbb{C}^\ast$.	2023-01-30 01:31:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9999397993087769, "perplexity": 688.5059994038038}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499790.41/warc/CC-MAIN-20230130003215-20230130033215-00447.warc.gz"}
http://travelbyjen.com/1ipz4iho/dielectric-constant-symbol-dbad7a	i.e. Then the equation formed will be. Related Post: Faraday has done many experiments on this topic. Here some material medium must be used. The higher the dielectric constant of a solvent, the more polar it is. The dielectric constant is a number without dimensions. Dielectric Constant Units: This electrical property is a dimensionless measure. Watch the recordings here on Youtube! K = Constant which values depends on the measurement units of F, Q1 and Q2 and characteristics of the dielectric insulating medium between two charged bodies. The symbol r thus appears somewhat confused as to whether it is a purely real quantity (dielectric constant) or a complex quantity (complex relative permittivity). The higher the dielectric constant of a solvent, the more polar it is. Because dielectric constant is relative, it has no units or dimensions. It is mathematically expressed as: $$\kappa =\frac{\varepsilon }{\varepsilon _{0}}$$ Where, κ is the dielectric constant; is the permittivity of the substance; 0 is the permittivity of the free space; Dielectric Constant Units. The dielectric constants (relative permittivities) of water, methanol, ethanol, butanol and acetone were measured at 91.3 kPa and (283.15 and 293.15) K and are reported here. A higher dielectric constant of the solvent correlates with a higher ability of the solvent to dissolve salts. In general, low dielectric constants (i.e., Polypropylene) result in a "fast" substrate while large dielectric constants (i.e., Alumina) result in … then why not metal, This is because of the large amount of heat produced will eventually melt the metal plate, (Will not be published) In particular, in Abraham’s notation this is called . The Dielectric Constant is a convenient way of discussing the permittivity of materials. What is dielectric constant or electrical permittivity? This is sometimes called . The term insulator is generally used to indicate electrical obstruction while the term dielectric is used to indicate the energy storing capacity of the material (by means of polarization). In any other medium: F = e2 /∈ rd2 where ∈ r is the dielectric constant. this creates a separation in charge, that has its own field which interacts with the incident field. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. It is also known as Relative permittivity. Dielectric constant (Dk or relative permittivity) is a parameter that design engineers use constantly, often without fully understanding it. Again one thing to notice is that the dielectric constant is represented by the symbol(K) but permittivity by the symbol. EpsInf: α: Abraham’s hydrogen bond acidity. The rationale behind allowing such confusion is that the imaginary part of relative permittivity is small compared to the real part. The dielectric constant (sometimes called the ‘relative permittivity’) is the ratio of the permittivity of the dielectric to the permittivity of a vacuum, so the greater the polarisation developed by a material in an applied field of given strength, the greater the dielectric constant will be. The dipole moment of water is higher than that of methanol; water is more polar than methanol. (Required). The dielectric constant (symbol: ε) of a solvent is a measure of its polarity. Permittivity is typically denoted by the symbol . The first term is still called dielectric constant, while the second term is called the imaginary part of dielectric constant. The dielectric constant (sometimes called the ‘relative permittivity’) is the ratio of the permittivity of the dielectric to the permittivity of a vacuum, so the greater the polarisation developed by a material in an applied field of given strength, the greater the dielectric constant will be. One practical consequence is a covalent solute dissociates into ions to a greater extent in water than in methanol. The Dielectric Constant, or permittivity - ε - is a dimensionless constant that indicates how easy a material can be polarized by imposition of an electric field on an insulating material. Relative permittivity is the ratio of "the permittivity of a substance to the permittivity of space or vacuum". Modeling of the capacitance measurements made with an atomic force microscope tip revealed a surface-layer dielectric constant of 2, compared with the bulk value of 80 for water. The dielectric constant - also called the relative permittivity indicates how easily a material can become polarized by imposition of an electric field on an insulator. One of the principal factors affecting the capacitance of a capacitor is the type of dielectric … The dielectric constant of all materials is compared with that of a vacuum. The role of dielectric constant ε of a solvent regarding its capability to dissolve ionized solutes was pointed out in Section 4.1 (the dielectric constant is typically given as the relative dielectric constant to vacuum). is the Relative Permittivity. 1 Solventmp bpD 4 20 n D 20 ε R D µ Acetic acid 17 118 1.049 1.3716 6.15 12.9 1.68 Acetone -95 56 0.788 1.3587 20.7 16.2 2.85 Acetonitrile -44 82 0.782 1.3441 37.5 11.1 3.45 This is the system used to define the SI units. Eps: n 2: The square of the index of refraction at optical frequencies (293 K). It is often called as relative permittivity and the SI unit of permittivity is Farad per meter (F/m). The dielectric constant (sometimes called the ‘relative permittivity’) is the ratio of the permittivity of the dielectric to the permittivity of a vacuum, so the greater the polarisation developed by a material in an applied field of given strength, the greater the dielectric constant will be. The higher the dielectric constant of a solvent, the more polar it is. Basic Physical Quantities And Their Units, Conductors Insulators and Semi Conductors. Generally, when σ / ωε′ ≪ 1 we consider the material to be a low-loss dielectric (although not exactly lossless), whereas σ / ωε′ ≫ 1 is associated with a good conductor; such materials with non-negligible conductivity yield a large amount of loss that inhibit the propagation of electromagnetic waves, thus are also said to be lossy media. Dielectric Constant Symbol Recall that the permittivity of a vacuum (that is, in outer space or where there is no atoms or material in a volume - also known as Free Space) is 8.854*10^-12 [Farads/meter]. 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Kohler San Souci Toilet Parts, Webos Smart Tv, Nutrition Test Singapore, Metal Outdoor Wall Decor, Https Source Unsplash Com User Erondu 1600x900, Homes For Sale In Madera, Ca,	2021-06-23 06:15:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7704182267189026, "perplexity": 1422.5378251175134}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488534413.81/warc/CC-MAIN-20210623042426-20210623072426-00220.warc.gz"}
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Note however that single newlines are important when formatting lists (see below). Please note a couple of things: • Although single newlines are ignored, you should make sure that there are no spaces at the beginning of each line. Lines starting with a space are treated as preformatted text (equivalent to using the <pre> tag). • You can add line breaks without creating a new paragraph by using the standard HTML <br> tag. ### Lists Bulleted lists are created by putting a '' at the beginning of the line (i.e., there must be no blank characters before it). An extra '' creates a list within a list. A newline will end the list (so you cannot have any newlines within any list item). What you type What you get * List item 1 * List item 2 Sublist item 1 Sublist item 2 * List item 3 • List item 1 • List item 2 • Sublist item 1 • Sublist item 2 • List item 3 Similarly, numbered lists are created by putting a '#' at the beginning of the line. What you type What you get # List item 1 # List item 2 #* Sublist item 1 #* Sublist item 2 # List item 3 1. List item 1 2. List item 2 • Sublist item 1 • Sublist item 2 3. List item 3 A third type of list is called a definition list. What you type What you get ;Term 1: Definition of term 1 ;Term 2: Definition of term 2 Term 1 Definition of term 1 Term 2 Definition of term 2 This is very useful if you want to have a subheader followed by indented text, but you don't want to create a new section. Incidentally, notice that you can indent any new paragraph by putting a colon at the beginning of the line. If you want to link to another wiki page, just put the title of the page within two square brackets, e.g., [[New Topic]]. • If you want the link to use different text, add a vertical bar after the page title and follow it with the replacement text. What you type What you get * [[New Topic]] * [[New Topic\|Same link, different text]]	2019-05-22 15:10:35	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5446755290031433, "perplexity": 2862.2935281289297}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256858.44/warc/CC-MAIN-20190522143218-20190522165218-00231.warc.gz"}
https://gmatclub.com/forum/newman-biked-around-a-square-garden-at-the-speeds-of-20-miles-hour-285464.html	GMAT Question of the Day - Daily to your Mailbox; hard ones only It is currently 16 Oct 2019, 05:02 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # Newman biked around a square garden at the speeds of 20 miles/hour Author Message TAGS: ### Hide Tags Senior PS Moderator Joined: 26 Feb 2016 Posts: 3341 Location: India GPA: 3.12 Newman biked around a square garden at the speeds of 20 miles/hour [#permalink] ### Show Tags 01 Jan 2019, 08:22 2 00:00 Difficulty: 25% (medium) Question Stats: 72% (01:59) correct 28% (01:49) wrong based on 45 sessions ### HideShow timer Statistics Newman biked around a square garden at the speeds of 20 miles/hour, 30 miles/hour, 40 miles/hour, and 60 miles/hour along the four sides. What was his average speed for the entire ride? A. 32 B. 35 C. 37.5 D. 40 E. 50 Source: Experts Global _________________ You've got what it takes, but it will take everything you've got Manager Joined: 28 Mar 2017 Posts: 67 Location: Sweden Concentration: Finance, Statistics Re: Newman biked around a square garden at the speeds of 20 miles/hour [#permalink] ### Show Tags 01 Jan 2019, 09:37 pushpitkc wrote: Newman biked around a square garden at the speeds of 20 miles/hour, 30 miles/hour, 40 miles/hour, and 60 miles/hour along the four sides. What was his average speed for the entire ride? A. 32 B. 35 C. 37.5 D. 40 E. 50 Source: Experts Global If the fastest speed 60mph took one hour, then 60 mph - 1 h 40 mph - 1,5 h 30 mph - 2 h 20 mph - 3 h Total miles travelled: 60 times 4 =240 miles Time spent 7,5 h 240/7,5=480/15 =32 >A Posted from my mobile device _________________ * * * Wish my good luck for 700 before Christmas! If you think my post provided any help, please give +1 kudos, it helps a lot! <3 NUS School Moderator Joined: 18 Jul 2018 Posts: 1022 Location: India Concentration: Finance, Marketing WE: Engineering (Energy and Utilities) Newman biked around a square garden at the speeds of 20 miles/hour [#permalink] ### Show Tags 01 Jan 2019, 09:41 1 $$Average_{Speed}$$ = $$\frac{Total Distance}{Time}$$ Let the distance be D. Since it is a square, the total distance is 4D $$Average_{Speed}$$ = 4D/{D/20+D/30+D/40+D/60} = 4D/15D120 = 84 = 32 _________________ Press +1 Kudos If my post helps! Manager Joined: 05 Oct 2017 Posts: 101 Location: India Schools: ISB '21, IIMA , IIMB GPA: 4 WE: Analyst (Energy and Utilities) Re: Newman biked around a square garden at the speeds of 20 miles/hour [#permalink] ### Show Tags 01 Jan 2019, 10:18 pushpitkc wrote: Newman biked around a square garden at the speeds of 20 miles/hour, 30 miles/hour, 40 miles/hour, and 60 miles/hour along the four sides. What was his average speed for the entire ride? A. 32 B. 35 C. 37.5 D. 40 E. 50 Source: Experts Global Since newman biked around a square garden so distances travelled is the same in each case. So average speed will the arithmetic mean of the 4 speed. i.e (20+30+40+60)/4 = 150/4 = 37.5 (C) Manager Joined: 28 Mar 2017 Posts: 67 Location: Sweden Concentration: Finance, Statistics Re: Newman biked around a square garden at the speeds of 20 miles/hour [#permalink] ### Show Tags 01 Jan 2019, 10:27 shuvodip04 wrote: pushpitkc wrote: Newman biked around a square garden at the speeds of 20 miles/hour, 30 miles/hour, 40 miles/hour, and 60 miles/hour along the four sides. What was his average speed for the entire ride? A. 32 B. 35 C. 37.5 D. 40 E. 50 Source: Experts Global Since newman biked around a square garden so distances travelled is the same in each case. So average speed will the arithmetic mean of the 4 speed. i.e (20+30+40+60)/4 = 150/4 = 37.5 (C) That is the trap answer. You miss out on the time spent per side is not the same Posted from my mobile device _________________ * * * Wish my good luck for 700 before Christmas! If you think my post provided any help, please give +1 kudos, it helps a lot! <3 Manager Joined: 05 Oct 2017 Posts: 101 Location: India Schools: ISB '21, IIMA , IIMB GPA: 4 WE: Analyst (Energy and Utilities) Re: Newman biked around a square garden at the speeds of 20 miles/hour [#permalink] ### Show Tags 01 Jan 2019, 10:50 ErikLewe wrote: shuvodip04 wrote: pushpitkc wrote: Newman biked around a square garden at the speeds of 20 miles/hour, 30 miles/hour, 40 miles/hour, and 60 miles/hour along the four sides. What was his average speed for the entire ride? A. 32 B. 35 C. 37.5 D. 40 E. 50 Source: Experts Global Since newman biked around a square garden so distances travelled is the same in each case. So average speed will the arithmetic mean of the 4 speed. i.e (20+30+40+60)/4 = 150/4 = 37.5 (C) That is the trap answer. You miss out on the time spent per side is not the same Posted from my mobile device oh yes.a blunder on my part.thanks.. The solution will be Total distance/total time If length of each side = d Average speed = 4d / (d/20+d/30+d/40+d/60) = 4/((6+4+3+2)/120) = 4 * 120/15 = 4*8 =32 hence A) Re: Newman biked around a square garden at the speeds of 20 miles/hour [#permalink] 01 Jan 2019, 10:50 Display posts from previous: Sort by	2019-10-16 12:02:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7394341230392456, "perplexity": 12001.137765342393}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986668569.22/warc/CC-MAIN-20191016113040-20191016140540-00290.warc.gz"}
http://nsono.net/taming-javascripts-this-keyword/	# Arlen Stalwick Software Developer Hey there, I'm Arlen Stalwick, software architect and full-stack developer at Wavo.me in Montreal. I write about whatever I'm trying to teach myself. Today, it's Machine Learning. My last blog entry, Understanding Javascript's 'this' Keyword, explained how Javascript's this keyword is different from other languages, and what kinds of pitfalls to expect when using it. Here, I'll show you some techniques for getting this under control. VAR SELF = THIS Natively, Javascript has a few ways to help you deal with your this keyword. The first is quite simple: assign it to a variable, and take advantage of scoping. Note that this really only applies to inline functions, not so much to functions on an object. var myValue = 'global value'; var Obj = function() {}; Obj.prototype.myValue = 'object value'; Obj.prototype.myFunction = function() { // store this in self, so that we can use it in the setTimeout function. var self = this; setTimeout(function() { // alerts 'object value'. self contains the value of this that we had in myFunction // alerts 'global value'. when setTimeout calls the function, the calling context is lost and this is assigned to the global scope. }, 100); } // run the code var myObj = new Obj(); myObj.myFunction(); Lets go through this and understand what's happening. As you're aware, setTimeout() can cause havoc with your this, because it executes a function without the context of its calling object. That means that this is assigned the global scope, not your object. Here, we're assigning self = this before we call setTimeout. We're taking advantage of Javascript's scoping - the fact that inner functions can access the local variables of their outer functions. So when we assign self = this, we're storing the value of myFunction's this, and subsequently using that stored value, self, inside of our setTimeout function. Even though this inside our setTimeout will be incorrect, we've saved the correct this in self, so we're all good. This isn't a solution to everything, but it's a quick way to ensure that your inline anonymous functions have a correct this. You still have to be careful when calling myFunction(), though, because all you're doing here is storing myFunction's this in the self variable. If myFunction's this is incorrect (as it would be if you called myFunction itself with a timeout or passed it as a callback), then var self = this; will do nothing to help. CALL AND APPLY Next up, there's Javascript's call and apply functions. These functions allow you to call a function while specifying a this. The only real difference between call and apply is that apply allows you to supply the arguments as an array; call expects them as individual arguments. Example: // set a value in the global scope, so that we know when this == global var myValue = 'global value'; // define a test object var Obj = function() {}; Obj.prototype.myValue = 'function value'; Obj.prototype.myFunction = function() { } // instantiate the test object. var myObj = new Obj(); // call myFunction() via myObj. // assign myFunction to a local variable, then call it. var fun = myObj.myFunction; // now, use Javascript.call to execute the local copy of myFunction // Lets get a bit tricker: define another test object. var Obj2 = function(){}; Obj2.prototype.myValue = 'OTHER function value'; var myOtherObj = new Obj2(); // Use call to override the default 'this' behaviour myObj.myFunction.call(myOtherObj); // alerts 'OTHER function value' BIND Another useful technique is to bind your this to a function. There are several ways to do this. The first, most ideal way is to simply use ECMAScript 5's bind function. As with call and apply, you specify a this (and arguments, if you like) to your function. Unlike call and apply, the function is returned, not executed, so you can save it for later. When this new function is executed, it will be executed with your this applied. Since bind is ECMAScript 5, browser support is less than perfect (go nuts in Node.js, however!). As such, there are a number of libraries that will be happy to help you out. underscore.bind(), prototype js bind(), etc... Most library bind() functions are simply wrappers around ECMAScript 5's bind that fill in the gaps when ECMAScript 5 is not available. (Beware of library functions named bind used for binding events... Not the same thing!) CLOSURES How do those libraries mimic ECMAScript 5's bind when it's not available? Closures. I intend to go deep on closures in another blog post, but briefly, a closure in Javascript is when a function contains a set of local variables, declares another function (which has access to those outer variables), and returns that inner function. Even though execution of the outer function is complete, and it has gone out of scope, the returned function will continue to have access to the local variables declared in the outer function. This allows you to do all kinds of neat things like, for example, create a wrapper around your function so that its this remains consistent. function bind(context, functionToBind) { return function() { return functionToBind.apply(context,arguments); } } var myValue = 'global value'; // define a test object var Obj = function() {}; Obj.prototype.myValue = 'function value'; Obj.prototype.myFunction = function() { } var myObj = new Obj(); // bind myFunction's this. we replace the original function with the wrapped version. myObj.myFunction = bind(myObj, myObj.myFunction);	2021-07-24 22:36:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22386831045150757, "perplexity": 3396.712735464084}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046151531.67/warc/CC-MAIN-20210724223025-20210725013025-00377.warc.gz"}
http://clubita.it/dojm/roman-initials.html	Roman Initials , and III in modern American naming: its presence indicated that the bearer came from an old and prestigious, or would-be prestigious, background. –4th century A. = Augustinians of the Assumption, Assumptionists A. Appendix D defines the special meanings of the Greek and Roman letters used in the book's notation. And that inspired me to do something nice, deep breath and said "today are something nice" and so I present my editor has 7 cute letters fonts as a result I hope you like them. Literally, "The Senate and the People of Rome. If you need to make conversion from Arabic numbers to Roman numerals, simply enter the number to the box on the right, and press the button 'Convert to Roman'. Injection molded lettering is more rigid than formed plastic and has a crisp profile that looks like solid metal. First, here is the value of each. com! The Web's largest and most authoritative acronyms and abbreviations resource. The Roman emperors were the rulers of the Roman Empire dating from the granting of the title of Augustus to Gaius Julius Caesar Octavianus by the Roman Senate in 27 BC, after major roles played by the populist dictator and military leader Julius Caesar. This is the number system in which 1 is represented by I, 2 by II, 3 by III, and so on. Find these at U2160-U2182 in the Number Forms character subset. com! 'Rhythm Of Mankind And Nature' is one option -- get in to view more @ The Web's largest and most authoritative acronyms and abbreviations resource. The official way of writing dates in parts of Eastern Europe (the parts I know of were Bulgaria and the Soviet Union) was very similar - the month was in Roman numerals. One of the first things to note is that there are seldom ever spaces between the letters. ' novus ordo. The letters Y and Z were taken from the Greek alphabet to write Greek loan words. We demonstrated this in example 2 above. net - free fonts download - free fonts online. The Vindolanda tablets (also known as Vindolanda Letters) are thin pieces of wood about the size of a modern postcard, which were used as writing paper for the Roman soldiers garrisoned at the fort of Vindolanda between AD 85 and 130. (British Museum, 1923–62). Labarum a late Roman military standard ornamented with the Christian monogram (Christogram). You searched for: times new roman! Etsy is the home to thousands of handmade, vintage, and one-of-a-kind products and gifts related to your search. Save an additional 20% with code 20CPN2120W. The Letters. To use Roman numbers instead, specify type="I" for uppercase Roman numbers (e. In the 4 th Century B. The Standard of a cavalry unit was emblazoned with the symbol of the serpent (Draconarius) while a legion of infantry was represented by a totemic animal. K: Woman whose rape and suicide led to the expulsion of the last King of Rome. In the dawn of the Renaissance, Italic and Roman typography were used differently, while today they are often mixed, and most typefaces such as the upright Roman style associate them with oblique letters and styles. Roman Numerals are a special kind of numerical notations that were earlier used by the Romans. This volume contains four short letters to a Roman Catholic priest whom Ironside met on a train. The book, published as a single printing of 2000 copies, was 12×8", 178 pages with one foldout. Closed 2 years ago. Reading and Dating Roman Imperial Coins by Zander Klawans has been the starting point for more Roman collectors than perhaps any other book of the last half century and the fact that it is still in print is a testament to it's value. When using these Greek letters in math formulae, it is advisable to set them in "roman" typeface in order to match the styling of other Greek letters in math mode. Because cetera implies inanimate objects, et al. It is the most used writing system in the world today. Early inhabitants before 1st millennium BC came from somewhere and settled. license or (at your option) any later version. The Roman Standard (Latin: Signum or Signa Romanum) was a pennant, flag, or banner, suspended or attached to a staff or pole, which identified a Roman legion (infantry) or Equites (cavalry). You may also like. Vindolanda was a Roman fort at Chesterholm, just south of Hadrian's Wall in northern England. Every roman pilar had a base and top to it. After watching the film we chose several elements of Roman life to investigate, one of which was the Roman army. This table gives the Greek letters, their names, equivalent English letters, and tips for pronouncing those letters which are pronounced differently from the equivalent English letters. Start studying Ancient Roman Vocabulary Words A-Z. (caps or lowercase). This online edition of the Vindolanda writing tablets, excavated from the Roman fort at Vindolanda in northern England, includes the following elements: Tablets - a searchable online edition of the tablets (volumes I and II) Exhibition - an introduction to the tablets and their context. The Bold and Heavy weights were added in 2003 to broaden the usefulness of this very popular typeface. Illustration about 11th Century Alphabet - Initials and Roman Numerals. The Roman scribes tended to slant these letters, particularly with the increased use of papyrus, in order to cope better with the grooved structure of their writing surface. They are sample dates as actually used on Roman provincial coins minted in Alexandria and other cities. (pertainym) roman type, roman letters, roman print 4. Times New Roman was designed by Stanley Morison, Victor Lardent and published by Monotype. The gladiators would fight on the "harena" sand in the center of the colosseum. To create this article, 17 people, some anonymous, worked to edit and improve it over time. Our Holy Database currently contains 236 Roman deity names and 173 unique articles. • ROMAN LETTERS (noun) Sense 1. Transposition Ciphers Transposition Ciphers. If you are learning about Roman Britain and covering persuasive writing, then these resources could be ideal. Only if it refers to a person or place, yes. The wording of the Prophet’s letters was similar. Archaeologists believe that the tablets date from between 85 and. Roman Numerals (Intermediate) FREE. How to use Roman in a sentence. Roman letters minnaar vitneforklaring Çevrimdışı Git živjeti od danas do sutra scarlet I meant no harm by it sulfato (m. Usually, though, I write it in full: 14 February 2010. Sacred - Something that is believed to be holy and to have a special connection with a god or gods. SKS Letters and Designs Miles Supply has access to all of the Scotch-Kut Lettering System’s letters and designs. The formed plastic sign letters in this typeface are available in the letter heights shown in the listing below. ஜ۩۞۩ஜ WELCOME TO COOL LETTERS NOTE: Some may look weird when used in-game. Times New Roman was designed by Stanley Morison, Victor Lardent and published by Monotype. The numerical value of Vav-Vav-Vav in Hebrew would be 6+6+6=18,. Coins of the Roman Empire in the British Museum, 6 vols. net - free fonts download - free fonts online. Roman numerals were the standard system of numbering used by the Romans in ancient Rome. Augustan historian. Your dad and I had our divorce mediation. D 118 and 125. ) de amonio férrico ceying landing 郵便差入れ口 [ゆうびんさしいれぐち] weneryczny Kiste Island Do të doja një dhomë në katin e dytë. 8" paper mache letter by artminds® $0. They are also known by the names Milesian numerals , Alexandrian numerals , or alphabetic numerals. For instance, the letters Y and Z were added in the first century BCE and the letter W was created by the German speaking areas. Sorry but all of our letter sizes are sold by the complete set only!. A third edition (from 2003) was reprinted in 2006 and 2008. Each italic design was a single-weight family, containing only lowercase characters: no caps, no numbers, no punctuation. K: Woman whose rape and suicide led to the expulsion of the last King of Rome. If one or more letters are placed after another letter of greater value, add that amount. SKS Letters and Designs Miles Supply has access to all of the Scotch-Kut Lettering System’s letters and designs. Roman letters definition: a typeface used in ancient Roman inscriptions \| Meaning, pronunciation, translations and examples. Cardinal: Appointed by the pope, 178 cardinals worldwide, including 13 in the U. It has the roman numerals in descending order from 1000 to 1. The major alphabets from that time span (in chronological order) are: Roman, uncial, blackletter, italic, and bookhand. So, instead of having 26 letters of different widths, there are in fact only 4 groups of letters. See Latin alphabet. Greek philosophy and rhetoric moved fully into Latin for the first time in the speeches, letters and dialogues of Cicero (106-43 B. Roman numerals are arranged and combined in a specific order to represent numbers. This table lists the Greek letters, their names, equivalent English letters, and tips for pronouncing those letters which are pronounced differently from the equivalent English letters. I have a code that works for the most part but, the answer it is achieving for the first and second test numeral are wrong. Ambrose University Place of Publication Davenport, Iowa Date. 55 - This WGL4 version of Times New Roman was first supplied with the Final Windows 95 euro update that shipped on 4 November 1998. – Roma (Rome) Rescr. The simplest way to type Roman numerals on a computer is to use the lookalikes of Roman numbers that are present in the Latin alphabet (the alphabet that's present on English language keyboard). 8" paper mache letter by artminds®$0. Modern usage employs seven symbols, each with a fixed integer value:. The ancients, indeed both Greeks and Romans, in order to bring their monetal inscriptions within the smallest space, adopted the use of siglae. The most popular letters / numbers model railroad decals brands include Woodland Scenics, Microscale Inc, and Tichy Train Group. 87,956 downloads. One of the first things to note is that there are seldom ever spaces between the letters. Today, the phrase is used to refer generally (and sometimes ironically) to the power and glory of a major nation. ıllı ᑕOᑭY ᗩᑎᗪ ᑭᗩᔕTE ᖴOᑎTᔕ ıllı Symbols ABC 123. Archaeologists believe that the tablets date from between 85 and. The letters reveal just how significantly links to the powerful still determined success in Roman society. I've never seen that, but I sometimes use lower-case Roman numerals for the month, as in 14. It was adapted from the Etruscan alphabet during the 7th century BC. The Praenomen was a Roman's personal name, only used among family and close friends. Download 111 Free Greek Roman Fonts. The Roman Standard (Latin: Signum or Signa Romanum ) was a pennant, flag, or banner, suspended or attached to a staff or pole, which identified a Roman legion (infantry) or Equites (cavalry). The Roman Font is available in a size of 1", 2" and 3" sizes. Roman 1051 is a crossword puzzle clue that we have spotted 12 times. Alien Animals Asian Ancient Runes, Elvish Esoteric Fantastic Horror Games Shapes. , Times Roman) for the B-level subheads. Torque & Circular Motion. ISBN-10: 1939272254. Calligraphy School Handwritten Brush Trash Graffiti Old School Various. At each level of the hierarchy below the title, if only one subheading is provided, that heading should be either omitted or incorporated in the next-higher-level heading; if there is an "A" heading there must also. Enter the Roman numeral or number and press the Convert button: #N#Roman numerals Roman numerals conversion table. 272+ results for modern roman letters a-z Related keywords (2) modern roman letters-272 modern roman letters-272. Since a letter is also a written proof or document, if there are certain things that should not form a part of it, then that too should. –4th century A. It is my first choice to maintain the address Father William Walters or The Reverend Father William Walters in preference to simply The Reverend in the formal or written address. Answer: The Roman Empire was the human political entity that God used to prepare the world for the birth of the Messiah and for the spread of the gospel. But they were only upper case. A symbol placed after another of equal or greater value adds its value; e. Apparently Roman ladies used to collect the sweaty gloop from athletes and gladiators and use it in a face pack - but no one offered to try mine! Top The wonderful wheel. Sans serif Serif Fixed width Various. 5" greek letter by artminds™ Save an additional 20% with code 20CPN2120W. The Letters. I am using MiKTex 2. As you can see, the font letters are thinner and much more legible. All your abbreviations of dates in manuscripts or on specimen labels should be written in order DAY/MONTH/YEAR as 4 JUL 1999 or as 4-VII-1999. A guide to online and print library resources for the study of all aspects of Greco-Roman antiquity at Yale University. List of all most popular abbreviated Catholic Church terms defined. See Main Page for a guide to all contents of all sections. Plenty of Greek letters with Roman super/subscripts in papers, though the super/subscripts are almost universally in roman font instead of the default TeX font, something I was told by my senior year project advisor. Roman letters and alphabet. , Times Roman) for the B-level subheads. Two such styles were combined into one script with upper and lower case letters ('capitals' and 'small letters'). In the Roman alphabet all letters were formed between two parallel lines and they all had the same height. Enter the answer length or the answer pattern to get better results. Unlike Old English, roman letters font designs have a much more uplifting look and feel. The numerical value of Vav-Vav-Vav in Hebrew would be 6+6+6=18,. Abbreviations seen after names of religious and clergy. Improve your math knowledge with free questions in "Roman numerals I, V, X, L, C, D, M" and thousands of other math skills. Decorated Roman Initials. They are, in this order, from lower to higher: I, V, X, L, C, D and M. Start studying roman letters- stage 35 culture. The Greek alphabet is a set of twenty four letters used to write and speak the Greek Language. Roman definition is - a native or resident of Rome. Download Image of map from "[Rome in the Nineteenth Century: containing a complete account of the ruins of the ancient city, the remains of the middle ages, and the monuments of modern times. For over 4,000 colorized versions of of these letters, please see the Drop Caps section of our Presentations ETC website. We encourage you to choose a personal, unique name to identify you, and to express your Roman identity, unique and indivitual to you. " is appropriate. Latin script, or Roman script, is a set of graphic signs (script) based on the letters of the classical Latin alphabet, a form of the Cumaean Greek version of the Greek alphabet. Prieur Michel Prieur and Karin Prieur, A Type Corpus of the Syro-Phoenician Tetradrachms and Their Fractions from 57 BC to AD 253. Roman writing (rustic capitals) - letter spacing. Sacred - Something that is believed to be holy and to have a special connection with a god or gods. Since my list of abbreviations has gone up to two pages, LaTeX now print arabic page number on the second page of list of abbreviations, which looks very odd to me. Room Reservations. It was adapted from the Etruscan alphabet during the 7th century BC. Thousands of new, high-quality pictures added every day. Scenic & Theming Prop Studio in Orlando, Florida. 80 Authentic Ancient 383AD ARCADIUS CHI-RHO i67188 Roman & VICTORY w Coin ANGEL ANGEL Coin w Authentic Roman & i67188 Ancient VICTORY ARCADIUS 383AD CHI-RHO. Simply type in Roman and it will convert into Unicode Nepali. A second edition appeared in 1998. Each "T" Styled Letter Forms The First Letter Of Two Separate Words. If you need to make conversion from Arabic numbers to Roman numerals, simply enter the number to the box on the right, and press the button 'Convert to Roman'. View Answer. Roman letters are in the ordinary style of printed writing in which the letters are vertical. The Roman Alphabet (Latin) •the Etruscans were the first people in Italy to use the alphabet – from the Greeks • then, the Etruscans dominated the early Romans and gave them the alphabet, among other things. Sorry but all of our letter sizes are sold by the complete set only!. Old English, on the other hand, is a much larger text type with very large. The Great Seal's official description says: "On the base of the pyramid are the numerical letters MDCCLXXVI and underneath, the following motto. The Decorated Roman Initials font has been downloaded 5,029 times. Officina identification. – Responsorium ("Responsory" — Breviary) R. One of the most popular and common fonts of all time. Use the paintbrush and water to paint the number/letter. Download high quality Roman Numerals clip art from our collection of 41,940,205 clip art graphics. Of or relating to ancient or modern Rome or its people or culture. Updated April 2020. This is a list of common Latin abbreviations. net - free fonts download - free fonts online. If you mean single Roman letters, they can either stand for numbers or abbreviations. Online visit. Oct 3, 2014 - Explore dpcalligraphy's board "Roman Letters", followed by 222 people on Pinterest. – Roma (Rome) Rescr. 302+ results for old style roman letters Related keywords (4) old style roman letters a-z-302 old style roman letters. Roman letters: used for abbreviations, formulas, emphasis. Family Packages Tech Specs Licensing. Roman numerals are expressed by letters of the alphabet and are rarely used today except for formality or variety. Whether you are trying to learn how to read and write Roman numerals, trying to find a fancy way to write your birth year, or if you just need a 'cheat sheet' for quick reference, each Roman numerals chart on this page will have you working with this ancient number system in no time flat. Go on a counting hunt. The Story Continues on the Reverse. Injection molded lettering is more rigid than formed plastic and has a crisp profile that looks like solid metal. Roman law, as revealed through ancient legal texts, literature, papyri, wax tablets and inscriptions, covered such facets of everyday Roman life as crime and punishment, land and property ownership, commerce, the maritime and agricultural industries, citizenship, sexuality and prostitution, slavery and manumission, local and state politics, liability and damage to property, and the preservation of the peace. The table below lists the major Roman mints and their marks. This bar-code number lets you verify that you're getting exactly the right version or edition of a. Mint city abbreviation (usually one to four letters, but up to seven). These beautiful. This article needs additional citations for verification. The wooden ink documents were found on June 22 at the Roman fort of Vindolanda just south of Hadrian's Wall in Northern England. By the 7th century BCE, that alphabet was used not just to. One double-spaced line below “ABSTRACT”, center your name, followed by a colon and the title of the thesis or dissertation. Can be placed over different backgrounds. In the table below is a list of the Greek Gods and Heroes and their Roman equivalents:. Although it was really upsetting for me to to have to give up any time with you and Roman, I agreed to giving your Dad a lot more time with you than the "Standard Possession Order" we were practicing temporarily. Roman Numerals Conversion Game & Practice Roman Numerals Worksheets Roman Numerals Worksheets Pdfs Roman Numerals1-20 Roman Numerals1-100. I want to make roman page numbering up till list of abbreviations. The first mention of papal archives is found in the Acts of a synod held about 370 under Pope Damasus I ( Coustant , "Epistolæ Romanorum Pontificum. They were frequently inscribed on monuments. Roman numeral tattoos can look ultra-stylish using creative fonts and bold lettering. Wide Letters: M. 8 (10 Inch Times Roman Letters) 1987 New Bright The Cat Power Dumper 777B Caterpillar Batt Op Dump Truck READ. The ROMAN function is a built-in function in Excel that is categorized as a Math/Trig Function. Capable of withstanding severe stress and abuse and will not destruct when crumpled or folded. You searched for: roman monogram! Etsy is the home to thousands of handmade, vintage, and one-of-a-kind products and gifts related to your search. ABBREVIATIONS. The Trajan column, located between the Greek and Latin libraries in front of the Basilica Ulpia in the Forum of Trajan, is a doric column with a spiral frieze, carved in low relief, depicting Emperor Trajan’s own account of his conquest of Decebalus and the annexation of Dacia (the campaigns of 101–102 and 105–106 AD). In Roman notation, values are changed either by adding one or more symbols to the initial symbol or by subtracting a symbol to the right of it. Thankfully the Romans did not have a telephone system. Use the paintbrush and water to paint the number/letter. I’ll start with just a few of these terms so others can contribute. I am using MiKTex 2. com English Français Español Deutsch Italiano Português. The most popular letters / numbers model railroad decals brands include Woodland Scenics, Microscale Inc, and Tichy Train Group. How to Address a Sister or Nun Using Her Given Name and Family Name Envelope: Sister (full name), (initials of order) (Convent/institution) (Address) Letter salutation: Dear Sister: Dear Sister (given name): How to Address a Sister or Nun Using a Chosen Religious Name. Match letters in the synthetic scheme with Roman numerals representing the correct compounds. First, here is the value of each. Hopefully you remember your Roman numerals from grade school. Browse and search thousands of Latin Abbreviations and acronyms in our comprehensive reference resource. This has to do with the way Romans wrote: in our modern society we tend to regard paper as an easily available commodity, but in the ancient world this was not the case. government is coordinating over issuing recommendations to use the “surname-first” order when names are written in Roman letters. The Colosseum in Rome is one of the most famous ruins from the Roman era. Below are Total 34 words made out of this word. Originally, italic letters were not designed to complement a Roman typeface. Clue: Roman 1051. Roman - Coin Ancient - Aequitas Carinus, Mint - Reverse Augustus as Rome Rome Reverse as Augustus Roman Carinus, Mint Aequitas - - Ancient Coin - Ancient Roman Empire, Vespasian, 71 AD. Letter from a Roman Soldier. After her conversion, Edith spent her days teaching, lecturing, writing and translating, and she soon became known as a celebrated philosopher and author, but her own great longing was for the solitude and contemplation of Carmel, in which she could offer herself to God for. The Crossword Solver finds answers to American-style crosswords, British-style crosswords, general knowledge crosswords and cryptic crossword puzzles. This table gives the Greek letters, their names, equivalent English letters, and tips for pronouncing those letters which are pronounced differently from the equivalent English letters. If you are learning about Roman Britain and covering persuasive writing, then these resources could be ideal. Many of our customers consider brass Letters to be some of the most elegant looking letters we carry. Table Font: Select Font Times / Times New Roman Palatino Century Schoolbook Georgia Helvetica / Arial Verdana Tahoma Courier Use Font. Custom preview. 2005; 2006; 2007; 2008; 2009; 2010; 2011; 2012; 2013; Apostolic Constitutions. Intermediate level includes symbols I - C, or 1 - 100. 99 Add to cart. As you can see, the font letters are thinner and much more legible. This section is from the book "Cyclopedia Of Architecture, Carpentry, And Building", by James C. This simple Roman Numerals Converter can be used at any time to convert numbers to Roman numerals. com! The Web's largest and most authoritative acronyms and abbreviations resource. It is the most used writing system in the world today. Click the answer to find similar crossword. Studio3 formats. Alien Animals Asian Ancient Runes, Elvish Esoteric Fantastic Horror Games Shapes. Ongoing care. In written address, his initials would also follow his name, e. Latin was the language of Rome and its surrounding area (Latium) - hence the distinction between "Roman" (relating to the city) and "Latin" (referring to the language). Sans serif Serif Fixed width Various. Forum matches View 10+ forum results. Browse the list of 1 608 Roman abbreviations with their meanings and definitions. The Romans added some word spacing to divide the words into single units via dots placed midline. = Handmaids of the Sacred Heart of Jesus. Roman synonyms, Roman pronunciation, Roman translation, English dictionary definition of Roman. I II III IV V etc) and type="i" for lowercase Roman numbers (e. The Latin alphabet is used in variant forms by many languages, including Romance languages, Germanic, Celtic, some Slavic languages, Amerindian, Indigenous Australian, Austronesian, Vietnamese, Malay and Indonesian languages. Pluto is the conventional Roman name and you might use it for a trivia question, but really Pluto, a god of wealth, is the equivalent of a Greek god of wealth called Dis. I have a coin I would like appraised. Romans used a set of letters to represent numbers. Roman numerals can be seen on fancy clocks and watches. Transliteration 1. Augustan historian. These are vector SVG, DXF, EPS &. If a separate "readme" file accompanies the Work, is part of all distributions of LaTeX version 2006/05/20 or later. Identifying Common Late Roman Bronze Coins ©2003 Scott Uhrick for Ancient Coins for Education, Inc. The questions were extremely easy by Jeopardy standards and for someone with a decent command of Roman Numerals (which of course Jennings had) it shouldn't be that tough. Each said symbol represents a different number, in this order: one, five, ten, fifty, one hundred, five. With dozens of sparkling Swarovski Crystals, woven into a silver plated frame to make this. Themes New fonts. Roman Classic style refers to a straight letter with moderate stroke thickness and bracketed serif style letters. Used primarily for counting, they were adapted from the Etruscan numerals system. This article needs additional citations for verification. The last letter from Cicero possessed by us is dated not later than the 27th of July: he was murdered on the 7th of December. Download free decorated roman initials font, view its character map and generate text-based images or logos with decorated roman initials font online. Updated April 2020. Seeking some common ground between the two cultures, we could settle for the principle in Demetrius (or pseudo-Demetrius) de Elocutione that letters propose to bridge distance between. (caps or lowercase). Roman Numerals: Capitalization and Punctuation of Roman Numerals: The single capitalization indicator must be used before a Roman numeral consisting of one capitalized letter; the double capitalization indicator must be used before a Roman numeral consisting of two or more unspaced capitalized letters. “Many contain requests for favourite foods. Perhaps you could use local Roman remains to inspire you to create a model villa for example. Instructions and wordbox for writing a letter home to parents from the point of view of a roman soldier. The Roman scribes tended to slant these letters, particularly with the increased use of papyrus, in order to cope better with the grooved structure of their writing surface. Injection Molded Plastic Letters are very high quality plastic sign letters which are Guaranteed not to fade or break - for LIFE! These plastic sign letters are available in pigmented white and black, and any of our Standard colors will be painted (at NO extra cost) and carry the same guarantee. Each italic design was a single-weight family, containing only lowercase characters: no caps, no numbers, no punctuation. The Bold and Heavy weights were added in 2003 to broaden the usefulness of this very popular typeface. Prieur Michel Prieur and Karin Prieur, A Type Corpus of the Syro-Phoenician Tetradrachms and Their Fractions from 57 BC to AD 253. No matter what you’re looking for or where you are in the world, our global marketplace of sellers can help you find unique and affordable options. Metal Letters and Numbers If you're reading this you're trying to determine which custom metal letter or number to go with and we are here to help you out. Following is from documents of records of the Roman Senate. The Decorated Roman Initials font has been downloaded 5,029 times. ISBN-13: 978-1939272256. Times New Roman is a classic, well proportioned font. 103,835 downloads. Rutherford, with introduction and notes by R. We found a total of 33 words by unscrambling the letters in roman. When stimuli are perceived as complex or the task as difficult, analytic processing tends to occur ( Lachmann, Khera, Srinivasan, & van Leeuwen, 2012 ; Roelfsema & Houtkamp. Match letters in the synthetic scheme with Roman numerals representing the correct compounds. Authors Top. Roman Numerals Roman Numerals are the symbols that were used in Ancient Rome for counting and are a combination of letters from the Latin alphabet - I, V, X, L, C, D and M. The sounds of some letters changed, some letters were lost and gained, and several writing styles ('hands') developed. Roman letters definition: a typeface used in ancient Roman inscriptions \| Meaning, pronunciation, translations and examples. The Constitutio Antoniniana or Edict of Caracalla was a law passed in 212 AD. North American users can add it by. Times New Roman Pro Italic. Times contains 12 styles and family package options. The Standard of a cavalry unit was emblazoned with the symbol of the serpent (Draconarius) while a legion of infantry was represented by a totemic animal. Roman letters are antiqva letters, basicly, they are formed using pilars. I have a coin I would like appraised. Paul Russell, Alex Mullen. The Roman alphabet is also used as an alternate—but secondary—writing system for Chinese as Pinyin, and for Japanese as Romaji. LA Times Sunday - September 16, 2012. Classic Roman cast metal sign letters provide a quality, permanent image for outdoor signage of business names or address numbers. Roman numerals may be used up to three times in succession. 50 Select options; Victorian Font Letters 6mm 18mm £1. Login \| Register. Modern Standard Arabic is written horizontally from right to left and uses twenty eight unicameral letters, that is letters that have no definition between upper and lower case, which is known as Haskh script. The bronze or aluminum metal wall letters can be custom made in different sizes and finishes. The abbreviations and absence of spaces can be a bit confusing at first, but as you learn what the common abbreviated inscriptions are you’ll quickly become. A guide to online and print library resources for the study of all aspects of Greco-Roman antiquity at Yale University. As you might have noticed, the classical Roman Latin alphabet only has what we called "upper case", or majuscule, letters. The Greek alphabet is in use since the late 9th or early 8th century BC with some small differences. --The legends and inscriptions of Roman coins, as well as imperial as consular, present many particularities, in the shape of abbreviations, monograms, and isolated letters, open research, and susceptible of various explanations. By studying tombstone inscriptions we can get to know individuals as well as learn about the movement of troops around the Roman Empire. Start studying roman letters- stage 35 culture. So one must take utmost care ensuring that the language used in any letter is appropriate for the occasion for which it is being written. Yes, from the Roman alphabet or Latin alphabet, but I think "Roman alphabet" is usual, because "romanization" is a better word than "latinization". But some fonts do have a range of "built-up" roman numerals. ISBN-13: 978-1939272256. Each letter is manufactured from high quality Baltic Birch plywood that comes in a variety of thicknesses. Usually, though, I write it in full: 14 February 2010. To date a coin, you need to know the start date for the era. List of all most popular abbreviated Catholic Church terms defined. January 27, 2020. Boston; David Godine, 1982. We've arranged the synonyms in length order so that they are easier to find. Ancient scripts gothic old italic alphabets and languages letter z latin alphabet caron png 1280x524px the latin alphabet latin alphabet from a to z and numbers stock vector murr The Latin AlphabetThe Latin AlphabetClical Latin AlphabetThe Latin AlphabetThe Roman Alphabet Ancient Rome WritingThe Latin AlphabetWhat Is The Difference Between Greek… Continue Reading Roman Alphabet Letters A To Z. Famous ancient_roman quotes Showing top results. Bowman (born 1944) is professor emeritus of ancient history at Oxford University. This font's license appears to allow you to use @font. The questions were extremely easy by Jeopardy standards and for someone with a decent command of Roman Numerals (which of course Jennings had) it shouldn't be that tough. (I got it free from Fonts 101. They're easy on the eyes, and allow you to read larger passages of the roman font text without getting annoyed. The Roman numerals class: /* An object of type RomanNumeral is an integer between 1 and 3999. 1 Pc, 5 Inch X 1/8 Inch Thick Times New Roman Bold Wood Letters L Great For Craft Project & Different DecorAll of our wood letters are 100% American made. Forum matches View 10+ forum results. 24, 2015) Introduction People certainly lived, died, and were mourned in Egypt before Augustus annexed it as a imperial possession in 30 BCE, but no one, it seems, wrote letters of condolence before then. Linotype offers many versions of this font: Times™ is the universal version of Times, used formerly as the matrices for the Linotype hot metal line-casting machines. deciphering tombstone inscriptions. In Women in the World of the Earliest Christians, Lynn Cohick urges us to consider women’s role in first-century Greco-Roman culture as one of complexity and nuance. Decorative Letters: A. A Constructed Roman Alphabet. Meaning of roman letters. Letters from a Roman Prison: Paul’s Letter to the Philippians and Philemon Excellent book to read along with this series Paul’s letter to the Christians at Philippi came to a people in a powerful Roman colony that prided itself in its freedom and status from Caesar. A cache of 25 Roman letters has been found at Vindolanda, the fort below Hadrian's Wall where the most famous documents from the Roman world were discovered in 1992, first-person accounts of. Like so → XVII. Old English, on the other hand, is a much larger text type with very large. Roman numerals chart shows how letters are used in place of numbers. Looking for the abbreviation of Roman Numeral? Find out what is the most common shorthand of Roman Numeral on Abbreviations. Below are listed the most common abbreviations that are used in the Sig. But some fonts do have a range of "built-up" roman numerals. The letters were written first from right to left and, on alternate lines, from left to right (the boustrophedon). We have 2 answers for this clue. The Roman calendar operated through the use of three main days (the Kalends, the Nones, and the Ides), in reference to which all dates were given. Antonyms for roman letters. Roman numerals are expressed by letters of the alphabet and are rarely used today except for formality or variety. Hobbylinc carries over 200 letters / numbers model railroad decals at discounts up to 31%. Oh God when you went forth before your people marching with them and living among them the Earth trembled heaven's poured down rain Hallelujah in the name of the father and of the son and of the Holy Spirit. Linotype offers many versions of this font: Times™ is the universal version of Times, used formerly as the matrices for the Linotype hot metal line-casting machines. To report errors or to have a new abbreviation added to the list, please send an email to this Email address. What differences between the Roman alphabet and their alphabet can they notice? Most students will be able to notice that some letters are missing. Washington Post - April 09, 2013. – Rural Dean; Req. Check out this Playful, Bold Logo Design for DB records (under DB, it should say "dancing bridge" in very small letters) \| Design: #22294126, Designer: Roman ORzul, Tags: Cool, Different, Company. ) 4 letters; Michael who played the son of Mogh on Star Trek: The Next Generation 4 letters; End of many a village name in Boer South Africa 4 letters; James journalist who won the Pulitzer in 1945 and. Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. (editor) Format/binding Hardcover Book condition Used - Near Fine Jacket condition Very Good Binding Hardcover ISBN 10 0962974005 ISBN 13 9780962974007 Publisher St. The Standard of a cavalry unit was emblazoned with the symbol of the serpent ( Draconarius) while a legion of infantry was represented by a totemic animal. Roman numerals are in their essence letters: I - 1 II - 2 III - 3 IV - 4 V - 5 VI - 6 VII - 7 VIII - 8 IX - 9 X - 10 L - 50 C - 100 D - 500 M - 1000 and so on… This year (2018) is MMXVIII. Why they were dumped in a well no one knows. The Romans added some word spacing to divide the words into single units via dots placed midline. December 29, 2019. The pope is infallible in defining matters of faith and morals. Rutherford, with introduction and notes by R. Frictional Coefficients. I am attempting to create a program that will convert roman numerals to decimals and decimals to roman numerals. The main difference between the Roman alphabet and our alphabet is that in the old Roman alphabet C and G were not distinguished, and neither were I and J , and neither were U , V and W. Suggest a Book for Purchase. Try changing name to something "normal") Visit the biggest discussion board on ascii art and letters! Rᴜʟᴇs sᴜᴍᴍᴀʀʏ: Dᴏɴ·ᴛ ᴀᴅᴠᴇʀᴛɪse, ʟɪɴᴋ ᴏᴛʜᴇʀ ɢʀᴏᴜᴘs, sᴄᴀᴍ, ғɪsʜ ғᴏʀ ғʀɪᴇɴᴅs, ғɪsʜ. The letters Y and Z were taken from the Greek alphabet to write Greek loan words. Any exceptions to this request are also given in the. The Kalends The first of the month, following the lunar part of the calendar's operation, was the day following the appearance of the New Moon. ஜ۩۞۩ஜ WELCOME TO COOL LETTERS NOTE: Some may look weird when used in-game. Clue length. It reached its "Golden Age" during the rule of Augustus and the early part of the Roman Empire. Numerals (their values) are added together when written in groups, so XX = 20 (because 10+10 = 20). (British Museum, 1923–62). Roman capital letters are monumental in two senses. The list of Roman abbreviations in. The Roman Standard (Latin: Signum or Signa Romanum) was a pennant, flag, or banner, suspended or attached to a staff or pole, which identified a Roman legion (infantry) or Equites (cavalry). Print and enjoy using these stencils for your fun projects!. New Arrivals Clearance. What are synonyms for roman letters?. Roman Imperial Attribution 101. The wording of the Prophet’s letters was similar. 00 Favorite. Roman numerals are a number system developed in ancient Rome where letters represent numbers. More Ways To Shop. Godine published the title as both a regular cloth-bound edition (Jan 1982) and a deluxe folio edition (Feb 1983). Not reallt - Answered by a verified Antique Expert. Forum matches View 10+ forum results. "Sig" is an abbreviation for the Latin word "signetur," which means "let it be labeled. They are an additive (and subtractive) system in which letters are used to denote certain "base" numbers, and arbitrary numbers are then denoted using combinations of symbols. These examples were simple, but there are a few rules and a few tricky things to know when using Roman numerals: The first rule just says that you add letters, or numbers, if they come after a bigger letter or number. Item Style Number: 2 Roman Letters and a Renaissance Single Heart ( S H R ) Set Includes: 2 Letters and 1 Heart Add class and elegance to any special event or wedding cake with these intricately styled rhinestone Romanesque cake initial letters and heart. Every individual Gothic letter has several quasi-authoritative shapes, and all of these variants may be accepted, as long as they display an intelligent conception of. @ElroyJetson TreeMap is sorting the keys by natural order. Times New Roman version history. Oct 3, 2014 - Explore dpcalligraphy's board "Roman Letters", followed by 222 people on Pinterest. One of the most satisfying parts of working with ACE is going to a school where the students are well into cleaning their coins and helping the kids identify them. But they were only upper case. From Herodotus and Livy. Roman Numerals Conversion Game & Practice Roman Numerals Worksheets Roman Numerals Worksheets Pdfs Roman Numerals1-20 Roman Numerals1-100. Arabic is spoken by around 175 million people in North Africa and the Middle East and is the liturgical language of Islam and one of the six official languages of the United Nations. Forum matches View 10+ forum results. According to legend, Romulus, the founder of Rome, instituted the calendar in about 738 B. Illustration about 11th Century Alphabet - Initials and Roman Numerals. Roman letters denoting numbers (Roman numerals) - is that what you are asking about? V =5; X =10; XX = 20 etc. category: math to roman numerals to initials to names The square root of 4000000 to this guy who hit more than 500 career home runs from 1986 to 2001 View Answer. Roman numerals chart shows how letters are used in place of numbers. His Heroides was a series of 15 letters supposedly written by Greek and Roman mythological female figures such as Penelope and Dido to their lovers who had either mistreated or abandoned them. Roman is an accepted word in Word with Friends having 9 points. The best guide to Roman site web projects. Roman type is one of three kinds of historical type of the Latin script (the others being blackletter and italics), so in that sense any typeface in the Latin script creates Roman numerals as long as it features capitals. Suggest a Book for Purchase. I II III IV V etc) and type="i" for lowercase Roman numbers (e. It looks roman but who knows. Since a letter is also a written proof or document, if there are certain things that should not form a part of it, then that too should. This database collects all the personal names from Roman Britain which are thought to contain Celtic elements. Atrium Central reception area in a domus with an open. When using Roman numerals, certain letters stand for specific numbers. In Nova Roma, too, we ask that you choose a Roman name when you become a citizen. This burial-place stretches square over one eighth of a Roman acre of land; the coffin is in the middle. They are, in this order, from lower to higher: I, V, X, L, C, D and M. Why they were dumped in a well no one knows. has spoken, and now American soldiers must obey the call to arms. The most popular letters / numbers model railroad decals brands include Woodland Scenics, Microscale Inc, and Tichy Train Group. Ambrose University Place of Publication Davenport, Iowa Date. Our thick lower case wood letters in Times New Roman are easy to paint and decorate for your craft projects. I am reading about Titus Flavius Sabinus (consul AD 47) and cannot find meaning of T. Women were not “cloistered in their homes” and contemporary Jewish leaders were not misogynists (24). After watching the film we chose several elements of Roman life to investigate, one of which was the Roman army. Keep the cursor where you need the Roman numeral inside the document 2. Room Reservations. Synonyms, crossword answers and other related words for ROMAN GOWN [tunic] We hope that the following list of synonyms for the word tunic will help you to finish your crossword today. This listing includes a zipped file with: (A-Z) letters in. Minor points follow capital letters. The Decorative Letters ClipArt gallery offers 861 examples of decorative letters in a variety of styles. It is used as the standard method of writing in most Western and Central European languages, as well as many languages from other parts of the world. They are an additive (and subtractive) system in which letters are used to denote certain "base" numbers, and arbitrary numbers are then denoted using combinations of symbols. For the final Seal, Charles Thomson put this date on the reverse side. DMC can mean Did mind coming, daily medical car, DeLorean Motor Company, Daimler. Each italic design was a single-weight family, containing only lowercase characters: no caps, no numbers, no punctuation. Farquharson, and A SELECTION FROM THE LETTERS OF MARCUS AND FRONTO. Latin IVLIVS CAESAR, English Julius Caesar). Quick Changing Roman Plastic Letters are Made for Grooved Vinyl and Felt Letter Boards and Directories They're ideal as quick change message boards for schools, offices, churches, hotel lobbies, coffee house, cafes, restaurants, food service, events, tradeshows, theaters, libraries ---any place where information needs to be posted and quickly. when the first stone structure called the "amphitheatrum flavium", or more commonly known as the colosseum, was constructed. (pertainym) roman type, roman letters, roman print 4. In Imperial Roman names, the praenomen functioned rather like the abbreviations Sr. 8" paper mache letter by artminds® \$0. This list is frequently updated — we are constantly discovering more! Abeona, Abundantia, Acca Larentia, Adeona, Aequitas, Aestas, Africus. At one end of the scale were large, formal inscriptions such as dedications to the gods or emperors, publications of official documents such as imperial letters and decrees, and, on a smaller scale, the names and titles of rulers minted on coins along with their portraits or the discharge papers, known as military diplomas, of Roman soldiers. Times New Roman Pro Italic. Insert a NEW PAGE section break after the material you wish to be Roman numbered. Roman numerals originated, as the name might suggest, in ancient Rome. These are vector SVG, DXF, EPS &. Vindolanda was a Roman fort at Chesterholm, just south of Hadrian's Wall in northern England. = Handmaids of the Sacred Heart of Jesus. Women were not “cloistered in their homes” and contemporary Jewish leaders were not misogynists (24). In recent years there has been a major push to record all coins found, not just those in good condition. The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are called Arabic numerals or sometimes Indo-Arabic numerals. Over 800 ABBREVIATIONS OF RELIGIOUS ORDERS. K: Woman whose rape and suicide led to the expulsion of the last King of Rome. Peace be upon him who follows the right path. Roman numbers take seven letters and work them into a multitude of combinations to create small and large numbers. It is based on five principles: The basic Roman alphabet, with no additional letters or diacritics used;; One spelling, with no phoneme (sound) written by two or more graphemes (letters);; Short vowels are spelled by single letters, while long vowels and diphthongs are spelled by two-letter combinations;. Artstor is a nonprofit organization committed to digital collection solutions for universities, museums, schools, and libraries worldwide. Amphitheatre A place where Romans went to watch entertainments. Roger Tomlin and Sophie Jackson […]. Each "T" Styled Letter Forms The First Letter Of Two Separate Words. What do these abbreviations stand for? Answer: “INRI” is an abbreviation for the Latin “ Iesus Nazarenus, Rex Iudaeorum ” (“Jesus the Nazarene, King of the Jews”), posted on the cross by order of the Roman procurator, Pontius Pilate. How do you create Roman numerals on MS Office 2007? by richabrams Aug 2, 2007 7:07AM PDT For the life of me, I cannot find a way to create a Roman numeral from MS Word 2007. They use a system of subtraction. You can use the alphabets on your keyboard and make it look like a Roman numeral. Enter the answer length or the answer pattern to get better results. This date was suggested by the first Great Seal committee (1776), who placed it in Roman numerals below their design. But today, in the ancient Roman town of Bath, I was wide awake. Book 6: 11, 14-20, 27, 31-32, 34-36, 39; Book 7: 19, 21-22, 27, 29, 38, 42. Originally, italic letters were not designed to complement a Roman typeface. An Etruscan pot found near Veii (a city which was sacked by Rome in the 5th century BCE) had the Etruscan abecedary inscribed on it, reminding the excavators of its Roman descendants. There were few praenomina in active use. Rutherford. ' novus ordo. Download the Decorated Roman Initials font by Dieter Steffmann. For the final Seal, Charles Thomson put this date on the reverse side. The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are called Arabic numerals or sometimes Indo-Arabic numerals. The letters J, U and W were much later additions well beyond the Roman influence at a time as the European languages evolved to include more sounds and words. Because cetera implies inanimate objects, et al. packaging and template manufactured by Stenso Lettering Company, 1101 E. " Example: I need to go to the store and buy some pie, milk, cheese, etc. Roman - Coin Ancient - Aequitas Carinus, Mint - Reverse Augustus as Rome Rome Reverse as Augustus Roman Carinus, Mint Aequitas - - Ancient Coin - Ancient Roman Empire, Vespasian, 71 AD. The Beginnings. Edith Stein: Letters to Roman Ingarden (Stein, Edith//the Collected Works of Edith Stein) by Edith Stein (Author), Hugh Candler Hunt (Author) 5. Lucius Lucilius, who was known as son of Quintus, grandson of Gnaeus, of the Claudian tribe; Gaius Lucilius Statius, freedman of Gaius. Total Number of words made out of Roman = 34 Roman is an acceptable word in Scrabble with 7 points. Developed from the Etruscan alphabet at some time before 600 bc, it can be traced through Etruscan, Greek, and Phoenician scripts to the North Semitic alphabet used in Syria and. Ambrose University Place of Publication Davenport, Iowa Date. Refunds are not available due to the nature of instant downloads. When using Roman numerals, certain letters stand for specific numbers. May 14, 2014 - Printable Times New Roman Alphabet Stencil K. THE COGNOMEN - The cognomen was the third of the three names of a Roman Citizen. This signifies that the signature was added by force, under duress, and was compelled. Find the perfect roman latin letters wall stock photo. A system such as this could have been used in the early stages of setting out an inscription prior to the brushing of the letters on to the stone. Ie: \Omega_{\rm{something}}. It is estimated that about one third of all Romans could read and write. Unlike Old English, roman letters font designs have a much more uplifting look and feel. -302 old style roman letters a - z-302 old style roman letters a --302. Let’s get Poetic! Write an acrostic poem. Roman handles everything from online evaluation to delivery of treatment and free ongoing care. The Trajan Inscription Capitalis Monumentalis. The bronze or aluminum metal wall letters can be custom made in different sizes and finishes. Alison Roman's Self-Quarantine Dating Life Is One Long Quest for Phone Sex. Mintmarks on gold coins often end with the letters OB (obryzium - refined or pure gold). Cute letters generator. He also questions the priest on consubstantiation and transubstantiation, the role of the Mass, the place of devotion to Mary, and. The routine at roman takes the address of a zero-terminated string in BC, and returns the value of the Roman number in that string as a 16-bit integer in HL. Report your symptoms and medical history to a US-licensed physician for evaluation. Old Roman cursive script, also called majuscule cursive and capitalis cursive, was the everyday form of handwriting used for writing letters, by merchants writing business accounts, by schoolchildren learning the Latin alphabet, and even emperors issuing commands. Roman forces are still commanded by general Vespasian. Unlike the Roman letters, which attained a complete and final development, Gothic letters never reached authoritative and definitive forms, any more than did Gothic architecture. List of all most popular abbreviated Roman Catholic Church terms defined. That formed the letter. Cut from mdf in your choice of material depth - 9, 18 or 25mm thick. Greek alpha α a Ligature ae Æ, æ ae Greek beta β b Eth ð d Greek gamma γ g Turkish i (undotted) ı i. Updated April 2020. 800 + 250 to this architect of a D. To convert Roman numerals greater than 3,999 use the table below for converter inputs. What differences between the Roman alphabet and their alphabet can they notice? Most students will be able to notice that some letters are missing. The Roman numerals are certain Latin script letters that may be used to represent numbers. sent hand-written letters to my friends and family, become the most hydrated woman in New York, met all my deadlines. Points to note: the longer strokes at the base of B, D, E, L, Q and to some extent T tend to descend more steeply than the shorter feet on other letters. This listing includes a zipped file with: (A-Z) letters in. These words are also the name of a papal blessing given from the balconies of major Roman basilicas at solemn occasions, such as papal coronations, the annual Easter blessing, etc. ” Then there was a formal salutation (from Latin word salutare, salus literally meaning “wish health to”): “Greetings and may you be healthy. Tele Tech Advanced Computer Services, Logo: Two Roman Style Bold Typed Letters Are Crossings Towards The Top. Commercial & Industrial --- Art & Entertainment. Mint city abbreviation (usually one to four letters, but up to seven). Our INITIALS SEARCH is particularly efficient when researching Silver or Jewelry marks which are made up of letters only. This item Woodland Scenics Roman Letters, Gold WOOMG703 Woodland Scenics Dry Transfer Decals Railroad Gothic Letters White Woodland Scenics R. Main point follows a Roman numeral. The above sample text would appear in Extended Basic Roman (non-rhotic version) as follows:. D 118 and 125. The Colosseum in Rome is one of the most famous ruins from the Roman era. As you can see, the font letters are thinner and much more legible. Used in the mintmarks of Roman coins to indicate officina 10, and in combination with other letters to indicate higher officina numbers. With so many different letters to choose from it really comes down to the type of look you’re looking to achieve. Each italic design was a single-weight family, containing only lowercase characters: no caps, no numbers, no punctuation. Not only are they good examples of persuasive letters, but they are historically accurate in detail.	2020-06-05 02:31:00	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.26513350009918213, "perplexity": 5048.935907482397}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590348492427.71/warc/CC-MAIN-20200605014501-20200605044501-00198.warc.gz"}
https://tjomega.net/inseparable-sentence-acjvq/is-bidirectional-search-complete-888db7	s value of a node {\displaystyle n} As in A* search, bi-directional search can be guided by a heuristic estimate of the remaining distance to the goal (in the forward tree) or from the start (in the backward tree). As a result, it is space bound in practice. {\displaystyle t} In normal graph search using BFS/DFS we begin our search in one direction usually from source vertex toward the goal vertex, but what if we start search form both direction simultaneously. {\displaystyle f=g+h} h The reverse search will always use the inverse cost (i.e. Bidirectional definition is - involving, moving, or taking place in two usually opposite directions. Front-to-Back is the most actively researched of the three categories. ( . The algorithm must be too efficient to find the intersection of the two search trees. Front-to-Front algorithms calculate the h value of a node n by using the heuristic estimate between n and some subset of to will give us ( p Since interfaces with is a bidirectional relationship, the search program searches for these occurrences: The source configuration item is … o Ira Pohl (1971) was the first one to design and implement a bi-directional heuristic search algorithm. A* (pronounced "A-star") is a graph traversal and path search algorithm, which is often used in many fields of computer science due to its completeness, optimality, and optimal efficiency. The bi-directional search terminates when both breadth-first searches "meet" at the same vertex. {\displaystyle \mathrm {OPEN} _{d'}} Here I introduce something theoretically faster than BFS, called Bidirectional Search. Thus, new nodes (i.e., children of a parent node) remain in the queue and old unexpanded node which are shallower than the new nodes, get expanded first. So bidirectional A* algorithm is basically the same as Bidirectional Dijkstra. The cost of moving from one city to another city is same. It runs two simultaneous searches: one forward from the initial state, and one backward from the goal, stopping when the two meet in the middle. Bidirectional search generally appears to be an efficient graph search because instead of searching through a large tree, one search is conducted backwards from the goal and one search is conducted forward from the start. And this area, covered by these two smaller circles, is roughly proportional to the number of vertices scanned during the bidirectional search. The reason for this approach is Now, we're going to join those two ideas to optimize the A* algorithm further. Once the search is over, the path from the initial state is then concatenated with the inverse of the path from the goal state to form the complete solution path. k Bidirectional search More formally, if Bidirectional search is an algorithm that uses two searches occurring at the same time to reach a target goal. The general search template given in Figure 2.7 can be considered as a combination of the two in Figures 2.4 and 2.6.One tree is grown from the initial state, and the other is grown from the goal state (assume again that is a singleton, ). {\displaystyle n} t to another state It runs two simultaneous searches: one forward from the initial state, and one backward from the goal, stopping when the two meet. A solution found by the uni-directional A* algorithm using an admissible heuristic has a shortest path length; the same property holds for the BHFFA2 bidirectional heuristic version described in de Champeaux (1983). When they meet, you should have a good path. t s Assume you have to travel from Arad city to Bucharest city. {\displaystyle s} and from The BHFFA algorithm fixed this defect Champeaux (1977). {\displaystyle p} Bidirectional-Search. So usually Bidirectional BFS is used in undirected unweighted graphs. not overestimating) heuristic estimate of the distance between nodes n and o. Front-to-Front suffers from being excessively computationally demanding. One major practical drawback is its () space complexity, as it stores all generated nodes in memory. Following is a road-map. , defined as being the cost from such that there exists some valid operator from each of the parent nodes to Similarly, for those edges that have inverse arcs (i.e. n . {\displaystyle H(n,o)} {\displaystyle t} Bidirectional search using BFS needs the edge weights to be same or non-existent. In BFS, goal test (a test to check whether the current … It runs two simultaneous searches: one forward from the initial state, and one backward from the goal, stopping when the two meet in the middle. Time and Space Complexity : Time and space complexity is O(b d/2). There remains multiple paths to reach Bucharest city from Arad city. 1 This has often been likened to a one-way street in the route-finding domain: it is not necessary to be able to travel down both directions, but it is necessary when standing at the end of the street to determine the beginning of the street as a possible route. simultaneously. or Bidirectional search #. P Bidirectional search Now that forward and backward search have been covered, the next reasonable idea is to conduct a bidirectional search. the cost of the arc in the forward direction). Bidirectional search is a graph search algorithm that finds a shortest path from an initial vertex to a goal vertex in a directed graph. ) Complete and Easy Bidirectional Typechecking for Higher-Rank Polymorphism Joshua Dunﬁeld Neelakantan R. Krishnaswami Max Planck Institute for Software Systems Kaiserslautern and Saarbrücken, Germany {joshua,neelk}@mpi-sws.org Abstract Bidirectional typechecking, in which terms either synthesize a type While it may seem as though the operators have to be invertible for the reverse search, it is only necessary to be able to find, given any node , searching from Bidirectional search is a graph search algorithm that finds a shortest path from an initial vertex to a goal vertex in a directed graph. Bidirectional search is a graph search algorithm which find smallest path form source to goal vertex. Search results; Bidirectional: A user searches for all configuration items with an interfaces with relationship to application Z. It is a simple search strategy where the root node is expanded first, then covering all other successors of the root node, further move to expand the next level nodes and the search continues until the goal node is not found. This helps focus the search. You desire to travel this route. n {\displaystyle s} to is a node with parent returns an admissible (i.e. It returns a valid list of operators that if applied to This involves calculating a heuristic estimate from n to every node in the opposing OPEN set, as described above. {\displaystyle h} Instead of searching from the start to the finish, you can start two searches in parallel―one from start to finish, and one from finish to start. These differ by the function used to calculate the heuristic. , The canonical example is that of the BHFFA (Bidirectional Heuristic Front-to-Front Algorithm),[2] where the h function is defined as the minimum of all heuristic estimates between the current node and the nodes on the opposing front. . d Every time a node n is put into the open list, its Bidirectional Search, as the name implies, searches in two directions at the same time: one forward from the initial state and the other backward from the goal. value must be calculated. , then Writing the code for Bidirectional BFS is easier if you have already written the code for Breadth First Search using queue. {\displaystyle n} {\displaystyle s} Bidirectional search still guarantees Bidirectional search is a graph search algorithm that finds a shortest path from an initial vertex to a goal vertex in a directed graph. How to use bidirectional in a sentence. The current best algorithm (at least in the Fifteen puzzle domain) is the BiMAX-BSF algorithm, created by Auer and Kaindl (Auer, Kaindl 2004). Bidirectional search isn’t feasible in chess. {\displaystyle n} k ) n 2 t n n {\displaystyle n} p {\displaystyle t} O , Bidirectional search is a graph search algorithm that finds a shortest path from an initial vertex to a goal vertex in a directed graph. Intel releases new Core M chips this year, Facebook launches website for cyber security, Differences Between Regular Programming And AI Programming. Search trees emanating from the start and goal nodes failed to meet in the middle of the solution space. Code. Front-to-Back algorithms calculate the But the search is not complete if l < d. Even if l > d, optimal solution is not guaranteed, as we could be eliminating some of the solutions at depths > l. ... Bidirectional Search. f n t It is important to realize that the first solution found may not be optimal, even if the two searches are both breadth-first; some additional search is required to make sure there isn't a shortcut across the gap. The OPEN sets increase in size exponentially for all domains with b > 1. Time and Space Complexity − Time and space complexity is O(b^{d/2}) + The reason for this approach is that in many cases it is faster: for instance, in a simplified model of search problem complexity in which … Bidirectional search is implemented by replacing the goal test with a check to see whether the frontiers of the two searches intersect; if they do, a solution has been found. In the previous lesson, you've learned that you can use a bidirectional search to optimize Dijkstra's algorithm. It’s a good idea that will help in some situations. This is usually done by expanding tree with branching factor b and the distance from start to goal is d. The search stops when searches from both directions meet in the middle. g It runs two simultaneous searches: one forward from the initial state, and one backward from the goal, stopping when the two meet. From Cracking the Coding Interview, 6th Edition, Page 108: "Bidirectional search is used to find the shortest path between a source and destination node. Bidirectional search still guarantees optimal solutions. {\displaystyle t} H This is usually done by expanding tree with branching factor b and the distance from start to goal is d. The, The merit of bidirectional search is its speed. s And to get the bidirectional A algorithm. BFS expands the shallowest (i.e., not deep) node first using FIFO (First in first out) order. ′ , the set of parent nodes of In given example, the same applies - it will produce output from one side, from the second it will stop on single vertex, so it will degrade to one-directional, therefore nothing makes bidirectional search unusable. The reason that this is faster is because the trees grow exponentially by their depth and therefore two smaller t… Definitions of Bidirectional_search, synonyms, antonyms, derivatives of Bidirectional_search, analogical dictionary of Bidirectional_search (English) s Bidirectional search is a brute-force search algorithm that requires an explicit goal state instead of simply a test for a goal condition. Completeness − Bidirectional search is complete if BFS is used in both searches. p When you cannot perform search - it does not matter whether it was bidirectional … E N Bidirectional algorithms can be broadly split into three categories: Front-to-Front, Front-to-Back (or Front-to-End), and Perimeter Search (Kaindl Kainz 1997). Approaches for Bidirectional Heuristic Search, Bidirectional Heuristic Front-to-Front Algorithm, Efficient Point-to-Point Shortest Path Algorithms, Artificial Intelligence: A Modern Approach, https://en.wikipedia.org/w/index.php?title=Bidirectional_search&oldid=895182301, Creative Commons Attribution-ShareAlike License, This page was last edited on 2 May 2019, at 14:52. n ( Optimality − It is optimal if BFS is used for search and paths have uniform cost. It is not always possible to search backward through possible states. and the root of the opposite search tree, But with the use of potentials. It runs two simultaneous searches: one forward from the initial state, and one backward from the goal, stopping when the two meet. I have implemented BFS the code is given below. Google has many special features to help you find exactly what you're looking for. A Bidirectional Heuristic Search is a state space search from some state The time complexity of Bidirectional Search is O(b^d/2) since each search need only proceed to half the solution path. ) h s {\displaystyle s} {\displaystyle p} = c. Bidirectional search is very useful, because the only successor of n in the reverse direction is Á(n/2) Â. {\displaystyle n} {\displaystyle s} {\displaystyle n} Completeness : Bidirectional search is complete if BFS is used in both searches. p Since at least one of the searches must be breadth-first in order to find a common state, the space complexity of bidirectional search is also O(b^d/2). So, let's denote the big circle by C1, and the two smaller circles by C2 and C3. What will happen in the directional search is we will be growing two circles of roughly the same radius until they touch. = Implementation of bidirectional search algorithm is difficult because additional logic must be included to decide which search tree to extend at each step. Search the world's information, including webpages, images, videos and more. Once the search is over, the path from the initial state is then concatenated with the inverse of the path from the goal state to form the complete solution path. BHFFA2 has, among others, more careful termination conditions than BHFFA. One should have known the goal state in advance. Optimality : It is optimal if BFS is used for search and paths have uniform cost. ... search in that it adds one complete layer of nodes before adding the next layer. It runs two simultaneous searches: one forward from the initial state, and one backward from the goal, stopping when the two meet in the middle. arcs going in both directions) it is not necessary that each direction be of equal cost. {\displaystyle k_{1}(p,n)=k_{2}(n,p)} Andrew Goldberg and others explained the correct termination conditions for the bidirectional version of Dijkstra’s Algorithm.[1]. Assuring that the comparisons for identifying a common state between the two frontiers can be done in constant time per node by hashing. About this video: In this video we will learn about Bidirectional Search Technique. . to {\displaystyle t} The reason for this approach is that in many cases it is faster: for instance, in a simplified model of search problem complexity in which both searches expand a tree with branching factor b, and the distance from start to goal is d, each of the two searches has complexity O(bd/2) (in Big O notation), and the sum of these two search times is much less than the O(bd) complexity that would result from a single search from the beginning to the goal. Bidirectional search is a brute-force search algorithm that requires an explicit goal state instead of simply a test for a goal condition. Balanced, bidirectional search Much better performance can usually be obtained by growing two RDTs, one from and the other from .This is particularly valuable for escaping one of the bug traps, as mentioned in Section 5.4.1.For a grid search, it is straightforward to implement a bidirectional search that ensures that the two trees meet. n Bidirectional search is a graph search algorithm that finds a shortest path from an initial vertex to a goal vertex in a directed graph. t (c)Copyrighted Artificial Intelligence, All Rights Reserved.Theme Design, Bidirectional Search, as the name implies, searches in two directions at the same time: one forward from the initial state and the other backward from the goal. n It operates by essentially running two simultaneous breadth-first searches, one from each node. Welcome to Golden Moments Academy (GMA). , . n Or, formally: where Bidirectional search is a graph search algorithm that finds a shortest path from an initial vertex to a goal vertex in a directed graph. def bfs(graph, start): path = [] queue = [start] while queue: vertex = queue.pop(0) if vertex not in path: path.append(vertex) queue.extend(graph[vertex]) return path. Sum of the time taken by two searches (forward and backward) is much less than the O(b. Below is very simple implementation representing the concept of bidirectional search using BFS. by using the heuristic estimate between (Auer Kaindl 2004). Taken by two searches occurring at the same vertex − it is optimal if BFS is used in unweighted! By C2 and C3 's information, including webpages, images, videos more. As bidirectional Dijkstra is given below the OPEN sets increase in size exponentially for all domains with >... Use the inverse cost ( i.e the three categories [ 1 ], among others, more careful termination for! Introduce something theoretically faster than BFS, called bidirectional search Technique calculating heuristic! Brute-Force search algorithm. is bidirectional search complete 1 ] vertex in a directed graph city to another city same. Edge weights to be same or non-existent s } will give us t { t. Applied to s { \displaystyle t } operators that if applied to s \displaystyle! Let 's denote the big circle by C1, and the two search trees ) since each need. Heuristic search algorithm that finds a shortest path from an initial vertex a! Of roughly the same radius until they touch drawback is its ( ) complexity! Complete if BFS is used for search and paths have uniform cost explicit goal state instead of a. Good path because additional logic must be included to decide which search tree to extend each... The world 's information, including webpages, images, videos and more multiple paths to reach Bucharest city Arad... Year, Facebook launches website for cyber security, Differences between Regular Programming and AI Programming bound practice... Of Dijkstra ’ s algorithm. [ 1 ] to design and implement a bi-directional heuristic search algorithm that two... \Displaystyle s } will give us t { \displaystyle t } conditions for the bidirectional version Dijkstra... Set, as it stores all generated nodes in memory 's algorithm. [ 1.... Optimality: it is optimal if BFS is easier if you have already the! Academy ( GMA ) it adds one complete layer of nodes before the... Node by hashing to optimize Dijkstra 's algorithm. [ 1 ] in.. Using BFS one city to Bucharest city differ by the function used to calculate heuristic... 'Ve learned that you can use a bidirectional is bidirectional search complete a common state between the search... Differences between Regular Programming and AI Programming is much less than the O b! The two smaller circles by C2 and C3 completeness − bidirectional search is an algorithm that a. The two search trees emanating from the start and goal nodes failed meet... Bound in practice, Facebook launches website for cyber security, Differences Regular! State between the two smaller circles, is roughly proportional to the number of vertices during! And goal nodes failed to meet in the reverse direction is Á ( n/2 ) Â correct termination conditions the. You have to travel from Arad city to Bucharest city from Arad city the first one design. Is a graph search algorithm. [ 1 ] is bidirectional search complete start and goal failed... Included to decide which search tree to extend at each step source to goal vertex in a directed....: time and space complexity, as described above proportional to the number of vertices scanned during the bidirectional of. Assume you have to travel from Arad city to another city is same already written the code is below... World 's information, including webpages, images, videos and more undirected unweighted graphs opposite.... Complete if BFS is easier if you have already written the code for bidirectional BFS is used for search paths! Some situations ( n/2 ) Â this defect Champeaux ( 1977 ) search. To a goal condition bi-directional heuristic search algorithm that finds a shortest path from an initial to! Search and paths have uniform cost one from each node that the comparisons for identifying a common between. Failed to meet in the middle of the solution path is bidirectional search complete the goal state in advance size exponentially all... If you have already written the code for bidirectional BFS is used in directions... Source to goal vertex in a directed graph each node layer of nodes adding! Comparisons for identifying a common state between the two frontiers can be done in constant time per by... Learned that you can use a bidirectional search is a brute-force search algorithm that uses two searches forward! Both searches the edge weights to be same or non-existent we 're going to join those ideas... Easier if you have already written the code for Breadth first search using.. And this area, covered by these two smaller circles, is roughly proportional to number... Or non-existent 1977 ) ( b d/2 ) the same as bidirectional Dijkstra to optimize the is bidirectional search complete! You find exactly what you 're looking for they meet, you learned. Extend at each step in advance careful termination conditions for the bidirectional version of Dijkstra ’ s a idea... Optimality: it is optimal if BFS is easier if you have already written the is... First search using BFS ira Pohl ( 1971 ) was the first one to design and implement bi-directional! Golden Moments Academy ( GMA ) is an algorithm that finds a shortest from! Paths to reach a target goal to another city is same in practice of n in the OPEN... Easier if you have already written the code for bidirectional BFS is easier if you have travel... Has many special features to help you find exactly what you 're looking for reach Bucharest city travel. Reverse direction is Á ( n/2 ) Â new Core M chips this year, Facebook launches for... Website for cyber security, Differences between Regular Programming and AI Programming 's information, including,! ) it is optimal if BFS is used for search and paths uniform. For the bidirectional version of Dijkstra ’ s algorithm. [ 1 ] of. Goldberg and others explained the correct termination conditions for the bidirectional version Dijkstra! In advance be done in constant time per node is bidirectional search complete hashing practical drawback is (! '' at the same time to reach Bucharest city from Arad city will in! Termination conditions than BHFFA usually bidirectional BFS is easier if you have to travel from city! I have implemented BFS the code is given below bi-directional heuristic search.! Videos and more bidirectional search is a graph search algorithm that finds a shortest path from an initial to!, Facebook launches website for cyber security, Differences between Regular Programming and AI Programming search to optimize a... B > 1 already written the code is given below to search backward through possible states the... At the same as bidirectional Dijkstra shortest path from an initial vertex to a vertex... Regular Programming and AI Programming direction ) in size exponentially for all domains with b > 1 for!, not deep ) node first using FIFO ( first in first out ).. } will give us t { \displaystyle t } s a good is bidirectional search complete will. Moving, or taking place in two usually opposite directions '' at the same radius until they touch help. Find the intersection of the time complexity of bidirectional search is complete BFS... Sum of the arc in the previous lesson, you should have known the goal in... Area, covered by these two smaller circles by C2 and C3 ( ) space complexity, described... Overestimating ) heuristic estimate of the arc in the directional search is a graph search that... The reverse search will always use the inverse cost ( i.e city is same for cyber,... Breadth first search using BFS complexity: time and space complexity: time space... Search algorithm that requires an explicit goal state in advance code for bidirectional BFS is in! Sum of the solution space be same or non-existent by hashing circles, is roughly proportional the... That each direction be of equal cost that you can use a search! In undirected unweighted graphs complete if BFS is used in undirected unweighted graphs bi-directional heuristic search that... Each direction be of equal cost circle by C1, and the two frontiers can be done constant! Two usually opposite directions multiple paths to reach a target goal very useful, because the only successor of in... City from Arad city to Bucharest city from Arad city to another city is same bhffa2 has, others... Shallowest ( i.e., not deep ) node first using FIFO ( in. Not always possible to search backward through possible states 1 ] Champeaux ( 1977 ) it is not possible! Andrew Goldberg and others explained the correct termination conditions than BHFFA to another city is same always to. Another city is same is - involving, moving, or taking place in two usually opposite.... Forward direction ) circles by C2 and C3 that will help in some.. Moments Academy ( GMA ) images, videos and more the time complexity of search! That finds a shortest path from an initial vertex to a goal condition termination conditions the... 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There remains multiple paths to reach Bucharest city from Arad city need only proceed to half solution!	2021-06-23 06:25:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5379770398139954, "perplexity": 1194.3495816906557}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488534413.81/warc/CC-MAIN-20210623042426-20210623072426-00309.warc.gz"}
https://brilliant.org/problems/some-of-them-are-waste/	# Some of them are waste! Algebra Level 5 $a^2+b^2+c^2+d^2+e^2+f^2=139$ $a^3+b^3+c^3+d^3+e^3+f^3=783$ $a^2+b^2+c^2+d^3+e^3+f^3=551$ $a^3+b^3+c^3+d^2+e^2+f^2=371$ $a^2+b^3+c^2+d^3+e^2+f^3=499$ $a+b+c+d+e+f=?$ Hint: This set ×	2017-05-28 12:37:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7200964093208313, "perplexity": 1833.8994518784289}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463609817.29/warc/CC-MAIN-20170528120617-20170528140617-00473.warc.gz"}
https://mcintyre.io/algebra/stardust/	# Stardust Under Construction How would you draw Earth on a computer? ## $\lambda$ Domain and Range Have you ever looked up at a sky full of stars? If you can find a spot free of light pollution you may even be able to observe our galaxy in all its splendor. The sight is truly awesome and it raises some big questions. Are we alone in the cosmos? Will we ever travel to another star system? And just how full is the sky, anyway? Let’s start with that last one. ### Discrete You’ve probably drawn countless graphs in math classes over the years. About the simplest thing you could draw is a single, solitary point like the one below. How much of the graph does the point take up? Certainly not much. To be precise, the point only takes up, well, a single point. Looking from left to right, it appears the $x$-values, or domain, of the graph is just $\{2\}$. The $y$-values, or range, is just $\{6\}$. Let’s expand the graph a little. It appears our domain and range have both grown! The domain is now $\{2, 4\}$ and the range is $\{6, 7\}$. Graphs that are comprised of collections, or sets, of individual points are known as discrete graphs. What happens when we connect the dots? ### Continuous The graph below connects the points $(2, 6)$ and $(4, 7)$ with a straight line. If you zoom in and move from left to right, you would find that the graph is filled in somewhere at every $x$-value between $2$ and $4$. You could do the same from top to bottom. When a graph’s domain and range take on every value in a given interval, we say it is continuous. In this case, the domain is $2 \leq x \leq 4$ and the range is $6 \leq y \leq 7$. ### Constellation The sky is full of more stars than you could hope to count. Take a walk outside tonight and try to spot a constellation. How would you describe the constellation’s position? You might start by saying, “Well, it’s in the sky over there, and I know the stars are really in outer space.” A mathematician would look at the situation and say something like, “I know the photons from those stars just arrived at my eye after traveling for a long time. From my perspective, the stars appear to be arranged in such-and-such shape. I bet I could describe each star’s position precisely—let me get a pencil.” The sketch below gives an example of drawing the Big Dipper. The statement let x = 5; declares that you’d like to use a variable named x and assign it the value $5$. Protip: Good variable names can make it much easier to understand what’s happening in your code. function setup() { createCanvas(400, 400); } function draw() { background(0); stroke(255); let alkaidX = 100; let alkaidY = 100; let mizarX = 150; let mizarY = 110; let aliothX = 170; let aliothY = 130; let megrezX = 190; let megrezY = 150; let phecdaX = 190; let phecdaY = 180; let merakX = 240; let merakY = 190; let dubheX = 260; let dubheY = 160; strokeWeight(5); point(alkaidX, alkaidY); point(mizarX, mizarY); point(aliothX, aliothY); point(megrezX, megrezY); point(phecdaX, phecdaY); point(merakX, merakY); point(dubheX, dubheY); strokeWeight(1); // look up "line" in the p5.js reference line(alkaidX, alkaidY, mizarX, mizarY); line(mizarX, mizarY, aliothX, aliothY); line(aliothX, aliothY, megrezX, megrezY); line(megrezX, megrezY, phecdaX, phecdaY); line(phecdaX, phecdaY, merakX, merakY); line(merakX, merakY, dubheX, dubheY); } Map out your own constellation on paper then in code. Research the proper names of the stars or come up with new ones. ### Domain and Range Review What are the domain and range of the stars in your constellation? How about when you connect them? ## Variables Take another walk outside tomorrow afternoon. Depending on your location and the weather, you may enjoy the sunshine on your face. Close your eyes and compare the feeling with the previous evening. With some exceptions, you’d probably notice it’s cooler at night than in the daytime. Our sensations of heat and cold depend primarily on the temperature of the air, how much radiation we absorb from the sun, and the wind. But why male models? — Derek Zoolander A mathematician would look at the situation and develop a general description, or model. She would say something like the following: “My sensation of heat changes and it depends on temperature, radiation, and wind.” ### Mathematics Sensations and other quantities that change are known as variables. You may be acquainted with variables like $x$ and $y$ from solving equations or graphing points and lines. Try to think of variables as containers, or buckets, or boxes—as you prefer. The big idea is that they hold values that can change, or vary. Math students often have trouble distinguishing between variables and the values they represent. $x$ may represent the number $5$ at a particular point along the graph of a line, so it’s easy to conclude (incorrectly) that $x=5$, now and always. In our model of sensation, heat is dependent upon the values of three variables that are independent, or taken as an input to the model. A mathematician would represent the heat model as a function using notation like $h(t, r, w)$. ### Computation At some point, you may have been asked to read a brief scenario and make up, or formulate, a model. Perhaps you were in science class and formulated a model like $d(v,t)=vt$ describing an object’s displacement with respect to velocity and time. If you ran in a straight line at $5$ meters per second for $10$ seconds, you would find yourself $5\times10=50$ meters away from your starting position. You could define the corresponding function in JavaScript like so. function d(v, t) { return vt; } v and t are part of our function definition, called parameters. If we called d(5, 10) somewhere in the sketch, the argument $5$ would be assigned to the parameter v and $10$ would be assigned to the parameter t. Parameters don’t change; the arguments we assign to them do. The following sketch of a meteor declares the variable velocity and assigns it the value $5$. This statement is like asking JavaScript to create a new box labelled “velocity” and putting a $5$ in it to start. time works similarly but with one big difference. Note that time is increased, or incremented, by $1$ when draw() executes. You could imagine this statement as taking the value held in the “time” box, adding $1$ to it, and putting the new value back in the box. The print() function is used to keep a running log of the current value of time. Protip: Printing values is a useful technique for debugging. // Variables you declare up here can be used // throughout the sketch. let velocity = 5; let time = 0; function setup() { createCanvas(400, 400); } function draw() { background(0, 25); stroke(255); fill(255); print(time); let x = d(velocity, time); let y = 200; circle(x, y, 10); time = time + 1; } function d(v, t) { return vt; } Look up fill() and circle() in the p5.js reference. The variables velocity and time were passed as arguments to d() where they were assigned to the parameters v and t, respectively. ### Variables Review What are variables? Describe how they relate to arguments and parameters. ## $\lambda$ Transformation Drawing collections of objects can get tricky fast. The sketch below draws a simple representation of Orion’s Belt. function setup() { createCanvas(400, 400); } function draw() { background(0); stroke(255); strokeWeight(5); // alnitak point(50, 100); // alnilam point(100, 75); // mintaka point(150, 50); } What if you wanted to shift everything to the left a little? Or how about representing the view from another location on Earth? It turns out realignments, or transformations, like these are easily handled by playing a little with our coordinate system. ### Translate How would you move the constellation $100$ pixels to the right? One way to go about it would be to comb through all your calls to point(), adding $100$ to the first argument. function setup() { createCanvas(400, 400); } function draw() { background(0); stroke(255); strokeWeight(5); // alnitak point(150, 100); // alnilam point(200, 75); // mintaka point(250, 50); } What if you wanted to move the constellation a smidge to the left? Or a little down? Each adjustment would require you to update all your calls to point() again. As you can imagine, this would quickly become a hassle for sketches composed of many objects. Shifting, or translating, is a transformation simplifies this sort of work. Add a call to translate() with two arguments for the amount left/right and up/down you’d like to shift your constellation. function setup() { createCanvas(400, 400); } function draw() { background(0); stroke(255); strokeWeight(5); translate(75, 50); // alnitak point(50, 100); // alnilam point(100, 75); // mintaka point(150, 50); } Move your call to translate() to the end of draw(). What happened? One way to think about transformations is as an adjustment to your coordinate system. Calling translate(1, 2) effectively shifts the origin from the top-left corner to $(1,2)$. Anything drawn afterward will appear relative to the new origin. Translations really start to shine when combined with other transformations. ### Rotate The Moon is Earth’s natural satellite and a source of inspiration for numerous creative works. Two notable examples are Nick Drake’s album Pink Moon and the Lua programming language. Let’s see if we can put the Moon in orbit around Earth. This sketch is a little more ambitious, so it will help to lay out some initial ideas. function setup() { createCanvas(400, 400); } function draw() { background(220); // drawEarth(); // drawMoon(); } Breaking a big problem down into manageable pieces is known as decomposition in computer science. The practice will serve you well in many walks of life. To keep things simple, draw Earth as a big, blue circle in the middle of your canvas. Remove the // when you’re ready to go. function setup() { createCanvas(400, 400); } function draw() { background(0); drawEarth(); // drawMoon(); } function drawEarth() { fill(0, 0, 255); circle(200, 200, 200); } You could make the drawEarth() function a bit more flexible by adding parameters for Earth’s position. Let’s make use of translate() as well. function setup() { createCanvas(400, 400); } function draw() { background(0); drawEarth(200, 200); // drawMoon(); } function drawEarth(x, y) { fill(0, 0, 255); translate(x, y); circle(0, 0, 200); } Pass mouseX and mouseY as arguments to drawEarth(). You’ll have the whole world in your hands. Not bad. Now we’ll draw the Moon as another circle some distance away from Earth. Transformations accumulate, so let’s take advantage of the fact that the origin is now at Earth’s center. function setup() { createCanvas(400, 400); } function draw() { background(0); drawEarth(200, 200); drawMoon(50); } function drawEarth(x, y) { fill(0, 0, 255); translate(x, y); circle(0, 0, 200); } fill(200); circle(0, 0, 50); } Switch the order in which you call drawEarth() and drawMoon(). What happened? The sketch works for the moment, but it’s worth getting ahead of some common sources of error. Transformations accumulate by default, but we have the ability to isolate groups of related transformations. It’s easy to rewrite drawEarth() and drawMoon() so that their transformations no longer impact one another—just call push() before the first transformation and pop() after the last piece of the object. We’ll rewrite drawMoon() so that it’s position is based on two transformations, first to Earth’s center and then to its place in orbit. function setup() { createCanvas(400, 400); } function draw() { background(0); drawEarth(200, 200); drawMoon(200, 200, 150); } function drawEarth(x, y) { push(); fill(0, 0, 255); translate(x, y); circle(0, 0, 200); pop(); } push(); fill(200); translate(centerX, centerY); circle(0, 0, 50); pop(); } Switch the order in which you call drawEarth() and drawMoon(). What happened this time? You can think of rotations as an adjustment to your coordinate system, just like translations. Rotating your coordinate system $45$ degrees clockwise would leave it looking something like the following. The $y$-axis label wound up off the canvas after rotating. Let’s try to get it back. You can compose a sequence of transformations to generate many interesting effects. Here’s the same $45$ degree clockwise rotation followed by a translation along the (rotated) positive $x$-axis. p5.js measures degrees in radians by default. They can be simpler to work with once you’ve gotten the hang of them, but we’ll stick to degrees for the time being. The Moon’s place in orbit will change, and when things change, it’s probably time to use variables. The following sketch adds the variable angle and uses it to control rotation. Note the call to angleMode() in setup(). let angle = 0; function setup() { createCanvas(400, 400); angleMode(DEGREES); } function draw() { background(0); drawEarth(200, 200); drawMoon(200, 200, 150, angle); angle = angle + 0.25; } function drawEarth(x, y) { push(); fill(0, 0, 255); translate(x, y); circle(0, 0, 200); pop(); } function drawMoon(centerX, centerY, radius, angle) { push(); fill(200); translate(centerX, centerY); rotate(angle); circle(0, 0, 50); pop(); } Rearrange the transformations in drawMoon() to see what happens. Breaking programs can provide great insight into how they work. ### Scale Earth and the Moon look good, but the sky is a little too empty, even by space standards. Let’s add our constellation to help fill out the sky. let angle = 0; function setup() { createCanvas(400, 400); angleMode(DEGREES); } function draw() { background(0); drawConstellation(20, 50); drawEarth(200, 200); drawMoon(200, 200, 150, angle); angle = angle + 0.25; } function drawConstellation(x, y) { push(); translate(x, y); stroke(255); strokeWeight(5); // alnitak point(50, 100); // alnilam point(100, 75); // mintaka point(150, 50); pop(); } function drawEarth(x, y) { push(); translate(x, y); fill(0, 0, 255); circle(0, 0, 200); pop(); } function drawMoon(centerX, centerY, radius, angle) { push(); translate(centerX, centerY); rotate(angle); fill(200); circle(0, 0, 50); pop(); } Write your own drawConstellation() function with parameters x and y. The stars are arranged just fine but they’re a little too big. We can adjust the overall size, or scale, of objects using the scale() function. let angle = 0; function setup() { createCanvas(400, 400); angleMode(DEGREES); } function draw() { background(0); drawConstellation(20, 50, 0.5); drawEarth(200, 200); drawMoon(200, 200, 150, angle); angle = angle + 0.25; } function drawConstellation(x, y, theScale) { push(); translate(x, y); scale(theScale); stroke(255); strokeWeight(5); // alnitak point(50, 100); // alnilam point(100, 75); // mintaka point(150, 50); pop(); } function drawEarth(x, y) { push(); translate(x, y); fill(0, 0, 255); circle(0, 0, 200); pop(); } function drawMoon(centerX, centerY, radius, angle) { push(); translate(centerX, centerY); rotate(angle); fill(200); circle(0, 0, 50); pop(); } Change the order you call functions in draw() to see what happens. Like translate() and rotate(), you can think of scale() visually as a stretch or compression of your coordinate system. The graph paper below was compressed by a factor of $0.5$ in the $x$-direction and stretched by a factor of $2$ in the $y$-direction. ### Transformation Review How do translate(), rotate(), and scale() work? ## Project Ideas Interactivity Rotate the Moon using your mouse. Analysis Use the print() function to log your constellation’s domain and range. Animation Fill out the surface terrains of Earth and the Moon using 2D Primitives.	2019-08-18 01:36:32	{"extraction_info": {"found_math": true, "script_math_tex": 43, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4179839789867401, "perplexity": 3050.0872297174137}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313536.31/warc/CC-MAIN-20190818002820-20190818024820-00486.warc.gz"}
https://www.physicsforums.com/threads/highly-localized-initial-psi-in-harmonic-well.876627/	# Highly localized initial psi in harmonic well ## Main Question or Discussion Point Say we start with a wavefunction inside a harmonic potential well, such that the initial $\psi(x)$ is confined to a central region much smaller than the ground state (hence $V(x)\approx0$).. and the expectation Kinetic Energy is equal to an energy eignenvalue $E_n$ of the system. Starting from here, will it ultimately converge over time into an energy eigenstate corresponding to $E_n$ ... OR.. will it slosh around forever in a complicated way? Related Quantum Physics News on Phys.org vanhees71 Gold Member 2019 Award It will slosh around forever in a complicated way. You can just solve the equation of motion by using the well-known energy-eigenstates. Given the wave function $\psi(t,\vec{x})$ at $t=0$ you define the corresponding coefficients $$\psi_j=\int_{-\infty}^{\infty} u_j^*(x) \psi(0,\vec{x}),$$ where $u_j(x)$ is the energy eigenfunction with eigenvalue $E_j=(j+1/2)\omega$, $j \in \{0,1,2,\ldots \}$. Then the wave function at any later time is given by $$\psi(t,x)=\sum_{j=0}^{\infty} \exp(-\mathrm{i} E_j t) \psi_j u_j(x).$$ This immediately shows that you never converge to an energy eigenfunction but that for any time all components of the initial wave function stay involved. This must be so, because only the energy eigenfunctions represent stationary states, i.e., if initially you don't have the system prepared in an energy eigenfunction the state can never become an energy eigenstate later. Thanks ! To paraphrase vanhees' answer more abstractly, the Schrödinger equation is linear and unitary (the eigenvalues are just phases) so any nontrivial linear combination of its eigenfunctions will never converge to a single eigenfunction.	2020-04-07 18:16:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9272280931472778, "perplexity": 443.90612205593953}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371803248.90/warc/CC-MAIN-20200407152449-20200407182949-00403.warc.gz"}
https://projecteuclid.org/euclid.aaa/1393511953	## Abstract and Applied Analysis ### Two New Efficient Iterative Regularization Methods for Image Restoration Problems #### Abstract Iterative regularization methods are efficient regularization tools for image restoration problems. The IDR($s$) and LSMR methods are state-of-the-arts iterative methods for solving large linear systems. Recently, they have attracted considerable attention. Little is known of them as iterative regularization methods for image restoration. In this paper, we study the regularization properties of the IDR($s$) and LSMR methods for image restoration problems. Comparative numerical experiments show that IDR($s$) can give a satisfactory solution with much less computational cost in some situations than the classic method LSQR when the discrepancy principle is used as a stopping criterion. Compared to LSQR, LSMR usually produces a more accurate solution by using the $L$-curve method to choose the regularization parameter. #### Article information Source Abstr. Appl. Anal., Volume 2013 (2013), Article ID 129652, 9 pages. Dates First available in Project Euclid: 27 February 2014 https://projecteuclid.org/euclid.aaa/1393511953 Digital Object Identifier doi:10.1155/2013/129652 Mathematical Reviews number (MathSciNet) MR3073472 Zentralblatt MATH identifier 1371.68315 #### Citation Zhao, Chao; Huang, Ting-Zhu; Zhao, Xi-Le; Deng, Liang-Jian. Two New Efficient Iterative Regularization Methods for Image Restoration Problems. Abstr. Appl. Anal. 2013 (2013), Article ID 129652, 9 pages. doi:10.1155/2013/129652. https://projecteuclid.org/euclid.aaa/1393511953 #### References • P. C. Hansen, J. G. Nagy, and D. P. O'Leary, Deblurring Images: Matries, Spectra and Filtering, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 2006. • R. J. 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Lewis, and L. Reichel, “Krylov subspace iterative methods for nonsymmetric discrete ill-posed problems in image restoration,” in Advanced Signal Processing: Algorithms, Architectures and Implementations XI, vol. 4474 of Proceedings of SPIE, pp. 224–233, The International Society for Optical Engineering, Bellingham, Wash, USA, August 2001. • P. Brianzi, P. Favati, O. Menchi, and F. Romani, “A framework for studying the regularizing properties of Krylov subspace methods,” Inverse Problems, vol. 22, no. 3, pp. 1007–1021, 2006. • P. Sonneveld and M. B. van Gijzen, “IDR(s): a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations,” SIAM Journal on Scientific Computing, vol. 31, no. 2, pp. 1035–1062, 2008. • D. C.-L. Fong and M. Saunders, “LSMR: an iterative algorithm for sparse least-squares problems,” SIAM Journal on Scientific Computing, vol. 33, no. 5, pp. 2950–2971, 2011. • M. B. van Gijzen and P. 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Serra-Capizzano, “On the regularizing power of multigrid-type algorithms,” SIAM Journal on Scientific Computing, vol. 27, no. 6, pp. 2053–2076, 2006. • M. Donatelli and S. Serra-Capizzano, “Filter factor analysis of an iterative multilevel regularizing method,” Electronic Transactions on Numerical Analysis, vol. 29, pp. 163–177, 2007/08. • V. A. Morozov, “On the solution of functional equations by the method of regularization,” Soviet Mathematics. Doklady, vol. 7, pp. 414–417, 1966. • P. C. Hansen, “Analysis of discrete ill-posed problems by means of the \emphL-curve,” SIAM Review, vol. 34, pp. 658–672, 1992. • G. Wahba, “Practical approximate solutions to linear operator equations when the data are noisy,” SIAM Journal on Numerical Analysis, vol. 14, no. 4, pp. 651–667, 1977. • I. Hnětynková, M. Plešinger, and Z. Strakoš, “The regularizing effect of the Golub-Kahan iterative bidiagonalization and revealing the noise level in the data,” BIT, vol. 49, no. 4, pp. 669–696, 2009. • L. Reichel and G. Rodriguez, “Old and new parameter choice rules for discrete ill-posed problems,” Numerical Algorithms, vol. 63, no. 1, pp. 65–87, 2013. • P. C. Hansen, T. K. Jensen, and G. Rodriguez, “An adaptive pruning algorithm for the discrete \emphL-curve criterion,” Journal of Computational and Applied Mathematics, vol. 198, no. 2, pp. 483–492, 2007. • P. C. Hansen, “The \emphL-curve and its use in the numerical treatment of inverse problems,” in Computational Inverse Problems in Electrocardiology, P. Johnston, Ed., pp. 119–142, WIT Press, Southampton, Ceremonial, 2001. • C. R. Vogel, “Non-convergence of the \emphL-curve regularization parameter selection method,” Inverse Problems, vol. 12, no. 4, pp. 535–547, 1996. • M. Hanke, “Limitations of the \emphL-curve method in ill-posed problems,” BIT, vol. 36, no. 2, pp. 287–301, 1996. • D. C. L. 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Hansen, “Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems,” Numerical Algorithms, vol. 6, no. 1-2, pp. 1–35, 1994.	2019-05-22 12:46:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 4, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21509839594364166, "perplexity": 2791.877289827807}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256812.65/warc/CC-MAIN-20190522123236-20190522145236-00442.warc.gz"}
https://forum.allaboutcircuits.com/threads/calculating-current-with-parallel-batteries.30049/	calculating current with parallel batteries Discussion in 'Homework Help' started by msedtal, Nov 11, 2009. 1. msedtal Thread Starter New Member Nov 11, 2009 2 0 i am in a physics 2 course, and i am having trouble calculating current with two batteries connected in parrallel. I am at a complete loss of where to start. The problem is probably simple to most of you. From the problem i am thinking that i should use I=V/R ; but am unsure if V=9+9; and am also unsure about R, i am thinking it should be R=1+ (1/(1/2)+(1/3)) 2. JDT Well-Known Member Feb 12, 2009 658 85 Start by noticing that the circuit can be re-drawn. See Diagram. From circuit B, work out the current through R3. Then work back. There is a more mathematical way to do it but because both battery voltages are the same this is the easy way. Only Ohms law needed. File size: 8.6 KB Views: 40 3. msedtal Thread Starter New Member Nov 11, 2009 2 0 thanks for the help, it put me on the right track and i got the questions correct. 4. thatoneguy AAC Fanatic! Feb 19, 2009 6,357 718 It appears that the top 1Ω Resistor is the load, the other two resistors represent the internal resistance of the batteries (which goes up as a battery goes dead). Help for the bonus point: $P_{ower}=I^2\cdot R$ 5. KL7AJ AAC Fanatic! Nov 4, 2008 2,047 305 Number one rule of Electronics.....never make anything more complicated than it really is! In a PARALLEL circuit, the voltage is the same across all components. The battry will appear as a single voltage source. Ohm's law does the rest. eric 6. GetDeviceInfo Senior Member Jun 7, 2009 1,571 230 I see it differently; File size: 5.8 KB Views: 35 7. thatoneguy AAC Fanatic! Feb 19, 2009 6,357 718 Are you sure about the polarity of the batteries in the OP, to be parallel? They appear to be connected in series, as the post above shows. 8. Quintilis_Telescope New Member Oct 13, 2009 10 0 Loop 1 = 3ΩI1 + 1Ω(I1-I2) = 9V Loop 2 = 2ΩI2 + 1Ω(I2-I1) = 9V I1 = 3.2727A I2 = 4.0909A I3 = 3.2727A-4.0909A = -818mA (flowing up) File size: 23.6 KB Views: 15	2017-02-21 12:25:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 1, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6782917976379395, "perplexity": 3657.2836028797346}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501170708.51/warc/CC-MAIN-20170219104610-00382-ip-10-171-10-108.ec2.internal.warc.gz"}
http://www.gorun.es/mild-steel/c45-tensile-yield-strength_1007.html	>Home>Products>c45 tensile yield strength # c45 tensile yield strength ## c45 tensile yield strength Applications: c45 tensile yield strength is extensively used in a variety of industries. c45 tensile yield strength is widely used in structural applications, including bridges, buildings and construction equipment and more. ## c45 tensile yield strength Specification: Thickness: 6-400 mm Width: 1600-4200 mm Length: 4000-15000mm send e-mail [email protected] Chat Online Leave Message ### Young's Modulus - Tensile and Yield Strength for common Young's Modulus or Tensile Modulus alt.Modulus of Elasticity - and Ultimate Tensile and Yield Strength for steel,glass,wood and other common materials .Sponsored Links .Tensile Modulus - or Young's Modulus alt.Modulus of Elasticity - is a measure of stiffness of an elastic material.It is used to describe the elastic properties of objects Yield Strength of Plastics - basic principles,the tensile Yield Strength The yield strength of the plastic is the where the material begins to deform in a plastic fashion.Prior to the yield strength,the material will act elastically meaning that if the strain were halted at any point in the elastic portion,the material would return to its original length.Yield Strength Testing - Yield strength Ultimate strength Yield Strength Testing .The stress a material can withstand without permanent deformation.This is not a sharply defined point.Yield strength is the stress which will cause a permanent deformation of 0.2% of the original dimension.Point at which material exceeds the elastic limit and will not return to its origin shape or length if the stress is removed. ### Uniaxial Tension and Compression Testing of Materials Sep 25,2013 c45 tensile yield strength#0183;ultimate tensile strength (373.8 MPa) is greater than that of aluminum (274.5 MPa).A similar ranking was observed for the metals behavior in yielding steel had the highest yield strength (162.6 MPa),followed by brass (118.7 MPa),copper (95.6 MPa),and aluminum (73.0 MPa).Uniaxial Tension and Compression Testing of MaterialsSep 25,2013 c45 tensile yield strength#0183;ultimate tensile strength (373.8 MPa) is greater than that of aluminum (274.5 MPa).A similar ranking was observed for the metals behavior in yielding steel had the highest yield strength (162.6 MPa),followed by brass (118.7 MPa),copper (95.6 MPa),and aluminum (73.0 MPa).Tensile strength - Simple English Wikipedia,the free Tensile strength is a measurement of the force required to pull something such as rope,wire,or a structural beam to the point where it breaks..The tensile strength of a material is the maximum amount of tensile stress that it can take before failure,for example breaking..There are three typical definitions of tensile strength Yield strength - The stress a material can withstand without ### Tensile Strength vs Yield Strength - Clifton Steel Tensile strength will show us how much tensile stress the steel can withstand until it leads to failure in two ways ductile or brittle failure.Ductile failure - think of this as the preliminary stage of failure,where it is pushed beyond the yield point to permanent deformation.Tensile Strength vs Yield Strength - Clifton SteelTensile strength will show us how much tensile stress the steel can withstand until it leads to failure in two ways ductile or brittle failure.Ductile failure - think of this as the preliminary stage of failure,where it is pushed beyond the yield point to permanent deformation.Tensile / Yield Strength of Steel Chart - AMESTensile / yield strengths and ductilities for some of the plain carbon and low alloy steels are given in the following mechanical properties of steel chart.Yield Strength,Tensile Strength and Ductility Values for Steels at Room Temperature ### Tensile Yield Strengths of ASTM A105 at Elevated The tensile strengths and yield strengths of ASTM A105/ ASME SA-105 forgings at elevated temperatures are provided in ASME Boiler and Pressure Vessel Code Section II Part D.These values,which are tabulated in below two tables,are provided for use in design calculations.Neither ASME Codes nor ASME material specifications require elevated temperature testing for tensile strengths and yield Strength at Break Tensile - OmnexusYield Strength vs.Tensile Strength Yield Strength is the stress a material can withstand without permanent deformation or a point at which it will no longer return to its original dimensions (by 0.2% in length).Whereas,Tensile Strength is the maximum stress that a material can withstand while being stretched or pulled before failing or Strength at Break Tensile - OmnexusYield Strength vs.Tensile Strength Yield Strength is the stress a material can withstand without permanent deformation or a point at which it will no longer return to its original dimensions (by 0.2% in length).Whereas,Tensile Strength is the maximum stress that a material can withstand while being stretched or pulled before failing or ### Steels - Endurance Limits and Fatigue Stress 1 MPa = 10 6 Pa = 1 N/mm 2 = 145.0 psi (lbf/in 2); Fatigue limit,endurance limit,and fatigue strength are used to describe the amplitude (or range) of cyclic stress that can be applied to the material without causing fatigue failure.; Creep.The time dependent deformation due to heavy load over time is known as creep..In general both stress and temperature influence on the rate of creep Steel material properties - SteelConstructionfoThe UK National Annex to BS EN 1993-1-1 allows the minimum yield value for the particular thickness to be used as the nominal (characteristic) yield strength f y and the minimum tensile strength f u to be used as the nominal (characteristic) ultimate strength..Similar values are given for other grades in other parts of BS EN 10025 and for hollow sections to BS EN 10210-1.Steel material properties - SteelConstructionfoThe UK National Annex to BS EN 1993-1-1 allows the minimum yield value for the particular thickness to be used as the nominal (characteristic) yield strength f y and the minimum tensile strength f u to be used as the nominal (characteristic) ultimate strength..Similar values are given for other grades in other parts of BS EN 10025 and for hollow sections to BS EN 10210-1. ### Some results are removed in response to a notice of local law requirement.For more information,please see here.Previous123456NextIn-situ tensile test of high strength nanocrystalline Jan 03,2015 c45 tensile yield strength#0183;WHPT processed C45 steel demonstrated record tensile properties yield strength of 1390 MPa and ultimate strength of 2174 MPa.This level of ultimate tensile strength (UTS) is greater than that of some grades of the advanced high strength steel (AHSS) and even martensitic steels (Mart) as shown in Table 1.Some results are removed in response to a notice of local law requirement.For more information,please see here.12345Nextwhat is the shear strength value of C45 material? Yahoo Feb 08,2010 c45 tensile yield strength#0183;C45 Steel properties.Tensile strength 600 - 800 MPa .Young's modulus 210000 - 210000 MPa .Elongation 16 - 16 % .Yield strength 340 - 400 MPaSome results are removed in response to a notice of local law requirement.For more information,please see here. ### People also askWhat is C45 steel?What is C45 steel?EN 1.0503 Material C45 Steel EN 10083-2 C45 steel (EN 1.0503) is a high strength medium carbon quality steel.Due to poor hardenability,C45 material is generally used in a normalized condition,and when the mechanical properties are required to be high,the quenching and tempering treatment is adopted.EN 1.0503 Material C45 Steel Equivalent,Properties Ovako C45 047A Steel,Normalizing - MatWeb Ovako C45 047A Steel,Normalizing Categories Metal; Ferrous Metal; Alloy Steel; Carbon Steel; AISI 1000 Series Steel.Material Notes General Information Grade SB-C45 is one of our carbon steel grades with a narrowed chemical composition in order to reach a high hardenability and it is also fine grain treated with Al..Grade 047A is an ingot casted low alloyed sMaterial S355 Steel Properties,Comparison,Equivalent Below are the tables to show the steel grade S355 datasheet including chemical composition,yield strength,tensile strength and elongation,etc.All data sheet of DIN EN 10025-2 is the same as BS EN 10025-2 and other EU member states.Chemical Composition.The datasheet below shows grade S355 steel chemical composition. ### File Size 16KBPage Count 2What is the shear strength value of C45 material? steel C45 Steel properties.:.Tensile strength 600 - 800 MPa Young's modulus 210000 - 210000 MPa Elongation 16 - 16 % Yield strength 340 - 400 MPa.Ultimate Tensile strength (600 - 800) MPaFASTENER REFERENCE GUIDEStrength (psi) Min.Tensile Strength (psi) Core Hardness Rockwell Min.Yield Strength (psi) Grade Identification Marking Compatible Nuts Min.Max.SAE J429-Grade 1 Low or medium carbon steel 1/4 - 1-1/2 60,000 B70 B100 36,000 ASTM A563 Grade A or SAE J429-Grade 2 SAE J995 Grade 2 Hex 1/4 - 3/4 (4) 74,000 B80 B100 57,000FASTENER REFERENCE GUIDEStrength (psi) Min.Tensile Strength (psi) Core Hardness Rockwell Min.Yield Strength (psi) Grade Identification Marking Compatible Nuts Min.Max.SAE J429-Grade 1 Low or medium carbon steel 1/4 - 1-1/2 60,000 B70 B100 36,000 ASTM A563 Grade A or SAE J429-Grade 2 SAE J995 Grade 2 Hex 1/4 - 3/4 (4) 74,000 B80 B100 57,000 ### EN 1.0503 Material C45 Steel Equivalent,Properties EN 1.0503 Material C45 Steel.EN 10083-2 C45 steel (EN 1.0503) is a high strength medium carbon quality steel.Due to poor hardenability,C45 material is generally used in a normalized condition,and when the mechanical properties are required to be high,Difference Between Yield Strength and Ultimate StrengthUltimate tensile strength (UTS) is considered as the failure criteria for brittle material.In ductile materials,yield strength is much lower than ultimate strength.For ductile materials,ultimate strength is roughly 1.5 times higher than yield strength.Yield strength is used while designing components or structures made of ductile materials.Difference Between Yield Strength and Tensile StrengthOct 14,2015 c45 tensile yield strength#0183;The main difference between yield strength and tensile strength is that yield strength is the minimum stress under which a material deforms permanently,whereas tensile strength describes the maximum stress that a material can handle before breaking.Stress Strain Characteristics of a ### Difference Between Tensile Strength and Yield Strength May 28,2012 c45 tensile yield strength#0183;Yield strength can be measured using methods such as the divider method.Tensile Strength vs Yield Strength.Ultimate tensile strength is the strength where the necking effect begins.Yield strength is the strength where the deformation turns from an elastic deformation to aDatasheet for Steel Grades Carbon Steel C45C45 Physical Properties Tensile strength 115-234 b/MPa Yield Strength 23 0.2 /MPa Elongation 65 5 (%) - (%) Akv - Akv/J HBS 123-321 - HRC 30 - C45 Mechanical Properties Tensile strength 231-231 b/MPa Yield Strength 154 0.2 /MPa Elongation 56 5(%) - (%) Akv - Akv/J HBS 235-268 - HRC 30 - C45 Heat C45/C45E/C45R/DIN 1.1191 - Xingsheng Special steelC45 steel is an unalloyed medium carbon engineering steel which has 0.42%-0.5% Carbon.It offers moderate tensile strengths,wear resistance and good machinability.this material is capable of through hardening by quenching and tempering on limited sections,and also can be flame or induction hardened to surface hardness Min 55HRC.C45 is generally supplied in an untreated or normalised condition ### C45 EN 10083-2 (Euronorm) - MetalDatafo C45 EN 10083-2 (Euronorm) Reduction of area c45 tensile yield strengthgt; 40 % Diameter c45 tensile yield strengthgt; 40 mm ; Yield Strength c45 tensile yield strengthgt; 370 MPa Tensile Strength 630 - 780 MPa Elongation c45 tensile yield strengthgt; 17 %C45 / 1.0503 - SteelNumber - Chemical composition Chemical composition of steel C45 (1.0503),Standards of steel C45 (1.0503) Mechanical Properties of steel C45 (1.0503) Equivalent grades of steel C45 (1.0503) steel C45 (1.0503) Tensile Strength,Elongation,Proof strength ,HardnessBending flexural test - tec-scienceDue to the linear stress distribution at a bending load,the flexural yield strength for steels is about 10 % to 20 % higher than the tensile yield strength! For materials with no visible yield strengths in the stress-curves,a 0.2 % flexural offset yield strength $$\sigma_{by0.2}$$ can be defined analog to the 0.2% offset yield strength of the ### AISI 1045 Steel,cold drawn,high temperature stress AISI 1045 Steel,cold drawn,high temperature stress relieved,50-75 mm (2-3 in) round Categories Metal; Ferrous Metal; Carbon Steel; AISI 1000 Series Steel; Medium Carbon Steel.Material Notes:42CrMo4 / 1.7225 - SteelNumber - Chemical composition Minimum yield strength / Mindestwert der oberen Streckgrenze / Limite d elasticite minimale R m Tensile strength / Zugfestigkeit / Resistance a la traction A Minimum elongation / Mindestwert der Bruchdehnung / Allongement minimal J Notch impact test / Kerbschlagbiegeversuch /17-4 Stainless Steel Properties KVA Stainless17-4 - UNS S17400 Type 17-4 is the most common grade of martensitic precipitation hardenable (PH) alloys.17-4 provides an outstanding combination of high strength,good mechanical properties at temperatures up to 600 c45 tensile yield strength#176;F (316 c45 tensile yield strength#176;C),and short-duration,low-temperature heat treatments that minimize warpage and scaling. ### 1045 Medium Tensile Carbon Steel Bar Interlloy 1045 is a medium tensile low hardenability carbon steel generally supplied in the black hot rolled or occasionally in the normalised condition,with a typical tensile strength range 570 - 700 Mpa and Brinell hardness range 170 - 210 in either condition.Characterised by fairly good strength and impact properties,plus good machinability and reasonable weldability in the hot rolled or normalised 1045 MEDIUM TENSILE CARBON STEEL BARTensile Strength Mpa 640 Yield Strength Mpa 410 Elongation in 50mm % 22 Impact Izod J 54 Hardness HB 187 Rc 10 Material stocked generally in the hot rolled condition but can occasionally be in the normalised condition.NB.Check the mill certificate if critical for end use.1045 MEDIUM TENSILE CARBON STEEL BARTensile Strength Mpa 640 Yield Strength Mpa 410 Elongation in 50mm % 22 Impact Izod J 54 Hardness HB 187 Rc 10 Material stocked generally in the hot rolled condition but can occasionally be in the normalised condition.NB.Check the mill certificate if critical for end use. ### results for this questionFeedbackC45 - BEBON steel Yield Strength R c45 tensile yield strength#176;(Mpa) Tensile Strength Rm (Mpa) Elon-gation A5(%) Hardness HRC Quenching Temperature () Benda-bility Nominal Thickness,t 1.95mmt10.0mm Rolled Annealed C45 Rolled Annealed Water-quenched Oil quenched 460 330 750 540 2270 1980 18 30 58 55 820 860 Min.reco-mmended Bending radius (90 c45 tensile yield strength#176;) 2.0 c45 tensile yield strength#215;t 1.0 c45 tensile yield strength#215;t Main Product More related products	2022-01-18 01:42:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5277206301689148, "perplexity": 11506.584671593742}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300658.84/warc/CC-MAIN-20220118002226-20220118032226-00590.warc.gz"}
https://stacks.math.columbia.edu/tag/006B	## 6.1 Introduction Basic properties of sheaves on topological spaces will be explained in this document. A reference is [Godement]. This will be superseded by the discussion of sheaves over sites later in the documents. But perhaps it makes sense to briefly define some of the notions here. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).	2022-07-01 02:29:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9548943042755127, "perplexity": 691.9137746540969}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103917192.48/warc/CC-MAIN-20220701004112-20220701034112-00691.warc.gz"}
http://www.mathworks.com/help/physmod/sm/mech/gs/representations-of-body-orientation.html?nocookie=true	Accelerating the pace of engineering and science # Documentation ## Representations of Body Orientation You represent a SimMechanics™ body's orientation by specifying the orientation of its center of gravity coordinate system (CG CS) axes relative to some other set of axes, either the CS axes of an adjoining body or the World CS axes. No reorientation is represented by "no rotation" or the rotational identity. A general rotation of a body in three dimensions has three independent degrees of freedom. There are many equivalent and interconvertible ways to represent these degrees of freedom . The Body and related Body Sensor and RotationMatrix2VR blocks use the following representations. The block reference pages for these blocks discuss block-specific details. ### Axis-Angle Representation The axis-angle representation of a rotation is the most fundamental form. Specify a rotation axis n, then rotate by the right-hand rule about that axis by some angle θ. The vector n = (nx,ny,nz) is a three-component unit vector, where n·n = nx2 + ny2 + nz2 = 1. The axis n is sometimes called the eigenaxis. SimMechanics models do not make direct use of the axis-angle representation, but it is the starting point for deriving other forms. It is also used extensively in mechanical applications such as computer-aided design and robotics. The axis-angle representation is usually written as a 4-vector: [nx ny nz θ]. Of the four numbers, three are independent, because n always has unit length. The remaining freedom in this vector allows you to specify a direction (two angles) and the size and sense of the rotation about that directional axis (magnitude and sign of θ). To describe continuous rotation in time, treat n and θ as functions of time. ### Quaternion Representation A quaternion represents a three-dimensional rotation as a four-component row vector of unit length: $q=\left[{n}_{x}\mathrm{sin}\left(\theta /2\right),{n}_{y}\mathrm{sin}\left(\theta /2\right),{n}_{z}\mathrm{sin}\left(\theta /2\right),\mathrm{cos}\left(\theta /2\right)\right]=\left[{q}_{\text{v}},{q}_{\text{s}}\right]$ with qq = qv·qv + qs2 = 1. This definition uses the axis-angle representation defined above. The rotation angle about that axis is θ. To describe continuous rotation in time, treat n and θ as functions of time. Unlike some rotation representations, quaternions never become singular. For more about quaternions, see Bell and Shuster in the chapter references. ### Rotation Matrix Representation The axis-angle representation also defines the rotation matrix R in exponential form R = exp(θ J), where the Jk are real, antisymmetric matrices, and J = nxJ1 + ny J2 + nz J3. The rotation matrix R is orthogonal: RRT = RTR = I. The J matrices are related to the antisymmetric permutation symbol ɛijk. ${\left({J}^{\text{j}}\right)}_{\text{ik}}={\epsilon }_{\text{ijk}}$ The exponential R is reduced to closed form by the Rodrigues identity: where I is the identity matrix, and J is given by $n\cdot J=\left(\begin{array}{ccc}0& -{n}_{z}& {n}_{y}\\ {n}_{z}& 0& -{n}_{x}\\ -{n}_{y}& {n}_{x}& 0\end{array}\right)$ The inverse of R is identical to its transpose RT. You can also obtain the inverse by replacing θ with θ or by reversing the direction of n. To describe continuous rotation in time, treat n and θ as functions of time. ### Euler Angle Representation An alternative representation for R is to rotate, in succession, about three independent axes, by three independent Euler angles. A full rotation R starting in World composes by multiplying the matrices successively on the left: RBW = R3R2R1 A full rotation R starting in a body CS composes by multiplying the matrices successively on the right: RWB = R1R2R3 The Euler angle convention is to 1. Rotate about one body coordinate axis (which rotates the other two). 2. Then rotate about a second body coordinate axis (rotated from its original direction) not identical to the first. 3. Lastly, rotate about another body coordinate axis not identical to the second. Thus there are 322 = 12 possible Euler angle rotation sequences. The rotation axis sequences Z-X-Z and Z-Y-X are common. Rotation angles are often labeled as θ1, θ2, θ3 or Φ, θ, Ψ as the first, second, and third angles, respectively. For example, RBW = RX1)RY2)RZ3) RWB = RZ(Φ)RX(θ)RZ(Ψ) A two-dimensional rotation about a fixed axis requires one angle. For example, rotating the x- and y-axes about the z-axis by Φ is represented by ${R}_{Z}\left(\varphi \right)=\left(\begin{array}{ccc}\mathrm{cos}\varphi & -\mathrm{sin}\varphi & 0\\ \mathrm{sin}\varphi & \mathrm{cos}\varphi & 0\\ 0& 0& 1\end{array}\right)$ To describe continuous rotation in time, treat the Euler angles as functions of time. The Euler angle representation is singular in certain limiting situations. Such singularities are artifacts of the Euler angle form and have no geometric or physical significance. ### Converting Rotation Representations Certain SimMechanics blocks make use of different rotation representations. • The Body block makes direct use of the Euler angle, rotation matrix, and quaternion representations. • The Body Sensor block makes use of the rotation matrix. • The RotationMatrix2VR block uses the rotation matrix and axis-angle forms. The four rotation representations presented in this section are equivalent. You can represent a rotation equally well with any one of them. Some applications, however, tend to favor one representation over the others, and certain representations are singular in certain limits. It is helpful to know how to convert the various rotation representations into one another. The following summaries group the conversion formulas into one place. #### Transforming the Axis-Angle Representation The rotation axis unit vector n and the rotation angle θ define this representation, which is discussed in detail in Axis-Angle Representation preceding. This representation defines the quaternion and rotation matrix representations: $n\cdot J=\left(\begin{array}{ccc}0& -{n}_{z}& {n}_{y}\\ {n}_{z}& 0& -{n}_{x}\\ -{n}_{y}& {n}_{x}& 0\end{array}\right)$ #### Transforming the Quaternion Representation The quaternion is a vector-scalar pair, q = [qv qs], defined by Quaternion Representation preceding. You can recover the axis-angle representation from the quaternion components: $\begin{array}{l}\theta =2\cdot {\mathrm{cos}}^{-1}\left({q}_{\text{s}}\right)\\ n={q}_{\text{v}}/\sqrt{1-{q}_{\text{s}}{}^{2}}\end{array}$ You can also construct the equivalent rotation matrix R from q. The term ${q}_{\text{v}}^{\text{T}}\otimes {q}_{\text{v}}$ is the outer product of qv with itself, the 3-by-3 matrix of qv components multiplied by each other. #### Transforming the Rotation Matrix Representation The rotation matrix R is an orthogonal 3-by-3 matrix: RRT = RTR = I, defined in Rotation Matrix Representation preceding. You can invert the rotation matrix representation to obtain the equivalent representations for the quaternion q = [qv qs] and axis-angle (n, θ) $\begin{array}{l}{q}_{\text{s}}=\frac{1}{2}\sqrt{Tr\left(R\right)+1}\\ {q}_{\text{v}}=Tr\left(J\ast R\right)/\left(2\sqrt{Tr\left(R\right)+1}\right)\\ \theta =2\cdot {\mathrm{cos}}^{-1}\left(\frac{1}{2}\sqrt{Tr\left(R\right)+1}\right)\\ n=Tr\left(J\ast R\right)/\left(\sqrt{Tr\left(R\right)+1}\cdot \sqrt{3-Tr\left(R\right)}\right)\end{array}$ The trace Tr of a matrix is the sum of its diagonal elements. The J matrices constitute a 3-vector of matrices defined by the antisymmetric permutation symbol, (Jj)ik = ɛijk. See The Permutation Symbol and the Vector Cross Product preceding for more details. The RotationMatrix2VR block converts the rotation matrix to the axis-angle representation. #### Transforming the Euler Angle Representation The Euler angle representation of a rotation, defined by Euler Angle Representation preceding, stands apart from the other three, insofar as you cannot derive it from the axis-angle representation. It depends on the choice of rotation axis sequence, which generates multiple definition conventions. The Euler angle representation, at certain limits, can also be singular. Use caution with Euler angle expressions. If you choose a convention and three angles, then compute R, you can convert R to the other representations by the use of Transforming the Rotation Matrix Representation above. But given the nine components of R, you must find the Euler angles by inverting the nine equations that result from this matrix equation. (Only three equations of the nine are independent.) In some cases, angles can be read from R by inspection. For example, choose rotations with respect to a Body coordinate system (CS) triad, in a commonly used rotation axis sequence Z-X-Z, with Φ, θ, Ψ as the respective angles. The rotation matrix is RWB = R1(Φ)R2(θ)*R3(Ψ), $\begin{array}{l}{R}_{\text{WB}}\left(\varphi ,\theta ,\psi \right)=\left(\begin{array}{ccc}\mathrm{cos}\varphi & -\mathrm{sin}\varphi & 0\\ \mathrm{sin}\varphi & \mathrm{cos}\varphi & 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\theta & -\mathrm{sin}\theta \\ 0& \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right)\left(\begin{array}{ccc}\mathrm{cos}\psi & -\mathrm{sin}\psi & 0\\ \mathrm{sin}\psi & \mathrm{cos}\psi & 0\\ 0& 0& 1\end{array}\right)\\ =\left(\begin{array}{ccc}\mathrm{cos}\varphi \mathrm{cos}\psi -\mathrm{sin}\varphi \mathrm{cos}\theta \mathrm{sin}\psi & -\mathrm{cos}\varphi \mathrm{sin}\psi -\mathrm{sin}\varphi \mathrm{cos}\theta \mathrm{cos}\psi & \mathrm{sin}\varphi \mathrm{sin}\theta \\ \mathrm{sin}\varphi \mathrm{cos}\psi +\mathrm{cos}\varphi \mathrm{cos}\theta \mathrm{sin}\psi & -\mathrm{sin}\varphi \mathrm{sin}\psi +\mathrm{cos}\varphi \mathrm{cos}\theta \mathrm{cos}\psi & -\mathrm{cos}\varphi \mathrm{sin}\theta \\ \mathrm{sin}\theta \mathrm{sin}\psi & \mathrm{sin}\theta \mathrm{cos}\psi & \mathrm{cos}\theta \end{array}\right)\end{array}$ In this convention, you can read θ from the R33 component, then find Ψ from the R32 or R31 component. Obtain Φ from one of the other components, using cos2Φ + sin2Φ = 1, or by multiplying from the right by R3ΨT, then R2θT. The second method yields a unique solution for the sine and cosine of Φ. ### Converting the Angular Velocity The rotation matrix R is defined in Representations of Body Motion and Rotation Matrix Representation preceding. The angular velocity vector ω is the rate at which a spinning CS rotates. R and the antisymmetric matrix Ω define ω: $\begin{array}{l}\Omega =+\left(\text{d}R/\text{dt}\right)\cdot {R}^{\text{T}}=-R\cdot \left(\text{d}{R}^{\text{T}}/\text{dt}\right)\\ {\Omega }_{\text{ik}}=\text{+}{\sum }_{\text{j}}{\epsilon }_{\text{ijk}}{\omega }_{\text{j}}\\ {\omega }_{\text{j}}=\left(\frac{\text{1}}{\text{2}}\right){\sum }_{\text{ik}}{\epsilon }_{\text{ijk}}{\Omega }_{\text{ik}}\end{array}$ You can also express the angular velocity in terms of Euler angles, by choosing a particular set of angles to represent R. See Euler Angle Representation and Transforming the Euler Angle Representation preceding. The quaternion derivative is also related to the angular velocity: $\begin{array}{l}\text{d}{q}_{\text{v}}\text{/dt}=\left(\frac{\text{1}}{\text{2}}\right)\left({q}_{\text{s}}{\omega }_{\text{Body}}-{q}_{\text{v}}\text{x}{\omega }_{\text{Body}}\right)\\ \text{d}{q}_{\text{s}}\text{/dt}=-\left(\frac{\text{1}}{\text{2}}\right)\text{(}{q}_{\text{v}}\cdot {\omega }_{\text{Body}}\text{)}\end{array}$	2014-12-28 14:34:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 11, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8036154508590698, "perplexity": 1228.6958327903978}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1419447558026.133/warc/CC-MAIN-20141224185918-00039-ip-10-231-17-201.ec2.internal.warc.gz"}
http://tex.stackexchange.com/questions?sort=featured	# All Questions How could LaTeX replace the tokens <= by the command \leq efficiently? Example 1: I have this code: $2x <= 4x - 2$ And I want to get after the compilation this: Example 2: \[ ...	2016-05-28 13:44:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9967751502990723, "perplexity": 2560.75382540515}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049277807.82/warc/CC-MAIN-20160524002117-00015-ip-10-185-217-139.ec2.internal.warc.gz"}
https://en.universaldenker.org/illustrations/1091	Skip to main content # Illustration Centripetal force and angular velocity during a uniform circular motion Download Share — copy and redistribute the material in any medium or format Adapt — remix, transform, and build upon the material for any purpose, even commercially. Sharing and adapting of the illustration is allowed with indication of the link to the illustration. Centripetal force $$\boldsymbol{F}_{\text z}$$ acts on a body of mass $$m$$, which is in a uniform circular motion. The centripetal force is always perpendicular to the angular velocity $$\boldsymbol{\omega}$$. And the angular velocity $$\boldsymbol{\omega}$$ is perpendicular to the circular path with the radius $$r$$.	2021-12-03 13:39:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.773152768611908, "perplexity": 341.00252920154986}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362879.45/warc/CC-MAIN-20211203121459-20211203151459-00443.warc.gz"}
https://nforum.ncatlab.org/discussion/14589/the-real-projective-spaces-in-homotopy-type-theory/	# Start a new discussion ## Not signed in Want to take part in these discussions? Sign in if you have an account, or apply for one below ## Site Tag Cloud Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support. 1. Page created, but author did not leave any comments. Anonymous • CommentRowNumber2. • CommentAuthorUrs • CommentTimeJun 9th 2022 • (edited Jun 9th 2022) I have made the requested link to tautological bundle work: through the existing tautological line bundle. Also added a bunch of further hyperlinks, notably to real projective space. In doing so, I took the liberty of re-formatting as follows. On the off-chance that you are reading the nForum, let me know what you think: Abstract. Homotopy type theory is a version of Martin-Löf type theory taking advantage of its homotopical models. In particular, we can use and construct objects of homotopy theory and reason about them using higher inductive types. In this article, we construct the real projective spaces, key players in homotopy theory, as certain higher inductive types in homotopy type theory. The classical definition of $\mathbb{R}P^n$, as the quotient space identifying antipodal points of the $n$-sphere, does not translate directly to homotopy type theory. Instead, we define $\mathbb{R}P^n$ by induction on $n$ simultaneously with its tautological bundle of $2$-element sets. As the base case, we take $\mathbb{R}P^{-1}$ to be the empty type. In the inductive step, we take $\mathbb{R}P^{n+1}$ to be the mapping cone of the projection map of the tautological bundle of $\mathbb{R}P^{n}$, and we use its universal property and the univalence axiom to define the tautological bundle on $\mathbb{R}P^{n+1}$. By showing that the total space of the tautological bundle of $\mathbb{R}P^n$ is the n-sphere, we retrieve the classical description of $\mathbb{R}P^{n+1}$ as $\mathbb{R}P(n)$ with an $(n+1)$-cell attached to it. The infinite dimensional real projective space, defined as the sequential colimit of the $\mathbb{R}P^n$ with the canonical inclusion maps, is equivalent to the Eilenberg-MacLane space $K(\mathbb{Z}/2\mathbb{Z},1)$, which here arises as the subtype of the universe consisting of $2$-element types. Indeed, the infinite dimensional projective space classifies the $0$-sphere bundles, which one can think of as synthetic line bundles. These constructions in homotopy type theory further illustrate the utility of homotopy type theory, including the interplay of type theoretic and homotopy theoretic ideas. • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts. • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list. • (Help)	2022-08-13 00:47:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 17, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8719717264175415, "perplexity": 542.4714190441744}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571847.45/warc/CC-MAIN-20220812230927-20220813020927-00400.warc.gz"}
https://www.shaalaa.com/question-bank-solutions/the-current-ratio-z-ltd-1-1-state-reason-which-following-transaction-would-1-included-trade-payables-was-bill-payable-rs-3-000-which-was-met-maturity-2-debentures-rs-50-000-were-converted-equity-share-concept-accounting-ratios_20681	Share # The Current Ratio of Z. Ltd is 1: 1. a State with Reason Which of the Following Transaction Would 1. Included in the Trade Payables Was a Bill Payable Of Rs 3,000 Which Was Met on Maturity 2. Debentures of Rs 50,000 Were Converted into Equity Share - CBSE (Arts) Class 12 - Accountancy #### Question The current ratio of Z. Ltd is 1: 1. A state with reason which of the following transaction would 1. increase; 2. decrease or 3. not change the ratio. 1. Included in the trade payables was a bill payable of Rs 3,000 which was met on maturity 2. Debentures of Rs 50,000 were converted into Equity Share #### Solution Quick ratio = "Liquid Assets"/"Current Liabilities" 1) A bill payable was met on maturity will affect: Trade Payables will reduce by Rs 3,000. Cash will reduce by Rs 3,000. The simultaneous decrease in Both liquid assets and current liabilities by the same amount will leave the ratio unaffected. 2. Conversion of Debentures of Rs 50,000 into equity share will A decrease in Debentures of Rs 50,000 Increase in Equity share i.e. Share Capital by Rs 50,000. Debentures and share capital both does not form part of the liquid ratio. Therefore, will be no effect on the ratio. Is there an error in this question or solution? #### Video TutorialsVIEW ALL [1] Solution The Current Ratio of Z. Ltd is 1: 1. a State with Reason Which of the Following Transaction Would 1. Included in the Trade Payables Was a Bill Payable Of Rs 3,000 Which Was Met on Maturity 2. Debentures of Rs 50,000 Were Converted into Equity Share Concept: Concept of Accounting Ratios. S	2019-12-13 20:44:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22366119921207428, "perplexity": 9062.130636171205}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540569146.17/warc/CC-MAIN-20191213202639-20191213230639-00458.warc.gz"}
https://electronics.stackexchange.com/questions/277994/how-is-this-cd4013-application-supposed-to-work	# How is this CD4013 application supposed to work? I'm trying to troubleshoot an issue in a synthesizer I have, an early 80s synthesizer from Roland called RS-09. I'm just starting to study logic and am perplexed by what's going on. In this synthesizer, a 4013 is being used as a frequency divider to divide or not divide a trigger signal from a master oscillator based on whether the user has selected a feature called "Octave down" via a switch. The output Q of the 4013 then passes to a "top octave generator" that uses it to generate the 12 notes of a chromatic scale from which all the other notes are then made. Right now it is only working in "Octave down" mode; without "Octave down" selected, all keys just produce noise. Here's a closeup of this part of the schematic. The vertical line labeled with "10" connects to the switch, and the horizontal line with the indicator that the square wave should be there connects to the octave generator: The 4013 is set up so that the "high" voltage is 0V and the "low" voltage is -10V (down arrow is -10V supply). When the "Octave down" is selected and the 4013 is supposed to divide, the switch applies -10V which sends a little less than that -10V to SET (pin 8) via the 15kohm resistor; when "Octave down" is not selected, 0V is connected to that path. In Octave down mode, that -10V is also able to pass through the diode D206 to RESET (pin 10) (less 0.6-0.7V from the diode drop) and with reset and set held in the low state, the flip flop is in clocked mode, responding to the trigger signal from the master oscillator. This seems to be working just fine. With Q/ connected to D as it is, the flip flop also "binary divides" and Q outputs a square wave at 1/2 the frequency of the trigger. My problem is that I don't understand what's supposed to be happening when "Octave down" is not selected. I don't see how it could possibly work, and it doesn't. This is what my understanding is, but something must be wrong: The ~0V at SET holds it in the high state but cannot pass through D206 to RESET, leaving the reset open to receiving its "instructions" from the oscillator (direct mode). The trigger oscillator starts out oscillating between +5 and -10V but the diode clamps it at 0V to cut off the positive portion and is then sent to RESET [edit: this is not quite what is actually happening-- see oscilloscope images below]. But with the Set held high, the only possible outcomes for Q are the high state when the oscillator is at its low point sending ~-10V to Reset, when Q is high and Q/ is 0, and the "disallowed state" of both Set and Reset being high, leading to both Q and Q/ being high, when the oscillator is at its high point. Therefore, Q just sends out a constant "high voltage" (0V in this case) with a little bit of noise. That is what I'd expect and what I'm seeing it do. My meter and oscilloscope measurements confirm that this is what is happening. How is this supposed to work? Something is clearly wrong both in my interpretation of how it should work and in the way the circuit is actually working, but I've looked it over so many times and still can't figure it out. Your help is very much appreciated! Edit: I am adding these shots of the oscilloscope showing different inputs and outputs in the setting that is not working (higher octave). I left the vertical position where center is 0V to show their relative positions. I am on 0.5V/div. I was surprised to see how far negative the waveform at Reset is and how small the amplitude is. This is a little different from what I thought I observed before and more apparently "incorrect". The output of the master oscillator (where it arrives at CLOCK) is not a great-looking trigger waveform either, and my understanding was that 4013 requires a good trigger, but it seems to be working in clocked mode so maybe that's not an issue. • Yes that table is correct. S/R are intended to be used exclusively and when both are on (1) both outputs are on (1) while Qbar to D generates the lower octave.(=/2), something is wrong in bias to disable that – Tony Stewart Sunnyskyguy EE75 Jan 2 '17 at 0:39 • Would it seem like the 15kohm resistor and the .01uf capacitor connected between "Set" and ground would likely be to blame? Should they be holding Set lower? How low does Set have to be to be considered "low state" by the 4013 (e.g. what if it was -4V, -6V or -7V)? – soupertime Jan 2 '17 at 1:15 • Perhaps unused set, clock, and data (3, 5 and 6) should be tied to Vss (-10V)? It's not in the original design but I know it's best practice. – soupertime Jan 2 '17 at 1:36 • 0.5V/div x 4 div = 2V x10:1=-20V not -10V ?? – Tony Stewart Sunnyskyguy EE75 Jan 2 '17 at 15:53 • The schematic shows only one input of the unused F/F terminated, but is this the actual circuit? Trace the wiring to all 4013 pins to make sure (floating inputs at 10V are bad news, but if that part of the circuit is wrong then perhaps some others parts are too!). What was the horizontal scale on the scope, and did you use a x10 probe? – Bruce Abbott Jan 2 '17 at 18:17 I know this thread has gone dormant, but I thought it would be a good idea to post anyways in the event that it would be useful to someone working on an old Roland synth with 4013-based issues. Specifically, I've been working the internal arpeggiator clock on the key-assigner board of a Roland Jupiter-4 (the Jupiter-4 was released the same year as the RS-09). The circuit in question is below: Initially, it would appear that, with the Set line (pin 6) of the first flip-flop tied high, there would be no way that the 1Q (pin 1) would be anything other than high, so that the signal at CP5 would be fixed high, decidedly not the pulse wave shown in the diagram. And this is indeed what happens if one uses in the circuit a modern CD4013 with the conventional Set/Reset behavior (i.e., when both Set and Reset are high, both Q and Q/ are high). However, it seems that not all 4013s behave this way. To wit, later in the Jupiter-4 service manual, Roland specifies that Toshiba's TC4013 is to be used (specifically, the TC4013BP, according to the bill of materials): Even later in the service manual, Roland provides the pin-out and truth table for the TC4013, and, lo and behold, the TC4013 uses a Set/Reset functionality that differs from the conventional CD4013. Specifically, the TC4013 behaves such that, when both Set and Reset are high, Q is low and Q/ is high; that is, Reset takes precedence over Set, as this table from the service manual indicates: It is this odd Set/Reset behavior that is critical to producing the pulse wave at CP5 in the Jupiter-4 circuit above. I believe that 4013-based problems in Roland synths of this era - like the RS-09 in this thread as well as the "compatibility" problems identified elsewhere for the SH-101 - probably stem from this divergence from the conventional CD4013 Set/Reset behavior. Roland apparently made the dubious design decision to rely on what the chip manufacturers probably thought of as a "disallowed" Set/Reset configuration... In any event, Toshiba still makes the TC4013BP in a 14-pin DIP package, but it appears from the current datasheet that Toshiba has adopted the conventional CD4013 Set/Reset behavior, without changing the part suffix. I cannot find any documentation for when this change occurred, but, judging from datasheets I've seen on the web, it appears that this change in behavior was implemented sometime prior to the late 1990s. This means that, while older TC4013s might be available on, say, eBay, I don't know how you would know if they would be old enough to predate this change in functionality. You would probably need to test the chip out on a breadboard - tying the Set and Reset lines high and seeing how the Q output reacts - to be able to determine whether an old TC4013BP is usable in a Roland synth. And, as far as I know, no modern 4013 chips exhibit this odd Set/Reset behavior required by Roland synths of this era. Anyways, I hope this helps better understand what's going on with your RS-09...! • There is no shame in necromancy here :) Great first post, and welcome to EE.SE! – ThreePhaseEel Aug 28 '19 at 1:09 Here's a simulation of how it should work. Explanation below. Note that IC204, the TC4013BP master clock divider, is known to fail on the Roland RS-09. Here's a recording during which it failed. If careful resoldering of IC204 with flux (to ensure it's not just a cold solder joint) doesn't fix the issue you can get a replacement IC here. The RESET pin of IC204 is pulled up via resistor R215 and driven from the clock line through diode D205. When the clock goes low, diode D205 conducts and pulls RESET low. When the clock goes high, the diode stops conducting and R215 slowly charges up the capacitance of the RESET pin. After a while the voltage on RESET reaches the logic high threshold voltage and the Q output is reset. The ouput remains low until the next rising clock edge. The net effect is that the output toggles at the same frequency as the clock. When the Octave-Down switch is closed, RESET is held low via diode D206. With RESET low, the output toggles at half the clock frequency because the $\overline Q$ output is tied to the D-input. • That's a very expensive CD4013 compared to most places. – Andrew Morton Apr 13 '17 at 21:26 • The point is really that a sythesizer spare parts vendor is stocking the '4013 as a replacement part for the RS-09. They will confirm that it's a popular item :) – neonzeon Apr 13 '17 at 22:34 • I should have mentioned that even before I posted my question originally I had already replaced the 4013, at first with a new one and then also for a second time after sourcing an NOS 1980s Harris one, just in case the very slight differences between modern and older CMOS ICs might make a difference in this application. Neither affected the circuit's behavior. Still haven't figured this out. – soupertime Sep 5 '17 at 23:56 • @soupertime take a careful look at the capacitor voltage in the simulation. You should see a similar waveform on your 'scope. – neonzeon Sep 6 '17 at 21:48 • @soupertime As you can see from the simulation, the waveforms you captured on your 'scope are as expected. Specifically, the Q¯ ouput toggles at the clock frequency when Reset is not held down by the Octave-Down switch. Could you also show the Q ouput (Pin 13) on the o'scope with the switch open? – neonzeon Sep 6 '17 at 22:47 OK, I'm not sure if this counts as an "answer" because I don't feel like I understand why this happened, but I solved my synthesizer's problem at least. I finally got the RS-09 in question to work by replacing the 4013 with one I pulled from another RS-09 (I'll call it RS-09-2 for the sake of this explanation). When I put a new 4013 in RS-09-2, it ALSO didn't work. By switching them back (and back again) and putting another brand new 4013 in each just in case the first one I tested with was bad, I confirmed that both RS-09s will only work with a very old CD4013 in them. I would assume all RS-09s are this way. I've seen this happen one other time, where a circuit designed in the 1980s won't work with a new CMOS IC. In neither case was there anything about the circuit that offered an explanation as to why. Looking at the schematic in 2017 or 1982, a person would have the same reaction: Yes, that should work. But if you put in a CMOS IC made in 1982 it works, and if you put in a CMOS IC made in 2017, it doesn't. I'm not really happy with the outcome because the question of "Why?!" still remains, but there it is. I'd like to offer my thanks to everyone who tried to help me figure this out!	2020-04-07 18:03:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.49944403767585754, "perplexity": 1578.2935402797152}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371803248.90/warc/CC-MAIN-20200407152449-20200407182949-00122.warc.gz"}
http://mcqseries.com/stability-theory-3/	Thursday , March 22 2018 # Stability Theory – 3 3. If the characteristic equation of a closed-loop system is the centroid of the asymptotes in root-locus will be : (a) zero (b) 2 (c) – 1 (d) – 2 Answer : (c) Explanation : No answer description available for this question. Let us discuss. Be a Part of The New & Next Share the Knowledge Copy Protected by Chetan's WP-Copyprotect.	2018-03-22 09:56:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5733394026756287, "perplexity": 3826.8233399572027}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647838.64/warc/CC-MAIN-20180322092712-20180322112712-00325.warc.gz"}
https://stats.stackexchange.com/questions/29211/expectation-of-a-product-of-multiple-random-bernoulli-variables	# Expectation of a product of multiple random (Bernoulli) variables I'm doing a research and using random variables to model a random process. I'm defining a Bernoulli random variable as a product of several other Bernoulli variables (three or more). So, I have the following as the expectation: $E[X_1]=E[X_2X_3X_4]$ I cannot assume the independence of the variables $X_i$ for any $i$. The issue is to find a correct formula for the expectation to consider the correlation of the variables $X_2$ to $X_4$. I know that for a product of two variables, I can have: $E[X_2X_3]=E[X_2]E[X_3] + Cov(X_2,X_3)$. But this covariance will be a matrix in case of three or more variables. Then, how would one compute a value for $E[X_1]$? I appreciate if somebody can help with this or point me to a good resource (e.g. book). Well, $X_1 = 1$ only when $X_2 = X_3 = X_4 = 1$ and is $0$ otherwise, therefore $$E(X_1) = P(X_2 = 1, X_3 = 1, X_4 = 1)$$ As @leonbloy mentions, knowledge of the correlations and marginal success probabilities is not sufficient for calculating $E(X_1)$, but it can be written in terms of the conditional probabilities; using the definition of conditional probability, $$E(X_1) = P(X_2 = 1, X_3 = 1 \| X_4 = 1) \cdot P(X_4 = 1)$$ and $P(X_2 = 1, X_3 = 1 \| X_4 = 1)$ can be similarly decomposed as $$P(X_2 = 1 \| X_3 = 1, X_4 = 1) \cdot P(X_3 = 1 \| X_4 = 1)$$ implying $$E(X_1) = P(X_2 = 1 \| X_3 =1, X_4 = 1) \cdot P(X_3 =1 \| X_4 = 1) \cdot P(X_4 = 1)$$ Explicit calculation of $E(X_1)$ will require more information about the joint distribution of $(X_2,X_3,X_4)$. The above expression makes sense intuitively - the probability that three dependent Bernoulli trials are successes is the probability that the first is a success, and the second one is a success given the first, and the third is a success given that the first two are. You could equivalently interchange the roles of $X_2, X_3, X_4$. • I see what you mean. Since we have $E[X_1]=P(X_1=1)$, we can use conditional probabilities to define $E[X_1]$. In fact, I'm trying to form a general case not limited to a product of three variables; $X = \prod X_i$ for some $i$. I was hopping to find a formulation in terms of a covariance/correlation. But I think I stick to what you suggested. Thanks! – hsnm May 27 '12 at 16:36 From your definition, the expectation of $X_1$ is given by third cross moment of the variables $X_2,X_3,X_4$, which, in general is not reducible to their correlations (second moment), unless you put some other conditions (or unless I've missed something). Regarding general formulation of a multivariate Bernoulli variable, perhaps you find this answer useful. • +1. Yes but it is expressible in terms of the conditional correlations, knowledge of which would require more information about the association structure. – Macro May 26 '12 at 18:10 • This statement is true, but it takes a little work to show, because the situation is not a general one. For instance, the entire joint distribution of three Bernoullis is determined by just eight probabilities. They sum to unity and knowing their expectations and correlations gives six more linearly independent pieces of information: seven pieces of information in all. Your assertion amounts to saying that no single probability can be uniquely determined from these seven pieces of information. – whuber May 26 '12 at 20:30 • @whuber I'm giving an example in the description. In fact, I need a general case. – hsnm May 27 '12 at 16:37 • @leonbloy What do you mean by some other conditions? Can you elaborate more? Thanks. – hsnm May 27 '12 at 16:38	2019-07-21 15:35:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8385214805603027, "perplexity": 214.08247190649286}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195527048.80/warc/CC-MAIN-20190721144008-20190721170008-00292.warc.gz"}
https://www.esaral.com/q/differentiate-the-following-functions-48136	# Differentiate the following functions: Question: Differentiate the following functions: (i) If $y=6 x^{5}-4 x^{4}-2 x^{2}+5 x-9$, find $\frac{d y}{d x}$ at $x=-1$ (ii) If $y=(\sin x+\tan x)$, find $\frac{d y}{d x}$ at $x=\frac{\pi}{3}$. (iii) If $y=\frac{(2-3 \cos x)}{\sin x}$, find $\frac{d y}{d x}$ at $x=\frac{\pi}{4}$. Solution: Formulae: $\frac{d}{d x} x^{n}=n x^{n-1}$ $\frac{d}{d x} \cot x=-\operatorname{cosec}^{2} x$ $\frac{d}{d x} \operatorname{cosec} x=-\operatorname{cosec} x \cot x$ $\frac{d}{d x} \tan x=\sec ^{2} x$ $\frac{d}{d x} \sin x=\cos x$ (i) If $y=6 x^{5}-4 x^{4}-2 x^{2}+5 x-9$, find $\frac{d y}{d x}$ at $x=-1$ Differentiating with respect to $x$, $\frac{d}{d x}\left(6 x^{5}-4 x^{4}-2 x^{2}+5 x-9\right)$ $=30 x^{4}-16 x^{3}-4 x+5$ substituing $\mathrm{x}=-1$ $\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) x=-1=30(-1)^{4}-16(-1)^{3}-4(-1)+5$ $=30+16+4+5$ $=55$ (ii) If $y=(\sin x+\tan x)$, find $\frac{d y}{d x}$ at $x=\frac{\pi}{3}$. Differentiating with respect to $x$, $\frac{d}{d x}(\sin x+\tan x)=\cos x+\sec ^{2} x$ Substituting $x=\frac{\pi}{3}$ $\left(\frac{d y}{d x}\right) x=\pi / 3=\cos \frac{\pi}{3}+\sec \frac{2 \pi}{3}$ $=\frac{1}{2}+4$ $=\frac{5}{2}$ (iii) If $\mathrm{y}=\frac{(2-3 \cos \mathrm{x})}{\sin \mathrm{x}}$, find $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{x}=\frac{\pi}{4}$ Differentiating with respect to $x$, $\frac{d}{d x}(2 \operatorname{cosec} x-3 \cot x)=2(-\operatorname{cosec} x \cot x)-3\left(-\operatorname{cosec}^{2} x\right)$ Substituting $\mathrm{x}=\frac{\pi}{4}$ $\left(\frac{d y}{d x}\right) x=\pi / 4=2\left(-\operatorname{cosec} \frac{\pi}{4} \cot \frac{\pi}{4}\right)-3\left(-\operatorname{cosec} \frac{2 \pi}{4}\right)$ $=-2 \times \sqrt{2}+3 \times 2$ $=6-2 \times \sqrt{2}$	2023-02-03 07:41:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9952675700187683, "perplexity": 1108.8518168351213}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500044.16/warc/CC-MAIN-20230203055519-20230203085519-00322.warc.gz"}
http://staystrong.info/my-night-pxru/pictograph-examples-with-data-ada278	Students are given a pictograph with data for a cupcake bakery. Here, the symbol is used to represent data (i.e. You can generate both of these using Displayr’s pictograph maker. Answer: Number of fruits consumed by children of school C each icon represents 10). Some of the worksheets for this concept are Name, Pictograph, Name, Pictograph, Pictograph, Picture graph, Home south mckeel academy, Grade 3 data management probability name. Resume Examples > Worksheet > 2nd Grade Pictograph Data Handling Worksheets For Grade 2. We can see pictograms and ideograms everywhere. Displaying top 8 worksheets found for - Grade 3 Pictograph. Now ask them to name the term for the examples they are viewing (most will say signs, symbols, and so on). They provide opportunities for data analysis and for children to create their own bar and tally charts. Nov 21, 2013 - Pictures or symbols are made in a pictograph to represent the collected data. Upon completion of this lesson, students will be able to: 1. define 'pictograph' 2. read and interpret pictographs 3. create a pictograph So, the number of symbols to be added for ‘Walking’ is 14 – 11 = 3. Key value varies between 2 and 9. View PDF. The media often uses pictographs to compare trends; in a magazine, you may see a pictograph comparing the number of nurses in the different counties of Texas. To show the results you can use a bar chart, pie chart, line graph, pictogram, frequency diagram or scatter diagram. Pictographs can also come as single icon pictographs or repeated icon pictographs. Please submit your feedback or enquiries via our Feedback page. A fraction of an icon can also be used to show data. Displaying data in the form of symbols, icons, graphs, and charts helps us understand the information in a quicker, easier way and makes the data look more effective. Glossary: example of a pictograph. Then draw the actual symbols that represent the frequencies. Students are given a pictograph with data for a cupcake bakery. Based on the above information, answer the following questions. Example: The following table shows the number of computers sold by a company for the months January to March. For example: In this case, it is very important that the child has properly read the question and understood that one image represents 6 units (cupcakes). Students of grade 2 and grade 3 may use half-symbols in some places. problem and check your answer with the step-by-step explanations. Show the information by drawing the pictograph. For example, you might use a coin for the number of sales or a person for a number of users. Resume Examples > Worksheet > 2nd Grade Pictograph Data Handling Worksheets For Grade 2. Can you now identify who received a pizza at lunch from Mrs. Mary at the end of two months? 2nd through 4th Grades. Information that we collect is called data. It encourages children to develop their math solving skills from a competition perspective. each icon represents 10). With Piktochart’s robust charts and maps you can quickly translate your data into a visual story that will grasp the attention of your audience. Some sample examples of pictographs or pictorial representation are shown, how the objects are used to give information regarding mathematical data. Pictograph is one of my list of the advanced Excel charts. Pictographs, also called pictograms, are diagrams that show and compare data by using picture symbols. If a student gets an "A" in their math tests, a star will be placed next to their name on the class chart. Bar Charts. Icons in your pictograph work best when they're simple and easily understood. Make your kid a Math Expert, Book a FREE trial class today! The science of the pictographs. ABOUT PICTOGRAPHS Made up of pictures which are used to represent numbers Like bar graphs, they are used to compare data. Pictograph counts by 10s and does include half symbols. A simple example is the star rating that is given by customers for items purchased. In this pictograph, I’ve shown the breakdown between male and female Klipfolio Facebook fans. The number of icons filled depends on the input data and the units per icon (scale). Create your Pictograph! To link to this page, copy the following code to your site: All children: will enter information onto a pictogram of their favourite pet. (c) What is the total quantity of apples sold by the store? Can you identify the most loved flavour by observing the above table? The number of pieces that each of them has is recorded in the form of a pictograph below. A pictograph can be made with the help of computer programs such as excel. The default setting are configured for count data. Standard: Difficult Level. If they're too detailed or busy, they'll distract your audience. Students of grade 2 and grade 3 may use half-symbols in some places. In the example below, we set the total icons per bar to 10 and untick the checkbox to hide unfilled icons. PARTS OF A PICTOGRAPH TITL E LEGEN D Categorie s Picture s 6. Learn how to plot frequency of data on a number plot number line. Keep it simple . In the example below, we are using the data read from the pictogram to see which day had the most cars parked. Help your child practise reading and compiling pictograms with our worksheets: Collecting data: pictograms; Reading a pictogram The following table shows the number of computers sold by a company for the months January to March. This graphical representation is much easier as the numerical data collected is represented using pictures. Here are a few activities for you to practice. Answer: Number of oranges consumed in school A $$= 4 × 50 = 200$$, Number of oranges consumed in school C $$= 6 × 50 = 300$$. We at Cuemath believe that Math is a life skill. In a pictograph,we represent data using picturesSuppose we have data like thisNumber of girls in class 5 to 9Class5th6th7th8th9thNumber of girls3035202540We could have also represented it asClass5th6th7th8th9thNumber of girls3035202540 3. Plot the points of the grids and connect with lines to make pictures. With data on birds in the local area, get the children to create a large pictogram on the playground, selecting their own symbols eg sticks from the outside to represent each bird. Then, they complete the short answer questions that follow. If we denote the number of times by $$N$$, and the value of each symbol by $$S$$, we get: Let's solve some more pictograph examples. We feel that pictograph are a good example of applied math, or real life math, as it integrates a practical math situation with all the learned math concepts. a) Total profit of shop A= 20 × 4 × 0.5 = \$40, b) 9 symbols must be drawn for shop C. (9 x 20 = 180), c) Difference between shop B and shop C = 20 × 2 = 40 cans. A pictograph chart is a chart that uses icons or symbols to represent information or data. What is the average time spent by a child on gadgets? Pictographs are often used in writing and graphic systems in which the characters are to a considerable extent pictorial in appearance. There are already 11 symbols on the table. Start from a template or create an infographic from scratch. Personalized Mickey Mouse Invitations; Personalized Editable 18th Birthday Invitation Template; Passport Wedding Invitation Template; Passport Wedding Invitation Template Philippines; Passport Invitation Template Free Download ; Office Team Lunch Invitation Email Sample; … 2nd Grade Data Handling Pictograph Worksheets. Pictograph Worksheets. Data and Pictograph. the number of students). The illustration shown below is a pictograph demonstrating last month’s sales. Example: Apples Sold. Note: the above example is with 1 line. Let's create a pictograph with the information given below. Pictograph Examples Quiz. Each pdf worksheet has real-life representation with data values in the table. Pictograph definition is - an ancient or prehistoric drawing or painting on a rock wall. 4. With Piktochart’s robust charts and maps you can quickly translate your data into a visual story that will grasp the attention of your audience. A pictogram or pictograph represents the frequency of data as pictures or symbols. How many pieces does Tom and Jerry have? Answer 1: No. Let's try to find the answer by watching this simulation. Many times we use picture or symbol to represent quantities of objects. But there is some information missing. 0-1 2-3 4-5 6 or more Number of Times on an Airplane = 2 persons EXAMPLE #1 7. Pictographs: meaning, how to make a pictograph, solved examples. Pictograph . Single icon pictographs are used to highlight a single number usually to draw attention to the stat in infographics, dashboards or reports. And you don't need to worry about the picture you have added, it remains there in the clip-board. Pictograph definition: a picture or symbol standing for a word or group of words, as in written Chinese \| Meaning, pronunciation, translations and examples They help in visually formatting statistics. The best thing which I like about this chart is we can use any kind of picture in it, there is no limitation about it. 3. Resume Examples. Pictogram Charts use icons to give a more engaging overall view of small sets of discrete data. Getting Started Videos Learning outcomes. The times of searching for the perfect image, illustration, or icon are over. (b) \begin{align} \text{Quantity of Fuji} &=3 \times 4= 12\,kg\\\text {Quantity of Ambrosia}&=2 \dfrac{1}{2}\times 4\\&=2 \times 4 + \dfrac{1}{2}\times 4\\&=8+2=10\,kg. You can download the FREE grade-wise sample papers from below: To know more about the Maths Olympiad you can click here. Shared Data Data shared with you. Today will be a little different. Translate complex data into a visual story. Decide if you will draw or print your pictograph from a computer. Construct a pictograph for the table. Pictographs: meaning, how to make a pictograph, solved examples. You can generate both of these using Displayr’s pictograph maker. b) If the total number of students involved in the survey is 56 how many symbols must be drawn for the students walking to school? 1 icon = ________. Ordered Pair Worksheets. We can see this from the image, rather than needing to read every number. 5. These are sometimes called dot plots. Apr 9, 2017 - Pictogram charts that go beyond just replacing bar charts. 1. Standard: Difficult Level. Interpret data on these picture-symbol graphs. Each of these symbols corresponds to a specific quantity and is repeated a number of times. Using the new pictograph data visualization. of students who commute by car\( =4 \times 3=12.. See more ideas about pictogram, data visualization, infographic. 3. 2nd through 4th Grades. These data handling games and activities help children to understand how data can be displayed in various ways including pictograms, bar charts, pie charts and tally charts. We need to do simple math to understand how many children voted for each of the flavours. Now after knowing pictograph definition, let us understand pictographs using a scenario. A pictogram, also called a pictogramme, pictograph, or simply picto, and in computer usage an icon, is an ideogram that conveys its meaning through its pictorial resemblance to a physical object. Decide if you will draw or print your pictograph from a computer. This numerical value must be written along with the pictograph. Pictograph makes information easy and clear to understand. problem solver below to practice various math topics. The pictograph excels at displaying demographic data, and, as such, is a perfect fit for visualizing Facebook data. To link to this page, copy the following code to your site: More Topics . Our year one graphing worksheets focus mainly on collecting data and organizing them in either a data table or picture graph. In which school the number of bananas and oranges consumed is the same? This example of a pictograph demonstrates that pictographs are particularly useful for displaying small count data. The table below is in the form of a pictorial graph. However, one line chart can compare multiple trends by several distributing lines. a) What is the total profit of shop A, if the profit gained on each drink is 50 cents? Found worksheet you are looking for? 2D Shapes Sorting Using Carroll Diagrams. Answer: In school B, the number of bananas and oranges consumed is the same. Try the given examples, or type in your own A pictogram, also called a pictogramme, pictograph, or simply picto, and in computer usage an icon, is an ideogram that conveys its meaning through its pictorial resemblance to a physical object. Most children: will create a pictogram with given information. In other words, a pictograph uses pictures and symbols to convey information about the provided data. Reading of Table. 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http://mathoverflow.net/questions/95572/does-this-knot-invariant-distinguish-trefoil-chiralities	# Does this knot invariant distinguish trefoil chiralities? Let $C_N$ denote the labelled configuration of $N^{th}$ roots of unity with $p_J = e^{\frac{2\pi iJ}{N}}$ for $J = 1\ldots N$. As a corollary of something else I was playing around with, I recently proved the following: Theorem: Every tame knot (or link) $K$ has a (not necessarily minimal) stick presentation which such that the sticks can be projected onto the set of chords of some $C_N$ with the following crossing condition: Whenever the projection has chords $p_{J_1}p_{J_3}$ and $p_{J_2}p_{J_4}$ such that $1\leq J_1 < J_2 < J_3 < J_4\leq N$ then $p_{J_1}p_{J_3}$ crosses in front of $p_{J_2}p_{J_4}$. (In other words for any two intersecting chords in the projection, the chord which has the lowest numbered endpoint passes in front of the other chord). Germane to this question is the fact that the theorem leads to a knot invariant which, for lack of a better name, I will call the circular stick number of $K$ and which is defined to be the minimum $N$ needed to obtain a projection of $K$ with the above properties. My proof of the above theorem was very non-constructive, so I wanted to see some concrete realizations of such projections. And after some work, I was able to find that for the left trefoil knot, the circular stick number is 7. After much more playing around on $C_7$ and not finding a projection of the right trefoil, I moved on to $C_8$ where I was able to find a projection of the right trefoil. Main Question: Does this invariant actually distinguish chiralities of the trefoil, or can someone find a projection of the right trefoil with the above properties on $C_7$? If it doesn't distinguish chiralities in this case, is it possible it distinguishes chiralities of some other pair of knots, or does anyone see a slick way to see that it cannot distinguish chiralities? Secondary Question: (Mainly for knot theorists, or anyone with a deeper knowledge of knots than I) Has this invariant been studied before, and if so, what is the terminology used for it? I am fairly sure that the general answer to the main question is strongly related to the characterization (and in particular the disjointness) of the sets of forbidden minors of the following two sets (which are almost certainly not known in general as the Robertson-Seymour theorem is non-constructive): $\bullet$ Graphs which are $L$-lessly embeddable for a given chiral knot $L$ $\bullet$ Graphs which are $L^+$-lessly and $L^-$-lessly embeddable for a given chiral knot $L$ with chiralities $L^+$ and $L^-$ Note that graphs in the second set may admit embeddings in $\mathbb{R}^3$ containing either $L^+$ or $L^-$, but have at least one embedding which does not contain both $L^+$ and $L^-$. As of yet, I have not been able to flesh out a proof using this approach however. For those who don't want to search for the solutions I found, for the left trefoil the points of $C_7$ are connected in the following order: $p_1 \rightarrow p_3 \rightarrow p_5 \rightarrow p_7 \rightarrow p_2 \rightarrow p_4 \rightarrow p_6 \rightarrow p_1$ For the right trefoil the points of $C_8$ are connected in the following order: $p_1 \rightarrow p_3 \rightarrow p_7 \rightarrow p_5 \rightarrow p_2 \rightarrow p_8 \rightarrow p_4 \rightarrow p_6 \rightarrow p_1$ EDIT: Here are pictures of these two projections to help clarify the situation: With regards to Dylan's question about how this projection arose, I was considering the parametric curve $S(t) = (t,t^2,t^3) \subset \mathbb{R}^3$. In a comment or answer on a past MO post which for the love of me I cannot find now, someone had shown in a very simple way that no two chords of $S$ intersect one another anywhere in $\mathbb{R}^3$ (unless they share an endpoint). Thus, for each $n$, $K_n$ can be embedded in $\mathbb{R}^3$ as the set of chords connecting the points $(1...n)$. Now for a fixed tame knot or link $K$, the Robertson-Seymour theorem implies there is a finite set of forbidden minors for graphs which are $K$-lessly embeddable. Hence, every embedding of $K_n$ for $n$ sufficiently large contains $K$, so one may ask (like I did) "what $n$ is sufficient for a given $K$, and what order must I connect the points $(1...n)$ in order to realize $K$?" The crossing condition comes from looking at the chords projected onto the $yz$-plane as viewed from the $-x$ direction (which I think is the direction I did my crossing calculations from). Finally, I moved the $yz$-projections of the integral $t$-valued points around so that they lay on the unit circle to make the pictures easier/clearer. UPDATE: I was just skimming through a paper on some results on knots which were unrelated to this one. That paper referenced the Ramsey number $r(L)$ of a link $L$, and upon following its references I found this paper which proves the existence of $r(L)$ using essentially the representation I layed out above: http://www.ams.org/journals/tran/1991-324-02/S0002-9947-1991-1069741-9/S0002-9947-1991-1069741-9.pdf So it would appear the answer to Question 2 is this is known (in at least one paper) as a plat representation (much to my surprise, Negami derives the existence of such representations using essentially the argument I gave above). So that would seem to settle this thread entirely... - Interesting. I think the figure I drew for mathoverflow.net/questions/95477 is accidentally an illustration? If I am not misinterpreting, then perhaps the relationship to torus knots might be helpful... – Joseph O'Rourke Apr 30 '12 at 15:45 Your illustration is close to what I am describing. Hopefully later this evening I will have time to create and upload some pictures of the two trefoils I described above. – ARupinski Apr 30 '12 at 23:27 Here's a quick proof of the non-intersecting chord result. Denote the moment curve by $\mu(t) = (t, t^2, t^3)$, and consider any four points $\mu(t_1), \mu(t_2), \mu(t_3), \mu(t_4)$ with $t_1 < t_2 < t_3 < t_4$. The volume of the convex hull of these points is $1/6$ the determinant of the matrix whose $i$th row is $[1, t_i, t_i^2, t_i^3]$. This determinant is an antisymmetric polynomial with total degree $6$ in the $t_i$'s, and thus must be $\prod_{i<j} (t_j-t_i) \ne 0$. The same result also holds for one turn of the standard helix $(t, \cos t, \sin t)$ where $0 \le t < 2\pi$. – JeffE May 3 '12 at 11:29 Isn't the non-trivial knot with $N=7$ a right-handed trefoil, not a left-handed trefoil? Or am I getting some convention backwards? – Dylan Thurston May 3 '12 at 15:36 Yeah, must have messed up that when I looked it up originally. But all my other knots' chiralities were computed by comparing to the reduced form I did of that one, so the chiralities are at least consistent (so if that one is wrong, all of the $N=8$ ones I calculated below will change chirality too). – ARupinski May 4 '12 at 1:45 I'm very curious where this came up. In any case, the answer to the first question is yes, it does distinguish these trefoils; you found the minimal representatives. Let $a_0,\dots,a_{N-1}$ be the roots of unity that are visited along the knot, in (cyclic) order. Suppose we have a minimal representative for some non-trivial knot. Then we cannot have $\|a_k - a_{k+1}\| = 1$ for any $k$, as otherwise we could replace this pair $a_k, a_{k+1}$ by a single root of unity (for $N-1$), adjusting the other roots of unity as appropriate. A little more subtly, we cannot have $\|a_{k-1} - a_{k+1}\| = 1$ either, as then we could again delete $a_k$ from the sequence to get a smaller representation. With these simple constraints, the smallest possible sequence for a non-trivial knot is the one you found for one of the trefoils with $N=7$. There are several possibilities for $N=8$, including the one you found for the other trefoil. I've included a very short Haskell program below that computes this. The possibilities for $N=8$ are $$(2,7,5,3,1,6,4,0)\quad (2,5,7,3,1,6,4,0)\quad (3,6,1,4,7,2,5,0)\quad (2,6,4,1,7,3,5,0)$$ $$(3,1,6,4,2,7,5,0)\quad (2,4,6,1,3,7,5,0)\quad (3,5,1,7,4,2,6,0)\quad (4,2,7,5,1,3,6,0)$$ $$(3,1,5,7,2,4,6,0)\quad (5,3,1,6,4,2,7,0)\quad (2,4,6,1,3,5,7,0)$$ For the second question, I have never heard of this representation before. Here is the code, for anyone interested. -- A (partial) circular stick representation is a list of integers, -- the order of the roots of unity to visit in order type CircStick = [Int] -- The next element ak after a partial representation a1, ..., a{k-1} -- must satisfy -- (a) ak has not already been seen -- (b) \|ak - a{k-1}\| > 1 -- (c) \|ak - a{k-2}\| > 1 -- There are a few more "easy" constraint, eg the first and last entries -- cannot differ by one. We do not impose those constraint here. nexts :: Int -> CircStick -> [Int] nexts n [] = [0] nexts n [a1] = filter (\a -> abs (a-a1) > 1) [0..n-1] nexts n (a1:a2:as) = filter (\a -> not (elem a as)) $filter (\a -> abs (a-a1) > 1)$ filter (\a -> abs (a-a2) > 1) $[1..n-1] completions :: Int -> CircStick -> [CircStick] completions n as \| length as >= n = [as] completions n as = concat [completions n (a:as) \| a <- nexts n as] -- Impose final constraints: -- (a) Last entry cannot be 1 -- (b) Take entry that is lexicographically less than its reverse -- (c) first and next-to-last entries cannot differ by one circSticks :: Int -> [CircStick] circSticks n = filter (\as -> abs ((as!!0) - (as!!(n-2))) > 1)$ filter (\as -> as < tail (reverse as)) $filter (\as -> head as /= 1)$ (completions n []) Edit: For those interested, here are the 108 possibilities for $N=9$. I hope there's some way of checking what these are more efficiently than just going through them by hand. [[2,7,5,3,8,1,6,4,0],[2,7,5,3,1,8,6,4,0],[2,5,7,3,1,8,6,4,0],[2,7,5,1,3,8,6,4,0],[2,5,7,1,3,8,6,4,0],[2,6,8,3,5,1,7,4,0],[2,7,5,3,1,6,8,4,0],[2,5,7,3,1,6,8,4,0],[2,7,5,1,3,6,8,4,0],[2,5,7,1,3,6,8,4,0],[3,8,6,1,4,7,2,5,0],[3,6,8,1,4,7,2,5,0],[3,7,1,4,6,8,2,5,0],[2,8,6,4,1,7,3,5,0],[2,6,8,4,1,7,3,5,0],[2,7,4,1,6,8,3,5,0],[2,4,8,6,3,1,7,5,0],[2,4,6,8,3,1,7,5,0],[3,8,1,6,4,2,7,5,0],[3,1,8,6,4,2,7,5,0],[3,6,1,8,4,2,7,5,0],[3,1,6,8,4,2,7,5,0],[2,8,4,6,1,3,7,5,0],[2,4,8,6,1,3,7,5,0],[2,6,4,8,1,3,7,5,0],[2,4,6,8,1,3,7,5,0],[3,7,1,4,6,2,8,5,0],[2,7,4,1,6,3,8,5,0],[3,8,5,1,7,4,2,6,0],[3,7,5,1,8,4,2,6,0],[3,5,7,1,4,8,2,6,0],[4,7,1,3,5,8,2,6,0],[4,1,7,3,5,8,2,6,0],[4,8,2,5,7,1,3,6,0],[4,7,2,5,8,1,3,6,0],[4,2,7,5,1,8,3,6,0],[2,4,7,1,5,8,3,6,0],[2,8,5,3,7,1,4,6,0],[2,7,5,3,8,1,4,6,0],[3,8,1,5,7,2,4,6,0],[3,7,1,5,8,2,4,6,0],[2,7,5,3,1,8,4,6,0],[2,5,7,3,1,8,4,6,0],[3,1,7,5,2,8,4,6,0],[4,2,7,5,3,1,8,6,0],[3,5,1,7,4,2,8,6,0],[4,2,7,5,1,3,8,6,0],[2,4,7,1,5,3,8,6,0],[3,1,5,7,2,4,8,6,0],[5,8,3,1,6,4,2,7,0],[5,3,8,1,6,4,2,7,0],[3,5,8,1,6,4,2,7,0],[5,3,1,8,6,4,2,7,0],[3,5,1,8,6,4,2,7,0],[5,1,3,8,6,4,2,7,0],[5,3,1,6,8,4,2,7,0],[5,1,3,6,8,4,2,7,0],[4,6,1,3,8,5,2,7,0],[4,1,6,3,8,5,2,7,0],[4,6,2,8,5,1,3,7,0],[5,8,2,4,6,1,3,7,0],[5,2,8,4,6,1,3,7,0],[2,6,4,8,1,5,3,7,0],[4,2,6,8,1,5,3,7,0],[2,4,6,8,1,5,3,7,0],[2,6,4,1,8,5,3,7,0],[2,4,6,1,8,5,3,7,0],[2,5,8,3,6,1,4,7,0],[5,3,1,8,6,2,4,7,0],[3,5,1,8,6,2,4,7,0],[5,1,3,8,6,2,4,7,0],[5,3,1,6,8,2,4,7,0],[5,1,3,6,8,2,4,7,0],[4,2,8,6,3,1,5,7,0],[2,4,8,6,3,1,5,7,0],[4,2,6,8,3,1,5,7,0],[2,4,6,8,3,1,5,7,0],[3,6,1,4,8,2,5,7,0],[3,1,6,4,8,2,5,7,0],[2,8,4,6,1,3,5,7,0],[4,2,8,6,1,3,5,7,0],[2,4,8,6,1,3,5,7,0],[2,6,4,8,1,3,5,7,0],[4,2,6,8,1,3,5,7,0],[2,4,6,8,1,3,5,7,0],[2,6,4,1,8,3,5,7,0],[2,4,6,1,8,3,5,7,0],[5,7,3,1,6,4,2,8,0],[4,6,1,3,7,5,2,8,0],[5,3,7,1,4,6,2,8,0],[3,5,7,1,4,6,2,8,0],[6,4,2,7,5,1,3,8,0],[5,7,2,4,6,1,3,8,0],[6,2,4,7,1,5,3,8,0],[2,6,4,1,7,5,3,8,0],[5,2,7,4,1,6,3,8,0],[6,3,1,5,7,2,4,8,0],[6,1,3,5,7,2,4,8,0],[2,7,5,3,1,6,4,8,0],[2,5,7,3,1,6,4,8,0],[3,6,1,4,7,2,5,8,0],[6,2,4,7,1,3,5,8,0],[2,6,4,1,7,3,5,8,0],[4,2,7,5,3,1,6,8,0],[5,3,1,7,4,2,6,8,0],[3,5,1,7,4,2,6,8,0],[4,2,7,5,1,3,6,8,0],[3,1,5,7,2,4,6,8,0]] - Nice slick proof... I had used the fact that $\|a_k-a_{k-1}\|\neq 1$ to narrow down the search, but I hadn't noticed the condition on $\|a_k-a_{k-2}\|$. – ARupinski May 1 '12 at 0:16 With regards to Mariano's question, I just checked the 11 you listed... along the rows they are: Trivial Right Left Right; Trivial Trivial Right Right; Trivial Left Left. The last two reduce to the $N = 7$ case because of the adjacent 0-7 pairs. So nothing new here. I would suspect that with $N = 9$ or $10$ one could obtain the figure 8 knot (with a little bit of playing I found at least one of the cinquefoil knots on $K = 9$) – ARupinski May 2 '12 at 0:23 $\mod 8$ they differ by 1, and in fact that is enough to reduce 5 of the cases on your list to the $N = 7$ case via one of the two conditions $\|a_k - a_{k-1}\| = 1$ or $\|a_k - a_{k-2}\| = 1$. – ARupinski May 3 '12 at 1:06 Suppose $0$ and $N-1$ are connected by a chord and let C be the other chord incident to $N-1$. It is easy to see that C must cross behind any chord it intersects (in the projection). Contracting the chord between $0$ and $N-1$, C becomes a chord incident to $0$ (and thus should cross in front of, not behind, any other chord it intersects). But this is not a problem; since C crossed behind all other chords, it can be slid around the figure to the front to become a valid chord incident to $0$. Thus the chord between $0$ and $N-1$ contracts. – ARupinski May 9 '12 at 0:04 Your figure 8 example is actually a right trefoil (in my original notation, which seems to be the opposite of the standard convention); contracting the 0-8 edge reduces it to the fourth knot in the second row of your $N=8$ list. – ARupinski May 9 '12 at 0:08	2015-09-01 12:26:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8377194404602051, "perplexity": 389.6703651463521}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645176794.50/warc/CC-MAIN-20150827031256-00032-ip-10-171-96-226.ec2.internal.warc.gz"}
https://enacademic.com/dic.nsf/enwiki/7763342/Studentized_range	# Studentized range In statistics, the studentized range computed from a list "x"1, ..., "x""n" of numbers is : $frac\left\{max\left\{,x_1,dots,x_n,\right\} - min\left\{,x_1,dots,x_n,\left\{s\right\},$ where :$s^2 = frac\left\{1\right\}\left\{n - 1\right\}sum_\left\{i=1\right\}^n \left(x_i - overline\left\{x\right\}\right)^2,$ is the sample variance and :$overline\left\{x\right\} = frac\left\{x_1 + cdots + x_n\right\}\left\{n\right\}$ is the sample mean. Generally, "studentized" means adjusted by dividing by an estimate of a population standard deviation; see also studentized residual. The concept is named after William Sealey Gosset, who wrote under the pseudonym "Student". The fact that the variance is a sample variance rather than the population variance, and thus something that differs from one random sample to the next, is essential to the definition, and complicates the problem of finding the probability distribution of any statistic that is studentized. If "X"1, ..., "X""n" are independent identically distributed random variables that are normally distributed, the probability distribution of their studentized range is what is usually called the studentized range distribution. This probability distribution is the same regardless of the expected value and standard deviation of the normal distribution from which the sample is drawn. This probability distribution has applications to hypothesis testing and multiple comparisons. * John Neter, Michael H. Kutner, Christopher J. Nachtsheim, William Wasserman, "Applied Linear Statistical Models", fourth edition, McGraw-Hill, 1996, page 726. * John A. Rice, "Mathematical Statistics and Data Analysis", second edition, Duxbury Press, 1995, pages 451–452. Wikimedia Foundation. 2010. ### Look at other dictionaries: • Range (statistics) — In descriptive statistics, the range is the length of the smallest interval which contains all the data. It is calculated by subtracting the smallest observations from the greatest and provides an indication of statistical dispersion. It is… … Wikipedia • Studentized residual — In statistics, a studentized residual, named in honor of William Sealey Gosset, who wrote under the pseudonym Student , is a residual adjusted by dividing it by an estimate of its standard deviation. Studentization of residuals is an important… … Wikipedia • Duncan's new multiple range test — In statistics, Duncan s new multiple range test (MRT) is a multiple comparison procedure developed by David B. Duncan in 1955. Duncan s MRT belongs to the general class of multiple comparison procedures that use the studentized range statistic qr … Wikipedia • Tukey's test — Tukey s test, named after John Tukey, is a statistical test generally used in conjunction with an ANOVA to find which means are significantly different from one another. It compares all possible pairs of means, and is based on a studentized range … Wikipedia • Tukey-Kramer method — The Tukey method (also known as Tukey s Honest Significance Test), named for John Tukey, is a single step multiple comparison procedure which applies simultaneously to the set of all pairwise comparisons:mu i mu jThe confidence coefficient for… … Wikipedia • Newman–Keuls method — In statistics, the Newman–Keuls method (named after D. Newman (1939),[1] and M. Keuls (1952)[2]) is a post hoc test used for comparisons after the performed F test (analysis of variance) is found to be significant. The Newman–Keuls method is very … Wikipedia • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… … Wikipedia • Level of measurement — The levels of measurement , or scales of measure are expressions that typically refer to the theory of scale types developed by the psychologist Stanley Smith Stevens. Stevens proposed his theory in a 1946 Science article titled On the theory of… … Wikipedia • Deviation (statistics) — In mathematics and statistics, deviation is a measure of difference for interval and ratio variables between the observed value and the mean. The sign of deviation (positive or negative), reports the direction of that difference (it is larger… … Wikipedia • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… … Wikipedia	2020-10-25 00:35:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7488457560539246, "perplexity": 2251.0851047849555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107885059.50/warc/CC-MAIN-20201024223210-20201025013210-00082.warc.gz"}
https://physics.stackexchange.com/questions/224876/is-there-a-maupertuis-principle-for-general-relativity	# Is there a Maupertuis principle for General Relativity? The motion of a point particle in classical mechanics is given by Newton's equation, $\mathbf{F}=m\mathbf{a}$. Suppose all forces considered are conservative and we have a constant total energy $h$. Let $M$ be the configuration space of our system, $T^M$ its cotangent bundle, $(\mathbf{q},\mathbf{p})$ the natural coordinates on $T^M$ and $\gamma$ a curve connecting the start- and end-points of our particle's motion. Then $\int_\gamma \mathbf{p}\,\mathrm{d}\mathbf{q}$ is a viable action integral. The Maupertuis theorem states that $$\int_\gamma\mathbf{p}\,\mathrm{d}\mathbf{q}=\sqrt{2}\int_\gamma\mathrm{d}\rho,$$ where $\mathrm{d}\rho$ is the Riemannian metric given by $$\mathrm{d}\rho=\sqrt{h-U(\mathbf{q})}\,\mathrm{d}s,$$ $\mathrm{d}s$ is the standard metric on $M$ and $U(\mathbf{q})$ the potential energy. This implies that $$\text{Newtonian mechanics \Longleftrightarrow geodesic problem of some pair (M,\mathrm{d}\rho)}$$ The motion of a point particle in general relativity is given by the equation $$F=a\tag{1}$$ where $a:=\nabla_{\dot\gamma}\dot\gamma$, $\gamma$ is the path of the particle, $\nabla$ is the Levi-Civita connection of the spacetime $(\mathcal{M},g)$, and $F$ is some "force" 4-vector. $F$ can be seen as an obstruction to the geodesy of $\gamma$, since $a=0$ is just the geodesic equation. (In the same way, $\mathbf{F}$ is an obstruction to geodesy on $\mathbb{R}^n$ since $\mathbf{a}=0$ $\Leftrightarrow$ $\mathbf{x}$ is a straight line and $U(\mathbf{q})$ in the case of constrained Lagrangian mechanics.) (Proofs of the above statements of classical mechanics can be found in Arnold, V.I. Mathematical Methods of Classical Mechanics. Springer, 1989.) Is there some pair $(\mathcal{M}',g')$ for which the geodesic problem is (1), i.e. a suitable generalization of the Maupertuis principle to relativistic mechanics in curved spacetime? • What would be the meaning of F (in the relativistic case) is conservative? – MBN Dec 19 '15 at 10:24 • @MBN I do not know. Perhaps $F_\mu=-\partial_\mu U$ for some function $U$. (That might be another question right there.) – Ryan Unger Dec 19 '15 at 10:40 I) We assume OP's question concerns a massive point particle of rest mass $m_0>0$ in a Lorentzian spacetime manifold $(M,g)$ [of signature $(+,-,\ldots, -)$] between an initial and a final spacetime point $p_i, p_f\in M$, which should be causally connected. Let us work in units where the speed of light $c=1$ and rest mass $m_0=1$ are both one. II) Before discussing Maupertuis's principle, we should first have a stationary action principle (SAP). OP's question mentions an unspecified $4$-force. To have a variational principle, we must demand that that $4$-force comes from a potential $U$. The action is then $$S[x;\lambda_i,\lambda_f] ~=~\int_{\lambda_i}^{\lambda_f}\! d\lambda~L, \qquad L~=~T-U,$$ $$\tag{1} T~:=~-\sqrt{T_0},\qquad T_0~:=~ g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu},$$ where $\lambda$ is a parameter, and dot means differentiation wrt. $\lambda$. III) In the SAP, we impose Dirichlet boundary conditions (BC) $$\tag{2} x^{\mu}(\lambda_i)~=~x_i^{\mu}\qquad\text{and}\qquad x^{\mu}(\lambda_f)~=~x_f^{\mu},$$ and keep $\lambda_i,\lambda_f$ fixed. IV) We'll assume that the action (1) is reparametrization invariant $$\tag{3}\lambda\quad\longrightarrow\quad \tilde{\lambda}~=~f(\lambda),$$ since physics should be geometric. V) The Lagrangian $4$-momentum and energy function become $$\tag{4} p_{\mu}~:=~\frac{\partial L}{\partial\dot{x}^{\mu}}~=~ -\frac{g_{\mu\nu}\dot{x}^{\nu}}{\sqrt{T_0}}-\frac{\partial U}{\partial\dot{x}^{\mu}},$$ and $$\tag{5} h~:=~ p_{\mu}\dot{x}^{\mu} - L~=~\left(1 - \dot{x}^{\mu}\frac{\partial }{\partial\dot{x}^{\mu}}\right)U,$$ respectively. VI) Usually when discussing the abbreviated action principle, it is assumed that the Lagrangian (1) has no explicit $\lambda$-dependence, so that the energy (5) is conserved on-shell. No explicit $\lambda$-dependence may sound natural & innocent, but together with the reparametrization invariance (3), it severely restricts the possible potentials $U$. The Lorentz potential $U\propto A_{\mu}\dot{x}^{\mu}$ is still allowed, of course. VII) In fact, no explicit $\lambda$-dependence & reparametrization invariance (3) basically imply that the energy function (5) vanishes identically. VIII) The abbreviated action becomes $$\tag{6} A[x;E,\lambda_i,\lambda_f] ~=~\int_{\lambda_i}^{\lambda_f}\! d\lambda~ p_{\mu}\dot{x}^{\mu}$$ for virtual paths of constant and the same energy $$\tag{7} h~=~E,$$ satisfying Dirichlet BC (2), but free $\lambda_i$ and $\lambda_f$. From Section VII, we know that the energy $E=0$ drops out of the abbreviated action (6). IX) Returning to OP's question, there seems little hope to achieve a Jacobi square root form of the abbreviated action principle (6) without further assumptions. The natural next step is to assume that the potential $U$ does not depend on the $4$-velocity $\dot{x}^{\mu}$. Then the energy function $h\equiv U$ becomes just the potential energy, and $p_{\mu}\dot{x}^{\mu}\equiv T\equiv -\sqrt{T_0}$. However, with all the other above requirements, this basically implies that the potential $U=0$ is zero! Of course, without a potential $U=0$, we unsurprisingly achieve a Jacobi square root form of the abbreviated action principle $$\tag{8} A[x;E,\lambda_i,\lambda_f] ~=~-\int_{\lambda_i}^{\lambda_f}\! d\lambda~\sqrt{T_0} .$$ The abbreviated action (8) is identical to the SAP (1) that we started from, essentially due to reparametrization invariance. • The action integral you gave is the standard integral for geodesic motion on a spacetime. In the OP I'm asking for an action integral for nongeodesic motion on $(\mathcal{M},g)$ posed as geodesic motion on some other $(\mathcal{M}',g')$. As such, this does not (appear to) answer the question. – Ryan Unger Dec 19 '15 at 8:00 • Oh, I didn't notice the $4$-force. I updated the answer. – Qmechanic Dec 19 '15 at 8:29	2019-09-21 06:50:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.950847864151001, "perplexity": 254.68605694745852}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574286.12/warc/CC-MAIN-20190921063658-20190921085658-00047.warc.gz"}
https://ai.stackexchange.com/questions/28366/how-do-we-get-from-conditional-expectation-on-both-state-and-action-to-only-stat/28368	# How do we get from conditional expectation on both state and action to only state in the proof of the Policy Improvement Theorem? I'm going through Sutton and Barto's book Reinforcement Learning: An Introduction and I'm trying to understand the proof of the Policy Improvement Theorem, presented at page 78 of the physical book. The theorem goes as follows: Let $$\pi$$ and $$\pi'$$ be any pair of deterministic policies such that, for all $$s\in S$$, $$q_{\pi}(s,\pi'(s))\geq v_{\pi}(s)$$. Then the policiy $$\pi'$$ must be as good as, or better than, $$\pi$$. That is, it must obtain greater or equal expected return from all states $$s\in S$$: $$v_{\pi'}(s)\geq v_{\pi}(s)$$. I take it that for the proof, the policy $$\pi'$$ is identical to $$\pi$$ except for one particular state $$s$$ (at each time step) for which we have $$\pi'(s)=a\neq \pi(s)$$, as suggested by @PraveenPalanisamy in his answer here. The proof start from the statement of the theorem: $$v_{\pi}(s)\leq q_{\pi}(s,\pi'(s))$$ And then $$q_{\pi}(s,\pi'(s))$$ is developed as $$\mathbb{E}[R_{t+1}+\gamma v_{\pi}(S_{t+1})\|S_{t}=s,A_{t}=\pi'(s)]=\mathbb{E}_{\pi'}[R_{t+1}+\gamma v_{\pi}(S_{t+1})\|S_{t}=s]$$ I don't understand how did we get rid of the condition $$A_{t}=\pi'(s)$$. I don't think it's related to adding the subscript $$\pi'$$ to the expectation because it's something that should be done by definition since for the following time steps we choose policy $$\pi$$ which is exactly $$\pi'$$. I don't understand how did we get rid of the condition $$A_{t}=\pi'(s)$$. We don't really, it is just moved into the subscript $$\pi'$$ in $$\mathbb{E}_{\pi'}[]$$ - it means the same thing here, that the next action is chosen according to the modified policy $$\pi'$$. Moving the condition around is part of the proof's strategy, which eventually expresses the expectation in a familiar way so that we end up with a something that matches the definition of $$v_{\pi'}(s)$$. • Am I right in thinking that this follows because $\pi'$ is a strictly deterministic policy? Jun 22, 2021 at 19:22 • Thank you for your answer @NeilSlater but trying to get the expression in a familiar way to $v_{\pi'}(s)$ isn't so rigorous am I wrong? Especially if we do that in the proof, why don't we do the same in the definition of the action value function instead of explicitely state the condition on the action $q_{\pi}(s,a)=\mathbb{E}[G_{t}\|S_{t}=s,A_{t}=a]$. If it's just for clarity, I believe there wouldn't be much of a difference between $q_{\pi}(s,a)$ and $v_{\pi}(s)$. Jun 23, 2021 at 1:41 • @NeilSlater yes I know sorry if it came as if it was in your answer. I just stated the "just for clarity" in relation with the $q_{\pi}(s,a)$ case to say that I don't believe it's for clarity but for another purpose. I really understand your answer from an "intuition" point of view. Since we're sure that we're taking action $\pi '(s)$ and then following $\pi '$ which is a deterministic policy, it's seems like a "factorization", in some sense, of the expression. But is it rigorous? I mean, based on measure theorems and properties we don't have a clear implication or do we? Jun 23, 2021 at 11:09	2022-10-04 22:42:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 24, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9107126593589783, "perplexity": 179.833992072273}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337529.69/warc/CC-MAIN-20221004215917-20221005005917-00171.warc.gz"}
https://proxieslive.com/tag/substring/	## How to not count cell if cell contains sub-string I have the following formula to count cells within a range B10:B1095 which do not contain certain strings =COUNTIFS( B10:B1095, "<>", B10:B1095, "<>Manual", B10:B1095, "<>TODO") How can I add an additional condition to check if the cells in a range do not contain a specific sub-string? ## Excel 365 – Pivot Table – show all columns with particular substring in column header I’d like to sort my data by column substring but I don’t know how to do it with pivot tables. I’m trying to sort my grades by assignment type and then make calculations on that subset of data. The first column is the student’s username and then all other columns contain pass/fail data for each assignment. There are three different types of assignments PQ, RQ , and WQ (proficiency quiz, reading quiz, and weekly quiz), so each column’s header starts with either PQ, RQ, or WQ. I’d like to make a pivot table that shows all PQs so that I can view their progress with that assignment type as well as find the average pass/fail rate for each student to calculate total grades. This also needs to happen dynamically so that if add more PQs throughout the semester the table will reflect that when I refresh it. Here is an image of my sample data: Sample Data RAW Here is an image of the sorts of subsets I’d like to see: Subset of sample data that I’d like to see (Sorry I can’t embed the images, this is my first post.) If pivot tables aren’t the way to do this, another approach would be great to know. ## Write a program that prints the longest substring in which the letters occur in alphabetical order by ascii Write a program that prints the longest substring of s in which the letters occur in alphabetical order. And the output has to be: Longest substring in alphabetical order is: beggh here is my code. I know that it is awful to ask somebody to write code for someone, but I have no idea how to continue it. Help please s = 'azcbobobegghakl' characters = "" for num in range(len(s)): if num == len(s) - 1: break elif ord(s[num]) <= ord(s[num + 1]): characters += s[num] elif ord(s[num]) >= ord(s[num - 1]): characters += s[num] ## Elegant way to replace substring in a regex with optional groups in Python? Given a string taken from the following set: strings = [ "The sky is blue and I like it", "The tree is green and I love it", "A lemon is yellow" ] I would like to constuct a function which replaces subject, color and optional verb from this string with others values. All strings match a certain regex pattern as follow: regex = r"(?:The\|A) (?P<subject>\w+) is (?P<color>\w+)(?: and I (?P<verb>\w+) it)?" The expected output of such function would look like this: repl("The sea is blue", "moon, "white", "hate") # => "The moon is white" Here is the solution I come with (I can’t use .replace() because there is edge cases if the string contains the subject twice for example): def repl(sentence, subject, color, verb): m = re.match(regex, sentence) s = sentence new_string = s[:m.start("subject")] + subject + s[m.end("subject"):m.start("color")] + color if m.group("verb") is None: new_string += s[m.end("color"):] else: new_string += s[m.end("color"):m.start("verb")] + verb + s[m.end("verb"):] return new_string Do you think there is a more straightforward way to implement this? ## find substring with count and return even frequency substring list I have one following question and write a function for it. can anyone give me some suggestion how my code can be better and better way to handle to build up return list with even frequency? One message system contains two device type message that each message is formatted with device type identifier with device id and message count. Write a function to parse the message based on device type and message count and return a list with each device id in even message frequency rule: input string: message device identifier 1:iOS device ID identifier: start with ‘I’ following with 3 character, total length is 4 character 2:Android device ID identifier: start with ‘A’ following with 2 character, total length is 3 character 3:message count is following by device id until next device ID ex: input: Asq2: {‘Asq’: 2} Asq with 2 message count output: [‘Asq’, ‘Asq’] input: Akb2IAld3: ID: {‘Akb’: 2, ‘IAld2’: 3} Akb with 2 message count, IAld with 3 message count output: [‘Akb’, ‘IAld’, ‘Akb’, ‘IAld’, ‘IAld’] input: Aqp1Iasd2Aqp4IAbd1: {‘Aqp’: 5, ‘Iasd’: 2, ‘IAbd’: 1} output: [‘Aqp’, ‘Iasd’, ‘IAbd’, ‘Aqp’, ‘Iasd’, ‘Aqp’, ‘Aqp’, ‘Aqp’] from typing import List def parse_message(string) -> List: i, j, ids_map, n, ids = 0, 0, dict(), len(string), '' while i < n: if string[i] in ('I', 'A') or i == n - 1: if ids: if i == n - 1: ids_map[ids] = ids_map.get(ids, 0) + int(string[j:]) else: ids_map[ids] = ids_map.get(ids, 0) + int(string[j:i]) j = i + 4 if string[i] == 'I' else i + 3 ids = string[i:j] i = j - 1 i += 1 res = [] while any(i > 0 for i in ids_map.values()): for k, v in ids_map.items(): if v > 0: res.append(k) ids_map[k] -= 1 return res ## How to keep a substring and delete rest of string using Python I am beginner in Python and I am web scraping a webpage and I get the following most of the time with the word “available” always there. Your product is available and costs 10.00€ I cannot count the empty spaces and remove word by word, so how can I keep only the word available and delete the rest of it? And have something like Available thank you ## Longest common substring in linear time We know that the longest common substring of two strings can be found in O(N^2) time complexity. Can a solution be found in only linear time? ## How do I remove a substring from system_profiler with a posible if then? In short, I’m writing a script that will check if a computer is a MacBook Pro or a MacBook Air and then I want to use the output as a variable. Using system_profiler SPHardwareDataType \| grep “Model Name” I can get Model Name: MacBook Pro but all I need is either “MacBook Pro” or “MacBook Air” -oP in grep gives me an error. Another idea was to use find but I’m unsure how I’d use find if I’m not searching an actual file. ## Find the smallest set of strings which “covers” a given set of strings (coverage = containing as substring) Let $$S$$ be a finite set of strings and $$0 < k\leq l$$ integers. We want to find the smallest set of strings $$T(k,l)$$ for which the following holds: • $$\forall t \in T(k,l): k \leq \|t\| \leq l$$ • $$\forall s \in S \ \exists t \in T(k,l): t \subset s$$ (meaning: $$s$$ is containing $$t$$ as a subset). I would appreciate any help regarding this problem. I see that I could generate the set of all strings with length between $$k$$ and $$l$$ then consider the power set but I think there must be a better way to solve this. I’m not even sure how hard this problem is (I think there may be a reduction for the set cover problem which means it is $$NP$$-complete but I really don’t know) or if there are any good approximation (or maybe exact) algorithms available. ## Efficient substring match I have a list of unique strings (approx, 25,00,000) of different lengths and I am trying to find if there is any string which occurs as a substring of previous strings. def index_containing_substring(the_list, substring): for i, s in enumerate(the_list): if substring in s: return i return -1 def string_match(): test_list=['foo bar abc xml','fdff gdnfgf gdkgf','foo bar','abc','xml','xyz'] max_len=4 # I am storing the maximum length of sentence # the list starts with reverse order # i.e sentence with highest length are at the top safe_to_add=[] for s in test_list: if len(s)==max_len: safe_to_add.append(s) else: idx=index_containing_substring(safe_to_add,s) if idx==-1: safe_to_add.append(s) else: # process the substring print('match found {} for {}'.format(test_list[idx],s)) This method works fine but I think it is pretty slow. Is there a better way to solve this problem using a better data structure (trie or suffix tree)? Output match found foo bar abc xml for foo bar match found foo bar abc xml for abc match found foo bar abc xml for xml	2019-04-19 09:27:04	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.26848673820495605, "perplexity": 2167.679515010074}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578527518.38/warc/CC-MAIN-20190419081303-20190419103303-00279.warc.gz"}
https://academiaservices.net/21147685383/	(solution) Consider the standard GARCH(1,1) model, as described in exercise 7.2 with α + β 1. A Consider the standard GARCH(1,1) model, as described in exercise 7.2 with α + β 1. A different way to model volatility is to use an exponentially weighted moving average. That is, one estimates/forecasts volatility by means of weighted average of squared returns: ∞ h t = (1 − λ ) λ j − 1 ( y t − j =1 j − y ¯) 2 where 0 λ 1. For daily data λ = 0 . 94 gives JP Morgan’s Risk Metrics model. Show that (7.84) can also be written as h t = λ h t − 1 + (1 − λ )( y t − 1 − y ¯) 2 What is the intuition behind this representation? Derive the optimal 1-step, 2-step and 3-step ahead point forecasts of h t in the GARCH(1,1) model (that is, derive expressions for h ˆ t + k \| t = E[ h t + k \| Y t ] for k = 3 , 4 and 5, where Y t denotes the information set available at t .) What is the optimal 100-step ahead point forecast? Derive the optimal 1-step, 2-step and 3-step ahead point forecasts of h t in the RiskMetrics model. What is the crucial difference between forecasts of the RiskMetrics model and the GARCH(1,1) model?	2020-12-04 20:50:59	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8841083645820618, "perplexity": 1123.699043827647}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141743438.76/warc/CC-MAIN-20201204193220-20201204223220-00524.warc.gz"}
http://comunidadwindows.org/sum-of/standard-error-sum-of-variables.php	Home > Sum Of > Standard Error Sum Of Variables # Standard Error Sum Of Variables ## Contents A transformation or function of a random variable is another random variable. Let X denote the first student's Math score, and let Y denote the second student's Verbal score.What isP(X > Y)? We realign the units with the variable by taking the square root of that variance, giving us the standard deviation (now in dollars again). Edit: I moved the short, to the point, answer up top. navigate here The random variables X1, X2, X3, …, Xn all have the same probability distribution, so they all have the same SE. That uncertainty involves three independent sources of error: (1) the line may be misplaced vertically because our sample mean only approximates the true mean of the response variable, (2) our sample Nice answer. –Placidia Jan 20 '13 at 18:46 1 I meant 20, my brain is random, no idea how I got to 6 –JohnPhteven Jan 20 '13 at 18:48 \| Here's how that plays out in my classroom. ## Sum Of Standard Deviations Then, $F_Z(z)$, the CDF of the variable $Z$, would give the probabilities associated with that random variable. Now we count the number of successes in n independent trials. Next we look at what happens when we add the variances: Voila! The standard error is the SD of the sample mean. Contents 1 Independent random variables 1.1 Proofs 1.1.1 Proof using characteristic functions 1.1.2 Proof using convolutions 1.1.3 Geometric proof 2 Correlated random variables 2.1 Proof 3 References 4 See also Independent Dave Bock has been a high school math teacher since 1969. These variables are independent because the trials are independent, so the SE of their sum, the number of successes in n independent trials each with probability p of success, is the Sum Of Variances Hey, if you want more bang for your buck, it looks like you should buy multiple one-pound bags of carrots, as opposed to one three-pound bag! ‹ Lesson 26: Random Functions Here I thought they'd also use $\dfrac{0.5}{\sqrt{20}}$, but instead they use $\sqrt{20} \times 0.5$. Sum Of Independent Random Variables Proof using convolutions [citation needed] For independent random variables X and Y, the distribution fZ of Z = X+Y equals the convolution of fX and fY: f Z ( z ) The maximum total is 24 + 13 = 37 ounces, and the minimum is 16 + 9 = 25 ounces -- a range of 12 ounces. The SE of an affine transformation of a random variable is related to the SE of the original variable in a simple way: It does not depend on the additive constant We always calculate variability by summing squared deviations from the mean. Variance Of Sum Of Independent Random Variables To get the standard deviation of the sum of the variables, we need to find the square root of the sum of the squared deviations from the mean. The expected value of the draw is 1×(1/4) + 3×(2/4) + 5×(1/4) = 3, which is also the average of the list of the numbers on the tickets: (1 + 3 The SE of the sample percentage φ of a random sample of size n with replacement from a 0-1 box is n−½×(p×(1−p))½, where p is the fraction of tickets in the ## Sum Of Independent Random Variables standard-deviation standard-error share\|improve this question edited Jan 20 '13 at 18:27 asked Jan 20 '13 at 17:26 JohnPhteven 12117 You should tag this as "homework" as well, since it this page The probability distribution for each outcome is provided by the following table: Outcome -$1.00$0.00 $3.00$5.00 Probability 0.30 0.40 0.20 0.10 The mean outcome for this game is calculated as Sum Of Standard Deviations Select two students at random. Sum Of Random Variables Variance Standard Error of an Affine Transformation of a Random Variable If Y = aX + b, where a and b are constants (i.e., if Y is an affine transformation of X), But notice that we're less certain about this remaining weight than we were about the weight before we poured out the bowlful. http://comunidadwindows.org/sum-of/standard-error-of-the-sum-of-two-random-variables.php Since $Z = X + Y$, then the mean of $Z$ is $E(Z) = 24+17 = 41$. Some of these results are derived directly; others are derived from each other using the rules about the SE of affine transformations and of sums of independent random variables. Soon someone gets it: "Add the variances!" Good idea, but can we add the variances? Standard Error Of Sum Of Two Variables The sample mean and sample sum are random variables: their values depend on the sample. Recall that an affine transformation consists of multiplying by a constant, then adding a constant: f(x)=ax+b. To calculate the SE of a random variable requires calculating the expected value of a transformation of the random variable. his comment is here One can think of a random variable as being a constant (its expected value) plus a contribution that is zero on average (i.e., its expected value is zero), but that differs That is, $$s = \frac{\sqrt{s_1^2 + s_2^2 + \ldots + s_{12}^2}}{\sqrt{12 \times n}}$$ share\|improve this answer edited Apr 11 '15 at 17:45 answered Apr 11 '15 at 17:33 Matteo Expected Value Of Sum Of Random Variables I give the students data from an experiment that tried both types of fuel in several cars (a situation involving matched pairs, but I don't point that out). We saw in that the expected value of an affine transformation of a random variable is just the same affine transformation applied to the expectation of the random variable. ## What's within our grasp here is the theorem's quantification of the variability in these sample means, and the key is (drum roll!) adding variances. Variances add for the sum and for the difference of the random variables because the plus-or-minus terms all dropped out along the way. The SE of a random variable is a measure of the width of its probability histogram; the SD of a list is a measure of the width of its histogram. Problem 1 is looking for a statement about the sample mean; Problem 2 is about the sum, since the weight of the package is the sum of the weights of individual Normal Distribution We return to the list of conditions and add one more: the independent groups condition. Because the SE of the sample mean of n draws with replacement shrinks as n grows, the sample mean is increasingly likely to be extremely close to its expected value, the References ^ Bennett Eisenberg and Rosemary Sullivan, Why is the Sum of Independent Normal Random Variables Normal, (\it Math. Proofs Proof using characteristic functions [citation needed] The characteristic function φ X + Y ( t ) = E ⁡ ( e i t ( X + Y ) ) {\displaystyle weblink The mean of the sum of two random variables X and Y is the sum of their means: For example, suppose a casino offers one gambling game whose mean winnings are This does not imply, however, that short term averages will reflect the mean. An experiment randomly assigns some pigs to one of two diets, identical except for the inclusion of the supplement in the feed for Group S but not for Group N. Let's derive that formula. The formula for the SE of a random variable with the hypergeometric distribution is the special case of the SE of the sample sum when the box is a 0-1 box. A random variable is its expected value plus chance variability Random variable = expected value + chance variability The expected value of the chance variability is zero. After a few weeks, we weigh the pigs; summaries of the weight gains appear in the table. Since the sample data all comes from the same population, the random variables will be identical. Consider tossing a fair coin 10 times: Let X be the number of heads in the first 6 tosses and let Y be the number of heads in the last 4 For a continuous random variable, the mean is defined by the density curve of the distribution. Standard Error (SE) of a Random Variable Just as the SD of a list is the rms of the differences between the members of the list and the mean of the Because of the radial symmetry, we have f ( x ) g ( y ) = f ( x ′ ) g ( y ′ ) {\displaystyle f(x)g(y)=f(x')g(y')} , and the	2018-07-16 15:53:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8550124168395996, "perplexity": 331.15706714040607}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589404.50/warc/CC-MAIN-20180716154548-20180716174548-00425.warc.gz"}
https://www.lawlegal.xyz/category/criminal-law/objectives-of-criminal-law	# Targets And Objectives For Criminal Justice ## Criminal Investigation Some hold a combined view that combines elements of those thought-about above (Alexander and Ferzan 2009, 3–19; Simester and von Hirsch 2011, 3–18; Tadros 2016, 159–172). One way to assemble such a view is by distinguishing between major and secondary functions. Primary functions are those that, when all else is equal, we now have most purpose to need the law to fulfil. Secondary capabilities are those we have cause to want the legislation to fulfil if it fails to fulfil its main features. Criminal law’s major features, it’s believable to assume, are preventive. Ceteris paribus, we have most cause to need felony law to convey a couple of world in which wrongs like theft or murder do not occur. ### Criminal Regulation Jurisdictions It is possible to imagine a world by which the law will get its method—by which people uniformly refrain from criminal conduct. If $$\phi$$ing is a criminal offense, cheap force could permissibly be used to forestall $$D$$ $$\phi$$ing. Police officers and private persons alike have powers to arrest $$D$$, and affordable pressure might permissibly be used to make arrests efficient. Civil law and criminal law are two broad and separate entities of regulation with separate sets of legal guidelines and punishments. It is also true that international society isn’t the identical as domestic society. The assertion of felony jurisdiction over a person is amongst probably the most coercive activities any society can undertake. Punishing a person entails conduct towards them which requires a deprivation of some … Read More # Felony Justice ## Public Interest Litigation Thus, of the four capabilities of the capitalist state recognized by Miliband the primary of them is the upkeep of regulation and order; what he dubs ‘the repressive operate’. The ‘state is at all times involved’ within the processes of criminal justice, Miliband argues, if only as a result of it defines the nature of `authorized norms and sanctions’. 4.eighty five Deterrence is derived from the utilitarian concept of punishment. There is widespread assist for the proposition that the mere existence of a felony justice system has the effect of deterring individuals from committing legal offences. There is a significant amount of academic literature on the underlying justification for punishment, largely dominated by two theories. The utilitarian theory of punishment justifies punishment on the basis that its advantages outweigh its detrimental effects. Proponents of this concept contemplate that punishment has the potential to reduce crime. ### Substantive Legal Legislation Similarly, the textual content states that the aim of legal prosecution is to “punish the defendant.” But as the textual content later explains, there are at least four targets that felony regulation seeks to serve. There are many different inaccurate statements throughout the e-book. The purpose for quoting Miliband’s evaluation is that it implies a somewhat totally different take on the aim and nature of felony justice. From a Marxist perspective – at least if we take Miliband as the reference level – the aim of criminal justice may be characterised as the continued upkeep of sophistication domination via … Read More # 1 2 Felony Regulation And Felony Process Precedent case legal guidelines and current instances will serve as a reference for discussion crime, its parts and related sentencing. Rehabilitation – Courts of legislation and the legal professional use this to rework criminals into useful members of society. Its chief goal is to persuade criminals of their mistaken doing thus preventing extra crimes from occurring. Civil law is worried with any harm or infringement of particular person rights. As against this, the criminal legislation is all about the acts which legislation defines as offenses. While a civil regulation is initiated by a plaintiff, i.e. the aggrieved celebration, in criminal legislation the petition is filed by the federal government. ## What Is The Major Objective Of Legal Regulation? Every country’s constitution enforces certain legal guidelines, for the purpose of maintaining order and protecting the society from crimes. The Civil law lays emphasis on resolving the dispute like family dispute, lease matters, disputes regarding the sale and so forth. On the other hand, Criminal law stresses on punishment to the offender, who breaches the law by acts corresponding to, homicide, rape, theft, smuggling, and so forth. ### Principles Of Felony Law This rise and crime has promoted a “get robust on crime” ambiance, harsher sentences, and a more “law and order” regime. Criminal Law is a basic course which focuses on the weather, causes and sentences associated with numerous legal offenses. As a legal mechanism of formal social control, the course examines a few of the restraints and privileges of crime as a … Read More	2022-09-28 18:46:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17086492478847504, "perplexity": 6912.091064514387}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335276.85/warc/CC-MAIN-20220928180732-20220928210732-00775.warc.gz"}
https://hero88.co/d4gpio6/7e3a43-what-is-standard-error-of-the-mean	# what is standard error of the mean Note: The Student's probability distribution is approximated well by the Gaussian distribution when the sample size is over 100. a) The standard error of the mean birth weight for a treatment group provides a measure of the precision of the sample mean as an estimate of the population parameter …. answer choices . N Calculating Standard Error of the Mean (SEM). So we know that the variance-- or we could almost say the variance of the mean or the standard error-- the variance of the sampling distribution of the sample mean is equal to the variance of our original … {\displaystyle {\bar {x}}} The setting was 327 villages in two rural counties in northwest China. , What is N? Confidence Interval: The two confidence intervals i.e. {\displaystyle \operatorname {SE} } It is used to make a comparison between sample means across the populations. Definition of standard error in the Definitions.net dictionary. x , R. A. Fisher names the limits of the confidence interval which contains the parameter as “fiduciary limits” and named the confidence placed in the interval as fiduciary probability. The standard deviation (often SD) is a measure of variability. {\displaystyle \operatorname {E} (N)=\operatorname {Var} (N)} What is the Standard Error? Standard Error of the Mean (a.k.a. Guide to Standard Error Formula. n If a number is added to a set that is far away from the mean, how does this affect standard deviation? is used, is to make confidence intervals of the unknown population mean. Statistics courses, especially for biologists, assume formulae = understanding and teach how to do statistics, but largely ignore what those procedures assume, and how their results mislead when those assumptions are unreasonable. Now learn Live with India's best teachers. , When you use VARMETHOD=TAYLOR, or by default if you do not specify the VARMETHOD= option, PROC SURVEYMEANS uses the Taylor series method to estimate the variance of the mean .The procedure computes the estimated variance as Standard Error of the Mean (a.k.a. When the sampling fraction is large (approximately at 5% or more) in an enumerative study, the estimate of the standard error must be corrected by multiplying by a ''finite population correction'':[10] Moreover, this formula works for positive and negative ρ alike. A t-test is a statistical method used to see if two sets of data are significantly different. Assuming a normal distribution, we can state that 95% of the sample mean would lie within 1.96 SEs above or below the population mean, since 1.96 is the 2-sides 5% point of the standard normal distribution. 1 A high standard error corresponds to the higher spreading of data for the undertaken sample. This means you're free to copy, share and adapt any parts (or all) of the text in the article, as long as you give appropriate credit … SE We are an Essay Writing Company Get an essay written for you for as low as $13/page simply by clicking the Place Order button! In other words, it's a numerical value that represents standard deviation of the sampling distribution of a statistic for sample mean x̄ or proportion p, difference between two sample means (x̄ 1 - x̄ 2) or proportions (p 1 - p 2) (using either standard deviation or p value) in statistical surveys & experiments. We do not capture any email address. x In 1893, Karl Pearson coined the notion of standard deviation, which is undoubtedly most used measure, in research studies. The standard deviation (SD) & standard error of the mean (SEM) are used to represent the characteristics of the sample data and explain statistical analysis results. x Psychology Definition of STANDARD ERROR OF THE MEAN: a standard deviation of the mean. , leading the following formula for standard error: (since the standard deviation is the square root of the variance). In such cases, the sample size sigma — standard deviation; n — sample size. Calculate deviation from the mean Calculate each measurement's deviation from the mean by subtracting the individual... 3. Therefore, the standard error of the mean is usually estimated by replacing The formula given above for the standard error assumes that the sample size is much smaller than the population size, so that the population can be considered to be effectively infinite in size. Calculating the ‘Standard Error of the mean’ or SEM is simple using Excel’s in-built functions. A trial with three treatment arms was used. {\displaystyle {\sigma }_{\bar {x}}} {\displaystyle {\bar {x}}} 900 seconds . Q. {\displaystyle X} For example, normally, the estimator of the population mean is the sample mean. N N = size of the sample data set answer explanation . Where: s = sample standard deviation x 1, ..., x N = the sample data set x̄. is simply given by. are taken from a statistical population with a standard deviation of Please note: your email address is provided to the journal, which may use this information for marketing purposes. [9] If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size, because the estimate of the population mean will improve, while the standard deviation of the sample will tend to approximate the population standard deviation as the sample size increases. Is SE just the abbreviation of SEM? and standard deviation The standard error of the mean, also called the standard deviation of the mean, is a method used to estimate the standard deviation of a sampling distribution. n independent observations from a population with mean (As we can rarely have the S.D. E {\displaystyle N} with estimator (c) What is the probability that you would have gotten this mean difference (see #24) or less in your sample? Standard error measures the precision of the estimate of the sample mean. alternatives . Control treatment was daily folic acid. N instead: As this is only an estimator for the true "standard error", it is common to see other notations here such as: A common source of confusion occurs when failing to distinguish clearly between the standard deviation of the population ( To the uninformed, surveys appear to be an easy type of research to design and conduct, but when students and professionals delve deeper, they encounter the answer … Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the sample size. Mathematically, the variance of the sampling distribution obtained is equal to the variance of the population divided by the sample size. Psychology Definition of STANDARD ERROR OF THE MEAN: a standard deviation of the mean. x X are the standard deviation of the sampling distribution of the sample mean! We can describe this using STANDARD ERROR of the MEAN (SEM) -> mathematically, SEM = SD/√(sample size). If the sampling distribution is normally distributed, the sample mean, the standard error, and the quantiles of the normal distribution can be used to calculate confidence intervals for the true population mean. with the sample standard deviation of a population, for σ we use the value of S.D. ¯ You can download a PDF version for your personal record. Outcome measures included birth weight. SE The SEM gets smaller as your samples get larger. Statology Study is the ultimate online statistics study guide that helps you understand all of the core concepts taught in any elementary statistics course and … Hence the estimator of Assuming a normal distribution, we can state that 95% of the sample mean would lie within 1.96 SEs above or below the population mean, since 1.96 is the 2-sides 5% point of the standard normal distribution. n Calculation of CI for mean = (mean + (1.96 x SE)) to (mean – (1.96 x SE)) ¯ … , σ Where: s = sample standard deviation x 1, ..., x N = the sample data set x̄. Definition of standard error in the Definitions.net dictionary. σ In this case, the observed values fall an average of 4.89 units from the regression line. This is basically a variant of standard deviation as both concepts correspond to the spread measures. S , then the mean value calculated from the sample Meaning of standard error. 25. Join courses with the best schedule and enjoy fun and interactive classes. The following expressions can be used to calculate the upper and lower 95% confidence limits, where A cluster randomised double blind controlled trial investigated the effects of micronutrient supplements during pregnancy. {\displaystyle \sigma _{x}} σ How can you calculate the Confidence Interval (CI) for a mean? which is simply the square root of the variance: There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. Here we discuss the formula for the calculation of standard error of mean with the examples and downloadable excel sheet.. observations Standard Deviation, is a measure of the spread of a series or the distance from the standard. The standard error is, by definition, the standard deviation of {\displaystyle nS_{X}^{2}+n{\bar {X}}^{2}} {\displaystyle \sigma } Now, this is where everybody gets confused, the standard error is a type of standard deviation for the distribution of the means. 4 . N 2 What is standard deviation? Put simply, the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean. Similar … How to calculate Standard Error Note the number of measurements (n) and determine the sample mean (μ). This is usually the case even with finite populations, because most of the time, people are primarily interested in managing the processes that created the existing finite population; this is called an analytic study, following W. Edwards Deming. Report an issue . In simple words, SD determines how the sample data represents the mean accurately. Standard deviation (SD) is the measure of dispersion of the individual data values. But if you mean you are interested in whether a particular data point is plausibly from the population you have modelled (eg to ask "is this number a really big outlier? a statistical index of the probability that a given sample mean is representative of the mean of the population from which the sample was drawn. Average birth weight was significantly higher in the multiple micronutrients group than in the control (folic acid) group (difference 42.3 g; P=0.019). {\displaystyle x_{1},x_{2},\ldots ,x_{n}} {\displaystyle {\widehat {\sigma _{\bar {x}}}}} Tags: Topics: Question 8 . While every effort has been made to follow citation style rules, there may be some discrepancies. The resulting misuse is, shall we say, predictable... Use and Misuse An interval estimate gives you a range of values where the parameter is expected to lie. Practically this tells us that when trying to estimate the value of a mean, due to the factor n 16 . 95% and 99% are in general use. When a … is equal to the standard error for the sample mean, and 1.96 is the approximate value of the 97.5 percentile point of the normal distribution: In particular, the standard error of a sample statistic (such as sample mean) is the actual or estimated standard deviation of the sample mean in the process by which it was generated. It gives an idea of the exactness and … Birth weight was available for analysis for 4421 live births. {\displaystyle {\bar {x}}} Gurland and Tripathi (1971) provide a correction and equation for this effect. It is the average of all the measurements. 2 ), you need to compare it to your estimate of the population mean and your estimate of the population standard deviation (not the sample mean's standard deviation, also known as SEM). , which is the most often calculated quantity, and is also often colloquially called the standard error). What does standard error mean? Small samples are somewhat more likely to underestimate the population standard deviation and have a mean that differs from the true population mean, and the Student t-distribution accounts for the probability of these events with somewhat heavier tails compared to a Gaussian. x ) Please refer to the appropriate style manual or other sources if you have any questions. n Standard Error (SE) provides, the standard deviation in different values of the sample mean. This often leads to confusion about their interchangeability. such that. ¯ … Although average birth weight was higher in the iron-folic acid group than in the control group, the difference was not significant (24.3 g; P=0.169). ¯ [5] See unbiased estimation of standard deviation for further discussion. In probability & statistics, the standard deviation of sampling distribution of a statistic is called as Standard Error often abbreviated as SE. σ ror of the mean (SEM), a statistical index of the probability that a given sample mean is representative of the mean of the population from which the sample was drawn. The SEM quantifies how accurately you know the true mean of the population. Determine how much each measurement varies from the mean. = mean value of the sample data set. You can easily calculate the standard error of the true mean using functions contained within the base R package. To estimate the standard error of a Student t-distribution it is sufficient to use the sample standard deviation "s" instead of σ, and we could use this value to calculate confidence intervals. . The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. Taylor Series Method. With n = 2, the underestimate is about 25%, but for n = 6, the underestimate is only 5%. x σ In regression analysis, the term "standard error" refers either to the square root of the reduced chi-squared statistic, or the standard error for a particular regression coefficient (as used in, say, confidence intervals). Introduction. If we plot the actual data points along with … σ This is because as the sample size increases, sample means cluster more closely around the population mean. If people are interested in managing an existing finite population that will not change over time, then it is necessary to adjust for the population size; this is called an enumerative study. is equal to the sample mean, What does standard error mean? ⁡ σ Understanding ‘Standard Error of the mean’ isn’t h Meaning of standard error. [12] See also unbiased estimation of standard deviation for more discussion. It is used to make statistical inferences about the population parameter, either through statistical hypothesis testing or through estimation by confidence intervals. x If you have a subscription to The BMJ, log in: Subscribe and get access to all BMJ articles, and much more. The effect of the FPC is that the error becomes zero when the sample size n is equal to the population size N. If values of the measured quantity A are not statistically independent but have been obtained from known locations in parameter space x, an unbiased estimate of the true standard error of the mean (actually a correction on the standard deviation part) may be obtained by multiplying the calculated standard error of the sample by the factor f: where the sample bias coefficient ρ is the widely used Prais–Winsten estimate of the autocorrelation-coefficient (a quantity between −1 and +1) for all sample point pairs. I recommend Snedecor and … Var is a random variable whose variation adds to the variation of [11]. when the probability distribution is unknown, This page was last edited on 5 February 2021, at 18:49. n Mean birth weight was 3153.7 g (n=1545; 95% confidence interval 3131.5 to 3175.9, standard deviation 444.9, standard error 11.32) in the control group, 3173.9 g (n=1470; 3152.2 to 3195.6, 424.4, 11.07,) in the iron-folic acid group, and 3197.9 g (n=1406; 3175.0 to 3220.8, 438.0, 11.68) in the multiple micronutrients group. An online standard error calculator helps you to estimate the standard error of the mean (SEM) from the given data sets and shows step-by-step calculations. How can you calculate the Confidence Interval (CI) for a mean? To summarize: SD measures variability in data we used to get 1 average (in this case, cell counts). ", "On the value of a mean as calculated from a sample", "Analysis of Short Time Series: Correcting for Autocorrelation", Multivariate adaptive regression splines (MARS), Autoregressive conditional heteroskedasticity (ARCH), https://en.wikipedia.org/w/index.php?title=Standard_error&oldid=1005049147, Creative Commons Attribution-ShareAlike License, in many cases, if the standard error of several individual quantities is known then the standard error of some. The standard error is defined as the error which arises in the sampling distribution while performing statistical analysis. When you look at scientific papers, sometimes the \"error bars\" on graphs or the ± number after means in tables represent the standard error of the mean, while in other papers they represent 95% confidence intervals. When the true underlying distribution is known to be Gaussian, although with unknown σ, then the resulting estimated distribution follows the Student t-distribution. In total, 5828 pregnant women were recruited. {\displaystyle \sigma } All the sample means which are normally distributed around M pop will lie between M pop + 3 SE M and M pop – 3 SE M . {\displaystyle \sigma } When you divide by a bigger number, you get a smaller number, so the more samples you have, the lower the SEM. Standard error of the mean is often abbreviated to standard error. The notation for standard error can be any one of SE, SEM (for standard error of measurement or mean), or SE. {\displaystyle \sigma } given by:[2]. In other words, it is the actual or estimated standard deviation of the sampling distribution of the sample statistic. The sampling distribution of a population mean is generated by repeated sampling and recording of the means obtained. This makes sense, because the mean of a large sample is likely to be closer to the true population mean than is the mean of a small sample. the standard deviation of the sampling distribution of the sample mean! ( Standard error of the mean tells you how accurate your estimate of the mean is likely to be. It is a measure of how far each observed value is from the mean. σ ¯ https://www.khanacademy.org/.../v/standard-error-of-the-mean x σ technical support for your product directly (links go to external sites): Thank you for your interest in spreading the word about The BMJ. Two interventions were investigated—daily iron with folic acid and daily multiple micronutrients (recommended allowance of 15 vitamins and minerals). of the entire population being sampled is seldom known. It is abbreviated as SEM. I prefer 95% confidence intervals. If the statistic is the sample mean, it is called the standard error of the mean (SEM).[2]. ( = mean value of the sample data set. This question is for testing whether or not you are a human visitor and to prevent automated spam submissions. 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A method used to evaluate the standard error of the total variance vitamins minerals. = 6, the population mean is one and to prevent automated spam submissions stratified by county, a! [ 5 ] See unbiased estimation of standard deviation of the sampling variation from the mean subtracting! Together and divide the sum total by the number of measurements ( n ) and determine the sample mean those! Measurement varies from the mean often abbreviated to standard error of the (. Question is for testing whether or not you are a human visitor and prevent! Supplements during pregnancy visitor and to prevent automated spam submissions = SD/√ ( sample size increases sample... By subtracting the individual... 3 = SD/√ ( sample size live births for... = sample standard deviation ( SD ) is the standard deviation of the mean more.. Error is the average distance that the observed values fall an average of units! And Tripathi ( 1971 ) provide a correction and equation for this effect a statistical used... R package number is added to a larger percentage of the mean calculate each measurement varies from the.... Statistic is called the standard error of the sampling distribution villages in two counties! Functions contained within the base R package interactive classes in simple words, it is used to if! An unbiased estimate of the population mean is one of samples 2 and prevent... Average ( in this article for 1 day for: £30 / 37! Mean tells you how accurate your estimate of the sampling distribution } of the temporal what is standard error of the mean we... Same as SEM mean or SEM is simple using Excel ’ s in-built.. Using functions contained within the base R package of 15 vitamins and minerals ) [... Made to follow citation style rules, there may be some discrepancies and much more and equation for effect... Get larger, Karl Pearson coined the notion of standard deviation ( SD.! Gaussian, and vary depending on the size of the mean ’ or SEM is simple using Excel ’ in-built... Data represents the mean is one correspond to the mean is one into account that spread of possible 's. Value of σ is unknown in 1893, Karl Pearson coined the notion of deviation..., for σ we use the value of S.D..., x n = 2, the underestimate about! A high standard error is the standard error of the mean Add all the samples and.: £30 / \$ 37 / €33 ( excludes VAT ). [ 2 ] small samples of <... Of an estimate of the mean ( SEM ) - > mathematically, SEM = (... As both concepts correspond to the higher spreading of data are significantly different and interactive classes abbreviated to error! Articles, and vary depending on the size of the sampling distribution, either through statistical hypothesis and interval.. In two rural counties in northwest China the sum total by the number of measurements ( n ) and the! Means cluster more closely around the population divided by the Gaussian distribution when the sample!. Is expected to lie < 20 page was last edited on what is standard error of the mean February 2021, at 18:49 means and. Add all the samples together and divide the sum total by the number of measurements ( n ) and the... Size increases, sample means of which our mean is a measure how! Of an estimate of the Student t-distribution and Tripathi ( 1971 ) a! Can you calculate the standard deviation, which may use this information for marketing.. Value of σ is unknown Gaussian distribution when the sample size ). [ 2.! Purpose, to express the reliability of an estimate of the spread of possible σ 's you spread. An average of 4.89 units from the standard deviation tells you how spread out the data....	2022-01-24 03:15:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8651596903800964, "perplexity": 891.6725495132763}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304471.99/warc/CC-MAIN-20220124023407-20220124053407-00151.warc.gz"}
https://dictionary.iucr.org/index.php?title=Absorption_edge&diff=1745&oldid=prev	# Difference between revisions of "Absorption edge" ### From Online Dictionary of Crystallography (Discontinuité, arête d'absorption (Fr). Absorptionskante (Ge). Canto de absorción (Sp). край поглощения (Ru). Spigolo di assorbimento (It) ## Definition An absorption edge is a sharp discontinuity in the absorption spectrum of X-rays by an element that occurs when the energy of the photon corresponds to the energy of a shell of the atom (K, LI, LII, LIII, etc.). ## Examples For gallium: $\lambda_{K}$ = 1.1958 Ǻ; $\lambda_{L_{I}}$ = 9.5446 Ǻ; $\lambda_{L_{II}}$ = 10.8414 Ǻ; $\lambda_{L_{III}}$= 11.1038 Ǻ; For arsenic: $\lambda_{K}$ = 1.0448 Ǻ; $\lambda_{L_{I}}$ = 8.1195 Ǻ; $\lambda_{L_{II}}$ = 9.1187 Ǻ; $\lambda_{L_{III}}$= 9.3617 Ǻ;	2020-08-04 05:52:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9713264107704163, "perplexity": 14843.708622083177}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735860.28/warc/CC-MAIN-20200804043709-20200804073709-00420.warc.gz"}
https://zbmath.org/?q=an%3A0959.34049	# zbMATH — the first resource for mathematics Set-valued pseudomonotone maps and degenerate evolution inclusions. (English) Zbl 0959.34049 The authors prove existence and uniqueness of a weak solution to the problem $(Bu)'+Au\ni f,\;u(0)=u_0,$ in a Banach space and then apply the theory to a frictional contact problem. $$B$$ is assumed to be linear and is allowed to vanish. $$A$$ is assumed to be a pseudomonotone set-valued operator. The proof of this theorem is based on the method of elliptic regularization from J. L. Lions [Quelques methods de resolution des problèmes aux limites nonlinéaires, Dunod, Paris (1969; Zbl 0189.40603)] adapted to the set-valued case. Notation, terminology and other preliminaries are given in section 2, while section 3 contains a number of technical lemmas needed for the proof of the main results, stated in theorems 4.3, 4.4 and 4.5. Sections 5-8 give a detailed explanation of the application, a problem of frictional contact between a deformable body and a moving rigid foundation. Theorems 4.4 and 4.5 are applied to the problem to obtain existence and uniqueness of a solution. The authors cite references for earlier work on this evolution inclusion and also information on pseudomonotone maps, plus give a number of references concerning related friction problems. ##### MSC: 34G25 Evolution inclusions 47H05 Monotone operators and generalizations 47H04 Set-valued operators 47N20 Applications of operator theory to differential and integral equations 49J40 Variational inequalities 74M15 Contact in solid mechanics 74M10 Friction in solid mechanics Full Text: ##### References: [1] DOI: 10.1016/0362-546X(91)90035-Y · Zbl 0722.73061 · doi:10.1016/0362-546X(91)90035-Y [2] Andrews K. T., Euro. J. Appl. Math. 8 pp 417– (1997) [3] DOI: 10.1016/S0020-7225(97)87426-5 · Zbl 0903.73065 · doi:10.1016/S0020-7225(97)87426-5 [4] DOI: 10.1016/0020-7225(95)00121-2 · Zbl 0900.73684 · doi:10.1016/0020-7225(95)00121-2 [5] DOI: 10.1016/0020-7225(94)E0042-H · Zbl 0899.73473 · doi:10.1016/0020-7225(94)E0042-H [6] Ionescu I. R., Eur. J. Mech. A/Solids 13 (4) pp 555– (1994) [7] DOI: 10.1016/0020-7225(88)90032-8 · Zbl 0662.73079 · doi:10.1016/0020-7225(88)90032-8 [8] DOI: 10.1016/0362-546X(86)90050-7 · Zbl 0603.47038 · doi:10.1016/0362-546X(86)90050-7 [9] DOI: 10.1016/0362-546X(95)00170-Z · Zbl 0865.73054 · doi:10.1016/0362-546X(95)00170-Z [10] DOI: 10.1016/0362-546X(87)90055-1 · doi:10.1016/0362-546X(87)90055-1 [11] DOI: 10.1007/BFb0086760 · doi:10.1007/BFb0086760 [12] DOI: 10.1512/iumj.1974.23.23056 · Zbl 0281.34061 · doi:10.1512/iumj.1974.23.23056 [13] DOI: 10.1016/0020-7683(95)00140-9 · Zbl 0926.74012 · doi:10.1016/0020-7683(95)00140-9 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.	2021-03-07 09:32:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4399368464946747, "perplexity": 2865.332607574051}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178376206.84/warc/CC-MAIN-20210307074942-20210307104942-00231.warc.gz"}
https://docs.microsoft.com/en-us/previous-versions/ms713271%28v%3Dvs.85%29	# spin The spin command starts spinning a disc or stops the disc from spinning. Videodisc devices recognize this command. To send this command, call the mciSendString function with the lpszCommand parameter set as follows. ``````_stprintf_s( lpszCommand, TEXT("spin %s %s %s"), lpszDeviceID, lpszUpDown, lpszFlags ); `````` Parameters lpszDeviceID Identifier of an MCI device. This identifier or alias is assigned when the device is opened. lpszUpDown One of the following flags. Value Meaning down Stops the disc from spinning. up Starts spinning the disc. lpszFlags Can be "wait", "notify", or both. For more information about these flags, see The Wait, Notify, and Test Flags. Return Values Returns zero if successful or an error otherwise. Remarks The following command starts spinning a videodisc device: ``````spin videodisc up `````` Requirements Windows NT/2000/XP: Included in Windows NT 3.1 and later. Windows 95/98/Me: Included in Windows 95 and later.	2020-03-30 22:23:13	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8215526342391968, "perplexity": 13681.904206897356}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370497309.31/warc/CC-MAIN-20200330212722-20200331002722-00004.warc.gz"}
https://machinelearningmastery.com/market-basket-analysis-with-association-rule-learning/	# Market Basket Analysis with Association Rule Learning Last Updated on September 15, 2021 The promise of Data Mining was that algorithms would crunch data and find interesting patterns that you could exploit in your business. The exemplar of this promise is market basket analysis (Wikipedia calls it affinity analysis). Given a pile of transactional records, discover interesting purchasing patterns that could be exploited in the store, such as offers and product layout. In this post you will work through a market basket analysis tutorial using association rule learning in Weka. If you follow along the step-by-step instructions, you will run a market basket analysis on point of sale data in under 5 minutes. Kick-start your project with my new book Machine Learning Mastery With Weka, including step-by-step tutorials and clear screenshots for all examples. Let’s get started. Photo by HealthGauge, some rights reserved. ## Association Rule Learning I once did some consulting work for a start-up looking into customer behavior in a SaaS app. We were interested in patterns of behavior that indicated churn or conversion from free to paid accounts. I spent weeks pouring over the data, looking at correlations and plots. I came up with a bunch of rules that indicated outcomes and presented ideas for possible interventions to influence those outcomes. I came up with rules like: “User Creates x widgets in y days and logged in n times then they will convert“. I ascribed numbers to the rules such as support (the number of records that match the rule out of all record) and lift (the % increase in predictive accuracy in using the rule to predict a conversion). It was only after I delivered and presented the report that I released what a colossal mistake I made. I had performed Association Rule Learning by hand, when there are off-the-shelf algorithms that could have done the work for me. I’m sharing this story so that it sticks in your mind. If you are sifting large datasets for interesting patterns, association rule learning is a suite of methods should be using. ### Need more help with Weka for Machine Learning? Take my free 14-day email course and discover how to use the platform step-by-step. Click to sign-up and also get a free PDF Ebook version of the course. ## 1. Start the Weka Explorer In previous tutorials, we have looked at running a classifier, designing and running an experiment, algorithm tuning and ensemble methods. If you need help downloading and installing Weka, please refer to these previous posts. Weka GUI Chooser Start the Weka Explorer. ## 2. Load the Supermarket Datasets Weka comes with a number of real datasets in the “data” directory of the Weka installation. This is very handy because you can explore and experiment on these well known problems and learn about the various methods in Weka at your disposal. Load the Supermarket dataset (data/supermarket.arff). This is a dataset of point of sale information. The data is nominal and each instance represents a customer transaction at a supermarket, the products purchased and the departments involved. There is not much information about this dataset online, although you can see this comment (“question of using supermarket.arff for academic research”) from the personal that collected the data. Supermarket dataset loaded in the Weka Explorer The data contains 4,627 instances and 217 attributes. The data is denormalized. Each attribute is binary and either has a value (“t” for true) or no value (“?” for missing). There is a nominal class attribute called “total” that indicates whether the transaction was less than \$100 (low) or greater than \$100 (high). We are not interested in creating a predictive model for total. Instead we are interested in what items were purchased together. We are interested in finding useful patterns in this data that may or may not be related to the predicted attributed. ## 3. Discover Association Rules Click the “Associate” tab in the Weka Explorer. The “Apriori” algorithm will already be selected. This is the most well known association rule learning method because it may have been the first (Agrawal and Srikant in 1994) and it is very efficient. In principle the algorithm is quite simple. It builds up attribute-value (item) sets that maximize the number of instances that can be explained (coverage of the dataset). The search through item space is very much similar to the problem faced with attribute selection and subset search. Click the “Start” button to run Apriori on the dataset. ## 4. Analyze Results The real work for association rule learning is in the interpretation of results. Results for the Apriori Association Rule Learning in Weka From looking at the “Associator output” window, you can see that the algorithm presented 10 rules learned from the supermarket dataset. The algorithm is configured to stop at 10 rules by default, you can click on the algorithm name and configure it to find and report more rules if you like by changing the “numRules” value. The rules discovered where: 1. biscuits=t frozen foods=t fruit=t total=high 788 ==> bread and cake=t 723 conf:(0.92) 2. baking needs=t biscuits=t fruit=t total=high 760 ==> bread and cake=t 696 conf:(0.92) 3. baking needs=t frozen foods=t fruit=t total=high 770 ==> bread and cake=t 705 conf:(0.92) 4. biscuits=t fruit=t vegetables=t total=high 815 ==> bread and cake=t 746 conf:(0.92) 5. party snack foods=t fruit=t total=high 854 ==> bread and cake=t 779 conf:(0.91) 6. biscuits=t frozen foods=t vegetables=t total=high 797 ==> bread and cake=t 725 conf:(0.91) 7. baking needs=t biscuits=t vegetables=t total=high 772 ==> bread and cake=t 701 conf:(0.91) 8. biscuits=t fruit=t total=high 954 ==> bread and cake=t 866 conf:(0.91) 9. frozen foods=t fruit=t vegetables=t total=high 834 ==> bread and cake=t 757 conf:(0.91) 10. frozen foods=t fruit=t total=high 969 ==> bread and cake=t 877 conf:(0.91) Very cool, right! You can see rules are presented in antecedent => consequent format. The number associated with the antecedent is the absolute coverage in the dataset (in this case a number out of a possible total of 4,627). The number next to the consequent is the absolute number of instances that match the antecedent and the consequent. The number in brackets on the end is the support for the rule (number of antecedent divided by the number of matching consequents). You can see that a cutoff of 91% was used in selecting rules, mentioned in the “Associator output” window and indicated in that no rule has a coverage less than 0.91. I don’t want to go through all 10 rules, it would be too onerous. Here are few observations: • We can see that all presented rules have a consequent of “bread and cake”. • All presented rules indicate a high total transaction amount. • “biscuits” an “frozen foods” appear in many of the presented rules. You have to be very careful about interpreting association rules. They are associations (think correlations), not necessary causally related. Also, short antecedent are likely more robust than long antecedent that are more likely to be fragile. Photo by goosmurf, some rights reserved. If we are interested in total for example, we might want to convince people that buy biscuits, frozen foods and fruit to buy bread and cake so that they result in a high total transaction amount (Rule #1). This may sound plausible, but is flawed reasoning. The product combination does not cause a high total, it is only associated with a high total. Those 723 transactions may have a vast assortment of random items in addition to those in the rule. What might be interesting to test is to model the path through the store required to collect associated items and seeing if changes to that path (shorter, longer, displayed offers, etc) have an effect on transaction size or basket size. ## Summary In this post you discovered the power of automatically learning association rules from large datasets. You learned that it is much more efficient approach to use an algorithm like Apriori rather than deducing rules by hand. You performed your first market basket analysis in Weka and learned that the real work is in the analysis of results. You discovered the careful attention to detail required when interpreting rules and that association (correlation) is not the same as causation. ## Discover Machine Learning Without The Code! #### Develop Your Own Models in Minutes ...with just a few a few clicks Discover how in my new Ebook: Machine Learning Mastery With Weka Covers self-study tutorials and end-to-end projects like: ### 51 Responses to Market Basket Analysis with Association Rule Learning 1. Deepak Babu March 18, 2014 at 5:50 pm # Nice article. I had written about using association rule mining using R in two parts, first part explaining concepts and the second part explaining implementation using R with visualizations. • jasonb March 22, 2014 at 9:27 am # Great post Deepak, thanks for sharing. • ia October 18, 2014 at 9:13 pm # thanks i have understand alot of concepts due to this example.But i m using weka 3.7.1 can u provide me an exmple with a small dataset in weka like weather.arff,.cpu.arff etc. i will be thankful 2. ia October 18, 2014 at 9:07 pm # Thanks alot sir for your cooperation But there is a problem that supermarket dataset have large size. I was try to perform the example but supermarket dataset was unavailable. Sir if you kindly provide example for other dataset like weather.arrf i will be thankful. 3. ibrar November 9, 2014 at 12:50 am # i m working on Arabic text, can your good self provide me the example in arabic dataset and then converted into CSV and ARFF format. I will be thankful 4. Faizan January 19, 2015 at 2:01 am # Dude, you really made my day. THANKS ALOT. I had another query, how to increase the size of heap? I have tried to edit the runWeka.bat file and runWeka.ini . I tried the xmx and maxheap=2014mb and everything else. But the problem is, I can’t save the edited file. • Jason Brownlee January 19, 2015 at 8:19 am # You can, but you may better off working with a representative sample of your source data. 5. Neeta March 26, 2015 at 7:52 pm # Is this weka tool is useful for find frequent and infrequent itemset and after that to find association rule? 6. Álvaro López López March 28, 2015 at 12:12 am # Hello Jason. Your article is great to introduce Association Rules with Weka’s Supermarket example. I would like to point that it is mistaken at this point about explaining the rules meaning: “The number in brackets on the end is the support for the rule (number of antecedent divided by the number of matching consequents). ” The figure is the value for Confidence metric, which also implies how the rules are ordered in the output. Best regards, Álvaro 7. maryam May 8, 2015 at 7:41 pm # i want to use association rule mining to consider count of purchase for each user. For example user A listened item a 3 times. Thanks ‘‘Click (CDi) and Length of Reading Time (High))___Click (CDj)’’ placement (CDj)’’ 8. YADAF September 14, 2017 at 3:09 am # Hie. Your article is great to introduce Association Rules with Weka’s Supermarket example. Best rules found: 1. biscuits=t frozen foods=t fruit=t total=high 788 ==> bread and cake=t 723 lift:(1.27) lev:(0.03) [155] conv:(3.35) 2. baking needs=t biscuits=t fruit=t total=high 760 ==> bread and cake=t 696 lift:(1.27) lev:(0.03) [149] conv:(3.28) 3. baking needs=t frozen foods=t fruit=t total=high 770 ==> bread and cake=t 705 lift:(1.27) lev:(0.03) [150] conv:(3.27) 4. biscuits=t fruit=t vegetables=t total=high 815 ==> bread and cake=t 746 lift:(1.27) lev:(0.03) [159] conv:(3.26) 5. party snack foods=t fruit=t total=high 854 ==> bread and cake=t 779 lift:(1.27) lev:(0.04) [164] conv:(3.15) 6. biscuits=t frozen foods=t vegetables=t total=high 797 ==> bread and cake=t 725 lift:(1.26) lev:(0.03) [151] conv:(3.06) 7. baking needs=t biscuits=t vegetables=t total=high 772 ==> bread and cake=t 701 lift:(1.26) lev:(0.03) [145] conv:(3.01) 8. biscuits=t fruit=t total=high 954 ==> bread and cake=t 866 lift:(1.26) lev:(0.04) [179] conv:(3) 9. frozen foods=t fruit=t vegetables=t total=high 834 ==> bread and cake=t 757 lift:(1.26) lev:(0.03) [156] conv:(3) 10. frozen foods=t fruit=t total=high 969 ==> bread and cake=t 877 lift:(1.26) lev:(0.04) [179] conv:(2.92) 9. Essam Mosallam October 18, 2017 at 8:42 am # i want to use association rule mining to consider count of purchase for each user. For example user A listened item a 3 times. Thanks and i want to analyze the result for True only not false for items, Thanks. 10. Sid October 31, 2017 at 12:14 am # What If it has Date ? If we need to find the patterns on when purchased and association between them • Jason Brownlee October 31, 2017 at 5:34 am # Yes, often the date can provide a lot of useful information. 11. Otto November 23, 2017 at 4:41 pm # Hello Jason! Thank you for such a cool website! I do have a question. What if my dataset is in binary (0,1)? I have a grocery store dataset where each column is an item and the 0 or 1 indicates whether it was bought or not. The row is the receipt, or the list of items bought by an individual. I wanted to find out what items where bought together by using the weka association but the top ten rules generated are always 0 (No). I want to find out the rules that only have 1 because it shows me which items where bought. As it is now the No rules generated in results are telling me what people are not buying. How do I get around this? I hope I am making sense. • Otto November 23, 2017 at 4:42 pm # examples 1. Vanilla Eclair=No Vanilla Meringue=No Chocolate Croissant=No Almond Bear Claw=No 63188 ==> Almond Tart=No 60738 lift:(1) lev:(0) [206] conv:(1.08) 2. Vanilla Eclair=No Vanilla Meringue=No Chocolate Croissant=No Blueberry Danish=No 63064 ==> Almond Tart=No 60618 lift:(1) lev:(0) [205] conv:(1.08) 3. Chocolate Eclair=No Vanilla Eclair=No Vanilla Meringue=No Chocolate Croissant=No 63181 ==> Almond Tart=No 60730 lift:(1) lev:(0) [205] conv:(1.08) 4. Vanilla Eclair=No Apricot Tart=No Vanilla Meringue=No Almond Bear Claw=No 63254 ==> Almond Tart=No 60797 lift:(1) lev:(0) [202] conv:(1.08) 5. Vanilla Eclair=No Apricot Tart=No Vanilla Meringue=No Blueberry Danish=No 63118 ==> Almond Tart=No 60666 lift:(1) lev:(0) [201] conv:(1.08) 6. Chocolate Eclair=No Vanilla Eclair=No Apricot Tart=No Vanilla Meringue=No 63234 ==> Almond Tart=No 60777 lift:(1) lev:(0) [201] conv:(1.08) 7. Vanilla Eclair=No Apricot Tart=No Vanilla Meringue=No Chocolate Croissant=No 63180 ==> Almond Tart=No 60725 lift:(1) lev:(0) [201] conv:(1.08) • Jason Brownlee November 24, 2017 at 9:34 am # It is also possible that there simply are not interesting patterns find, keep that in mind. • Jason Brownlee November 24, 2017 at 9:34 am # It’s a good question. Perhaps the data is not rich enough? Perhaps you can collect more features to describe each outcome? 12. Tom February 22, 2018 at 9:44 pm # Hi jason..how can i use those rules if i want to create personalized push notification for my customers? I already have a dataset which includes the customer’s past purchases. What i am finding difficult is how to send each of them in one click the personalized advert. • Jason Brownlee February 23, 2018 at 11:57 am # It really depends on your application and data. 13. Jesús Martínez March 13, 2018 at 12:41 am # Very good. Thanks for sharing this analysis. 14. Vinod March 14, 2018 at 11:24 pm # Hi, is it mandatory to have the basket size greater than 1, i mean we need to consider only those invoices were we have more than 1 product purchased to get product association ? • Jason Brownlee March 15, 2018 at 6:31 am # Yes, unless you are looking across purchases for a customer. 15. Vinod March 15, 2018 at 4:44 pm # Thanks Jason, yes across purchases for a customer can through some good insights as well, will have a look at it. 16. Vinod April 4, 2018 at 10:55 pm # Hi, i am done with processing the market basket report and its now time to get some insights out of it and recommend. My question is having looked at Confidence and Lift parameter, which one is the better metric to look at ? secondly i can see some Antecedents have decent support and lift > 1 but the confidence matrix is low with around 15%. I know that recommendations are very subjective and majorly upto how business looks at each parameters. But need some directional views on how do we go about in recommending things back to client. • Jason Brownlee April 5, 2018 at 6:01 am # Perhaps find out the goals of the stakeholders and use that to motivate the interpretation. Or take different stances in the report and then interpret results. E.g. if x is important then the method found that …, otherwise if y is important, the method found that… I hope that helps. 17. steamwash April 15, 2019 at 3:12 pm # Nice Blog! Thanks for sharing this useful post. 18. Carlos Trujillo Almeida April 21, 2019 at 7:11 am # Hi Jason For the same example and without changing any parameters, what is the support of the second rule obtained, and how is it obtained? 19. Karimi August 29, 2019 at 7:44 pm # Hi Jason, how do i select the attributes using association rule to be input for a classifier like MLP? • Jason Brownlee August 30, 2019 at 6:17 am # I’m not sure it is an appropriate approach. 20. stellamaris August 29, 2019 at 11:00 pm # Thanks Jason, is it possible to select best attributes from the generated rules? 21. Karimi August 29, 2019 at 11:31 pm # Hi Jason, i used wbc dataset that has 9 attributes and class attributes to create association rule model the rules generated are having only 6 attributes and a class attribute. Are the 3 attributes not appearing irrelevant? 22. Esther Mead February 4, 2020 at 5:33 am # jason, thank you so much for all of your great tutorials! my question is, can the apriori algorithm be implemented in python for predicting housing prices (say, for the common boston housing prices dataset)? 23. Misha March 26, 2020 at 2:14 am # Very interesting article Jason! I think one thing to add is that there is a lot of algorithms that essentially get the same rules given a set of parameters and they differ in computational speed. So depending how wide your dataset is you may want to consider using ECLAT or F-P Growth. I have recently posted an article with some theory and Python implementation on my blog, so feel free to take a look and let me know what you think: https://pyshark.com/market-basket-analysis-using-association-rule-mining-in-python/ 24. Sundeep March 29, 2020 at 12:52 pm # Hi Jason Thank you for a great article. I’m working on a school assignment using weka with supermarket dataset. I’m confused about one question and hope you can help ASAP. Question: Let’s assume that 80% of values are missing in this dataset. Then what is the number of items which average customer’s basket contains (hint: this means that 20% of attributes have the value “t” in an instance on average)? Thanks 25. Ash May 17, 2020 at 3:27 pm # HI , This is great. Thank you for this. Can you please tell me how to find the highest rank association rule in this? • Jason Brownlee May 18, 2020 at 6:08 am # Sorry, I don’t have more tutorials on association rules. Perhaps in the future. 26. Mark November 4, 2020 at 9:09 pm # “The number in brackets on the end is the support for the rule” “Support” are the numbers close to ‘promise’ part and ‘consequent’ part of the rule. Next article about apriori where author does not know how to explain results. 27. Ammar Azlan August 24, 2021 at 11:56 am # Hi Jason, I love your article. I just want to point out some typo. I hope it helps. “I’m sharing this story so that it sticks in your mind. If you are sifting large datasets for interesting patterns, association rule learning is a suite of methods should should be using.” • Adrian Tam August 24, 2021 at 11:58 am # Good catch! Thank you. 28. Deep Learner September 15, 2021 at 1:29 am # Hi Jason, Thanks for the post always excellent. We love your work! Just a note, I found the link you have provided that is meant to give some info about the data is dead, however, I looked and found another discussion where there are some details about the data set right here:	2022-12-07 16:14:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19346129894256592, "perplexity": 3945.691762180806}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711200.6/warc/CC-MAIN-20221207153419-20221207183419-00258.warc.gz"}
https://www.emathhelp.net/pt/square-feet-to-square-yards/	# Square feet to square yards This free conversion calculator will convert square feet to square yards, i.e. sq ft to sq yd. Correct conversion between different measurement scales. A fórmula é $Á_{\text{sq yd}} = 0.111111111111111 Á_{\text{sq ft}}$, onde $Á_{\text{sq ft}} = 15$. Portanto, $Á_{\text{sq yd}} = 1.666666666666667$. Resposta: $15 \text{sq ft} = 1.666666666666667 \text{sq yd}$.	2022-05-16 14:27:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9481421113014221, "perplexity": 14453.707047481219}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662510138.6/warc/CC-MAIN-20220516140911-20220516170911-00021.warc.gz"}
https://kb.osu.edu/dspace/handle/1811/15878?show=full	# THE MICROWAVE SPECTRUM OF GLYCOLALDEHYDE. Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/15878 Files Size Format View 1969-U-07.jpg 98.83Kb JPEG image dc.creator Simons, Margaret A. en_US dc.creator Woods, R. C. en_US dc.date.accessioned 2006-06-15T16:54:51Z dc.date.available 2006-06-15T16:54:51Z dc.date.issued 1969 en_US dc.identifier 1969-U-7 en_US dc.identifier.uri http://hdl.handle.net/1811/15878 dc.description Author Institution: Department of Chemistry, University of Wisconsin en_US dc.description.abstract Glycolaldehyde $(CHOCH_{2}OH)$ has been found to exist in an intramolecularly hydrogen bonded ring structure with a plane of symmetry. A large number of transitions have been assigned in the frequency range 8-18GHz for the normal species and for $CHOCH_{2}OD$. The rotational constants derived from these data are $A = 18,446.28 MHz, B = 6525.92, C = 4969.23$ for the normal and $A = 17491.15, B = 6499.57, C = 4883.10$ for the deuterated form. Five sets of lines due to excited vibrational states have also been assigned. Preliminary Stark effect measurements indicate $\mu_{a} = 0.50D$ and $\mu_{b} =2.38D$ are the dipole moment components. en_US dc.format.extent 101203 bytes dc.format.mimetype image/jpeg dc.language.iso English en_US dc.publisher Ohio State University en_US dc.title THE MICROWAVE SPECTRUM OF GLYCOLALDEHYDE. en_US dc.type article en_US	2016-06-24 20:28:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.32432788610458374, "perplexity": 10335.99399869619}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783391519.0/warc/CC-MAIN-20160624154951-00108-ip-10-164-35-72.ec2.internal.warc.gz"}
https://www.quizover.com/online/course/9-3-use-properties-of-angles-triangles-and-the-pythagorean-by-openstax?page=3	# 9.3 Use properties of angles, triangles, and the pythagorean theorem (Page 4/15) Page 4 / 15 We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths. $\text{Δ}ABC$ and $\text{Δ}XYZ$ are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle. ## Solution Step 1. Read the problem. Draw the figure and label it with the given information. The figure is provided. Step 2. Identify what you are looking for. The length of the sides of similar triangles Step 3. Name. Choose a variable to represent it. Let a = length of the third side of $\Delta ABC$ y = length of the third side $\Delta XYZ$ Step 4. Translate. The triangles are similar, so the corresponding sides are in the same ratio. So $\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}$ Since the side $AB=4$ corresponds to the side $XY=3$ , we will use the ratio $\frac{\mathrm{AB}}{\mathrm{XY}}=\frac{4}{3}$ to find the other sides. Be careful to match up corresponding sides correctly. Step 5. Solve the equation. Step 6. Check: Step 7. Answer the question. The third side of $\Delta ABC$ is 6 and the third side of $\Delta XYZ$ is 2.4. $\text{Δ}ABC$ is similar to $\text{Δ}XYZ.$ Find $a.$ 8 $\text{Δ}ABC$ is similar to $\text{Δ}XYZ.$ Find $y.$ 22.5 ## Use the pythagorean theorem The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around $500$ BCE. Remember that a right triangle has a $\text{90°}$ angle, which we usually mark with a small square in the corner. The side of the triangle opposite the $\text{90°}$ angle is called the hypotenuse , and the other two sides are called the legs . See [link] . The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. ## The pythagorean theorem In any right triangle $\text{Δ}ABC,$ ${a}^{2}+{b}^{2}={c}^{2}$ where $c$ is the length of the hypotenuse $a$ and $b$ are the lengths of the legs. To solve problems that use the Pythagorean Theorem, we will need to find square roots. In Simplify and Use Square Roots we introduced the notation $\sqrt{m}$ and defined it in this way: $\text{If}\phantom{\rule{0.2em}{0ex}}m={n}^{2},\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}\sqrt{m}=n\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}n\ge 0$ For example, we found that $\sqrt{25}$ is $5$ because ${5}^{2}=25.$ We will use this definition of square roots to solve for the length of a side in a right triangle. Use the Pythagorean Theorem to find the length of the hypotenuse. ## Solution Step 1. Read the problem. Step 2. Identify what you are looking for. the length of the hypotenuse of the triangle Step 3. Name. Choose a variable to represent it. Let $c=\text{the length of the hypotenuse}$ Step 4. Translate. Write the appropriate formula. Substitute. Step 5. Solve the equation. Step 6. Check: Step 7. Answer the question. The length of the hypotenuse is 5. Use the Pythagorean Theorem to find the length of the hypotenuse. 10 Use the Pythagorean Theorem to find the length of the hypotenuse. 17 Use the Pythagorean Theorem to find the length of the longer leg. ## Solution Step 1. Read the problem. Step 2. Identify what you are looking for. The length of the leg of the triangle Step 3. Name. Choose a variable to represent it. Let $b=\text{the leg of the triangle}$ Label side b Step 4. Translate. Write the appropriate formula. Substitute. Step 5. Solve the equation. Isolate the variable term. Use the definition of the square root. Simplify. Step 6. Check: Step 7. Answer the question. The length of the leg is 12. how to know photocatalytic properties of tio2 nanoparticles...what to do now it is a goid question and i want to know the answer as well Maciej Do somebody tell me a best nano engineering book for beginners? what is fullerene does it is used to make bukky balls are you nano engineer ? s. what is the Synthesis, properties,and applications of carbon nano chemistry Mostly, they use nano carbon for electronics and for materials to be strengthened. Virgil is Bucky paper clear? CYNTHIA so some one know about replacing silicon atom with phosphorous in semiconductors device? Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure. Harper Do you know which machine is used to that process? s. how to fabricate graphene ink ? for screen printed electrodes ? SUYASH What is lattice structure? of graphene you mean? Ebrahim or in general Ebrahim in general s. Graphene has a hexagonal structure tahir On having this app for quite a bit time, Haven't realised there's a chat room in it. Cied what is biological synthesis of nanoparticles what's the easiest and fastest way to the synthesize AgNP? China Cied types of nano material I start with an easy one. carbon nanotubes woven into a long filament like a string Porter many many of nanotubes Porter what is the k.e before it land Yasmin what is the function of carbon nanotubes? Cesar I'm interested in nanotube Uday what is nanomaterials and their applications of sensors. what is nano technology what is system testing? preparation of nanomaterial Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it... what is system testing what is the application of nanotechnology? Stotaw In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google Azam anybody can imagine what will be happen after 100 years from now in nano tech world Prasenjit after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments Azam name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world Prasenjit how hard could it be to apply nanotechnology against viral infections such HIV or Ebola? Damian silver nanoparticles could handle the job? Damian not now but maybe in future only AgNP maybe any other nanomaterials Azam Hello Uday I'm interested in Nanotube Uday this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15 Prasenjit can nanotechnology change the direction of the face of the world how did you get the value of 2000N.What calculations are needed to arrive at it Privacy Information Security Software Version 1.1a Good Berger describes sociologists as concerned with What is the expressiin for seven less than four times the number of nickels How do i figure this problem out. how do you translate this in Algebraic Expressions why surface tension is zero at critical temperature Shanjida I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason s. Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)= . After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?	2018-10-17 03:00:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 32, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7858358025550842, "perplexity": 960.0177951958808}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583510969.31/warc/CC-MAIN-20181017023458-20181017044958-00090.warc.gz"}
http://frontlines.comixiade.org/159rv4pg/4733ce-nadph2-is-also-known-as	{{courseNav.course.mDynamicIntFields.lessonCount}} lessons Similarly to the mammalian inner membrane bound mitochondrial or bacterial cytoplasmic β-oxidations, the peroxisomal system includes MFE which catalyzes the second and the third reactions of the pathway ().The simultaneous existence of MFE-1 (also known as L-bifunctional protein, LBP) and MFE-2 (D-bifunctional protein, DBP) in mammalian peroxisomes catalyzing the same set of reactions … NADPH is the major player involved in the light-independent reactions in chloroplasts where it is converted back to NADP+. The formation of organic molecules from carbon dioxide is called... carbon fixation. Ordinary C-3 plants form a 3-carbon compound called … This is because with a ribose sugar and phosphate group Nick became a nucleotide, since a nucleotide is a sugar, phosphate, and nitrogen base. Specifically, it can act as an electron carrier for a group of enzymes known as dehydrogenases that often catalyze oxidation-reduction reactions (or redox reactions). Just like respiration O 2 use and CO 2 release in some step of photorespiration. such as oxygen in the photosynthesis reaction When an H is present, NADH and NADPH are in a reduced form because whenever a molecule receives a hydrogen or electron it's said, in chemistry, to have been reduced. © copyright 2003-2021 Study.com. New questions in Biology. When Nick visits cells after being absorbed into them by the stomach, he likes to borrow a ribose and phosphate from phosphoribosyl pyrophosphate, more often called by the acronym PRPP. NADPH oxidase 2 (Nox2), also known as cytochrome b(558) subunit beta or Cytochrome b-245 heavy chain, is a protein that in humans is encoded by the NOX2 gene (also called CYBB gene). Enrolling in a course lets you earn progress by passing quizzes and exams. What is generic jojo game (Also known as GJJG) Generic Jojo's Bizarre Adventure is a game IN DEVELOPMENT based on the Jojo's Bizarre Adventure Series, but the stands take a Feminine side. Mitochondria: Site of cellular respiration, generating ATP from the catabolism of sugars, fats, etc. NADPH stands for nicotinamide adenine dinucleotide phosphate hydrogen. These twins had characteristics of both Atty and Nick in his nicotinate ribonucleotide form. The addition of a hydrogen creates NADPH, the reduced form of NADP+. So the twin derivatives became known mostly by their acronyms (since their names were long and arduous): NAD+ (nicotinamide adenine dinucleotide) and NADP+ (nicotinamide adenine dinucleotide phosphate). Have their own ribosomes. This lesson will explore NADPH--where it comes from, what it means, what it looks like, and how it works. NADPH is an electron carrier. It is the reason why C4 plants are also known as cool-season or temperate plants. First, let's learn about Nick. Conflict Between Antigone & Creon in Sophocles' Antigone, Quiz & Worksheet - Desiree's Baby Time & Place, Quiz & Worksheet - Metaphors in The Outsiders, Quiz & Worksheet - The Handkerchief in Othello. Niacin (vitamin B3) is changed in the cell to nicotinate ribonucleotide, where it combines with ATP and an amide group is added to form NADP+ (nicotinamide adenine dinucleotide phosphate). It helps your muscles to work, which lets you run, walk, sit, stand, and breathe. Darla has taught undergraduate Enzyme Kinetics and has a doctorate in Basic Medical Science. first two years of college and save thousands off your degree. Often, NADPH can trade places with its twin NADH during nitrogen fixation. Public users are able to search the site and view the abstracts and keywords for each book and chapter without a subscription. The only difference was that one twin had a phosphate group (PO4) instead of an alcohol (OH) group. Nicotinamide Adenine Dinucleotide (NAD+) is a coenzyme present in biological systems. These reactions requires 12 NADPH2 and 18 ATP molecules to synthesis a one glucose molecule..... HOPE IT HELPS!!! - Structure, Composition & Facts, Quiz & Worksheet - History & Branches of Chemistry, DNA Replication - Processes and Steps: Help and Review, The Transcription and Translation Process: Help and Review, California Sexual Harassment Refresher Course: Supervisors, California Sexual Harassment Refresher Course: Employees. Define also known as. Alcohol dehydrogenase [NADP+] also known as aldehyde reductase or aldo-keto reductase family 1 member A1 is an enzyme that in humans is encoded by the AKR1A1 gene. The enzyme, RUBISCO, acts as a catalyst in this reaction. All other trademarks and copyrights are the property of their respective owners. Is also known as the suicide sac because the leakage of its enzymes (that function best at a pH of 5) can destroy a cell through auto digestion. The first stable compound in C4 cycle is 4-carbon compound. All Rights Reserved. B. Their main job in the cell is to shuttle electrons around. Thus, both were called nicotinamide adenine dinucleotide. 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(If they lose a hydrogen or electron, the molecule is oxidized.). Once upon a time, Nick and Atty met up in the cell after Nick had borrowed a ribose and phosphate from PRPP and when Atty was waiting around to provide the cell with some extra energy. In the dark reactions of photosynthesis (also known as the Calvin Cycle), carbon dioxide (CO 2) is converted into glucose through a series of complicated reactions involving ATP (adenosine triphosphate) and NADPH 2 (nicotinamide adenine dinucleotide phosphate), two essential compounds synthesized during the light reactions of daylight. a. NADP + ; Reduced b. NADPH; Oxidized c. NAD + ; Reduced d. NAD + ; Oxidized Question. ATP and NADPH2 from the light reactions are used to convert 3-phosphoglycerate to glyceraldehyde 3-phosphate, the three-carbon carbohydrate precursor to glucose and other sugars. NADPH is important in the formation of: Get access risk-free for 30 days, It was as nicotinate ribonucleotide that Nick met Atty. In terms of period of occurrence, light reactions only take place during the day when there is light from the sun. just create an account. ATP is a very important molecule in the body. This enzyme was formerly classified as EC 1.1.1.171. They got an adenine from Atty, so their middle name was adenine, and because both were ultimately composed of 2 nucleotides (one from Atty and one from Nick as nicotinate ribonucleotide) their last names were dinucleotide. lessons in math, English, science, history, and more. The topic of interstitial cystitis (IC), also known as painful bladder syndrome (PBS), and folate/one carbon metabolism has previously been unaddressed in research. ), A Dictionary of Food and Nutrition (3 ed. Homocysteine is also known to increase oxidative stress, disrupt cross-linking of collagen molecules, and increase levels of advanced glycation end products, … High serum homocysteine has been shown to have detrimental effects on neural cells, vascular endothelial cells, osteoblasts, and osteoclasts. Try refreshing the page, or contact customer support. Create your account. NADPH is used in biosynthesis. Log in here for access. alternative name for nicotinamide‐adenine dinucleotide phosphate (reduced), the oxidized form being NADP. NADPH oxidase (nicotinamide adenine dinucleotide phosphate oxidase) is a membrane-bound enzyme complex that faces the extracellular space. Log in or sign up to add this lesson to a Custom Course. Like many almost identical twins, NAD+ and NADP+ can take each other's place with almost no other cellular component the wiser. Plus, get practice tests, quizzes, and personalized coaching to help you The C4 plants are much more efficient in utilizing nitrogen and gathering carbon dioxide from the soil and atmosphere. succeed. Thus NADPH is a reducing agent (meaning that it has antioxidant properties) that can protect the cell membrane and other cellular structures from becoming oxidized. How the products of the light reactions, ATP and NADPH, are used to fix carbon into sugars in the second stage of photosynthesis. Consider the following reaction: NADH+FAD+H^{+}\rightarrow NAD^{+}+FADH_{2} Which of the following statements is correct? It is sometimes called molecular currency because it is used in many different processes such as cellular respiration and fermentation. These electrons are given in the form of a hydride ion (H–), a hydrogen … Both function as electron carriers. (banner made by Yamete_Sama) Checklist. Do anabolic pathways use NADP+ to remove electrons? (c) Copyright Oxford University Press, 2021. Visit the AP Biology: Help and Review page to learn more. and career path that can help you find the school that's right for you. Adv. What is the Main Frame Story of The Canterbury Tales? \| {{course.flashcardSetCount}} All rights reserved. The internal membrane of the chloroplast is also known as... thylakoid membrane. They were named after both Nick and Atty. Actions. You could not be signed in, please check and try again. Decisions Revisited: Why Did You Choose a Public or Private College? Do you choose to FIGHT or SIMP? Please subscribe or login to access full text content. Methylenetetrahydrofolate Reductase (NADPH2): A flavoprotein amine oxidoreductase that catalyzes the reversible conversion of 5-methyltetrahydrofolate to 5,10-methylenetetrahydrofolate. All these NAD+, NADH and NADPH are important co-factors in biological reactions. Speaking of enzymes (proteins that speed up chemical reactions), NADPH also acts as a co-enzyme. A. NADH is the oxidant. NADH is the reduced form of NAD+. Now despite being very similar, NAD+ and NADP+ are used very differently by the cells. {{courseNav.course.topics.length}} chapters \| Flashcards - Real Estate Marketing Basics, Flashcards - Promotional Marketing in Real Estate, Common Core Worksheets \| Printable Math & English Worksheets. NADPH is used in the biosynthesis (production) of lipids (fatty acids and cholesterols), neurotransmitters, nucleotides and amino acids. Create an account to start this course today. ). How Do I Use Study.com's Assign Lesson Feature? NADPH: Abbreviation for nicotinamide adenine dinucleotide phosphate (reduced form). But oxidants can be used by the cell as a signaling molecule, so it's not all bad. This results in the formation of 2 molecules of 3-phosphoglyceric acid. Hatch Slack Cycle or C4 Cycle: This is an alternative to the Calvin cycle that occurs during dark reaction in photosynthesis. credit by exam that is accepted by over 1,500 colleges and universities. Nicotinamide adenine dinucleotide phosphate, abbreviated NADP+ or, in older notation, TPN (triphosphopyridine nucleotide), is a cofactor used in anabolic reactions, such as the Calvin cycle and lipid and nucleic acid syntheses, which require NADPH as a reducing agent. courses that prepare you to earn Not a part of the endomembrane system. So it is possible that one hydrogen gets added in the reduction, but the other hydrogen is donated in the next oxidation, and therefore deuterium tracing studies of NAD(P)H will give both +1 and +2 deuterium isotopes. Photorespiration (also known as the oxidative photosynthetic carbon cycle, or C2 photosynthesis) is a process in plant metabolism which attempts to … Vitamin B3 is niacin. Thus, both were called nicotinamide adenine dinucleotide. NADPH is a product of the first stage of photosynthesis and is used to help fuel the reactions that take place in the second stage of photosynthesis. Photosynthetic pigments in the chloroplast are organized into distinct clusters known as... photosystems. This molecule plays a crucial role in some of the chemical reactions that make up the process of photosynthesis. For questions on access or troubleshooting, please check our FAQs, and if you can''t find the answer there, please contact us. credit-by-exam regardless of age or education level. An error occurred trying to load this video. These can be perennial or annual.The perfect temperature to grow for these plants is 90-95°F. Water is also broken apart in this process so the electrons can … Thus, NAD+ becomes NADH and NADP+ becomes NADPH. ... Access to the complete content on Oxford Reference requires a subscription or purchase. It works as a reducing agent in lipid and nucleic acid synthesis. This gene encodes a member of the aldo/keto reductase superfamily, which consists of more than 40 known enzymes and proteins. This narrative review highlights a potential connection for those with mast cell-related IC and histamine-mediated pain that is explore … It is also known as Hatch Slack pathway. Light Reaction (also known as light dependent reaction) The light reaction uses chlorophyll (which is the main photosynthetic pigment) to capture light, and then uses the light energy to make ATP and NADPH. They play a vital role in enzyme-catalyzed metabolic r… 2. The protein content is low as compared to C3 plants. imaginable degree, area of According to research, about 85% of plants on the universe are C3 plants. But two different molecules can't have the same name, even if they were derived from the … But they both refer tot he same thing, the reduced form of NADP+... Hope it helps in clearing your doubt. Has the main job of carrying around electrons for use by the cells,... 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Some metabolic reactions the body and Atty RuBisCo shows the Oxygense activity at day then. Printable Math & English Worksheets NADH, and source of electrons in nitrogen fixation of chemical. From which of the chemical reactions that make up the process of photosynthesis of carrier.. Or education level dioxide from the catabolism of sugars, fats, etc species ROS... Preferred except in cases where the... access to the Calvin cycle that occurs during dark reaction in,!, Pyrosequencing derives its name from which of the following as well distinguish. Noun that means anything produced in the formation of: get access risk-free 30... Can act as an antioxidant, co-enzyme, and personalized coaching to help you succeed to access full content... Get access risk-free for 30 days, just create an account add this lesson you must a... Other names like Nicotinic acid, niacin or vitamin B3 cases where...! Where the... access to the Calvin cycle that occurs during dark reaction in photosynthesis use! 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Marketing in Real Estate Marketing Basics, flashcards - Promotional Marketing in Real Estate, Common Core \|! Soil and atmosphere property of their respective owners forms reactive oxygen species ( ROS ) info you need find... Is 90-95°F requires a subscription or purchase membrane of the following superfamily, which lets you earn progress passing. All these NAD+, NADH and NADP+ can take each other 's place almost... 'S place with almost no other cellular component the wiser similar, becomes. One more thing they both like to carry, and that 's hydrogen be! Hatch Slack cycle or C4 cycle is 4-carbon compound … C4 plants are also known RuBisCo... Enzyme complex that faces the extracellular space both refer tot he same thing, the oxidized form being.... Had a phosphate group ( PO4 ) instead of an alcohol ( )... A course lets you run, walk, sit, stand, and they typically... Nadh during nitrogen fixation is important in the light-independent reactions, NAD+ and NADP+ can take other... Textbooks refer to NADPH addition, it is sometimes called Molecular currency because is... Of two molecules which we 'll follow the interactions of two molecules which we follow... A course lets you earn progress by passing quizzes and exams reactions ), the oxidized form being.... The addition of a hydrogen creates NADPH, the nadph2 is also known as form of NADP+ that make up process! Or purchase RuBisCo shows the Oxygense activity at day time then occurs allosteric for...: get access risk-free for 30 days, just create an account Molecular currency because it the! Ros ) places with its twin NADH, visit our Earning Credit page as RuBisCo and donor in light! Get practice tests, quizzes, and they are typically segregated into organelles and cytosol from to. Why C4 plants are also known as translation, English Dictionary definition also! And view the abstracts and keywords for each book and chapter without a subscription had phosphate! Atp is a very important molecule in the course of making another thing addition, it is converted to! Oh ) group Story of the following pigments in the Crucible molecules ca n't have the parent..., generating ATP from the same name, even if they lose a hydrogen or,! Another thing please check and try again electrons for use by the cell is to electrons.	2021-09-21 12:10:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30275073647499084, "perplexity": 7800.1280754284235}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057202.68/warc/CC-MAIN-20210921101319-20210921131319-00529.warc.gz"}
https://www.physicsforums.com/threads/question-about-this-proof.530242/	1. Sep 14, 2011 flyingpig 1. The problem statement, all variables and given/known data Prove that for all integers $$n \geq 1$$, one has $$1 + 2 + ... + n = \frac{n(n+1)}{2}$$ (1) S(1) = 1, true (2) Let n = k + 1 $$1 + 2 + ... + k + (k + 1) = \frac{(k+`1)(k + 2)}{2}$$ 3. The attempt at a solution Why is the last series $$1 + 2 + ... + k + (k +1)$$ instead of $$1 + 2 +...+ (k + 1)$$? 2. Sep 14, 2011 rock.freak667 Because the (k+1)th term to each side of the equation, which happens to be 'k+1'. The right side simplifies to (k+1)(k+2)/2 3. Sep 14, 2011 Staff: Mentor You're skipping a step here, again. There are three things you have to establish in an induction proof: 1) Base case (typically for n = 1) 2) The induction hypothesis - you assume that the statement is true for n = k 3) The induction step (I think that's what it's called) - you use the statement for n = k to show that the statement is also true for n = k + 1. These two are exactly the same. Each one represents the sum of the integers from 1 through k + 1. The first expression explicitly shows k, and the other one doesn't, but we can infer that the second expression doesn't skip from k - 1 to k + 1 in the sum. 4. Sep 14, 2011 flyingpig Is it bad that I don't show it? 5. Sep 14, 2011 Staff: Mentor If your professor is a stickler, or if he/she isn't convinced that you know what it should be, it is. In an induction proof, you should ALWAYS write down your induction hypothesis.	2018-01-22 23:10:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7033513784408569, "perplexity": 447.93351304460526}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084891543.65/warc/CC-MAIN-20180122213051-20180122233051-00047.warc.gz"}
https://socratic.org/questions/how-do-you-simplify-the-square-root-of-1-96-without-a-calculator	# How do you simplify the square root of 1.96 without a calculator? ##### 1 Answer Jun 18, 2018 See a solution process below: #### Explanation: We can rewrite this expression as: $\sqrt{1.96} \implies \sqrt{\frac{100}{100} \times 1.96} \implies \sqrt{\frac{196}{100}} \implies \frac{\sqrt{196}}{\sqrt{100}} = \frac{\sqrt{196}}{10}$ Next, we can rewrite the number under the radical as: $\frac{\sqrt{196}}{10} \implies \frac{\sqrt{4 \times 49}}{10} \implies \frac{\sqrt{4} \times \sqrt{49}}{10} \implies \frac{2 \times 7}{10} \implies \frac{14}{10} \implies 1.4$	2020-08-14 00:42:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 2, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9543595910072327, "perplexity": 2101.9896069218566}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439739104.67/warc/CC-MAIN-20200813220643-20200814010643-00428.warc.gz"}
https://www.computer.org/csdl/trans/tc/2000/07/t0727-abs.html	<p><b>Abstract</b>—A very-high radix algorithm and implementation for circular CORDIC is presented. We first present in depth the algorithm for the vectoring mode in which the selection of the digits is performed by rounding of the control variable. To assure convergence with this kind of selection, the operands are prescaled. However, in the CORDIC algorithm, the coordinate <tmath>$x$</tmath> varies during the execution so several scalings might be needed; we show that two scalings are sufficient. Moreover, the compensation of the variable scale factor (including the CORDIC scale factor and the prescaling factors) is done by computing the logarithm of the scale factor and performing the compensation by an exponential. Then, we combine, in a unified unit, the proposed vectoring algorithm and the very-high radix rotation algorithm, which was previously proposed by the authors. We compare with low-radix implementations in terms of latency and hardware complexity. Estimations of the delay for 32-bit precision show a speedup of about two with respect to the radix-4 case with redundant addition. This speedup is obtained at the cost of an increase in the hardware complexity, which is moderate for the pipelined implementation. We also compare at the algorithmic level with other very-high radix proposals, demonstrating the advantages of our algorithms.</p>	2018-06-24 05:15:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38528960943222046, "perplexity": 970.9346241508082}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267866358.52/warc/CC-MAIN-20180624044127-20180624064127-00004.warc.gz"}
http://adam.chlipala.net/cpdt/repo/rev/f8bcd33bdd91?revcount=240	### changeset 215:f8bcd33bdd91 Port DataStruct author Adam Chlipala Wed, 11 Nov 2009 14:00:04 -0500 768889c969e9 d1464997078d Makefile src/DataStruct.v src/DepList.v src/Generic.v 4 files changed, 116 insertions(+), 111 deletions(-) [+] line wrap: on line diff --- a/Makefile Wed Nov 11 12:21:28 2009 -0500 +++ b/Makefile Wed Nov 11 14:00:04 2009 -0500 @@ -7,7 +7,6 @@ MODULES := $(MODULES_NODOC)$(MODULES_DOC) VS := $(MODULES:%=src/%.v) VS_DOC :=$(MODULES_DOC:%=%.v) -GLOBALS := .coq_globals TEMPLATES := $(MODULES_CODE:%=templates/%.v) .PHONY: coq clean doc dvi html templates install cpdt.tgz @@ -17,13 +16,13 @@ Makefile.coq: Makefile$(VS) coq_makefile $(VS) \ - COQC = "coqc -I src -dump-glob$(GLOBALS)" \ + COQC = "coqc -I src" \ COQDEP = "coqdep -I src" \ -o Makefile.coq clean:: Makefile.coq make -f Makefile.coq clean - rm -f Makefile.coq .depend $(GLOBALS) cpdt.tgz \ + rm -f Makefile.coq .depend cpdt.tgz \ latex/.sty latex/cpdt. templates/.v rm -f .aux .dvi .log @@ -51,7 +50,6 @@ html: Makefile$(VS) src/toc.html mkdir -p html cd src ; coqdoc --interpolate $(VS_DOC) \ - --glob-from ../$(GLOBALS) \ -d ../html cp src/toc.html html/ --- a/src/DataStruct.v Wed Nov 11 12:21:28 2009 -0500 +++ b/src/DataStruct.v Wed Nov 11 14:00:04 2009 -0500 @@ -1,4 +1,4 @@ * * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 @@ -32,22 +32,22 @@ \| Nil : ilist O \| Cons : forall n, A -> ilist n -> ilist (S n). - (** We might like to have a certified function for selecting an element of an [ilist] by position. We could do this using subset types and explicit manipulation of proofs, but dependent types let us do it more directly. It is helpful to define a type family [index], where [index n] is isomorphic to [{m : nat \| m < n}]. Such a type family is also often called [Fin] or similar, standing for "finite." ) + (* We might like to have a certified function for selecting an element of an [ilist] by position. We could do this using subset types and explicit manipulation of proofs, but dependent types let us do it more directly. It is helpful to define a type family [fin], where [fin n] is isomorphic to [{m : nat \| m < n}]. The type family names stands for "finite." ) ( EX: Define a function [get] for extracting an [ilist] element by position. ) ( begin thide ) - Inductive index : nat -> Set := - \| First : forall n, index (S n) - \| Next : forall n, index n -> index (S n). + Inductive fin : nat -> Set := + \| First : forall n, fin (S n) + \| Next : forall n, fin n -> fin (S n). - (* [index] essentially makes a more richly-typed copy of the natural numbers. Every element is a [First] iterated through applying [Next] a number of times that indicates which number is being selected. + (** [fin] essentially makes a more richly-typed copy of the natural numbers. Every element is a [First] iterated through applying [Next] a number of times that indicates which number is being selected. Now it is easy to pick a [Prop]-free type for a selection function. As usual, our first implementation attempt will not convince the type checker, and we will attack the deficiencies one at a time. [[ - Fixpoint get n (ls : ilist n) {struct ls} : index n -> A := - match ls in ilist n return index n -> A with + Fixpoint get n (ls : ilist n) : fin n -> A := + match ls with \| Nil => fun idx => ? \| Cons _ x ls' => fun idx => match idx with @@ -58,13 +58,13 @@ ]] - We apply the usual wisdom of delaying arguments in [Fixpoint]s so that they may be included in [return] clauses. This still leaves us with a quandary in each of the [match] cases. First, we need to figure out how to take advantage of the contradiction in the [Nil] case. Every [index] has a type of the form [S n], which cannot unify with the [O] value that we learn for [n] in the [Nil] case. The solution we adopt is another case of [match]-within-[return]. + We apply the usual wisdom of delaying arguments in [Fixpoint]s so that they may be included in [return] clauses. This still leaves us with a quandary in each of the [match] cases. First, we need to figure out how to take advantage of the contradiction in the [Nil] case. Every [fin] has a type of the form [S n], which cannot unify with the [O] value that we learn for [n] in the [Nil] case. The solution we adopt is another case of [match]-within-[return]. [[ - Fixpoint get n (ls : ilist n) {struct ls} : index n -> A := - match ls in ilist n return index n -> A with + Fixpoint get n (ls : ilist n) : fin n -> A := + match ls with \| Nil => fun idx => - match idx in index n' return (match n' with + match idx in fin n' return (match n' with \| O => A \| S _ => unit end) with @@ -80,13 +80,13 @@ ]] - Now the first [match] case type-checks, and we see that the problem with the [Cons] case is that the pattern-bound variable [idx'] does not have an apparent type compatible with [ls']. We need to use [match] annotations to make the relationship explicit. Unfortunately, the usual trick of postponing argument binding will not help us here. We need to match on both [ls] and [idx]; one or the other must be matched first. To get around this, we apply a trick that we will call "the convoy pattern," introducing a new function and applying it immediately, to satisfy the type checker. + Now the first [match] case type-checks, and we see that the problem with the [Cons] case is that the pattern-bound variable [idx'] does not have an apparent type compatible with [ls']. We need to use [match] annotations to make the relationship explicit. Unfortunately, the usual trick of postponing argument binding will not help us here. We need to match on both [ls] and [idx]; one or the other must be matched first. To get around this, we apply the convoy pattern that we met last chapter. This application is a little more clever than those we saw before; we use the natural number predecessor function [pred] to express the relationship between the types of these variables. [[ - Fixpoint get n (ls : ilist n) {struct ls} : index n -> A := - match ls in ilist n return index n -> A with + Fixpoint get n (ls : ilist n) : fin n -> A := + match ls with \| Nil => fun idx => - match idx in index n' return (match n' with + match idx in fin n' return (match n' with \| O => A \| S _ => unit end) with @@ -94,7 +94,7 @@ \| Next _ _ => tt end \| Cons _ x ls' => fun idx => - match idx in index n' return ilist (pred n') -> A with + match idx in fin n' return ilist (pred n') -> A with \| First _ => fun _ => x \| Next _ idx' => fun ls' => get ls' idx' end ls' @@ -104,10 +104,10 @@ There is just one problem left with this implementation. Though we know that the local [ls'] in the [Next] case is equal to the original [ls'], the type-checker is not satisfied that the recursive call to [get] does not introduce non-termination. We solve the problem by convoy-binding the partial application of [get] to [ls'], rather than [ls'] by itself. ) - Fixpoint get n (ls : ilist n) {struct ls} : index n -> A := - match ls in ilist n return index n -> A with + Fixpoint get n (ls : ilist n) : fin n -> A := + match ls with \| Nil => fun idx => - match idx in index n' return (match n' with + match idx in fin n' return (match n' with \| O => A \| S _ => unit end) with @@ -115,7 +115,7 @@ \| Next _ _ => tt end \| Cons _ x ls' => fun idx => - match idx in index n' return (index (pred n') -> A) -> A with + match idx in fin n' return (fin (pred n') -> A) -> A with \| First _ => fun _ => x \| Next _ idx' => fun get_ls' => get_ls' idx' end (get ls') @@ -131,27 +131,26 @@ (* A few examples show how to make use of these definitions. ) Check Cons 0 (Cons 1 (Cons 2 Nil)). -(* [[ - -Cons 0 (Cons 1 (Cons 2 Nil)) +(** %\vspace{-.15in}% [[ + Cons 0 (Cons 1 (Cons 2 Nil)) : ilist nat 3 ]] ) + ( begin thide ) Eval simpl in get (Cons 0 (Cons 1 (Cons 2 Nil))) First. -(* [[ - +(** %\vspace{-.15in}% [[ = 0 : nat ]] ) + Eval simpl in get (Cons 0 (Cons 1 (Cons 2 Nil))) (Next First). -(* [[ - +(** %\vspace{-.15in}% [[ = 1 : nat ]] ) + Eval simpl in get (Cons 0 (Cons 1 (Cons 2 Nil))) (Next (Next First)). -(* [[ - +(** %\vspace{-.15in}% [[ = 2 : nat ]] ) @@ -163,15 +162,15 @@ Variables A B : Set. Variable f : A -> B. - Fixpoint imap n (ls : ilist A n) {struct ls} : ilist B n := - match ls in ilist _ n return ilist B n with + Fixpoint imap n (ls : ilist A n) : ilist B n := + match ls with \| Nil => Nil \| Cons _ x ls' => Cons (f x) (imap ls') end. (* It is easy to prove that [get] "distributes over" [imap] calls. The only tricky bit is remembering to use the [dep_destruct] tactic in place of plain [destruct] when faced with a baffling tactic error message. ) - Theorem get_imap : forall n (idx : index n) (ls : ilist A n), + Theorem get_imap : forall n (idx : fin n) (ls : ilist A n), get (imap ls) idx = f (get ls idx). ( begin thide ) induction ls; dep_destruct idx; crush. @@ -182,7 +181,7 @@ (* * Heterogeneous Lists ) -(* Programmers who move to statically-typed functional languages from "scripting languages" often complain about the requirement that every element of a list have the same type. With fancy type systems, we can partially lift this requirement. We can index a list type with a "type-level" list that explains what type each element of the list should have. This has been done in a variety of ways in Haskell using type classes, and it we can do it much more cleanly and directly in Coq. ) +(* Programmers who move to statically-typed functional languages from "scripting languages" often complain about the requirement that every element of a list have the same type. With fancy type systems, we can partially lift this requirement. We can index a list type with a "type-level" list that explains what type each element of the list should have. This has been done in a variety of ways in Haskell using type classes, and we can do it much more cleanly and directly in Coq. ) Section hlist. Variable A : Type. @@ -213,8 +212,8 @@ We can use [member] to adapt our definition of [get] to [hlists]. The same basic [match] tricks apply. In the [MCons] case, we form a two-element convoy, passing both the data element [x] and the recursor for the sublist [mls'] to the result of the inner [match]. We did not need to do that in [get]'s definition because the types of list elements were not dependent there. ) - Fixpoint hget ls (mls : hlist ls) {struct mls} : member ls -> B elm := - match mls in hlist ls return member ls -> B elm with + Fixpoint hget ls (mls : hlist ls) : member ls -> B elm := + match mls with \| MNil => fun mem => match mem in member ls' return (match ls' with \| nil => B elm @@ -254,14 +253,13 @@ MCons 5 (MCons true MNil). Eval simpl in hget someValues MFirst. -( [[ - +( %\vspace{-.15in}% [[ = 5 : (fun T : Set => T) nat ]] ) + Eval simpl in hget someValues (MNext MFirst). -(* [[ - +(** %\vspace{-.15in}% [[ = true : (fun T : Set => T) bool ]] ) @@ -288,7 +286,6 @@ Inductive exp : list type -> type -> Set := \| Const : forall ts, exp ts Unit - ( begin thide ) \| Var : forall ts t, member t ts -> exp ts t \| App : forall ts dom ran, exp ts (Arrow dom ran) -> exp ts dom -> exp ts ran @@ -310,8 +307,8 @@ ( EX: Define an interpreter for [exp]s. ) ( begin thide ) -Fixpoint expDenote ts t (e : exp ts t) {struct e} : hlist typeDenote ts -> typeDenote t := - match e in exp ts t return hlist typeDenote ts -> typeDenote t with +Fixpoint expDenote ts t (e : exp ts t) : hlist typeDenote ts -> typeDenote t := + match e with \| Const _ => fun _ => tt \| Var _ _ mem => fun s => hget s mem @@ -322,33 +319,32 @@ (* Like for previous examples, our interpreter is easy to run with [simpl]. ) Eval simpl in expDenote Const MNil. -(* [[ - +(** %\vspace{-.15in}% [[ = tt : typeDenote Unit ]] ) + Eval simpl in expDenote (Abs (dom := Unit) (Var MFirst)) MNil. -(* [[ - +(** %\vspace{-.15in}% [[ = fun x : unit => x : typeDenote (Arrow Unit Unit) ]] ) + Eval simpl in expDenote (Abs (dom := Unit) (Abs (dom := Unit) (Var (MNext MFirst)))) MNil. -(* [[ - +(** %\vspace{-.15in}% [[ = fun x _ : unit => x : typeDenote (Arrow Unit (Arrow Unit Unit)) ]] ) + Eval simpl in expDenote (Abs (dom := Unit) (Abs (dom := Unit) (Var MFirst))) MNil. -(* [[ - +(** %\vspace{-.15in}% [[ = fun _ x0 : unit => x0 : typeDenote (Arrow Unit (Arrow Unit Unit)) ]] ) + Eval simpl in expDenote (App (Abs (Var MFirst)) Const) MNil. -(* [[ - +(** %\vspace{-.15in}% [[ = tt : typeDenote Unit ]] ) @@ -376,16 +372,16 @@ (* We say that a list of length 0 has no contents, and a list of length [S n'] is a pair of a data value and a list of length [n']. ) - Fixpoint findex (n : nat) : Set := + Fixpoint ffin (n : nat) : Set := match n with \| O => Empty_set - \| S n' => option (findex n') + \| S n' => option (ffin n') end. (* We express that there are no index values when [n = O], by defining such indices as type [Empty_set]; and we express that, at [n = S n'], there is a choice between picking the first element of the list (represented as [None]) or choosing a later element (represented by [Some idx], where [idx] is an index into the list tail). ) - Fixpoint fget (n : nat) : filist n -> findex n -> A := - match n return filist n -> findex n -> A with + Fixpoint fget (n : nat) : filist n -> ffin n -> A := + match n with \| O => fun _ idx => match idx with end \| S n' => fun ls idx => match idx with @@ -394,8 +390,9 @@ end end. - (* Our new [get] implementation needs only one dependent [match], which just copies the stated return type of the function. Our choices of data structure implementations lead to just the right typing behavior for this new definition to work out. ) + (* Our new [get] implementation needs only one dependent [match], and its annotation is inferred for us. Our choices of data structure implementations lead to just the right typing behavior for this new definition to work out. ) ( end thide ) + End filist. (* Heterogeneous lists are a little trickier to define with recursion, but we then reap similar benefits in simplicity of use. ) @@ -423,14 +420,13 @@ \| x :: ls' => (x = elm) + fmember ls' end%type. - (* The definition of [fmember] follows the definition of [findex]. Empty lists have no members, and member types for nonempty lists are built by adding one new option to the type of members of the list tail. While for [index] we needed no new information associated with the option that we add, here we need to know that the head of the list equals the element we are searching for. We express that with a sum type whose left branch is the appropriate equality proposition. Since we define [fmember] to live in [Type], we can insert [Prop] types as needed, because [Prop] is a subtype of [Type]. + (** The definition of [fmember] follows the definition of [ffin]. Empty lists have no members, and member types for nonempty lists are built by adding one new option to the type of members of the list tail. While for [index] we needed no new information associated with the option that we add, here we need to know that the head of the list equals the element we are searching for. We express that with a sum type whose left branch is the appropriate equality proposition. Since we define [fmember] to live in [Type], we can insert [Prop] types as needed, because [Prop] is a subtype of [Type]. We know all of the tricks needed to write a first attempt at a [get] function for [fhlist]s. [[ - Fixpoint fhget (ls : list A) : fhlist ls -> fmember ls -> B elm := - match ls return fhlist ls -> fmember ls -> B elm with + match ls with \| nil => fun _ idx => match idx with end \| _ :: ls' => fun mls idx => match idx with @@ -444,7 +440,7 @@ Only one problem remains. The expression [fst mls] is not known to have the proper type. To demonstrate that it does, we need to use the proof available in the [inl] case of the inner [match]. ) Fixpoint fhget (ls : list A) : fhlist ls -> fmember ls -> B elm := - match ls return fhlist ls -> fmember ls -> B elm with + match ls with \| nil => fun _ idx => match idx with end \| _ :: ls' => fun mls idx => match idx with @@ -458,13 +454,14 @@ (* By pattern-matching on the equality proof [pf], we make that equality known to the type-checker. Exactly why this works can be seen by studying the definition of equality. ) Print eq. - (* [[ - + (** %\vspace{-.15in}% [[ Inductive eq (A : Type) (x : A) : A -> Prop := refl_equal : x = x + ]] -In a proposition [x = y], we see that [x] is a parameter and [y] is a regular argument. The type of the constructor [refl_equal] shows that [y] can only ever be instantiated to [x]. Thus, within a pattern-match with [refl_equal], occurrences of [y] can be replaced with occurrences of [x] for typing purposes. All examples of similar dependent pattern matching that we have seen before require explicit annotations, but Coq implements a special case of annotation inference for matches on equality proofs. ) +In a proposition [x = y], we see that [x] is a parameter and [y] is a regular argument. The type of the constructor [refl_equal] shows that [y] can only ever be instantiated to [x]. Thus, within a pattern-match with [refl_equal], occurrences of [y] can be replaced with occurrences of [x] for typing purposes. ) (* end thide ) + End fhlist. Implicit Arguments fhget [A B elm ls]. @@ -489,7 +486,7 @@ Variable f : A -> B -> B. Variable i : B. - Fixpoint ifoldr n (ls : ilist A n) {struct ls} : B := + Fixpoint ifoldr n (ls : ilist A n) : B := match ls with \| Nil => i \| Cons _ x ls' => f x (ifoldr ls') @@ -520,21 +517,23 @@ ============================ ifoldr (fun (t' : tree nat) (n0 : nat) => sum t' + n0) 0 (imap inc i) >= ifoldr (fun (t' : tree nat) (n0 : nat) => sum t' + n0) 0 i + ]] We are left with a single subgoal which does not seem provable directly. This is the same problem that we met in Chapter 3 with other nested inductive types. ) Check tree_ind. - ( [[ - -tree_ind + ( %\vspace{-.15in}% [[ + tree_ind : forall (A : Set) (P : tree A -> Prop), (forall a : A, P (Leaf a)) -> (forall (n : nat) (i : ilist (tree A) n), P (Node i)) -> forall t : tree A, P t + ]] The automatically-generated induction principle is too weak. For the [Node] case, it gives us no inductive hypothesis. We could write our own induction principle, as we did in Chapter 3, but there is an easier way, if we are willing to alter the definition of [tree]. ) + Abort. Reset tree. @@ -544,42 +543,39 @@ Section tree. Variable A : Set. - (* [[ - + (** %\vspace{-.15in}% [[ Inductive tree : Set := \| Leaf : A -> tree \| Node : forall n, filist tree n -> tree. - ]] - -[[ - Error: Non strictly positive occurrence of "tree" in "forall n : nat, filist tree n -> tree" + ]] The special-case rule for nested datatypes only works with nested uses of other inductive types, which could be replaced with uses of new mutually-inductive types. We defined [filist] recursively, so it may not be used for nested recursion. - Our final solution uses yet another of the inductive definition techniques introduced in Chapter 3, reflexive types. Instead of merely using [index] to get elements out of [ilist], we can %\textit{%#<i>#define#</i>#%}% [ilist] in terms of [index]. For the reasons outlined above, it turns out to be easier to work with [findex] in place of [index]. ) + Our final solution uses yet another of the inductive definition techniques introduced in Chapter 3, reflexive types. Instead of merely using [fin] to get elements out of [ilist], we can %\textit{%#<i>#define#</i>#%}% [ilist] in terms of [fin]. For the reasons outlined above, it turns out to be easier to work with [ffin] in place of [fin]. ) Inductive tree : Set := \| Leaf : A -> tree - \| Node : forall n, (findex n -> tree) -> tree. + \| Node : forall n, (ffin n -> tree) -> tree. - (** A [Node] is indexed by a natural number [n], and the node's [n] children are represented as a function from [findex n] to trees, which is isomorphic to the [ilist]-based representation that we used above. ) + (* A [Node] is indexed by a natural number [n], and the node's [n] children are represented as a function from [ffin n] to trees, which is isomorphic to the [ilist]-based representation that we used above. ) + End tree. Implicit Arguments Node [A n]. -(* We can redefine [sum] and [inc] for our new [tree] type. Again, it is useful to define a generic fold function first. This time, it takes in a function whose range is some [findex] type, and it folds another function over the results of calling the first function at every possible [findex] value. ) +(* We can redefine [sum] and [inc] for our new [tree] type. Again, it is useful to define a generic fold function first. This time, it takes in a function whose range is some [ffin] type, and it folds another function over the results of calling the first function at every possible [ffin] value. ) Section rifoldr. Variables A B : Set. Variable f : A -> B -> B. Variable i : B. - Fixpoint rifoldr (n : nat) : (findex n -> A) -> B := - match n return (findex n -> A) -> B with + Fixpoint rifoldr (n : nat) : (ffin n -> A) -> B := + match n with \| O => fun _ => i \| S n' => fun get => f (get None) (rifoldr n' (fun idx => get (Some idx))) end. @@ -608,7 +604,7 @@ crush. Qed. -Lemma sum_inc' : forall n (f1 f2 : findex n -> nat), +Lemma sum_inc' : forall n (f1 f2 : ffin n -> nat), (forall idx, f1 idx >= f2 idx) -> rifoldr plus 0 f1 >= rifoldr plus 0 f2. Hint Resolve plus_ge. @@ -639,11 +635,11 @@ \| BConst : bool -> exp' Bool ( begin thide ) -\| Cond : forall n t, (findex n -> exp' Bool) - -> (findex n -> exp' t) -> exp' t -> exp' t. +\| Cond : forall n t, (ffin n -> exp' Bool) + -> (ffin n -> exp' t) -> exp' t -> exp' t. ( end thide ) -(* A [Cond] is parameterized by a natural [n], which tells us how many cases this conditional has. The test expressions are represented with a function of type [findex n -> exp' Bool], and the bodies are represented with a function of type [findex n -> exp' t], where [t] is the overall type. The final [exp' t] argument is the default case. +(** A [Cond] is parameterized by a natural [n], which tells us how many cases this conditional has. The test expressions are represented with a function of type [ffin n -> exp' Bool], and the bodies are represented with a function of type [ffin n -> exp' t], where [t] is the overall type. The final [exp' t] argument is the default case. We start implementing our interpreter with a standard type denotation function. ) @@ -660,8 +656,8 @@ Variable A : Set. Variable default : A. - Fixpoint cond (n : nat) : (findex n -> bool) -> (findex n -> A) -> A := - match n return (findex n -> bool) -> (findex n -> A) -> A with + Fixpoint cond (n : nat) : (ffin n -> bool) -> (ffin n -> A) -> A := + match n with \| O => fun _ _ => default \| S n' => fun tests bodies => if tests None @@ -677,17 +673,14 @@ (* Now the expression interpreter is straightforward to write. ) -Fixpoint exp'Denote t (e : exp' t) {struct e} : type'Denote t := - match e in exp' t return type'Denote t with - \| NConst n => - n - \| Plus e1 e2 => - exp'Denote e1 + exp'Denote e2 +Fixpoint exp'Denote t (e : exp' t) : type'Denote t := + match e with + \| NConst n => n + \| Plus e1 e2 => exp'Denote e1 + exp'Denote e2 \| Eq e1 e2 => if eq_nat_dec (exp'Denote e1) (exp'Denote e2) then true else false - \| BConst b => - b + \| BConst b => b \| Cond _ _ tests bodies default => ( begin thide ) cond @@ -705,8 +698,8 @@ Variable default : exp' t. Fixpoint cfoldCond (n : nat) - : (findex n -> exp' Bool) -> (findex n -> exp' t) -> exp' t := - match n return (findex n -> exp' Bool) -> (findex n -> exp' t) -> exp' t with + : (ffin n -> exp' Bool) -> (ffin n -> exp' t) -> exp' t := + match n with \| O => fun _ _ => default \| S n' => fun tests bodies => match tests None return _ with @@ -747,8 +740,8 @@ (* Like for the interpreters, most of the action was in this helper function, and [cfold] itself is easy to write. ) -Fixpoint cfold t (e : exp' t) {struct e} : exp' t := - match e in exp' t return exp' t with +Fixpoint cfold t (e : exp' t) : exp' t := + match e with \| NConst n => NConst n \| Plus e1 e2 => let e1' := cfold e1 in @@ -779,7 +772,7 @@ (* To prove our final correctness theorem, it is useful to know that [cfoldCond] preserves expression meanings. This lemma formalizes that property. The proof is a standard mostly-automated one, with the only wrinkle being a guided instantation of the quantifiers in the induction hypothesis. ) Lemma cfoldCond_correct : forall t (default : exp' t) - n (tests : findex n -> exp' Bool) (bodies : findex n -> exp' t), + n (tests : ffin n -> exp' Bool) (bodies : ffin n -> exp' t), exp'Denote (cfoldCond default tests bodies) = exp'Denote (Cond n tests bodies default). induction n; crush; @@ -802,8 +795,8 @@ (* It is also useful to know that the result of a call to [cond] is not changed by substituting new tests and bodies functions, so long as the new functions have the same input-output behavior as the old. It turns out that, in Coq, it is not possible to prove in general that functions related in this way are equal. We treat this issue with our discussion of axioms in a later chapter. For now, it suffices to prove that the particular function [cond] is %\textit{%#<i>#extensional#</i>#%}%; that is, it is unaffected by substitution of functions with input-output equivalents. ) -Lemma cond_ext : forall (A : Set) (default : A) n (tests tests' : findex n -> bool) - (bodies bodies' : findex n -> A), +Lemma cond_ext : forall (A : Set) (default : A) n (tests tests' : ffin n -> bool) + (bodies bodies' : ffin n -> A), (forall idx, tests idx = tests' idx) -> (forall idx, bodies idx = bodies' idx) -> cond default tests bodies @@ -831,6 +824,19 @@ ( end thide ) +(* * Choosing Between Representations ) + +(* It is not always clear which of these representation techniques to apply in a particular situation, but I will try to summarize the pros and cons of each. + + Inductive types are often the most pleasant to work with, after someone has spent the time implementing some basic library functions for them, using fancy [match] annotations. Many aspects of Coq's logic and tactic support are specialized to deal with inductive types, and you may miss out if you use alternate encodings. + + Recursive types usually involve much less initial effort, but they can be less convenient to use with proof automation. For instance, the [simpl] tactic (which is among the ingredients in [crush]) will sometimes be overzealous in simplifying uses of functions over recursive types. Consider a function [replace] of type [forall A, filist A n -> fin n -> A -> filist A n], such that [replace l f x] should substitute [x] for the element in position [f] of [l]. A call to [replace] on a variable [l] of type [filist A (S n)] would probably be simplified to an explicit pair, even though we know nothing about the structure of [l] beyond its type. In a proof involving many recursive types, this kind of unhelpful "simplification" can lead to rapid bloat in the sizes of subgoals. + + Another disadvantage of recursive types is that they only apply to type families whose indices determine their "skeletons." This is not true for all data structures; a good counterexample comes from the richly-typed programming language syntax types we have used several times so far. The fact that a piece of syntax has type [Nat] tells us nothing about the tree structure of that syntax. + + Reflexive encodings of data types are seen relatively rarely. As our examples demonstrated, manipulating index values manually can lead to hard-to-read code. A normal inductive type is generally easier to work with, once someone has gone through the trouble of implementing an induction principle manually with the techniques we studied in Chapter 3. For small developments, avoiding that kind of coding can justify the use of reflexive data structures. ) + + (* * Exercises ) (* remove printing * ) @@ -849,6 +855,7 @@ t ::= bool \| t + t p ::= x \| b \| inl p \| inr p e ::= x \| b \| inl e \| inr e \| case e of [p => e] \| _ => e + ]] [x] stands for a variable, and [b] stands for a boolean constant. The production for [case] expressions means that a pattern-match includes zero or more pairs of patterns and expressions, along with a default case. --- a/src/DepList.v Wed Nov 11 12:21:28 2009 -0500 +++ b/src/DepList.v Wed Nov 11 14:00:04 2009 -0500 @@ -31,14 +31,14 @@ Implicit Arguments icons [n]. - Fixpoint index (n : nat) : Type := + Fixpoint fin (n : nat) : Type := match n with \| O => Empty_set - \| S n' => option (index n') + \| S n' => option (fin n') end. - Fixpoint get (n : nat) : ilist n -> index n -> A := - match n return ilist n -> index n -> A with + Fixpoint get (n : nat) : ilist n -> fin n -> A := + match n return ilist n -> fin n -> A with \| O => fun _ idx => match idx with end \| S n' => fun ls idx => match idx with --- a/src/Generic.v Wed Nov 11 12:21:28 2009 -0500 +++ b/src/Generic.v Wed Nov 11 14:00:04 2009 -0500 @@ -217,7 +217,7 @@ Definition datatypeDenoteOk := forall P : T -> Prop, (forall c (m : member c dt) (x : nonrecursive c) (r : ilist T (recursive c)), - (forall i : index (recursive c), P (get r i)) + (forall i : fin (recursive c), P (get r i)) -> P ((hget dd m) x r)) -> forall v, P v.	2018-01-21 01:01:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5127370953559875, "perplexity": 8914.530638223316}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889798.67/warc/CC-MAIN-20180121001412-20180121021412-00466.warc.gz"}
https://stats.stackexchange.com/questions/1812/fa-choosing-rotation-matrix-based-on-simple-structure-criteria	# FA: Choosing Rotation matrix, based on “Simple Structure Criteria” One of the most important issues in using factor analysis is its interpretation. Factor analysis often uses factor rotation to enhance its interpretation. After a satisfactory rotation, the rotated factor loading matrix L' will have the same ability to represent the correlation matrix and it can be used as the factor loading matrix, instead of the unrotated matrix L. The purpose of rotation is to make the rotated factor loading matrix have some desirable properties. One of the methods used is to rotate the factor loading matrix such that the rotated matrix will have a simple structure. L. L. Thurstone introduced the Principle of Simple Structure, as a general guide for factor rotation: ## Simple Structure Criteria: 1. Each row of the factor matrix should contain at least one zero 2. If there are m common factors, each column of the factor matrix should have at least m zeros 3. For every pair of columns in the factor matrix, there should be several variables for which entries approach zero in the one column but not in the other 4. For every pair of columns in the factor matrix, a large proportion of the variables should have entries approaching zero in both columns when there are four or more factors 5. For every pair of columns in the factor matrix, there should be only a small number of variables with nonzero entries in both columns The ideal simple structure is such that: 1. each item has a high, or meaningful, loading on one factor only and 2. each factor have high, or meaningful, loadings for only some of the items. The problem is that, trying several combinations of rotation methods along with the parameters that each one accepts (especially for oblique ones), the number of candidate matrices increases and it is very difficult to see which one better meets the above criteria. When I first faced that problem I realized that I was unable to select the best match by merely 'looking' at them, and that I needed an algorithm to help me decide. Under the stress of project's deadlines, the most I could do was to write the following code in MATLAB, which accepts one rotation matrix at a time and returns (under some assumptions) whether each criterion is met or not. A new version (If I would ever tried to upgrade it) would accept a 3d matrix (a set of 2d matrices) as an argument, and the algorithm should return the one that better fits the above criteria. How would you extract an algorithm out of those criteria? I am just asking for your opinions (I also think that there's been criticism over the usefulness of the method by itself) and perhaps better approaches to the rotation matrix selection problem. Also, I would like to know what software do you prefer to perform FA. If it's R, what package do you use? (I must admit that if I had to do FA, I would turn to SPSS again). If someone wants to provide some code, I would prefer R or MATLAB. P.S. The above Simple Structure Criteria formulation can be found in the book "Making Sense of Factor Analysis" by PETT, M., LACKEY, N., SULLIVAN, J. P.S.2 (from the same book): "A test of successful factor analysis is the extent to which it can reproduce the original corr matrix. If you also used oblique solutions, among all select the one that generated the greatest number of highest and lowest factor loadings." This sounds like another constraint that the algorithm could use. P.S.3 This question has also been asked here. However, I think it fits better on this site. function [] = simple_structure_criteria (my_pattern_table) %Simple Structure Criteria %Making Sense of Factor Analysis, page 132 disp(' '); disp('Simple Structure Criteria (Thurstone):'); disp('1. Each row of the factor matrix should contain at least one zero'); disp( '2. If there are m common factors, each column of the factor matrix should have at least m zeros'); disp( '3. For every pair of columns in the factor matrix, there should be several variables for which entries approach zero in the one column but not in the other'); disp( '4. For every pair of columns in the factor matrix, a large proportion of the variables should have entries approaching zero in both columns when there are four or more factors'); disp( '5. For every pair of columns in the factor matrix, there should be only a small number of variables with nonzero entries in both columns'); disp(' '); disp( '(additional by Pedhazur and Schmelkin) The ideal simple structure is such that:'); disp( '6. Each item has a high, or meaningful, loading on one factor only and'); disp( '7. Each factor have high, or meaningful, loadings for only some of the items.'); disp('') disp('Start checking...') %test matrix %ct=[76,78,16,7;19,29,10,13;2,6,7,8]; %test it by giving: simple_structure_criteria (ct) ct=abs(my_pattern_table); items=size(ct,1); factors=size(ct,2); my_zero = 0.1; approach_zero = 0.2; several = floor(items / 3); small_number = ceil(items / 4); large_proportion = 0.30; meaningful = 0.4; some_bottom = 2; some_top = floor(items / 2); % CRITERION 1 disp(' '); disp('CRITERION 1'); for i = 1 : 1 : items count = 0; for j = 1 : 1 : factors if (ct(i,j) < my_zero) count = count + 1; break end end if (count == 0) disp(['Criterion 1 is NOT MET for item ' num2str(i)]) end end % CRITERION 2 disp(' '); disp('CRITERION 2'); for j = 1 : 1 : factors m=0; for i = 1 : 1 : items if (ct(i,j) < my_zero) m = m + 1; end end if (m < factors) disp(['Criterion 2 is NOT MET for factor ' num2str(j) '. m = ' num2str(m)]); end end % CRITERION 3 disp(' '); disp('CRITERION 3'); for c1 = 1 : 1 : factors - 1 for c2 = c1 + 1 : 1 : factors test_several = 0; for i = 1 : 1 : items if ( (ct(i,c1)>my_zero && ct(i,c2)<my_zero) \|\| (ct(i,c1)<my_zero && ct(i,c2)>my_zero) ) % approach zero in one but not in the other test_several = test_several + 1; end end disp(['several = ' num2str(test_several) ' for factors ' num2str(c1) ' and ' num2str(c2)]); if (test_several < several) disp(['Criterion 3 is NOT MET for factors ' num2str(c1) ' and ' num2str(c2)]); end end end % CRITERION 4 disp(' '); disp('CRITERION 4'); if (factors > 3) for c1 = 1 : 1 : factors - 1 for c2 = c1 + 1 : 1 : factors test_several = 0; for i = 1 : 1 : items if (ct(i,c1)<approach_zero && ct(i,c2)<approach_zero) % approach zero in both test_several = test_several + 1; end end disp(['large proportion = ' num2str((test_several / items)100) '% for factors ' num2str(c1) ' and ' num2str(c2)]); if ((test_several / items) < large_proportion) pr = sprintf('%4.2g', (test_several / items) 100 ); disp(['Criterion 4 is NOT MET for factors ' num2str(c1) ' and ' num2str(c2) '. Proportion is ' pr '%']); end end end end % CRITERION 5 disp(' '); disp('CRITERION 5'); for c1 = 1 : 1 : factors - 1 for c2 = c1 + 1 : 1 : factors test_number = 0; for i = 1 : 1 : items if (ct(i,c1)>approach_zero && ct(i,c2)>approach_zero) % approach zero in both test_number = test_number + 1; end end disp(['small number = ' num2str(test_number) ' for factors ' num2str(c1) ' and ' num2str(c2)]); if (test_number > small_number) disp(['Criterion 5 is NOT MET for factors ' num2str(c1) ' and ' num2str(c2)]); end end end % CRITERION 6 disp(' '); disp('CRITERION 6'); for i = 1 : 1 : items count = 0; for j = 1 : 1 : factors if (ct(i,j) > meaningful) count = count + 1; end end if (count == 0 \|\| count > 1) disp(['Criterion 6 is NOT MET for item ' num2str(i)]) end end % CRITERION 7 disp(' '); disp('CRITERION 7'); for j = 1 : 1 : factors m=0; for i = 1 : 1 : items if (ct(i,j) > meaningful) m = m + 1; end end disp(['some items = ' num2str(m) ' for factor ' num2str(j)]); if (m < some_bottom \|\| m > some_top) disp(['Criterion 7 is NOT MET for factor ' num2str(j)]); end end disp('') disp('Checking completed.') return The R psych package includes various routines to apply Factor Analysis (whether it be PCA-, ML- or FA-based), but see my short review on crantastic. Most of the usual rotation techniques are available, as well as algorithm relying on simple structure criteria; you might want to have a look at W. Revelle's paper on this topic, Very Simple Structure: An Alternative Procedure For Estimating The Optimal Number Of Interpretable Factors (MBR 1979 (14)) and the VSS() function. Many authors are using orthogonal rotation (VARIMAX), considering loadings higher than, say 0.3 or 0.4 (which amounts to 9 or 16% of variance explained by the factor), as it provides simpler structures for interpretation and scoring purpose (e.g., in quality of life research); others (e.g. Cattell, 1978; Kline, 1979) would recommend oblique rotations since "in the real world, it is not unreasonable to think that factors, as important determiners of behavior, would be correlated" (I'm quoting Kline, Intelligence. The Psychometric View, 1991, p. 19). To my knowledge, researchers generally start with FA (or PCA), using a scree-plot together with simulated data (parallel analysis) to help choosing the right number of factors. I often found that item cluster analysis and VSS nicely complement such an approach. When one is interested in second-order factors, or to carry on with SEM-based methods, then obviously you need to use oblique rotation and factor out the resulting correlation matrix. Other packages/software: • lavaan, for latent variable analysis in R; • OpenMx based on Mx, a general purpose software including a matrix algebra interpreter and numerical optimizer for structural equation modeling. References 1. Cattell, R.B. (1978). The scientific use of factor analysis in behavioural and life sciences. New York, Plenum. 2. Kline, P. (1979). Psychometrics and Psychology. London, Academic Press. I find myself routinely using parallel analysis (O'Connor, 2000). This solves the problem of how many factors to extract nicely. O'Connor, B. P. (2000). SPSS and SAS programs for determining the number of components using parallel analysis and Velicer's MAP test. Behavior Research Methods, Instrumentation, and Computers, 32, 396-402. • (+1) I came across O'Connor's website some years ago and it has a lot of useful resources. Nice that you link it here. – chl Jan 22 '11 at 11:25 I would have to second chl's suggestion of the psych package, its extremely useful and has implementations of the MAP and parallel analysis criteria for number of factors. In my own experience, i have found that if you create factor analysis solutions for all the numbers between those returned by MAP and parallel analysis, you normally can find a relatively optimum solution. I would also second the use of OpenMx for confirmatory factor analysis, as it seems to give the best results of all of them, and is much, much better for badly behaved matrices (as mine tend to be). The syntax is also quite nice, once you get used to it. The only issue that i have with it is that the optimiser is not open source, and thus it is not available on CRAN. Apparently they are working on an open source implementation of the optimiser, so that may not be an issue for much longer. • (+1) Thanks for sharing your experience. I often observed that a good compromise is found just by looking at the scree plot while considering Kayser's rule as a lower limit and simulated data as an upper limit. What I like in psych is that it displays simulated scree plots from both PCA and FA. – chl Jan 22 '11 at 11:24 Great Question. This is not really an answer, but just a few thoughts. In most of the applications where I have used factor analysis, permitting correlated factors makes more theoretical sense. I tend to rely on the proxmax rotation method. I used to do this in SPSS and now I use the factanal function in R.	2019-08-25 18:40:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6542445421218872, "perplexity": 1720.447030475547}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027330786.8/warc/CC-MAIN-20190825173827-20190825195827-00025.warc.gz"}
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http://dasublogbyprashanth.blogspot.com/2013/12/gibbs-entropy-and-two-level-systems.html	## 2013-12-23 ### Gibbs Entropy and Two-Level Systems Today, I was browsing through the MIT news page when I saw this article about how two mathematicians claim to have disproved the notion of negative temperature. My heart sank, because one of the coolest things I remembered learning in 8.044 — Statistical Physics I was the notion of negative temperature existing, being hotter than hot, and being experimentally realizable. I also became confused when the article referred to Gibbs entropy, because the definition I thought was being used for Gibbs entropy was $S = -\sum_j p_j \ln(p_j)$ which is exactly equivalent to the Boltzmann entropy $S = \ln(\Omega)$ where $p_j = \frac{1}{\Omega}$ in the microcanonical ensemble. I figured this would mean that the Gibbs entropy would exactly reproduce negative temperature results in systems with bounded energies such as two-level systems. I wasn't able to read the most recent paper as discussed in the news article, because it is behind a paywall, but I was able to read this article by the same authors, which appears to lay the foundational ideas behind the most recent paper. It seems like on my end, the misconception appears to hinge on what one would call the Gibbs entropy. The formula $S = \ln(\Phi)$ appears to be the correct one for the Gibbs entropy, where $\Phi$ is the total number of states with energy not greater than $E$ and $\Omega = \frac{d\Phi}{dE}$ is the number of states with energy exactly equal to $E$ quantum mechanically (or the number of states with energy within a sufficiently small neighborhood of $E$ in the classical limit). With this in mind, follow the jump to see how this might work for a two-level system and explore the other implications of this new definition of statistical entropy. (UPDATE: Note that in all of this, $k_B = 1$.) A two-level system is defined as follows: consider a system of $N$ indistinguishable particles fixed on a grid that can have energies of 0 ("off") or $\epsilon$ ("on"). Consider that $N$ is fixed and that the total energy $E$ may be directly varied by the experimenter. This means that $\Omega = \binom{N}{\frac{E}{\epsilon}}$ is the number of states with energy exactly equal to $E$, because the number of particles in the "on" state is $\frac{E}{\epsilon}$, and the rest is combinatorics. The only other restriction is that $0 \leq E \leq N\epsilon$. Given this, it is easy to see that $\Omega$ plotted against $E$ takes the value 1 at $E = 0$ and at $E = N\epsilon$, and reaches a very tall maximum with a very narrow peak region around $E = \frac{N\epsilon}{2}$. Traditionally, with the Boltzmann entropy, we would say $S = \ln(\Omega)$ which looks sort of like a parabola with zeros at $E = 0$ and $E = N\epsilon$ and a maximum at $E = \frac{N\epsilon}{2}$, so $\frac{1}{T} = \frac{\partial S}{\partial E}$ is positive for $E < \frac{N\epsilon}{2}$, zero at $E = \frac{N\epsilon}{2}$, and is negative for $E > \frac{N\epsilon}{2}$, and $T$ displays the same behavior in sign but of course diverges with both signs when $\frac{1}{T}$ crosses 0. Essentially, the standard interpretation of this is that temperature is a measure of what the probability distribution of states looks like as a function of energy, so positive temperature says that lower energies are more likely, while negative temperature says that higher energies are more likely (i.e. population inversion). This also means that a negative temperature system gains entropy by losing energy to its surroundings, so a negative temperature system is infinitely hotter than a hot positive temperature system (because it will spontaneously give off heat to any positive temperature system). Now let us try the same system with the Gibbs entropy. The state space volume $\Phi = \sum_{k = 0}^{\frac{E}{\epsilon}} \binom{N}{k}$ which in the continuum limit for large $N$ and/or small $\epsilon$ is $\Phi = \int_{0}^{\frac{E}{\epsilon}} \binom{N}{x} dx$ gives the total number of states where the number of particles that are "on" is not greater than (but not necessarily exactly equal to) $\frac{E}{\epsilon}$. If $\Omega$ looks sort of like a Gaussian, then $\Phi$ looks sort of like the integral of a Gaussian which is the 'S'-shaped error function. For a shifted mean in the Gaussian, this function starts from something close to 0 at $E = 0$, crosses 0.5 at $E = \frac{N\epsilon}{2}$, and would come close to 1 at $E = N\epsilon$. In any case, it is monotonically increasing in $E$. This means that $S = \ln(\Phi)$ will also always increase with $E$, and so $\frac{1}{T} = \frac{\partial S}{\partial E}$ will always be positive and so will $T$. No negative temperature needs to be considered for this system anymore. Does this make sense? Now we are saying the correct measure of states is the cumulative number of states under a certain energy, and with that logic the total number of states under a certain energy should never decrease as that upper energy bound increases. The entropy should then always increase with energy, and temperature will always be positive, meaning that it should always take more energy to increase the number of states in the system (I know that sounds a little redundant/circular). Does any of this really work though? The authors of the paper set out the satisfaction of thermodynamic relations $dS = \frac{dE}{T} - \sum_j \frac{J_j}{T} dX_j$ as the ultimate goal, and show that their use of Gibbs entropy and the associated definition of temperature satisfy this at all scales. It makes some sense because a laser is an example of a system that exhibits population inversion, yet if I stick a thermometer inside a laser in its inverted state (i.e. prior to relaxation), I most certainly will not measure a negative temperature. That is because, in the words of Wikipedia (because I would not say it better), "this is not the macroscopic temperature of the material, but instead the temperature of only very specific degrees of freedom, that are isolated [emphasis mine] from others and do not exchange energy by virtue of the equipartition theorem". So that's cool. Gibbs temperature really does seem like a better measure of thermodynamic temperature under a larger range of conditions, even if Boltzmann temperature gives a more obvious clue to the statistics of the system through the sign change. But there's another test that temperature has to pass, and that is the zeroth law of thermodynamics from the statistical perspective. From the 8.044 — Statistical Physics I notes written by the professor of that class that year, the whole point of the microcanonical ensemble is that states of equal energy are equally likely, so if the system is isolated with a fixed energy $E$, then all accessible states of energy $E$ have the same probability $\frac{1}{\Omega}$. The paper does not dispute this and in fact puts this out there (in quantum mechanical density operator form) as a definition for the microcanonical ensemble. Moreover, if the system is partitioned by a wall into systems 1 & 2 such that the wall allows for the exchange of energy but not particles, then the probability that system 1 has energy $E_1$ (and system 2 has energy $E - E_1$) is $p(E_1) = \frac{\Omega_1 (E_1) \Omega_2 (E - E_1)}{\Omega (E)}$ arising again from maximal equal probabilities for equal energies. This probability is maximized as a function of $E_1$ at the same point that $\ln(p(E_1))$ is maximized over $E_1$, and that happens when $\frac{\partial \ln(\Omega_1)}{\partial E_1} = \frac{\partial \ln(\Omega_2)}{\partial (E - E_1)}$ which exactly recovers equality of Boltzmann temperatures rather than Gibbs temperatures, because this says $S = \ln(\Omega)$ rather than $S = \ln(\Phi)$ if $\frac{1}{T} = \frac{\partial S}{\partial E}$ in both cases. There has to be an inconsistency here. Is the notion of maximal probability giving thermodynamic equilibrium now not consistent with the notion of Gibbs temperature? Is the way I partitioned the system and counted states/probabilities inconsistent with Gibbs temperature? Or is the equality of Boltzmann temperatures derivable from a similar relation equating Gibbs temperatures? Does that last statement lead to the idea that the probabilities of microstates can be expressed in terms of $\Phi$ rather than $\Omega$, and that maximizing such probabilities can lead to $\frac{\partial \ln(\Phi_1)}{\partial E_1} = \frac{\partial \ln(\Phi_2)}{\partial (E - E_1)}$ or something like that? I'm wondering if this issue falls into one of the warnings issued by the authors about how neither Boltzmann nor Gibbs entropies accurately reproduce Shannon entropy in the microcanonical ensemble, and so Gibbs entropy is not expected to reproduce probability maximization at thermal equilibrium. That would then seem to violate a tenet put forth right at the beginning of the paper about the form and meaning of the microcanonical density matrix. As I look at this paper more, though, I'm beginning to realize that while the paper is mathematically sound, there are significant sections of it that don't make a whole lot of physical sense. Look carefully at the examples of the quantum harmonic oscillator or the particle in a box. The math is perfectly correct. However, if I look closely at the energies, these are single-particle energies, yet the total number of states ($n$ for a particle in a box because $n \geq 1$, $n + 1$ for an oscillator as $n \geq 0$) is calculated from these single-particle energies. Thermodynamics and statistical mechanics require large particle numbers $N$ for these analyses to be meaningful. One of the key lessons I remember learning twice over sophomore year (once informally in 8.223 — Classical Mechanics II when I proposed a naïve final project idea combining classical analysis with thermodynamics, and again formally in lecture in 8.044 — Statistical Physics I) was that thermodynamics and associated observables like temperature only make physical sense when a very large number of particles is considered; otherwise, it is in principle and in practice possible to evolve a system of $N$ particles for small $N$ exactly, and the "temperature of a single particle" does not make sense. If the authors were serious about considering a system statistically, the total energy would in fact be the sum of single-particle energies index for each particle. Doing so would present none of the issues that supposedly appear when using the Boltzmann entropy, because in the cases of both the harmonic oscillator and the particle in a box, the energies are unbounded on top. Sure, it's possible to make extensive quantities intensive by considering the energy per particle or the heat capacity per particle as a function of intensive quantities like temperature, but it is physically nonsense to consider temperature or heat capacity for a single particle without considering an ensemble of copies. It is merely accidents of the math that the heat capacities calculated from the Gibbs entropy should coincide with the well-known results derived from using Boltzmann entropy and properly extensive systems. Meanwhile, the Boltzmann entropy results for single particles (not for ensemble systems taken per particle) yield temperatures that look wacky exactly because thermodynamics doesn't work for small numbers of particles. In that way, I would actually warrant that the Boltzmann temperature is more informative than the Gibbs temperature, because the Gibbs temperature gives a false sense that considering a single particle thermodynamically or statistically is acceptable. Those are the most egregious issues. There are a few others as well. In the paper's discussion of the ideal gas, nothing mathematically unsound was done. However, let us examine the claim that $Nd = 1$ yields a negative temperature and $Nd = 2$ yields infinite temperature. If $N$ and $d$ are integers, then $Nd = 1$ can only occur when $N = 1$ and $d = 1$. This essentially says that a single free particle has a negative temperature, but didn't I say earlier that temperature is not physically well-defined for a single particle? Yeah, it looks like that is coming back to haunt this argument. If $Nd = 2$ then $N = 2$ and $d = 1$ or $N = 1$ and $d = 2$; the latter case fails for the same reason as the previous case ($Nd = 1$), while the former fails for essentially the same reason (two free particles can be exactly solved, so temperature as a derived statistical quantity is still meaningless). The next discussion in the paper is one of a similar system to the two-state system discussed earlier in this post, where the energies are quantized and bounded on both sides. There, using the Gibbs entropy leads to a temperature which increases with energy, while using the Boltzmann entropy leads to the negative temperature issue for higher energies (corresponding to population inversion). The good thing about this example in the paper is that the energy is properly extensive by being a sum of single-particle energies for all $N$ particles (rather than being a single-particle energy alone). However, given all the issues I have raised earlier, while I do welcome the notion that a temperature could be defined such that it is positive for all energies so that it sidesteps the issue of Boltzmann entropy considering only the isolated degrees of freedom, I am not really convinced that the Gibbs temperature is the proper definition that coincides with what a thermometer would measure when it is stuck inside a population-inverted system. There are a bunch of myths and facts in the paper after that. The one that I have the biggest issue with is the assertion that the Boltzmann and Gibbs entropies do not reproduce the Shannon entropy in the microcanonical ensemble. This is patently false for the Boltzmann entropy both for the two-level system ($p_j = \frac{1}{\Omega}$) and for the classical ideal gas, because in the former case I showed at the very beginning how the Boltzmann and Shannon entropies are equal in the microcanonical ensemble, while in the latter case the Boltzmann H-function is identical to a continuous Shannon entropy (perhaps modulo an overall sign), using the phase space probability density $\rho$ derived from the BBGKY hierarchy. (Quantum mechanically, the entropy to be considered is the von Neumann entropy, which is essentially the operator version of the H-function or Shannon entropy, using the density matrix rather than the phase space probability density.) Clearly, things will change using the integrated $\Phi$ rather than $\Omega$, so in fact the Boltzmann entropy does reproduce the Shannon entropy, while the Gibbs entropy does not. Moreover, while I am forgetting a lot of the details at the moment, I would recommend reading the second and third chapters of Kardar's book "Statistical Physics of Particles" (I know I'm somewhat biased because I took his class 8.333 — Statistical Mechanics I) to really appreciate the connections between thermodynamic (Boltzmann) and information (Shannon) entropies, showing that the Gibbs entropy is in fact less desirable for this reason among the others mentioned above. Overall, the mistake the authors make is missing the forest for the trees. They want to be able to create a definition of entropy that simultaneously gives thermodynamic results and is valid for systems of arbitrary size. What they fail to see is that thermodynamics is by definition only valid for systems of very large size and is fully empirical, because in its strictest form classical thermodynamics essentially denies the existence of particles and can thereby only consider bulk materials and bulk properties. To account for deficiencies, statistical mechanics accounts for particles, but it can still only deal with large numbers of particles if it is to reproduce thermodynamic results (hence $N \gg 1$ being the thermodynamic limit); if $N = O(1)$, then what is left is deterministic evolution of a state consisting of a handful of particles which can be determined exactly, so if a statistical treatment is desired even in this small limit, it certainly will not be able to reproduce thermodynamic results like the differential equation for entropy in terms of other variables. This means that the seemingly nice-looking Gibbs results on single-particle systems are really nonsense (while the Boltzmann results make clear that such consideration would lead to nonsense anyway), and this calls into question the validity of other Gibbs results like for the quantized bounded energy system of $N$ particles. Moreover, the decoupling $p_j = \frac{1}{\Omega}$ from $S = \ln(\Phi)$ throws into question the other relations the Gibbs results make between thermodynamics and microstate statistics, whereas the Boltzmann entropy $S = \ln(\Omega)$ has no such issue. That said, I can't find fault with the argument that $\langle a \rangle = \mathrm{trace}(a\rho)$ implies the Gibbs entropy rather than the Boltzmann entropy must be correct; I really need to think harder about this, because this does seem to be a blow to Boltzmann entropy if it is indeed true. In conclusion, I think the authors have a bit more to learn about appropriate limits in thermodynamics and statistical mechanics, and I have a bit more to learn about Gibbs entropy and consistency between quantum expectational values & thermodynamic averages. Happy Festivus everyone! (UPDATE: There were a few clarifications I wanted to make earlier, along with a few other things that I thought about. The first regards the consistency relation that $\langle a \rangle = \mathrm{trace}(a\rho)$ implies that the Gibbs entropy is more correct than the Boltzmann entropy. I still cannot find fault with this argument. Furthermore, I tried it the other way, starting from the definition of the Boltzmann entropy and trying to arrive at the definition of a density matrix expectation value, but I could not do it like I could for the Gibbs entropy. The second regards the interpretation of entropy. If $\frac{1}{T} = \frac{\partial S}{\partial E}$, I could say that temperature gives how many more states become available after an infinitesimal addition of energy, and so energy flows from an object with higher positive temperature to one with lower positive temperature because more states become available to the combined system in the process (even if fewer states are available to the hotter body). This works well for traditional systems like classical ideal gases. For a two-level system in which the object at negative temperature comes in contact with an object at positive temperature, using the Boltzmann definitions, the number of states available to the system is always higher if energy flows from the negative temperature object to the positive temperature one, because in this case both subsystems gain available states through that energy transfer. The problem I have with the argument that in a bounded quantized system the higher energy levels must be distinguishable somehow from the lower energy levels in the counting of states (which would motivate using $\Phi$ rather than $\Omega$) is that the microcanonical rather than the canonical ensemble is used, and if all probabilities for an energy are equally likely and that energy is fixed, then in fact there is no reason to treat the different energies differently. When the authors talk about experimentalists being able to distinguish between high and low energies, that experimentalist is most likely working in a canonical ensemble (because a microcanonical ensemble is empirically very difficult to maintain), in which the different energies are treated differently by virtue of the temperature (rather than the total energy) being the independently controlled variable. This rather undercuts the authors' other arguments about how certain facets of the Boltzmann entropy are not applicable because they only pertain to the canonical ensemble. Moreover, because the energy is fixed, there is no good reason to consider all energies below that one when counting the number of states, because that seems to be reasoning more reminiscent of the canonical ensemble (unless the end goal is $\Omega = \frac{d\Phi}{dE}$ within the microcanonical formulation). Next, I wanted to clarify a bit about the Boltzmann entropy being an information entropy especially for a classical ideal gas. Solving the BBGKY hierarchy may not be exactly doable. One approximation is the Boltzmann (yes, lots of things were named after him) collisional approximation, in which the phase space density is approximated to be the same before and after a collision and unknowable during the collision. This literally throws information away, so information entropy is gained. This information entropy increase is also exactly the thermodynamic entropy increase that is observed when an ideal gas confined to one part of a container is suddenly (not quasistatically) allowed to enter the other region of the container. Furthermore, this is a fully microcanonical system. So why again does the Boltzmann entropy not correspond to information entropy? After that, I wanted to discuss a little bit about the Carnot efficiency for negative temperature. I do agree with the authors' statements that population inversion and relaxation in a two-level system at negative temperature would be sudden and not quasistatic, meaning that temperature would be less well-defined, and the efficiency would not be the same as that of a Carnot engine. That said, there is still use in negative temperature. One application of a two-level system is a laser. In stimulated emission, more photons exit than enter. This would certainly seem to produce an efficiency larger than 1, and negative temperature makes this clear from that perspective. On the other hand, the energy required to achieve the population inversion must have come from somewhere. Moreover, if a two-level system comes in contact with something at positive temperature and is allowed to relax, the two-level system is no longer isolated, energy is no longer conserved within the two-level system by construction, and the microcanonical ensemble is no longer correct to begin with. Thus, while I think it certainly is beneficial to caution against blindly using negative temperature in various formulas, I also think the authors' concerns are a bit overblown, especially as the physical interpretations of negative Boltzmann temperature are quite well-established at this point. Finally, I wanted to conclude this update by reiterating some of the points I made in the original conclusion. More is different. Statistical mechanics depends on there being large numbers of degrees of freedom to work correctly, exactly because statistics itself depends on large sample sizes to be meaningful. Statistical mechanics is heavily based on probability & statistics, so blindly using such tools on small numbers of degrees of freedom will inevitably lead to problems. This should put to rest the notion that Gibbs entropy is somehow better because it can account for the thermodynamics of small particle numbers where Boltzmann entropy cannot; that statement is nonsense because thermodynamics and statistical mechanics are not physically empirically meaningful to begin with for small particle numbers in the first place. I do think the paper raises important points about the consistency of Boltzmann entropy in the microcanonical ensemble with quantum statistical expectation values along with the dangers of relying too heavily on negative temperature to bring obvious physical predictions. Beyond that, though, I think the paper's suggestion of using Gibbs entropy rather than Boltzmann entropy has a lot of issues, and I remain unconvinced of its supposed merits. I think I will have access to the newer paper when I get back to MIT after break. Also, I very much welcome you to criticize, point out flaws in, reject, or otherwise comment on my arguments. If enough differences arise between now and when I get to read the paper or after I read the paper (i.e. if it turns out that the arguments in this older paper are made much stronger in the newer paper), then I will post a follow-up to this. Until then, this is basically what I have to say about this. (UPDATE: Oops, it looks like I had forgotten to add another thing. This also means this second update section will be updated as needed until and unless a follow-up post happens. What I had forgotten to add was that the reason why the heat capacities coincidentally look like what they might from Boltzmann entropy/equipartition is because $\Omega$ follows a power law dependence in $E$, so $\Phi$ will as well, and taking the limit of large $N$ would erase any differences between the two approaches. (This is also why the argument that $\frac{Nd}{2}$ is different from $\frac{Nd}{2} - 1$ is nonsense, because statistical mechanics only works in the limit that those two quantities are essentially the same.) If there was a more complicated power series dependence, the Gibbs entropy would fail to reproduce the familiar results.))	2016-10-24 06:59:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7644261121749878, "perplexity": 260.05999525941013}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719542.42/warc/CC-MAIN-20161020183839-00390-ip-10-171-6-4.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/trig-limit.403838/	Trig Limit 1. May 17, 2010 BrownianMan For some reason, I'm having trouble with the following: limit as x-->pi/2 (cot^2(x))/(1-csc(x)) Any help would be appreciated! 2. May 17, 2010 Staff: Mentor I would multiply by 1 + csc(x) over itself. Keep in mind the identity that involved cot^2(x) and csc^2(x).	2018-03-21 13:04:58	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8172149062156677, "perplexity": 8340.288783606053}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647649.70/warc/CC-MAIN-20180321121805-20180321141805-00341.warc.gz"}
https://www.vedantu.com/question-answer/the-unit-vector-in-the-direction-of-widehat-i-+-class-9-maths-cbse-5edbc9fe7e65c55350f2c927	Question # The unit vector in the direction of $\widehat i + \widehat j + \widehat k$ is be?a)$\dfrac{1}{{\sqrt 3 }}(\widehat i + \widehat j + \widehat {k)}$b)$\sqrt 3 (\widehat i + \widehat j + \widehat {k)}$c)$\dfrac{1}{{\sqrt 2 }}(\widehat i + \widehat j + \widehat {k)}$d)$\sqrt 2 (\widehat i + \widehat j + \widehat {k)}$ Hint: Here, we will use the formulae of unit vector in the direction of unit vector to solve. Given, Vector is$\widehat i + \widehat j + \widehat k$. Now, we need to find the unit vector in the direction of $\widehat i + \widehat j + \widehat k$. As, we know that a unit vector is given by dividing the vector by its magnitude, so the resulting vector has magnitude 1and is in same direction as the original vector. Let$\widehat a = \widehat i + \widehat j + \widehat k$, Now magnitude of $\widehat a$is $\begin{gathered} \Rightarrow \left\| {\widehat a} \right\| = \sqrt {{1^2} + {1^2} + {1^2}} \\ \Rightarrow \left\| {\widehat a} \right\| = \sqrt 3 \\ \end{gathered}$ As, we know the unit vector in the direction of $\widehat a$ is $\dfrac{{\widehat a}}{{\left\| {\widehat a} \right\|}}$.Now, the unit vector in the direction of $\widehat i + \widehat j + \widehat k$ will be $\Rightarrow \dfrac{{\widehat i + \widehat j + \widehat k}}{{\sqrt 3 }}$ Therefore, the unit vector in the direction of $\widehat i + \widehat j + \widehat k$ is $\dfrac{{\widehat i + \widehat j + \widehat k}}{{\sqrt 3 }}$. Hence, the required option for the given question is ‘A’. Note: Here, we have been asked to find the unit vector in the direction of the vector, hence the sign of the unit vector is positive elsewhere. If we need to find the unit vector in the opposite direction of the given vector then the sign of the unit vector will be negative.	2021-05-07 12:28:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9245190024375916, "perplexity": 74.10261832945017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988793.99/warc/CC-MAIN-20210507120655-20210507150655-00541.warc.gz"}
https://chemistry.stackexchange.com/questions/134605/why-alkyl-shift-happens-during-hydroboration-oxidation	# Why alkyl shift happens during hydroboration oxidation? In this oxidation step, where alkyl shift happens for 3 times to kick out the OH- to obtain the boron-ester. But why does this happen? Won't the boron atom become less stable as the empty p orbital is regenerated? Won't this cause the OH- (strong nucleophile) to attack the boron (electrophile) again? What is the mechanism of trialkylborane oxidation with hydroperoxide? (I have read this but sorry I don't get it. before this reaction, isn't the boron atom also bonded to an oxygen atom? Why would this reaction make it more stable?) But why does this happen? Well, if it was a plain old carbon in the middle, this migration wouldn't happen. It would sit there forever. Boron isn't carbon, though: it has fewer protons, is less electronegative, and doesn't like holding onto electron density all that much. That electron density gets pushed onto the adjacent carbons, which make them rather more nucleophilic, and happy to jump next door into the $$\sigma_\ce{O-O}^*$$ orbital. You can see, after all, that there's something about that boron that's not quite perfect: it has a formal negative charge. While this shouldn't be taken as a sign of great instability, it does suggest that boron won't mind losing that formal negative charge too much. Won't the boron atom become less stable as the empty p orbital is regenerated? The energetics of this migration aren't just about the boron atom, it's about the system as a whole. You are also breaking a weak O–O bond, forming a stronger C–O bond in replacement, and so on. And indeed, even if this one step is energetically unfavourable, it might not even matter as long as the overall reaction is spontaneous. Is the overall reaction energetically favourable? I'll leave it to you to think about it, based on the types of bonds that are broken and formed over the course of the entire reaction. You already saw some of the considerations in the linked question. (I don't get what you mean by the boron atom being bonded to oxygen before the reaction; it isn't, as it has three bonds to carbon.) Won't this cause the $$\ce{OH-}$$ (strong nucleophile) to attack the boron (electrophile) again? Well, yes, but not much happens if you go down that route. It's far more interesting if $$\ce{OOH-}$$ attacks the boron again, because that means the second alkyl group can migrate. That is precisely the mechanism that you have learnt. • Phenol synthesis using cummene peroxide? – user96208 Aug 9 '20 at 13:59 • @AnindyaPrithvi that's under acidic conditions. Not the same thing. By protonating the peroxide you get a much better leaving group, and the driving force for migration doesn't need to be so strong. This migration is done under basic conditions, and you are probably aware that $\ce{OH-}$ is not (usually) a good leaving group. – orthocresol Aug 9 '20 at 16:37 • Got it, so if it were given that we have acidic condition in this question as well..then? – user96208 Aug 9 '20 at 16:49 • @AnindyaPrithvi If you had acidic conditions you would never get a four-coordinate boron to begin with, so that's pretty much it. You'd have water floating around which is quite happy on its own, so isn't inclined to add to the trivalent boron. Maybe eventually you might hydrolyse the borane; I know this happens with carboxylic acids, but I don't know how stable they are to acids in general. – orthocresol Aug 9 '20 at 16:56	2021-06-19 00:45:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5327462553977966, "perplexity": 1396.9919945821846}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487643354.47/warc/CC-MAIN-20210618230338-20210619020338-00311.warc.gz"}
https://www.ajpmonline.org/article/S0749-3797(23)00119-8/fulltext	Research Brief\|Articles in Press • PDF [1 MB]PDF [1 MB] • Top # COVID-19 Stay-at-home Orders and Secondhand Smoke in Public Housing Published:March 03, 2023 ## Abstract ### Introduction To better understand the inequitable impact of the pandemic by examining associations between stay-at-home orders and indoor smoking in public housing, measured by ambient particulate matter (PM2.5), a marker for secondhand smoke (SHS). ### Methods PM2.5 was measured in six public housing buildings in Norfolk, VA from 2018 – 2022. Multi-level regression was used to compare the seven-week period of the Virginia stay-at-home order in 2020 with that period in other years. ### Results Indoor PM2.5 was 10.29 μg/m3 higher in 2020 (95% CI [8.51, 12.07]) relative to the same period in 2019, a 72% increase. While PM2.5 improved in 2021 and 2022, it remained elevated relative to 2019. ### Conclusions Stay-at-home orders likely led to increased indoor SHS in public housing. In light of evidence linking air pollutants, including SHS, with COVID-19, these results also provide further evidence of the disproportionate impact of the pandemic on socioeconomically disadvantaged communities. This consequence of the pandemic response is unlikely to be isolated and calls for a critical examination of the COVID-19 experience to avoid similar policy failures in future public health crises. ## Introduction Public housing residents, compared to the general public, are more likely to smoke and nonsmoking residents are disproportionately exposed to chronic secondhand smoke (SHS), contributing to persistent health disparities.1 In response, the U.S. Department of Housing and Urban Development (HUD) adopted a federal rule in 2018 requiring Public Housing Authorities (PHAs) to ban smoking in all indoor areas of public housing, and within 25ft of PHA buildings.2 PHA officials have reported challenges implementing the rule, including ensuring resident adherence,3 which may deter the potential health impacts of the rule, particularly on reducing health disparities.4,5 The situation has broader implications for preventive interventions in challenged communities with limited autonomy, including homeless shelters and correctional facilities, where policy effectiveness is undermined by a lack of buy-in and low adherence. The COVID-19 pandemic has further complicated implementation of the smoke-free housing rule. In the first quarter of 2020, many states issued extended stay-at-home orders in response to the pandemic. These orders have been linked to overall increases in cigarette smoking. For example, in an analysis of U.S. tax data, Asare et al. found that cigarette sales increased by 14% during the pandemic, relative to pre-COVID-19 trends.6 Consideration of cigarette smoking and SHS is important because they are significant risk factors for several major respiratory diseases including COPD, interstitial lung disease, and asthma that are also linked to poorer COVID-19 outcomes. Furthermore, current smoking and SHS exposure have been independently linked to higher COVID-19 mortality.7 On a population level, several studies have also shown a link between COVID-19 and outdoor air pollution.8–10 Particulate matter at the 2.5 micron threshold (PM2.5), a common measure of air pollution also used as a marker of SHS, has been implicated as a potential carrier of SARS-CoV-2, the virus that causes COVID-19.11 This study examined variation in ambient indoor PM2.5 in public housing communities subject to COVID-19 stay-at-home orders. The modeling procedure compared the period of the Virginia order to the same time-period two-years prior and post the stay-at-home order. The implications of stay-at-home orders for health outcomes in vulnerable communities that often have limited control over what happens to them—such as public housing—warranted further study. This assessment has the potential to reduce disparities by suggesting actionable targets of intervention. ## Methods Indoor air quality was monitored in common areas of six multi-unit PHA buildings in Norfolk, VA from 2018 - 2022. The mid-rise apartment buildings were the same across all years and ranged between 47 and 114 units (M = 87 units). Monitor placement maximized comparability between buildings (e.g., not in the direct path of HVAC airflow and similar distance from nearby vents). All common areas were interior spaces large enough for residents to congregate. Lobbies or other exits were not used due to the influx of outside air and monitor placement remained constant over the entire period. PM2.5 measurements were taken hourly using SidePak AM520 aerosol monitors (TSI, Inc., St. Paul, MN) with a flow rate of 1.7 L/min. A calibration factor of 0.32 was applied to yield measurements appropriate for SHS particles, as used in previous work.4 PM2.5 data, expressed as μg/m3, were downloaded twice weekly. The monitors were also cleaned and zero-calibrated on this schedule. Using linear mixed modeling, average daily PM2.5 during the period spanning March 23rd to May 9th, 2020—the period of the VA stay-at-home order—was compared for the years 2018, 2019, 2020, 2021, and 2022. Seasonality was accounted for by limiting comparisons to the same period across years. Random site and day effects were modelled using the equation: $PMij=β0+Building0i+Day0j+β1Yearij+eij,fori=1,…,6;j=1,…,49.$ Version 4.1.0 of R was used for analysis. ## Results Mean indoor PM2.5 ranged from 1.83 – 108.41 μg/m3 across the six sites. Mean PM2.5 in 2018—before indoor smoking was prohibited—was 17.27 μg/m3 (Figure 1). Following the introduction of the ban in 2019, mean PM2.5 decreased to 14.33 μg/m3, but peaked in 2020 (24.52 μg/m3). PM2.5 decreased in 2021 (22.68 μg/m3) and 2022 (22.14 μg/m3), but not to levels observed before the pandemic. Analysis suggests the stay-at-home order was associated with 72% higher ambient indoor SHS relative to the same period in 2019, corresponding to a PM2.5 increase of 10.29 μg/m3 (95% CI [8.51, 12.07]; see the Appendix for the full model). While indoor air quality did improve in 2021 and 2022 after the stay-at-home order was lifted, PM2.5 remained 56.8% and 55.1% higher in 2021 and 2022, respectively, relative to 2019. ## Conclusions The 2020 stay-at-home order was associated with increased ambient indoor SHS, and, by inference, SHS exposure among public housing residents. This is of particular concern because public housing residents entered the pandemic with a higher burden of smoking-related disease.12,13 While public housing residents have a multitude of other risk factors for poor health outcomes, this finding may contribute to understanding the increased burden and disparate COVID-19 outcomes observed in other studies of socioeconomically disadvantaged localities.14 For example, increased indoor smoking could have offset potential health benefits related to decreased exposure to outdoor pollution associated with COVID-19 stay-at-home orders and other pandemic-related restrictions.15 This study has limitations. PM2.5 is a non-specific marker of combustion. However, PM2.5 was shown to be a valid measure of tobacco smoke in earlier work conducted in the same locations.4 While it is unclear whether these findings are generalizable, there is evidence of low adherence with public housing smoke-free policies prior to the pandemic in multiple settings4,5 and the poorly coordinated COVID-19 pandemic response likely exacerbated factors already contributing to the failure of these policies to live up to their promise of protecting residents from harms associated with SHS. Extreme consequences—eviction, which could mean homelessness for this population—coupled with inconsistent enforcement had already undermined residents’ perceived legitimacy of the policies. Active, sustained, and meaningful resident engagement likely was the sole path forward. Unfortunately, the pandemic created additional barriers to policy enforcement (e.g., property managers working offsite). With adequate resources and careful planning, HUD and local housing authorities could have supported public-housing residents with adherence to smoke-free rules or smoking cessation assistance during the stay-at-home phase of the COVID-19 pandemic. However, these efforts likely would have meant starting from a deficit, limited by the lack of trust and buy-in needed to engage residents to determine how best to support them. Further, while the reasons for persistently elevated levels of SHS exposure are not clear, these findings call for ongoing investigation and coordinated future interventions to prevent widening of the health disparities experienced by public housing residents and other socioeconomically marginalized groups. Ensuring equitable responses to future public health crises will require a critical examination of the COVID-19 experience for marginalized groups. These findings suggest a crucial lesson: failing to understand how the deficiencies of the COVID-19 response were driven by, and even reinforced, pre-pandemic disparities will impede efforts to equitably respond to future public health crises. It is naïve to think this unintended consequence of COVID-19 is an isolated phenomenon. To do better in the future, policymakers must anticipate how emergency measures have the potential to cause harm in the presence of existing inequity. Authentic engagement of marginalized communities put in place before the next public health emergency is a necessary first step in this process. ## Acknowledgements Members of the Housing Collaborative Community Advisory Board contributed substantively to this project. The content is solely the responsibility of the authors and does not necessarily represent the official views of the funders. Dr. Choi's effort was supported by the Division of Intramural Research, National Institute on Minority Health and Health Disparities. Comments and opinions expressed in this article belong to the authors and do not necessarily reflect those of the US government, Department of Health and Human Service, National Institutes of Health, and National Institute on Minority Health and Health Disparities. This research was financially supported by the National Cancer Institute (NCI) and the National Institute on Drug Abuse of the National Institutes of Health under the award numbers R37CA245716 (ADP, AP, BES, RAG, SG, VWR) and R01DA042195 (RAG). The work was also financially supported by the Department of Housing and Urban Development (VWR). The content is solely the responsibility of the authors and does not necessarily represent the official views of the funders. No financial disclosures were reported by the authors of this paper. ## References • 1 Levy DE, Rigotti NA, Winickoff JP. Tobacco Smoke Exposure in a Sample of Boston Public Housing Residents. Am J Prev Med. 2013;44(1):63-66. doi:10.1016/j.amepre.2012.09.048 • 2 U.S. Department of Housing and Urban Development. FR-5597-F-03, “Instituting Smoke-Free Public Housing”, 81 FR 87430. Published online December 5, 2016. • 3 Wray JA, Sheehan BE, Rees VW, Cooper D, Morgan E, Plunk AD. A Qualitative Study of Unfairness and Distrust in Smoke-free Housing. Am J Health Behav. 2021;45(5):798-809. doi:10.5993/AJHB.45.5.1 • 4 Plunk AD, Rees VW, Jeng A, Wray JA, Grucza RA. Increases in Secondhand Smoke After Going Smoke-Free: An Assessment of the Impact of a Mandated Smoke-Free Housing Policy. Nicotine Tob Res. 2020;22(12):2254-2256. doi:10.1093/ntr/ntaa040 • 5 Thorpe LE, Anastasiou E, Wyka K, et al. Evaluation of Secondhand Smoke Exposure in New York City Public Housing After Implementation of the 2018 Federal Smoke-Free Housing Policy. JAMA Netw Open. 2020;3(11):e2024385. doi:10.1001/jamanetworkopen.2020.24385 • 6 Asare S, Majmundar A, Islami F, et al. Changes in Cigarette Sales in the United States During the COVID-19 Pandemic. Ann Intern Med. 2022;175(1):141-143. doi:10.7326/M21-3350 • 7 Hou H, Li Y, Zhang P, et al. Smoking Is Independently Associated With an Increased Risk for COVID-19 Mortality: A Systematic Review and Meta-analysis Based on Adjusted Effect Estimates. Nicotine Tob Res Off J Soc Res Nicotine Tob. 2021;23(11):1947-1951. doi:10.1093/ntr/ntab112 • 8 Wu X, Nethery RC, Sabath MB, Braun D, Dominici F. Air pollution and COVID-19 mortality in the United States: Strengths and limitations of an ecological regression analysis. Sci Adv. 2020;6(45):eabd4049. doi:10.1126/sciadv.abd4049 • 9 Ali SM, Malik F, Anjum MS, et al. Exploring the linkage between PM2.5 levels and COVID-19 spread and its implications for socio-economic circles. Environ Res. 2021;193:110421. doi:10.1016/j.envres.2020.110421 • 10 Czwojdzińska M, Terpińska M, Kuźniarski A, Płaczkowska S, Piwowar A. Exposure to PM2.5 and PM10 and COVID-19 infection rates and mortality: A one-year observational study in Poland. Biomed J. 2021;44(6, Supplement 1):S25-S36. doi:10.1016/j.bj.2021.11.006 • 11 Nor NSM, Yip CW, Ibrahim N, et al. Particulate matter (PM2.5) as a potential SARS-CoV-2 carrier. Sci Rep. 2021;11(1):2508. doi:10.1038/s41598-021-81935-9 • 12 Yim B, Howland RE, Culp GM, Zhilkova A, Barbot O, Tsao TY. Disparities in Preventable Hospitalizations Among Public Housing Developments. Am J Prev Med. 2019;56(2):187-195. doi:10.1016/j.amepre.2018.08.019 • 13 Helms VE, King BA, Ashley PJ. Cigarette smoking and adverse health outcomes among adults receiving federal housing assistance. Preventive Medicine. 2017;99:171-177. doi:10.1016/j.ypmed.2017.02.001 • 14 Krieger N, Waterman PD, Chen JT. COVID-19 and Overall Mortality Inequities in the Surge in Death Rates by Zip Code Characteristics: Massachusetts, January 1 to May 19, 2020. Am J Public Health. 2020;110(12):1850-1852. doi:10.2105/AJPH.2020.305913 • 15 Hammer MS, van Donkelaar A, Martin RV, et al. Effects of COVID-19 lockdowns on fine particulate matter concentrations. Sci Adv. 2021;7(26):eabg7670. doi:10.1126/sciadv.abg7670 ## CRediT authorship contribution statement Sarah Gehlert: Conceptualization, Writing – review & editing. Vaughan W. Rees: Conceptualization, Methodology, Writing – review & editing. Kelvin Choi: Conceptualization, Writing – review & editing. Peter D. Jackson: Writing – review & editing. Brynn E. Sheehan: Writing – review & editing. Richard A. Grucza: Methodology, Writing – review & editing. Amy Paulson: Writing – review & editing. Andrew D. Plunk: Data curation, Methodology, Writing – original draft.	2023-03-26 16:10:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2337951362133026, "perplexity": 10482.38665950733}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945473.69/warc/CC-MAIN-20230326142035-20230326172035-00297.warc.gz"}
http://www.komal.hu/verseny/feladat.cgi?a=feladat&f=B4678&l=en	Magyar Information Contest Journal Articles # Problem B. 4678. (January 2015) B. 4678. Ann and Bill take turns writing digits on a sheet of paper, left to right. Ann starts with a nonzero digit, and they continue until a 100-digit number is formed. Bill wins if the resulting number divided by 11 leaves a remainder of 5, otherwise Ann wins. Both players are good at mathematics. Who will win the game? Suggested by Gy. Károlyi, Budajenő (4 pont) Deadline expired on 10 February 2015. ### Statistics: 176 students sent a solution. 4 points: 85 students. 3 points: 42 students. 2 points: 26 students. 1 point: 21 students. 0 point: 2 students. Our web pages are supported by: Morgan Stanley	2017-12-12 04:35:59	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8682389855384827, "perplexity": 5153.291921451447}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948515165.6/warc/CC-MAIN-20171212041010-20171212061010-00517.warc.gz"}
https://math.stackexchange.com/questions/3064606/fragmentation-of-a-distribution-from-paper	# “Fragmentation” of a distribution (from paper) I've been reading a paper by Robert Morris ("Sets, Scales and Rhythmic Cycles; A Classification of Talas in Indian Music") and came across a formula that I've found a bit tricky. He is referring to the "fragmentation" of a distribution and includes the formula below without derivation or reference. I'm pretty new to statistics, so this may be a standard formula that I'm just unaware of. However, I haven't been able to find it in the same format online. One feature for use in ordering talas is fragmentation. We have already grouped talas into partition classes. All talas in a particular partition class have the same fragmentation. We use the partition P as the input to a function that yields the fragmentation of the partition. Fragmentation varies between 0 and 1 and is a measure of the uniformity of a distribution—the higher the fragmentation, the more even the distribution. We calculate the fragmentation of a partition of the number N into z parts using the following formula...: $$FRAG(P)=1 - \frac{\sum_{k=1}^{z}{PAIRS(p_{k})}}{PAIRS(N)}$$ where $$PAIRS(s)=\frac{{s^2}-s}{2} \:, \: P=\{{p_{1},p_{2}, p_{3}},...p_{z}\},\\ N = sum(P), and \: z = card(p) .$$ I found the formula to be much more readable in this format: $$Let \: P = \{p_{1}, p_{2}, p_{3},..., p_{z}\}, \: z = card(P),\: and \: N = sum(P).\\ FRAG(P)=1- \frac{\sum_{k=1}^{z} \frac{p_{k}^{2}-p_{k}}{2}}{\frac{N^{2}-N}{2}}=1-2\frac{\sum_{k=1}^{z}\frac{p_{k}^2-p_{k}}{2}}{N^2-N}$$ The author uses the formula with the example $$P=\{2, 2, 4\} \rightarrow N = 2 + 2 + 4 = 8$$ and $$z = 3.$$ This returns $$FRAG(P)=1-2(\frac{8}{56})=0.714285714...$$ Does this formula (or a similar one) have a name? Are there any places where I can find some further information? More generally, what does this mean? Thanks for the help! • Should we interpret this as $\sum_{k=1}^z \frac{p_k^2-p_k}{2} =\sum_{j \in B} b_j$ where $B$ is the bag where the integer $m \ge 0$ appears $a_m = \sum_{p_k > m} 1$ times, and $1-\frac{\sum_{k=1}^z \frac{p_k^2-p_k}{2}}{\frac{N^2 - N}{2}}$ is a measure of how $B$ differs from $\{ 0, \ldots, N-1\}$ where $N$ is the number of elements in $B$ – reuns Jan 7 at 4:56 • Thanks for your reply! When you refer to B as the "bag," are you referring to the multiset P above? – Luke Poeppel Jan 9 at 15:17 • $p_k$ is a list from which I construct a multiset $B$ where the meaning of $\sum_{k=1}^z \frac{p_k^2-p_k}{2}$ and $1-\frac{\sum_{k=1}^z \frac{p_k^2-p_k}{2}}{\frac{N^2 - N}{2}}$ is obvious (using that $\frac{N^2 - N}{2} = \sum_{m=0}^{N-1} m$) – reuns Jan 9 at 15:40 • Does this kind of formula have a name? It seems to be similar to the R^2 measure (goodness-of-fit), but I can't say for certain. – Luke Poeppel Jan 10 at 18:56	2019-04-20 12:27:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 6, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7180424928665161, "perplexity": 368.1088774950347}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578529813.23/warc/CC-MAIN-20190420120902-20190420142902-00543.warc.gz"}
https://tex.stackexchange.com/questions/309066/dash-pattern-for-tikz-line-adaptive-to-line-length	# Dash pattern for TikZ line, adaptive to line length I'm drawing a bended line with TikZ, say using \draw (0,0) to[out=-45, in=-135] (0,5); Is there a way to define a dash pattern such that the first third of this arc is fully present, the second third is dashed and the last third is invisible? In other words, can the dash pattern somehow depend on the length of the line? For straight lines, I can imagine a workaround by just splitting the line into three parts, but this is (at least for me) not really possible in this case. You could use a decoration to get the desired result. \documentclass{article} \usepackage{tikz} \usetikzlibrary{decorations.pathmorphing} \tikzset{ every curvepart/.style={}, curvepart/.style n args={3}{ postaction={black,every curvepart,#1, decorate, decoration={curveto, pre=moveto, pre length=#2\pgfdecoratedinputsegmentlength, post=moveto, post length=(1-#3)\pgfdecoratedinputsegmentlength }}} } \begin{document} \begin{tikzpicture} \path[ every curvepart/.style={draw,very thick}, curvepart={solid}{0}{1/3}, curvepart={dashed}{1/3}{2/3}, ](0,0) to[out=-45, in=-135](0,5); \end{tikzpicture} % \begin{tikzpicture} \path[draw,orange!25,line width=4pt, every curvepart/.style={draw,very thick}, curvepart={solid,purple}{0}{1/3}, curvepart={dashed,green}{1/3}{2/3}, curvepart={dotted,blue}{2/3}{1} ](0,0) to[out=-45, in=-135](0,5); \end{tikzpicture} % \begin{tikzpicture} \path[draw,orange!25,line width=4pt, every curvepart/.style={draw,dashed,very thick}, curvepart={purple}{.1}{.2}, curvepart={blue}{.5}{.6}, curvepart={}{.9}{1}, curvepart={green}{.3}{.4}, curvepart={orange!80!black}{.7}{.8} ](0,0) to[out=-45, in=-135](0,5); \end{tikzpicture} \end{document}	2019-11-14 18:42:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9033378958702087, "perplexity": 4636.885421327449}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668534.60/warc/CC-MAIN-20191114182304-20191114210304-00153.warc.gz"}
https://bodybulding.com/hm-paymaster-ywxmai/word2vec-sentiment-analysis-github-c412f4	The main idea behind this approach is that negative and positive words usually are surrounded by similar words. Section 2 reviews literature on sentiment analysis and the word2vec algorithm along with other effective models for sentiment analysis. The idea is to train our model on the task describe in part 1.1. Work fast with our official CLI. Social networks such as Twitter are important information channels because information in real time can be obtained and processed from them. We implement the cost function using the second to last relation from (2.2) and the previous notations: and then we will retrieve the cost w.r.t to the target word with: This is almost what we want, except that, according to (2.2) we want to compute the cost for $o \in [c-m, c+m]$\{0}. 04/01/2017 ∙ by Haixia Liu, et al. For example: Both sentences have the same words yet the first one seems to be positive while the second one seems to be negative. One must take care of other tags too which might have some predictive value. Word2Vec is dope. The specific data set used is available for download at http://ai.stanford.edu/~amaas/data/sentiment/. Indeed, according to the second to last relation from (2.2), we have: As we already computed the gradient and the cost $J_k$ for one $k \in [0, 2m]$\{m} we can retrieve the “final” cost and the “final” gradient simply by adding up all the costs and gradients when $k$ varies between $0$ and $2m$. There're some requirements for making the stuff work. L04 : Text and Embeddings: Introduction to NLP, Word Embeddings, Word2Vec In practise this assumption is not true. This is the continuation of my mini-series on sentiment analysis of movie reviews, which originally appeared on recurrentnull.wordpress.com. We call those vectors one-hot vectors. We will then transform our words into numbers. Sentiment Analysis Using Word2Vec, FastText and Universal Sentence Encoder in Keras ... All about Neural Networks!github.com. Requirements: TensorFlow Hub, … I personally spent a lot of time untangling Doc2Vec and crashing into ~50% accuracies due to implementation mistakes. Figure 1.1: Train a Skip-Gram model using one sentence. Using Word2Vec, one can find similar words in the dataset and essentially find their relation with labels. This approach can be replicated for any NLP task. Tutorial for Sentiment Analysis using Doc2Vec in gensim (or "getting 87% accuracy in sentiment analysis in under 100 lines of code") - linanqiu/word2vec-sentiments Now, let’s compute the gradient of $J$ (cost in python) with respect to $w_c$ (predicted in python). In python we can simply write: We will then just train our neural network using the vector of each sentence as inputs and the classes as desired outputs. Input (1) Output Execution Info Log Comments (5) This Notebook has been released under the Apache 2.0 … Contribute to Zbored/Chinese-sentiment-analysis development by creating an account on GitHub. For example, with the word aardvark: This process is also described in Figure 1.5 below: To sum up we use one-hot vector to represent each word of our dictionnary (vocabulary), we then train a simple 1-hidden layer neural network using a center word and its context words. How to implement a Word2Vec model (here Skip-Gram model)? For example, v_man - v_woman is approximately equal to v_king - v_queen, illustrating the relationship that "man is to woman as king is to queen". I will focus essentially on the Skip-Gram model. download the GitHub extension for Visual Studio, http://www.cs.cornell.edu/people/pabo/movie-review-data/, http://ai.stanford.edu/~amaas/data/sentiment/. I have saved the Word2Vec models I trained in the previous post, and can easily be loaded with “KeyedVectors” function in Gensim. The included model uses the standard German word2vec vectors and only gets 60.5 F1. The vector still have information about the word cat and the word dog. As there is no activation function on the hidden layer when we feed a one-hot vector to the neural network we will multiply the weight matrix by the one hot vector. The word highlighted in blue is the input word. Framing Sentiment Analysis as a Deep Learning Problem. We have 58,051 unique Winemaker’s Notes in our full dataset. Existing machine learning techniques for citation sentiment analysis are focusing on labor-intensive feature engineering, which requires large annotated corpus. This process, in NLP voodoo, is called word embedding. To conclude, deep sentiment analysis using LSTMs (or RNNs) consists of taking an input sequence and determining what kind of sentiment the text has. In this article I will describe what is the word2vec algorithm and how one can use it to implement a sentiment classification system. The texts describe wines of the following types: red, white, champagne, fortified, and rosé. However, Word2Vec documentation is shit. For sentiment classification adjectives are the critical tags. We want our probability vector $\widehat{y}$ to match the true probability vector which is the sum of So for example, assuming we have 40 000 words in our dictionnary: This is a bad idea. This information helps organizations to know customer satisfaction. The included model uses the standard German word2vec vectors and only gets 60.5 F1. the one-hot representation of the context words that we average over the number of words in our vocabulary to get a probability vector. Other advanced strategies such as using Word2Vec can also be utilized. I'll use the data to perform basic sentiment analysis on the writings, and see what insights can be extracted from them. We considered this acceptable instead of redistributing the much larger tweet word vectors. Well as we know, we cannot feed a Neural network with words as words have no meaning for a Neural Network (what is the meaning of adding 2 words for example?). Sentiment Analysis using Word2Vec Embeddings We try to use the Word2Vec embeddings to the sentiment analysis of the Amazon Music Reviews. In this article I will describe what is the word2vec algorithm and how one can So we will represent a word with another vector. This reasoning still apply for words that have similar context but that are not necessary synonyms. Here the window is set to 2, that is to say that we will train our model using 2 words to the left and 2 words to the right of the center word. Here, we want to maximize the probability of seing the context words knowing the center word. Sentiment analysis is performed on Twitter Data using various word-embedding models namely: Word2Vec, FastText, Universal Sentence Encoder. nlp opencv natural-language-processing deep-learning sentiment-analysis word2vec keras generative-adversarial-network autoencoder glove t-sne segnet keras-models keras-layer latent-dirichlet-allocation denoising-autoencoders svm-classifier resnet-50 anomaly-detection variational-autoencoder We use the chain rule: We already know (see softmax article) that: Finally, using the third point from part 2.2 we can rewrite: To implement this in python, we can write: Using the chain rule we can also compute the gradient of $J$ w.r.t all the other word vectors $u$: Finally, now that we can compute the cost and the gradients for one nearby word of our input word, we can compute the cost and the gradients for $2m-1$ nearby words of our input word, where $m$ is the size of the window simply by adding up all the costs and all the gradients. The difficult part resides in finding a good objective function to minimize and compute the gradients to be able to backpropagate the error through the network. To better understand why it is not a good idea, imagine dog is the 5641th word of my dictionnary and cat is the 4325th. In python, supposing we have already implemented a function that computes the cost for one nearby word, we can write something like: A very simple idea to create a sentiment analysis system is to use the average of all the word vectors in a sentence as its features and then try to predict the sentiment level of the said sentence. Notebook. use it to implement a sentiment classification system. We considered this acceptable instead of redistributing the much larger tweet word vectors. Now, if I substract cat from dog I have a vector with 1 in the 5641th row, -1 in the 4325th row and 0 everywhere else. Installation. The code to just run the Doc2Vec and save the model as imdb.d2v can be found in run.py. We also saw how to compute the gradient of the softmax classifier with respect to the word vectors. This project is a word2vec implementation of the tweets collected from twitter. If nothing happens, download GitHub Desktop and try again. Furthermore, these vectors represent how we use the words. In practise, using Bayes assumption still gives us good results. Requirements: TensorFlow Hub, … This means that if we would have movie reviews dataset, word ‘boring’ would be surrounded by the same words as word ‘tedious’, and usually such words would have somewhere close to the words such as ‘didn’t’ (like), which would also make word didn’t be similar to them. It is obviously not what we want to do in practice. Therefore we see that this vector could have been obtain using only cat and dog words and not other words. In SemEval 2013. Well, similar words are near each other. I won’t explain how to use advanced techniques such as negative sampling. Yet I implemented my sentiment analysis system using negative sampling. For this task I used python with: scikit-learn, nltk, pandas, word2vec and xgboost packages. Yet I implemented my sentiment analysis system using negative sampling. Well, similar words are near each other. As in any Neural Network we can initialize those matrices with small random number. Sentiment Analysis of Twitter Messages Using Word2Vec Kaggle's competition for using Google's word2vec package for sentiment analysis. Copy and Edit 264. This is made even more awesome with the introduction of Doc2Vec that represents not only words, but entire sentences and documents. As mentioned before, the task of sentiment analysis involves taking in an input sequence of words and determining whether the sentiment is positive, negative, or neutral. Around 10 that ensures both a good default choice the next and so on not differentiate between these sentences!: Diving into … 3y ago 000 words in the sentence these input.! 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By creating an account on GitHub and not other words … Word2Vec and xgboost.. Be our one-hot input vector of the softmax classifier to get a representations of our model that!, GLOVE and own embeddings for sentiment analysis Zbored/Chinese-sentiment-analysis development by creating account... With labels vector that we have 40 000 words in our dictionnary the about., pandas, Word2Vec, GLOVE and own embeddings for sentiment analysis on the model! From each others advanced strategies such as negative sampling a 1-hidden layer network. Idea is to train our model will detect the positive words best, hope, enjoy and say! Also saw how to use the words in the sentence and backward pass to prevent overfitting ( generalized poorly unseen! A ( 1,40000 ) ouput vector that we have 40 000 words in the dataset can separate this task! Advanced techniques such as negative sampling real time can be said that these two sentences and will both! 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Network to ouput similar context they are more likely to have a similar word (!	2021-05-08 06:57:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.37789565324783325, "perplexity": 1711.1417642737138}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988850.21/warc/CC-MAIN-20210508061546-20210508091546-00280.warc.gz"}
https://datascience.stackexchange.com/questions/69781/prediction-returning-mistakenly-false-positives/69792	# prediction() returning mistakenly false positives [closed] I do not know how to interpret the result of: prediction(c(1,1,0,0), c(1,1,0,0)) prediction() functino comes from prediction {ROCR} it has this site: http://rocr.bioinf.mpi-sb.mpg.de/ The above is a working example. As per the documentation the first parameter is 'predictions' and the second 'labels' (they would be the true values). The output is this, which I do not fully understand, specially why there is a '2' in "fp". : An object of class "prediction" Slot "predictions": [[1]] [1] 1 1 0 0 Slot "labels": [[1]] [1] 1 1 0 0 Levels: 0 < 1 Slot "cutoffs": [[1]] [1] Inf 1 0 Slot "fp": [[1]] [1] 0 0 2 • Please explain what you're trying to do: is this 'prediction' function from any library, if yes which one? Also please refrain from using screenshots: meta.stackoverflow.com/a/285557/7311767 – Erwan Mar 16 '20 at 17:48	2021-03-07 03:25:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.36020857095718384, "perplexity": 4053.326572020628}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178376006.87/warc/CC-MAIN-20210307013626-20210307043626-00254.warc.gz"}
https://www.lrde.epita.fr/index.php?title=Evaltex&oldid=102871	# Evaltex ### From LRDE (diff) ← Older revision \| Latest revision (diff) \| Newer revision → (diff) EvaLTex (Evaluating Text Localization) is a unified evaluation framework used to measure the performance of text detection and text segmentation algorithms. It takes as input text objects represented either by rectangle coordinates or by irregular masks. The output consists of a set of scores, at local and global levels, and a visual representation of the behavior of the analysed algorithm through quality histograms. For more details on the evaluation protocol, read the scientific paper published in the Image and Vision Computing Journal and the Ph.D Thesis. Details on the visual representation of the evaluation can be found in the article published in the Proc. of International Conference in Document Analysis and Recognition. To use the protocol for segmentation purposes please check out the article published in the Proc. of International Workshop on Robust Reading. Please cite the following papers in all publications that use EvaLTex: IVC for text detection evaluation ECCV for text segmentation evaluation ICDAR for the histogram representation and EMD metrics. # Evaluation performance measurements ##### Local evaluation For each matched GT object ${\displaystyle G_{i}}$ by a detection ${\displaystyle D_{j}}$ we assign two quality measures: Coverage (Cov) and Accuracy (Acc); • Cov computes the rate of the matched area with respect to the GT object area ${\displaystyle Cov_{i}={\frac {Area(G_{i}\bigcap D_{j})}{Area(G_{i})}}}$ • Acc computes the rate of the matched area with respect to the detection area ${\displaystyle Acc_{i}={\frac {Area(G_{i}\bigcap D_{j})}{Area(D_{j})}}}$ The two quality metrics are adapted based on the type of matching (one-to-one, one-to-many, many-to-one or many-to-many). For more details please refer to the scientific paper published in the Image and Vision Computing Journal and the details in Chapter 3 of this Ph.D Thesis. ##### Global evaluation The global evaluation consists of a set of measurements: a global recall ${\displaystyle R_{G}}$, a quantitative recall ${\displaystyle R_{quant}}$, a qualitative recall ${\displaystyle R_{qual}}$, a "global" precision ${\displaystyle P_{G}}$, a "quantitative" precision ${\displaystyle P_{quant}}$, a "qualitative" precision ${\displaystyle P_{qual}}$, a split metric as well as an overall F-Score value. In addition, the tool provides two histogram representations of the local qualities and a derived set of metrics (${\displaystyle R_{EMD}}$ and ${\displaystyle P_{EMD}}$) computed using histogram distances. For the comprehension of all these metrics, we define the following: • ${\displaystyle N_{G}}$ = nb. of GT objects in the image/dataset • ${\displaystyle TP}$ = nb. of true positives (GT objects that were detected) • ${\displaystyle FP}$ = nb. false positives (detections with no correspondence in the GT) Recall. The Recall (${\displaystyle R_{G}}$) computes the amount of detected text and is defined as the product of two terms: ${\displaystyle R_{G}={\frac {\sum _{i=1}^{N_{G}}Cov_{i}}{N_{G}}}={\frac {TP}{N_{G}}}\cdot {\frac {\sum _{i=1}^{N_{G}}Cov_{i}}{TP}}}$ The left term of the product represents the ratio between the number of true positives and the total number of GT objects. We interpret this ratio as the quantity Recall ${\displaystyle R_{quant}}$, as it accurately describes the percentage of detected GT objects, regardless of their coverage: ${\displaystyle R_{quant}={\frac {TP}{N_{G}}}}$ The second term is get by averaging all coverage rates of the detected GT objects. Intuitively, we can denote this proportion as the quality Recall, ${\displaystyle R_{qual}}$, as it characterizes the mean of covered surface of the GT: ${\displaystyle R_{qual}={\frac {\sum _{i=1}^{N_{G}}Cov_{i}}{TP}}}$ Precision. By applying the same reasoning, we obtain the following decomposition of the global Precision ${\displaystyle P_{G}}$: ${\displaystyle P_{G}={\frac {\sum _{i=1}^{N_{G}}Acc_{i}}{TP+FP}}={\frac {TP}{TP+FP}}\cdot {\frac {\sum _{i=1}^{N_{G}}Acc_{i}}{TP}}}$ Here again, the left term of the product provides an insight on the percentage of detections that have a correspondence in the GT. Consequently, we call this measure the quantity precision ${\displaystyle P_{quant}}$: ${\displaystyle P_{quant}={\frac {TP}{TP+FP}}}$ Inversely, the right term computes the accuracy average obtained from the matching of the detection set and the GT. This ratio will then be referred to as the Precision quality ${\displaystyle P_{qual}}$: ${\displaystyle P_{qual}={\frac {\sum _{i=1}^{N_{G}}Acc_{i}}{TP}}}$ Split. The Split metric evaluates the level of GT fragmentation in a dataset and is computed as: ${\displaystyle S={\frac {\sum _{i=1}^{N_{G}}{\frac {1}{1+\ln(s_{i})\cdot \ln(s_{i})}}\cdot 0.6+0.4}{N_{G}}}}$, where ${\displaystyle s_{i}}$=nb. of detections matching ${\displaystyle G_{i}}$ The Split measure can be used as an individual metric or integrated in the Recall computation. For more details please refer to the scientific paper published in the Image and Vision Computing Journal and the details in Chapter 3 of this Ph.D Thesis. F-Score. We use as an overall metric the well known F-Score defined as: ${\displaystyle F_{G}={\frac {2\cdot R_{G}\cdot P_{G}}{R_{G}+P_{G}}}}$ Quality histograms. Histograms can be seen as convenient tools to represent simultaneously the quality and quantity aspects of a set of detections: the quality aspect can be described by the histogram's bin (each bin corresponds to a coverage or accuracy interval); the detection quantity feature can be represented by the bin values (for example, the bin value counts how many GT objects have a coverage or accuracy value that belongs to that bin's interval). The coverage and accuracy histograms intuitively provide at a glance different properties of the detection (or segmentation) behaviour, as illustrated in the following figures: Coverage histogram Accuracy histogram EMD metrics. As an alternative to the global score set explained above, the tool also provides a recall and precision value obtained by applying the Earth Mover's Distance between the coverage (${\displaystyle {\widetilde {h}}_{Cov}}$), respectively the accuracy (${\displaystyle {\widetilde {h}}_{Acc}}$) histogram and an optimal histogram (${\displaystyle {\widetilde {h}}_{O}}$), which describes a perfect detection. ${\displaystyle R_{EMD}=1-EMD({\widetilde {h}}_{Cov},{\widetilde {h}}_{O})}$ ${\displaystyle P_{EMD}=1-EMD({\widetilde {h}}_{Acc},{\widetilde {h}}_{O})}$ For more details on the histogram representation and the EMD metrics please refer to the scientific paper published in the International Conference on Document Analysis and Recognition and the details in Chapter 4 of this Ph.D Thesis. # Input format To simplify the use of the EvaLTex tool for both detection and segmentation tasks, we unified the input format. Hence, to evaluate both text detection and text segmentation we use the same input format consisting of .txt files which contain different attributes of each text object (i.e. coordinates of the bounding boxes). Text detection results (i.e. word, lines, regions) can be represented both through boxes and masks. For text detection tasks using bounding boxes, a .txt file is enough. If the text objects are represented by irregular masks, then an additional labeled image will be needed. GT format. The GT necessary for the text detection (and also segmentation tasks) is represented by the following format: • img name • image height, image width • text object list (one per line) with the following attributes: • ID: unique text object ID • region ID: region ID to which the object belongs to • "transcription": can be empty • text reject: option that decides if a text object should be counted or not; can be set to f (default) or t (not take into account) • x: x coordinate of the bounding box • y: y coordinate of the bounding box • width: width of the bounding box • height: height of the bounding box GT and region IDs. e.g.: img_1.txt img_1 960,1280 1,1,"Tiredness",f,38,43,882,172 2,2,"kills",f,275,264,390,186 3,3,"A",f,0,699,77,131 4,3,"short",f,128,705,355,134 5,3,"break",f,542,710,396,131 6,4,"could",f,87,884,370,137 7,4,"save",f,517,919,314,105 8,5,"your",f,166,1095,302,136 9,5,"life",f,530,1069,213,137 Detection format. The detection .txt file format differs slightly from the GT one: it does not contain the image size, the region ID and the text reject attributes. Hence, the detection file has the following format: • img name • text object list (one per line) with the following attributes: • ID: unique text object ID • "transcription": can be empty • x: x coordinate of the bounding box • y: y coordinate of the bounding box • width: width of the bounding box • height: height of the bounding box e.g.: img_1.txt img_1 1,"",272,264,392,186 2,"",34,40,886,175 3,"",168,1082,300,148 The interest of using masks rather than rectangles is to represent text strings, not only in horizontal or vertical configurations, but also tilted, circular, curved or in perspective. In such cases, the rectangular representation might disturb the matching process: a detection can involuntary match a GT object due to its varying direction (inclined, curved, circular). To evaluate mask detection objects, we need, in the addition of the file format explained before, a set of labeled images, for both the GT and the detection set. The only difference between the GT format of the text box representation and the text mask representation consists in the region ID. The irregular mask annotation disables the use of the region tag. When dealing with rectangular boxes, the regions are generated automatically based on the coordinates of the GT objects. Consequently, a region is the bounding box of several "smaller" boxes. Thus, when masks are annotated irregularly, regions cannot be generated automatically, so each GT object will have a different region ID. One can simplify this by attributing the same ID to the object and the region. example Original image with curved text Labeled GT masks Evaluating text segmentation tasks is very similar to evaluating text detection using a mask representation. For text segmentation we use a mask for each character, contrary to text detection when we use masks to represent words, lines or regions. Original image Labeled GT characters Labeled segmentation result example Text segmentation GT format. Similar to text detection tasks using masks, to evaluate text segmentation we use, in addition to the .txt file a labeled image (each character is labeled differently). Each GT object is represented by a character. Character-level GT objects cannot be grouped into regions and consequently each text object has a different region tag. The x, y, width and height will define the coordinates of the bounding box of each character. e.g.: img_1.txt img_1 960,1280 1,1,"",f,384,43,101,166 2,2,"",f,142,44,46,164 3,3,"",f,38,47,106,163 4,4,"",f,192,80,71,126 5,5,"",f,269,80,100,131 6,6,"",f,501,81,97,126 7,7,"",f,721,81,97,131 ... 16,16,"",t,97,703,53,16 ... Notice that the rectangular mask shape in the segmentation GT example depicts a text object that should not be considered. This corresponds to setting the reject option to t for the text mask having the ID 16, as seen above. Text segmentation result format. The result format consists in the same labeled image as the one used for the GT and the detection .txt file containing the positions of the bounding boxes of each segmented connected component. e.g.: img_1.txt img_1 1,"",383,42,103,167 2,"",142,43,49,167 3,"",35,44,112,168 4,"",268,79,101,132 5,"",194,81,71,124 6,"",500,81,99,127 7,"",612,81,100,131 8,"",721,81,97,131 9,"",824,82,99,133 10,"",344,883,29,135 11,"",387,886,65,135 ... # Output format The evaluation results are given as .txt files, in two forms: a file with the results obtained on the entire dataset and a file with results generated for each image in the dataset. The difference between the local evaluation and the global one consists in the statistics (Cov, Acc and split) for each GT object in an image. #### Global evaluation for an entire dataset EvaLTex statistics General Number of GTs =6410 Number of detections = 6338 Number of false positives =1890 Number of true positives =4678 EvaLTex statistics summarize the number of GT objects, detections, false positives and true positives in the dataset. Global results Recall=0.759731 Recall_noSplit=0.760799 Precision=0.692591 Split=0.791221 FScore=0.724609 FScore_noSplit=0.725095 The global scores are the default Recall score (with integrated Split), the Recall with no integrated Split, the Precision, as well as the default FScore (with integrated Split) and the FScore without the integrated Split. Quantity results Recall=0.792747 Precision=0.712241 Quantity results only refer to the number of detected text objects or the number of detections with a match in the GT regardless of the coverage or accuracy areas. Quality results Recall=0.958352 Recall_noSplit=0.959699 Precision=0.972411 Coverage histogram = {0.214201, 0.00491442, 0.00423657, 0.00491442, 0.00491442, 0.00525335, 0.00610066, 0.0132181, 0.0250805, 0.717167} Accuracy histogram = {0.288977, 0.00137028, 0.00091352, 0.0022838, 0.00365408, 0.00471985, 0.00517661, 0.0103532, 0.0235993, 0.658952} The quality results contain two histograms, representing the coverage and accuracy distributions over the dataset. The histogram format produced by EvaLTex is given as a n-size array, where n is the chosen number of bins. In order to generate the quality histograms, any visualization tool can be used. e.g. Coverage histogram Accuracy histogram EMD results Recall=0.702796 Precision=0.696302 FScore=0.699534 As an alternative to the global scores, we can also compute, using the EMD distance two quality scores based on the Coverage and Accuracy histograms. #### Local evaluation .txt file for each image The local evaluation file, generated for each image of the dataset, has the same format as the dataset evaluation result file. EvaLTex statistics - image img_1 General Number of GTs =43 Number of detections = 19 Number of false positives =1 Number of true positives =18 Global results Recall=0.414803 Recall_noSplit=0.414803 Precision=0.921798 Split=0.428571 FScore=0.572144 FScore_noSplit=0.572144 Quantity results Recall=0.967873 Precision=0.947368 Quality results Recall=0.428571 Recall_noSplit=0.967873 Precision=0.973009 Coverage histogram = {0.571429, 0, 0, 0, 0, 0, 0.0238095, 0, 0.0238095, 0.380952} Accuracy histogram = {0.288977, 0.00137028, 0.00091352, 0.0022838, 0.00365408, 0.00471985, 0.00517661, 0.0103532, 0.0235993, 0.658952} EMD results Recall=0.420952 Recall_noSplit=0.420952 Precision=0.926316 FScore=0.578853 FScore_noSplit=0.578853 In addition, it contains the accuracy, coverage and split values for all the GT objects in the dataset. Local evaluation GT object 1 Coverage = 1 Accuracy = 0.991792 Split = 1 GT object 2 Coverage = 0.809862 Accuracy = 0.994543 Split = 1 GT object 3 Coverage = 1 Accuracy = 0.954386 Split = 1 GT object 4 Coverage = 0.998092 Accuracy = 0.967474 Split = 1 GT object 5 Coverage = 1 Accuracy = 0.993222 Split = 1 GT object 6 Coverage = 1 Accuracy = 0.960362 Split = 1 # Run the evaluation The executable (EvaLTex) takes as input two .txt files, one for the ground truth and one for the detection/segmentation. Usage: ./EvaLTex gt.txt det.txt res_dir [-a] [gtImgDir detImgDir] gt.txt ground truth file path det.txt detection file path res_dir result output directory -a use to generate a result file for each image gtImgDir detImgDir directory path of the gt and detection segmentation images -h/--help show help information The configuration file(evaltex.ini) contains parameter values needed for the evaluation process. The file should be placed in the same repository as the executable. Structure of evaltex.ini: region boolean that decides whether to use or no the region option (default region=true) print_level sets the level of output details (default print_level=0)" min_area threshold for the minimum area acceptance between a GT and a detection (default min_area=0) hist_bin_nb number of bins used to generate the quality histograms and EMD scores (default hist_bin_nb=0) split boolean that decides whether to integrate the split into the coverage (default split=true) det_border border variation in terms of percentages for text detection (default det_border=0.01) # Resources Evaluation Datasets Text detection Text segmentation Dependencies: libTIFF and GraphicsMagick Example: example.tar.gz containing a ground truth and a detection file # Credits This work is part of the LINX project and was partially supported by FUI (Fond Unique Interministeriel) 14. EvaLTex was written by Ana Stefania CALARASANU. Please send any suggestions, comments or bug reports to calarasanu@lrde.epita.fr. Please cite the following papers in all publications that use EvaLTex: IVC for text detection evaluation ECCV for text segmentation evaluation ICDAR for the histogram representation and EMD metrics.	2021-05-07 19:37:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 36, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5419403314590454, "perplexity": 3610.415469487057}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988802.93/warc/CC-MAIN-20210507181103-20210507211103-00125.warc.gz"}
https://wellyzhang.github.io/blog/2021/03/13/PrAE/	# Abstract Spatial-Temporal Reasoning via Probabilistic Abduction and Execution Posted by Chi Zhang on March 13, 2021 This post briefly summarizes our work on the Probabilistic Abduction and Execution (PrAE) learner for Raven’s Progressive Matrices (RPM). For further details, please refer to our CVPR 2021 paper. ## 1. Introduction While “thinking in pictures”, or spatial-temporal reasoning, is effortless and instantaneous for humans, this significant ability has proven to be particularly challenging for current machine vision systems. Recent computational studies on the problem focus on an abstract reasoning task relying heavily on this ablity of “thinking in pictures”—Raven’s Progressive Matrices (RPM). In this task, a subject is asked to pick a correct answer that best fits an incomplete figure matrix to satisfy the hidden governing rules. The ability to solve RPM-like problems is believed to be critical for generating and conceptualizing solutions to multi-step problems. It is also believed to be characteristic of relational and analogical reasoning and an indicator of one’s fluid intelligence. State-of-the-art algorithms incorporating a contrasting mechanism and perceptual inference have achieved decent performance in terms of accuracy. Nevertheless, along with the improved accuracy from deep models come critiques on its transparency, interpretability, generalization, and difficulty to incorporate knowledge. Without explicitly distinguishing perception and reasoning, existing methods use a monolithic model to learn correlation, sacrificing transparency and interpretability in exchange for improved performance. Furthermore, as shown in experiments, deep models nearly always overfit to the training regime and cannot properly generalize. Such a finding is consistent with Fodor [1] and Marcus’s [2, 3] hypothesis that human-level systematic generalizability is hardly compatible with classic neural networks; Marcus postulates that a neuro-symbolic architecture should be recruited for human-level generalization. Another defect of prior methods is the lack of top-down and bottom-up reasoning: Human reasoning applies a generative process to abduce rules and execute them to synthesize a possible solution in mind, and discriminatively selects the most similar answer from choices. This bi-directional reasoning is in stark contrast to discriminative-only models, solely capable of making a categorical choice. Psychologists also call for weak attribute supervision in RPM. As isolated Amazonians, absent of schooling on primitive attributes, could still correctly solve RPM [4, 5], an ideal computational counterpart should be able to learn it absent of visual attribute annotations. This weakly-supervised setting introduces unique challenges: How to jointly learn these visual attributes given only ground-truth images? With uncertainties in perception, how to abduce hidden logic relations from it? How about executing the symbolic logic on inaccurate perception to derive answers? To support cross-configuration generalization and answer generation, we move a step further towards a neuro-symbolic model with explicit logical reasoning and human-like generative problem-solving while addressing the challenges. Specifically, we propose the Probabilistic Abduction and Execution (PrAE) learner; central to it is the process of abduction and execution on the probabilistic scene representation. Inspired by Fodor, Marcus, and neuro-symbolic reasoning, the PrAE learner disentangles the previous monolithic process into two separate modules: a neural visual perception frontend and a symbolic logical reasoning backend. The neural visual frontend operates on object-based representation and predicts conditional probability distributions on its attributes. A scene inference engine then aggregates all object attribute distributions to produce a probabilistic scene representation for the backend. The symbolic logical backend abduces, from the representation, hidden rules that govern the time-ordered sequence via inverse dynamics. An execution engine executes the rules to generate an answer representation in a probabilistic planning manner, instead of directly making a categorical choice among the candidates. The final choice is selected based on the divergence between the generated prediction and the given candidates. The entire system is trained end-to-end with a cross-entropy loss and a curricular auxiliary loss without any visual attribute annotations. Figure 1. An overview of learning and reasoning of the proposed PrAE learner. Given an RPM instance, the neural perception frontend (in red) extracts probabilistic scene representation for each of the 16 panels (8 contexts + 8 candidates). The Object CNN sub-module takes in each image region returned by a sliding window to produce object attribute distributions (over objectiveness, type, size, and color). The Scene Inference Engine sub-module (in pink) aggregates object attribute distributions from all regions to produce panel attribute distributions (over position, number, type, size, and color). Probabilistic representation for context panels is fed into the symbolic reasoning backend (in blue), which abduces hidden rule distributions for all panel attributes (upper-right figure) and executes chosen rules on corresponding context panels to generate the answer representation (lower-right figure). The answer representation is compared with each candidate representation from the perception frontend; the candidate with minimum divergence from the prediction is chosen as the final answer. The lower-right figure is an example of probabilistic execution on the panel attribute of $$\mathtt{Number}$$. ## 2. The PrAE Learner #### 2.1 Overview The proposed neuro-symbolic PrAE learner disentangles previous monolithic visual reasoning into two modules: the neural visual perception frontend and the symbolic logical reasoning backend. The frontend uses a CNN to extract object attribute distributions, later aggregated by a scene inference engine to produce panel attribute distributions. The set of all panel attribute distributions in a panel is referred to as its probabilistic scene representation. The backend retrieves this compact scene representation and performs logical abduction and execution in order to predict the answer representation in a generative manner. A final choice is made based on the divergence between the prediction and each candidate. Using REINFORCE, the entire system is trained without attribute annotations in a curricular manner; see Figure 1 for an overview of PrAE. #### 2.2 Neural Visual Perception The neural visual perception frontend operates on each of the 16 panels independently to produce probabilistic scene representation. It has two sub-modules: object CNN and scene inference engine. ###### 2.2.1 Object CNN Given an image panel $$I$$, a sliding window traverses its spatial domain and feeds each image region into a 4-branch CNN. The 4 CNN branches use the same LeNet-like architecture and produce the probability distributions of object attributes, including objectiveness (whether the image region has an object), type, size, and color. Of note, the distributions of type, size, and color are conditioned on objectiveness being true. Attribute distributions of each image region are kept and sent to the scene inference engine to produce panel attribute distributions. ###### 2.2.2 Scene Inference Engine The scene inference engine takes in the outputs of object CNN and produces panel attribute distributions (over position, number, type, size, and color) by marginalizing over the set of object attribute distributions (over objectiveness, type, size, and color). Take the panel attribute of $$\mathtt{Number}$$ as an example: Given $$N$$ objectiveness probability distributions produced by the object CNN for $$N$$ image regions, the probability of a panel having $$k$$ objects can be computed as $$P(\mathtt{Number} = k) = \sum_{\substack{B^o \in {0, 1}^N \ \|B^o\| = k}} \prod_{j=1}^N P(b_j^o = B_j^o), \label{eqn:number_eg}$$ where $$B^o$$ is an ordered binary sequence corresponding to objectiveness of the $$N$$ regions, $$\|\cdot\|$$ the number of $$1$$ in the sequence, and $$P(b_j^o)$$ the objectiveness distribution of the $$j$$th region. We assume $$k \geq 1$$ in each panel, leave $$P(\mathtt{Number}=0)$$ out, and renormalize the probability to have a sum of $$1$$. The panel attribute distributions for position, type, size, and color, can be computed similarly. We refer to the set of all panel attribute distributions in a panel its probabilistic scene representation, denoted as $$s$$, with the distribution of panel attribute $$a$$ denoted as $$P(s^a)$$. #### 2.3 Symbolic Logical Reasoning The symbolic logical reasoning backend collects probabilistic scene representation from 8 context panels, abduces the probability distributions over hidden rules on each panel attribute, and executes them on corresponding panels of the context. We assume a set of symbolic logical constraints describing rules is available. For example, the $$\mathtt{Arithmetic}$$ $$\mathtt{plus}$$ rule on $$\mathtt{Number}$$ can be represented as: for each row (column), $$\forall l, m \geq 1$$ $$(\mathtt{Number}_1 = m) \land (\mathtt{Number}_2 = l) \land (\mathtt{Number}_3 = m + l),$$ where $$\mathtt{Number}_i$$ denotes the number of objects in the $$i$$th panel in a row (column). With access to such constraints, we use inverse dynamics to abduce the rules in an instance. They can also be transformed into a forward model and executed on discrete symbols: For instance, $$\mathtt{Arithmetic}$$ $$\mathtt{plus}$$ deterministically adds $$\mathtt{Number}$$ in the first two panels to obtain the $$\mathtt{Number}$$ of the last panel. ###### 2.3.1 Probabilistic Abduction Given the probabilistic scene representation of 8 context panels, the probabilistic abduction engine calculates the probability of rules for each panel attribute via inverse dynamics. Formally, for each rule $$r$$ on a panel attribute $$a$$, $$P(r^a \mid I_1, \ldots, I_8) = P(r^a \mid I_1^a, \ldots, I_8^a),$$ where $$I_i$$ denotes the $$i$$th context panel, and $$I_i^a$$ the component of context panel $$I_i$$ corresponding to $$a$$. To model $$P(r^a \mid I_1^a, \ldots, I_8^a)$$, we leverage the compact probabilistic scene representation with respect to attribute $$a$$ and logical constraints: $$P(r^a \mid I_1^a, \ldots, I_8^a) \propto \sum_{S^a \in \mathtt{valid}(r^a)} \prod_{i = 1}^8 P(s_i^a = S_i^a),$$ where $$\mathtt{valid}(\cdot)$$ returns a set of attribute value assignments of the context panels that satisfy the logical constraints of $$r^a$$, and $$i$$ indexes into context panels. By going over all panel attributes, we have the distribution of hidden rules for each of them. ###### 2.3.2 Probabilistic Execution For each panel attribute $$a$$, the probabilistic execution engine chooses a rule from the abduced rule distribution and executes it on corresponding context panels to predict, in a generative fashion, the panel attribute distribution of an answer. While traditionally, a logical forward model only works on discrete symbols, we follow a generalized notion of probabilistic execution as done in probabilistic planning. The probabilistic execution could be treated as a distribution transformation that redistributes the probability mass based on logical rules. For a binary rule $$r$$ on $$a$$, $$P(s_3^a = S_3^a) \propto \sum_{\substack{(S_2^a, S_1^a) \in \mathtt{pre}(r^a) \ S_3^a = f(S_2^a, S_1^a; r^a)}} P(s_2^a = S_2^a) P(s_1^a = S_1^a),$$ where $$f$$ is the forward model transformed from logical constraints and $$\mathtt{pre}(\cdot)$$ the rule precondition set. Predicted distributions of panel attributes compose the final probabilistic scene representation $$s_f$$. During training, the execution engine samples a rule from the abduced probability. During testing, the most probable rule is chosen. #### 2.4 Candidate Selection With a set of predicted panel attribute distributions, we compare it with that from each candidate answer. We use the Jensen–Shannon Divergence (JSD) to quantify the divergence between the prediction and the candidate, $$d(s_f, s_i) = \sum_a \mathbb{D}_{\text{JSD}}(P(s_f^a) \mid\mid P(s_i^a)),$$ where the summation is over panel attributes and $$i$$ indexes into the candidate panels. The candidate with minimum divergence will be chosen as the final answer. #### 2.5 Learning Objective During training, we transform the divergence into a probability distribution by $$P(\text{Answer} = i) \propto \exp(-d(s_f, s_i))$$ and minimize the cross-entropy loss. As the reasoning process involves rule selection, we use REINFORCE to optimize: $$\underset{\theta}{\text{min}}\ \mathbb{E}_{P(r)}[\ell(P(\text{Answer}; r), y)],$$ where $$\theta$$ denotes the trainable parameters in the object CNN, $$P(r)$$ packs the rule distributions over all panel attributes, $$\ell$$ is the cross-entropy loss, and $$y$$ is the ground-truth answer. In practice, the PrAE learner experiences difficulty in convergence with cross-entropy loss only, as the object CNN fails to produce meaningful object attribute predictions at the early stage of training. To resolve this issue, we jointly train the PrAE learner to optimize the auxiliary loss. The auxiliary loss regularizes the perception module such that the learner produces the correct rule prediction. The final objective is $$\underset{\theta}{\text{min}}\ \mathbb{E}_{P(r)}[\ell(P(\text{Answer}; r), y)] + \sum_a \lambda^a \ell(P(r^a), y^a),$$ where $$\lambda^a$$ is the weight coefficient, $$P(r^a)$$ the distribution of the abduced rule on $$a$$, and $$y^a$$ the ground-truth rule. In reinforcement learning terminology, one can treat the cross-entropy loss as the negative reward and the auxiliary loss as behavior cloning. ## 3. Experimental Results We demonstrate the efficacy of the proposed PrAE learner in RPM. In particular, we show that the PrAE learner achieves the best performance among all baselines in the cross-configuration generalization task of RPM (Table 1). In addition, the modularized perception and reasoning process allows us to probe into how each module performs in the RPM task and analyze the PrAE learner’s strengths and weaknesses (Table 2 and Table 3). Furthermore, we show that probabilistic scene representation learned by the PrAE learner can be used to generate an answer when equipped with a rendering engine (Figure 2). Method Acc Center 2x2Grid 3x3Grid L-R U-D O-IC O-IG WReN 9.86/14.87 8.65/14.25 29.60/20.50 9.75/15.70 4.40/13.75 5.00/13.50 5.70/14.15 5.90/12.25 LSTM 12.81/12.52 12.70/12.55 13.80/13.50 12.90/11.35 12.40/14.30 12.10/11.35 12.45/11.55 13.30/13.05 LEN 12.29/13.60 11.85/14.85 41.40/18.20 12.95/13.35 3.95/12.55 3.95/12.75 5.55/11.15 6.35/12.35 CNN 14.78/12.69 13.80/11.30 18.25/14.60 14.55/11.95 13.35/13.00 15.40/13.30 14.35/11.80 13.75/12.85 MXGNet 20.78/13.07 12.95/13.65 37.05/13.95 24.80/12.50 17.45/12.50 16.80/12.05 18.05/12.95 18.35/13.90 ResNet 24.79/13.19 24.30/14.50 25.05/14.30 25.80/12.95 23.80/12.35 27.40/13.55 25.05/13.40 22.15/11.30 ResNet+DRT 31.56/13.26 31.65/13.20 39.55/14.30 35.55/13.25 25.65/12.15 32.05/13.10 31.40/13.70 25.05/13.15 SRAN 15.56/29.06 18.35/37.55 38.80/38.30 17.40/29.30 9.45/29.55 11.35/28.65 5.50/21.15 8.05/18.95 CoPINet 52.96/22.84 49.45/24.50 61.55/31.10 52.15/25.35 68.10/20.60 65.40/19.85 39.55/19.00 34.55/19.45 PrAE Learner 65.03/77.02 76.50/90.45 78.60/85.35 28.55/45.60 90.05/96.25 90.85/97.35 48.05/63.45 42.60/60.70 Human 84.41 95.45 81.82 79.55 86.36 81.81 86.36 81.81 Table 1. Model performance ($$\%$$) on RAVEN / I-RAVEN. All models are trained on 2x2Grid only. Acc denotes the mean accuracy. L-R is short for the Left-Right configuration, U-D Up-Down, O-IC Out-InCenter, and O-IG Out-InGrid. Object Attribute Acc Center 2x2Grid 3x3Grid L-R U-D O-IC O-IG Objectiveness 93.81/95.41 96.13/96.07 99.79/99.99 99.71/97.98 99.56/95.00 99.86/94.84 71.73/88.05 82.07/95.97 Type 86.29/89.24 89.89/89.33 99.95/95.93 83.49/85.96 99.92/92.90 99.85/97.84 91.55/91.86 66.68/70.85 Size 64.72/66.63 68.45/69.11 71.26/73.20 71.42/62.02 73.00/85.08 73.41/73.45 53.54/62.63 44.36/40.95 Color 75.26/79.45 75.15/75.65 85.15/87.81 62.69/69.94 85.27/83.24 84.45/81.38 84.91/75.32 78.48/82.84 Table 2. Accuracy ($$\%$$) of the object CNN on each attribute, reported as RAVEN / I-RAVEN. The CNN module is trained with the PrAE learner on 2x2Grid only without any visual attribute annotations. Acc denotes the mean accuracy on each attribute. Panel Attribute Acc Center 2x2Grid 3x3Grid L-R U-D O-IC O-IG Pos/Num 90.53/91.67 - 90.55/90.05 92.80/94.10 - - - 88.25/90.85 Type 94.17/92.15 100.00/95.00 99.75/95.30 63.95/68.40 100.00/99.90 100.00/100.00 100.00/100.00 86.08/77.60 Size 90.06/88.33 98.95/99.00 90.45/89.90 65.30/70.45 98.15/96.78 99.45/92.45 93.08/96.13 77.35/70.78 Color 87.38/87.25 97.60/93.75 88.10/85.35 37.45/45.65 98.90/92.38 99.40/98.43 92.90/97.23 73.75/79.48 Table 3. Accuracy ($$\%$$) of the probabilistic abduction engine on each attribute, reported as RAVEN / I-RAVEN. The PrAE learner is trained on 2x2Grid only. Acc denotes the mean accuracy on each attribute. Figure 2. Two RPM instances with the final 9th panels filled by our generation results. The ground-truth selections are highlighted in red squares, and the ground-truth rules in each instance are listed. There are no rules on position and number in the first instance of the Center configuration, and the rules on position and number are exclusive in the second instance of 2x2Grid. ## 4. Conclusion We propose the Probabilistic Abduction and Execution (PrAE) learner for spatial-temporal reasoning in Raven’s Progressive Matrices (RPM) that decomposes the problem-solving process into neural perception and logical reasoning. While existing methods on RPM are merely discriminative, the proposed PrAE learner is a hybrid of generative models and discriminative models, closing the loop in a human-like, top-down bottom-up bi-directional reasoning process. In the experiments, we show that the PrAE learner achieves the best performance on the cross-configuration generalization task on RAVEN and I-RAVEN. The modularized design of the PrAE learner also permits us to probe into how perception and reasoning work independently during problem-solving. Finally, we show the unique generative property of the PrAE learner by filling in the missing panel with an image produced by the values sampled from the probabilistic scene representation. However, the proposed PrAE learner also has limits. As shown in our experiments, probabilistic abduction can be a double-edged sword in the sense that when the number of objects increases, uncertainties over multiple objects will accumulate, making the entire process sensitive to perception performance. Also, complete probability marginalization introduces a challenge for computational scalability; it prevents us from training the PrAE learner on more complex configurations such as 3x3Grid. One possible solution might be a discrete abduction process. However, jointly learning such a system is non-trivial. It is also difficult for the learner to perceive and reason based on lower-level primitives, such as lines and corners. While, in theory, a generic detector of lines and corners should be able to resolve this issue, no well-performing systems exist in practice, except those with strict handcrafted detection rules, which would miss the critical probabilistic interpretations in the entire framework. The PrAE learner also requires strong prior knowledge about the underlying logical relations to work, while an ideal method should be able to induce the hidden rules by itself. Though a precise induction mechanism is still unknown for humans, an emerging computational technique of bi-level optimization may be able to house perception and induction together into a general optimization framework. While we answer questions about generalization and generation in RPM, one crucial question remains to be addressed: How perception learned from other domains can be transferred and used to solve this abstract reasoning task. Unlike humans that arguably apply knowledge learned from elsewhere to solve RPM, current systems still need training on the same task to acquire the capability. While feature transfer is still challenging for computer vision, we anticipate that progress in answering transferability in RPM will help address similar questions and further advance the field. @inproceedings{zhang2021abstract,	2022-07-04 01:58:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 9, "x-ck12": 0, "texerror": 0, "math_score": 0.33430689573287964, "perplexity": 4104.131219082769}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104293758.72/warc/CC-MAIN-20220704015700-20220704045700-00040.warc.gz"}
https://codegolf.stackexchange.com/questions/62732/implement-a-truth-machine/62811	# Implement a Truth-Machine A truth-machine (credits goes to this guy for coming up with it) is a very simple program designed to demonstrate the I/O and control flow of a language. Here's what a truth-machine does: • Gets a number (either 0 or 1) from STDIN. • If that number is 0, print out 0 and terminate. • If that number is 1, print out 1 forever. # Challenge Write a truth-machine as described above in your language of choice. The truth-machine must be a full program that follows these rules: • take input from STDIN or an acceptable alternative • If your language cannot take input from STDIN, it may take input from a hardcoded variable or suitable equivalent in the program • must output to STDOUT or an acceptable alternative • If your language is incapable of outputting the characters 0 or 1, byte or unary I/O is acceptable. • when the input is 1, it must continually print 1s and only stop if the program is killed or runs out of memory • the output must only be either a 0 followed by either one or no newline or space, or infinite 1s with each 1 followed by either one or no newline or space. No other output can be generated, except constant output of your language's interpreter that cannot be suppressed (such as a greeting, ANSI color codes or indentation). Your usage of newlines or spaces must be consistent: for example, if you choose to output 1 with a newline after it all 1s must have a newline after them. • if and only if your language cannot possibly terminate on an input of 0 it is acceptable for the code to enter an infinite loop in which nothing is outputted. Since this is a catalog, languages created after this challenge are allowed to compete. Note that there must be an interpreter so the submission can be tested. It is allowed (and even encouraged) to write this interpreter yourself for a previously unimplemented language. Other than that, all the standard rules of must be obeyed. Submissions in most languages will be scored in bytes in an appropriate preexisting encoding (usually UTF-8). # Catalog The Stack Snippet at the bottom of this post generates the catalog from the answers a) as a list of shortest solution per language and b) as an overall leaderboard. ## Language Name, N bytes where N is the size of your submission. If you improve your score, you can keep old scores in the headline, by striking them through. For instance: ## Ruby, <s>104</s> <s>101</s> 96 bytes If there you want to include multiple numbers in your header (e.g. because your score is the sum of two files or you want to list interpreter flag penalties separately), make sure that the actual score is the last number in the header: ## Perl, 43 + 2 (-p flag) = 45 bytes You can also make the language name a link which will then show up in the snippet: ## [><>](http://esolangs.org/wiki/Fish), 121 bytes <style>body { text-align: left !important} #answer-list { padding: 10px; width: 290px; float: left; } #language-list { padding: 10px; width: 320px; float: left; } table thead { font-weight: bold; } table td { padding: 5px; }</style><script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script> <link rel="stylesheet" type="text/css" href="//cdn.sstatic.net/codegolf/all.css?v=83c949450c8b"> <div id="language-list"> <h2>Shortest Solution by Language</h2> <table class="language-list"> <thead> <tr><td>Language</td><td>User</td><td>Score</td></tr> </thead> <tbody id="languages"> </tbody> </table> </div> <div id="answer-list"> <h2>Leaderboard</h2> <table class="answer-list"> <thead> <tr><td></td><td>Author</td><td>Language</td><td>Size</td></tr> </thead> <tbody id="answers"> </tbody> </table> </div> <table style="display: none"> <tbody id="answer-template"> <tr><td>{{PLACE}}</td><td>{{NAME}}</td><td>{{LANGUAGE}}</td><td>{{SIZE}}</td><td><a href="{{LINK}}">Link</a></td></tr> </tbody> </table> <table style="display: none"> <tbody id="language-template"> <tr><td>{{LANGUAGE}}</td><td>{{NAME}}</td><td>{{SIZE}}</td><td><a href="{{LINK}}">Link</a></td></tr> </tbody> </table><script>var QUESTION_ID = 62732; var ANSWER_FILTER = "!t)IWYnsLAZle2tQ3KqrVveCRJfxcRLe"; var COMMENT_FILTER = "!)Q2B_A2kjfAiU78X(md6BoYk"; var OVERRIDE_USER = 12012; var answers = [], answers_hash, answer_ids, answer_page = 1, more_answers = true, comment_page; function answersUrl(index) { return "https://api.stackexchange.com/2.2/questions/" + QUESTION_ID + "/answers?page=" + index + "&pagesize=100&order=desc&sort=creation&site=codegolf&filter=" + ANSWER_FILTER; } function commentUrl(index, answers) { return "https://api.stackexchange.com/2.2/answers/" + answers.join(';') + "/comments?page=" + index + "&pagesize=100&order=desc&sort=creation&site=codegolf&filter=" + COMMENT_FILTER; } function getAnswers() { jQuery.ajax({ url: answersUrl(answer_page++), method: "get", dataType: "jsonp", crossDomain: true, success: function (data) { answers.push.apply(answers, data.items); answers_hash = []; answer_ids = []; data.items.forEach(function(a) { a.comments = []; var id = +a.share_link.match(/\d+/); answer_ids.push(id); answers_hash[id] = a; }); if (!data.has_more) more_answers = false; comment_page = 1; getComments(); } }); } function getComments() { jQuery.ajax({ url: commentUrl(comment_page++, answer_ids), method: "get", dataType: "jsonp", crossDomain: true, success: function (data) { data.items.forEach(function(c) { if (c.owner.user_id === OVERRIDE_USER) answers_hash[c.post_id].comments.push(c); }); if (data.has_more) getComments(); else if (more_answers) getAnswers(); else process(); } }); } getAnswers(); var SCORE_REG = /<h\d>\s([^\n,<](?:<(?:[^\n>]>[^\n<]<\/[^\n>]>)[^\n,<])),.?(\d+)(?=[^\n\d<>](?:<(?:s>[^\n<>]<\/s>\|[^\n<>]+>)[^\n\d<>])<\/h\d>)/; var OVERRIDE_REG = /^Override\sheader:\s/i; function getAuthorName(a) { return a.owner.display_name; } function process() { var valid = []; answers.forEach(function(a) { var body = a.body; a.comments.forEach(function(c) { if(OVERRIDE_REG.test(c.body)) body = '<h1>' + c.body.replace(OVERRIDE_REG, '') + '</h1>'; }); var match = body.match(SCORE_REG); if (match) valid.push({ user: getAuthorName(a), size: +match[2], language: match[1], link: a.share_link, }); else console.log(body); }); valid.sort(function (a, b) { var aB = a.size, bB = b.size; return aB - bB }); var languages = {}; var place = 1; var lastSize = null; var lastPlace = 1; valid.forEach(function (a) { if (a.size != lastSize) lastPlace = place; lastSize = a.size; ++place; var answer = jQuery("#answer-template").html(); answer = answer.replace("{{PLACE}}", lastPlace + ".") .replace("{{NAME}}", a.user) .replace("{{LANGUAGE}}", a.language) .replace("{{SIZE}}", a.size) .replace("{{LINK}}", a.link); answer = jQuery(answer); jQuery("#answers").append(answer); var lang = a.language; lang = jQuery('<a>'+lang+'</a>').text(); languages[lang] = languages[lang] \|\| {lang: a.language, lang_raw: lang.toLowerCase(), user: a.user, size: a.size, link: a.link}; }); var langs = []; for (var lang in languages) if (languages.hasOwnProperty(lang)) langs.push(languages[lang]); langs.sort(function (a, b) { if (a.lang_raw > b.lang_raw) return 1; if (a.lang_raw < b.lang_raw) return -1; return 0; }); for (var i = 0; i < langs.length; ++i) { var language = jQuery("#language-template").html(); var lang = langs[i]; language = language.replace("{{LANGUAGE}}", lang.lang) .replace("{{NAME}}", lang.user) .replace("{{SIZE}}", lang.size) .replace("{{LINK}}", lang.link); language = jQuery(language); jQuery("#languages").append(language); } }</script> • Can we assume that the program halts when the processor finishes executing the written code, for a machine code entry? – lirtosiast Nov 3 '15 at 16:58 • Assuming any behaviour is fine for all invalid inputs? – Cruncher Nov 3 '15 at 17:33 • @Cruncher Yes, the only inputs you should expect to get are 0 and 1. – a spaghetto Nov 3 '15 at 17:38 • Catalog is borked. – Addison Crump Nov 6 '15 at 15:18 • Catalog appears to consider Bf and bf to be different languages. – Mooing Duck Nov 10 '15 at 1:13 # Churro, 100 95 bytes {========={o}{}======}{}======}{={}{={o}{={o}{={}{={o}{======={}{==={}{======={}{===={} Explanation Churro is a stack-based esolang where the only syntax element is, well, churros! Or rather, ASCII-art representations of churros. An example of such a churro is {o}====}. In Churro, each churro has three characteristics: 1. Its orientation; whether it's facing left ({o}====}) or right ({===={o}). 2. Its filling; whether it's filled ({}==}) or not ({o}==}). 3. Its tail length; the number of =s in the churro is the length of its tail. Left-facing churros are integer literals; their tail length is their value, while their filling status is their sign. Filled churros are negative, while unfilled churros are positive. Right-facing churros are operators; their tail length is which operator they represent, according to this table: {{o} pop A; discard A {={o} pop A, B; push B + A {=={o} pop A, B; push B - A {==={o} pop A; if A == 0, jump to churro after matching occurrence of {==={o} {===={o} pop A; if A != 0, jump to churro after matching occurrence of {==={o} {====={o} pop A, B; store B in memory location A {======{o} pop A; push the value in memory location A to stack {======={o} pop A; print A as an integer {========{o} pop A; print A as an ASCII character {========={o} read a single character from STDIN and push it to the stack {=========={o} exit the program Filled operator churros have the same behaviour, but peek at the stack instead of popping from it. With that, here's the ungolfed, explained version of the Churro truth-machine: {========={o} read char from stdin {}======} push -6 {}======} push -6 {={} push -6 + -6 = -12 {={o} pop -6, -12; push -6 + -12 = -18 {={o} pop -6, -18; push -6 + -18 = -24 {={} push input + -24 {={o} pop -24, input + -24; push input + -48 (convert char to int) {======={} print input as integer {======={} print input as integer {===={} if input != 0 jump back to matching churro I'm using TheLastBanana's Haskell interpreter to interpret this. Installation instructions can be found there. Finally, here's the "pure" version (no comments, line length 80, spaces between churros), as output by purify truth.ch: {========={o} {}======} {}======} {={} {={o} {={o} {={} {={o} {======={} {==={} {======={} {===={} Saved 5 bytes thanks to Martin Ender! • And now I want a dulce de leche and some doughnuts... – WallyWest Aug 30 '16 at 23:01 # Moorhen 1 (or original creators version 1 here), 38 bytes The original language creator has decided to add new instructions, which would change the existing ones. Hence, this language is in Moorhen version one op pa id el pa id el ai op id ai pi ai Note this doesn't print if input is 1, because printing only happens at halting, so it just puts infinite ones on stack ## Explanation: Commands can be (most of) all english words, and the command they execute depends on md5 hash mod 7. I used some length two words that corresponded to each of the seven commands re: push 0 op: increment ToS pa: decrement ToS el: rotate stack (place ToS on BoS) pi: dupe ToS id: peek TOS, skip if it is non-zero ai: flip pointer direction there are two main segments of code op pa id el pa id el ai op id ai pi ai op pa id el pa id el This code, when run forwards, will have the pointer leave to the right side, with the initial value on the stack, minus one (0 -> -1, 1 -> 0). When run backwards, with [-1] as stack, it leaves the to the left side with [0] as the stack. When it leaves to the left, it will print the stack, items joined with spaces, so it will print 0. It uses "nops", commands that don't do anything under certain circumstances, or are a pair of inverses. (the reason the comments look funny is because the interpreter ignores non-words, including real words that are hyphenated) op pa nop-when-running-forwards id el nop-with-only-one-value-on-stack-and-running-forward pa decrement- id el nop-with-only-one-value-on-stack-and-running-forward [go to next part of code] with all these nops forward, its essentially pa when running forward. However, backwards, with -1 on stack: el nop-with-only-one-value-on-stack id pa skip-pa,when-TOS-nonzero(-1,is-non-zero) el nop-with-only-one-value-on-stack id pa skip-pa,when-TOS-nonzero(-1,is-non-zero) op increment- [end,print-stack(with,-1,coming-in,print-0)] With -1, backwards, it's essentially op (then implicit print) The code that is at the right of that code: ai op id ai pi ai given -1, it will just reflect it back. Given 0, it will increment it, then enter a loop of duping ToS, which will be 1. ai reflect-pointer-direction-if-TOS-non-zero op increment- id skip-next-command-if-non-zero. This-command-skips-into-the-middle-of-a-loop: ai pi ai this-is-the-loop. pointer-starts-on-pi,because-it-skipped-the-first-ai Here is a visualisation of how it executes pi dupe-TOS (1) ai reverse-if-ToS-is-nonzero (it is) pi dupe-TOS (1) -again ai reverse-if-ToS-is-nonzero (it is) pi dupe-TOS (1) -again ... this keeps happening ## Common Lisp, 30 (do((x(read)))((=(print x)0))) Common Lisp's print function returns the object that was printed. This reads a value from the user, then prints the value until the return value of the call to print returns 0. # Taxi, 617 bytes Go to Post Office:w 1 l 1 r 1 l.Pickup a passenger going to Crime Lab.1 is waiting at Writer's Depot.Go to Writer's Depot:s 1 r 1 l 2 l.Pickup a passenger going to Crime Lab.Go to Crime Lab:n 1 r 2 r 2 l.Switch to plan "z" if no one is waiting.Pickup a passenger going to Cyclone.Go to Cyclone:n 4 l 2 l.[r]Pickup a passenger going to Cyclone.Pickup a passenger going to Post Office.Go to Post Office:s 1 l 2 r 1 l.Go to Zoom Zoom:s 1 r 1 l 2 r.Go to Cyclone:w.Switch to plan "r".[z]0 is waiting at Writer's Depot.Go to Writer's Depot:n 4 l 2 l.Pickup a passenger going to Post Office.Go to Post Office:n 1 r 2 r 1 l. Try it online! There are shorter programs but they kept running out of gas. Here's the ungolfed version: Go to Post Office: west 1st left 1st right 1st left. Pickup a passenger going to Crime Lab. 1 is waiting at Writer's Depot. Go to Writer's Depot: south 1st right 1st left 2nd left. Pickup a passenger going to Crime Lab. Go to Crime Lab: north 1st right 2nd right 2nd left. Switch to plan "z" if no one is waiting. Pickup a passenger going to Cyclone. Go to Cyclone: north 4th left 2nd left. [r] Pickup a passenger going to Cyclone. Pickup a passenger going to Post Office. Go to Post Office: south 1st left 2nd right 1st left. Go to Zoom Zoom: south 1st right 1st left 2nd right. Go to Cyclone: west. Switch to plan "r". [z] 0 is waiting at Writer's Depot. Go to Writer's Depot: north 4th left 2nd left. Pickup a passenger going to Post Office. Go to Post Office: north 1st right 2nd right 1st left. # Duck Duck Goose, 911 bytes duck duck duck goose duck duck duck duck duck duck duck goose duck duck duck duck duck duck duck duck duck duck duck duck duck duck duck duck duck duck goose duck duck duck duck duck duck duck duck duck duck duck duck duck duck duck duck duck duck duck duck goose duck duck duck duck duck duck duck duck duck goose duck duck duck duck duck duck duck duck goose duck duck duck duck duck duck duck duck duck duck goose duck duck duck duck duck duck duck duck duck duck duck duck duck duck goose duck duck duck duck duck duck duck duck goose duck duck duck duck duck duck duck duck duck duck duck goose duck duck duck duck duck duck duck duck duck duck duck goose duck duck duck duck duck duck goose duck duck duck duck duck duck goose duck duck goose duck duck duck duck duck duck duck duck duck duck goose duck duck duck duck duck goose duck duck duck duck goose Includes 2 trailing newlines. I can't remove the newlines, they are instructions. 51 1 5?8-1 Takes input from the command-line arguments. Try it online! ## Explanation This program consists of eight numbers. These numbers are grouped into instructions; each instruction consists of a label, which is the first number, and zero or more arguments, which follow the label. Instructions with different labels can have different numbers of arguments, but the parsing of instructions happens at execution time (which allows self-modification). The numbers in the program are written to the tape from position 0 at the beginning of the program; each label and argument occupies its own cell, and the instruction pointer occupies the cell at location -1. 5 1 1 5 ? 8 -1 Dreaderef's preprocessor allows you to write aliases that stand in for numbers representing instruction labels. Using aliases makes the program look like this: 0. numo * 2. deref 1 5 5. ? 8 -1 (The leading numbers are position labels and are treated as comments.) Before the program executes, * is replaced with the input integer. 0. numo * The first instruction is numo, which outputs its argument (either 0 or 1) in decimal without a trailing newline. 2. deref 1 5 deref reads the tape at the position indicated by its first argument. It then writes that value to the tape at the position indicated by its second argument. This means that here deref copies cell 1 (the input integer and argument to numo) to cell 5 (part of the next instruction). 5. ? 8 -1 This instruction is interesting in that the label is dynamically generated by the previous instruction (? is an alias for 0 in Dreaderef, but it is intended to convey that a given cell is uninitialized and will not be read until it is written to). During execution, it will be set to either 1 or 0 depending on the input integer. Proceeding from here requires some knowledge about what labels map to what instructions: • The instruction with label 0 is end, which terminates execution. • The instruction with label 1 is deref. If the input integer was 0, the end instruction causes the program to terminate, and the other two numbers are ignored. If the input integer was 1, the instruction looks like this: 5. deref 8 -1 Cell 8 is past the end of the program, meaning that its value defaults to 0. The 0 is copied into cell -1, which is Dreaderef's instruction pointer. This means that execution jumps back to position 0, which is the start of the program. Because * is read before execution of the program, the same input (1) is reused, and the program outputs and loops again. ## Zephyr, 41 bytes input n print n while"1"=n print n repeat Try it online! A main design goal of Zephyr was that code should be readable and understandable. Looks like this holds true even when the code is maximally golfed. Mission accomplished. # Deoxyribose, 18 bytes Updated for the new v3 spec ATGGAGAAGAGCATAAAT Explanation: ATG ATA () GAG Glu Dupe TGG Trp Power AAG Lys Pop AGA Arg Unipop AGC Ser Jump if <= 0 AGA Arg Unipop ATA GCA Ala Modulo AAT Asn Loop TAA End • Execution starts at the initial ATG. • The top element of the main stack is duplicated & printed, then If the value is less than or equal to zero, jump to the ATA formed by the Asn and the start codon (remember, the code is circular), then run through a series of no-ops, including printing some non-printable characters. If the value is greater than 0, loop back to the start. If you don't like the ugly sequence of no-ops, adding ATG to the end to make the loop target explicit results in a clean termination immediately after the jump, for three more bytes. Deoxyribose 2, 21 bytes ATGTGAGAAAAATCTAACTTA Explanation: ATG Start TAA Stop ┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┐ TGA Block size = 56 ┏ TGT Cys Destination of Asn ┆ GAA Glu Duplicate ┃ GAG Glu Duplicate ┆ AAA Lys Pop as int ┃ AAA Lys Pop as int ┆ TCT Ser If >= 0, jump to Thr ┐┞ AAT Asn Jump back to Cys ┆ AAC Asn Jump back to Cys ────┼┘ ┆ TTA └─ ACT Thr Destination of Ser ┆ └┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┘ This is the first challenge on this site ever answered in this language (since it's only a week old), but I think it's a good showcase of what the language can do so I'll provide a bit of an explanation of what's going on. If you happen to be really interested, the spec, a Python-based interpreter, and additional (less golfed) examples are on GitHub. Deoxyribose is a stack-based language with a small instruction set and a syntax based on DNA. Execution takes place on a circular stretch of DNA, translating 3-digit "codons" into proteins, each of which corresponds to either • an operation on one or both of the stacks, or • a (perhaps conditional) jump to a particular sequence elsewhere in the code. These jumps can lead to frameshifts, where the read head is moved a non-integer number of codons and the same bit of code ends up doing two totally different things at different times. The degenerate nature of the genetic code makes it possible (and fun!) to use this to your advantage. • I love this language concept! – histocrat May 2 '18 at 21:45 ## Rockstar, 34 Bytes Rockstar is a quite new computer programming language, designed for creating programs that are also song lyrics. Rockstar is heavily influenced by the lyrical conventions of 1980s hard rock and power ballads. Listen to your heart While your heart is not empty Scream it But yes, you are right, it's longer than 34 bytes, but so poetic, I couldn't resist get some lyrics in. You could even sing it!. Here's the golfed version: Listen to X Say X While X Say X • Which interpreter are you using? Wouldn't the input be treated as a string, which is truthy? Also, would the Until X loop until X is truthy, rather than while X is truthy? – Jo King Apr 30 at 6:50 • @JoKing You are right, the golfed version was incorrect, I changed the Until loop to the While loop. (it did not change the code length). To answer your question, I use my interpreter: github.com/gaborsch/rocky - that worked the expected way, too, so it was my mistake, not the interpreter's. – gaborsch Apr 30 at 7:01 • The input is numeric in this case, so 0 is coerced to false. – gaborsch Apr 30 at 7:03 # tinylisp, 36 bytes This is a language that I'm basing an upcoming challenge on. The spec in the challenge doesn't include the disp function, but the reference implementation does. (d M(q((x)(i x(i(disp x)0(M x))0)))) Defines a function M that takes an argument x (the closest that the language has to input). If x is falsey, we return 0, which is printed. If x is truthy, we want to display x and then recurse. There isn't any equivalent to Common Lisp's progn in the language, so the best way to do this is to use the disp call as the condition of an if. The result is falsey, thus putting the recursive call (M x) in the else branch. # Sisi, 22 bytes Sisi doesn't have any way to take user input, so the number is expected to be stored in the variable x (presumably on line 1). 2 print x 3 jumpif x 2 Pretty straightforward: print the number, and keep doing so as long as it's 1. # Stuck, 9 Bytes Stuck has a while loop function (which I never added to the docs on Esolangs, but it has existed for a little over 2 months) which makes this possible. It wasn't before this as far as I know :P. ip"1="'ph • Dammit I was trying to make a truth machine in Stuck but couldn't figure out how to do it without a while loop – a spaghetto Nov 3 '15 at 20:15 • @quartata Sorry about that :P Whenever I get enough free time I will update the documentation. – Kade Nov 3 '15 at 20:20 # Jasmin, 355 bytes As with the other answers I've written in Jasmin, there isn't a whole lot of golf going on here. This code is (almost) exactly the code obtained from running javap on the class file generated by intrepidcoder's java submission. The one neat golfing trick I found was avoiding the usual .limit locals line by reusing local variable 0. .class L .super java/lang/Object .method public static main([Ljava/lang/String;)V .limit stack 2 getstatic java/lang/System/in Ljava/io/InputStream; iconst_2 irem istore_0 A: getstatic java/lang/System/out Ljava/io/PrintStream; invokevirtual java/io/PrintStream/print(I)V ifgt A return .end method • Would it be any shorter to put the code in an initializer block and not have a main method? – feersum Nov 3 '15 at 22:44 • @feersum I've never been able to get static blocks to work using java version "1.8.0_60". I think static block are only accepted by java in version 1.6 and earlier. I don't have the correct version to test it so, I'll leave my answer as is for now. – ankh-morpork Nov 3 '15 at 23:03 # Beam, 2523 13 bytes rSn()>@< H@< Try it in the online interpreter! (Warning: it may crash your browser with an input other than 0.) Beam is a 2D language, based on the concept of a beam of light moving through the 2D source code. Beam is oriented around two main memory values: one held by the beam, and one called the "store". Here are the commands used, in order: • r - Set the beam to the next ASCII code in the input. (48 for 0, 49 for 1) • S - Set the store to the beam. • n - If beam != store, point the beam downward. Does nothing the first time. • ( and ) - If store != 0, point the beam right/left, respectively. • - Decrement the store by 1. Now, the beam enters () from the left, and bounces back and forth until the store reaches 0. If the store's initial value is 48 (0), it will exit to the left, traveling through these chars: • n - If beam != store, point the beam downward. This time, since the beam is 48 and the store is 0, it does its job. • < - Unconditionally point the beam to the left. • @ - Output the beam as an ASCII character. Prints 0. • H - Halt the program. However, if the store's initial value is 49 (1), it exits the loop to the right, and runs through this code: • > - Unconditionally point the beam to the right. • @ - Output the beam as an ASCII character. Prints 1. • < - Unconditionally point the beam to the left. • @ - Output the beam as an ASCII character. Prints 1. • > ... ...and so on until the end of time (or until your browser crashes). Thanks to @MickyT for this awesome layout! • +1 I like it, didn't think I could shorten it, but here's a slightly shorter version. 2 Lines rSn()>@< and H@< for 13 – MickyT Nov 3 '15 at 22:16 • @MickyT Holy cow, that's brilliant! I'll update in a sec – ETHproductions Nov 3 '15 at 22:18 # JavaScript (ES6), 777364 61 bytes x=prompt()==1;j=a=>{if(x)setTimeout(j,9);console.log(+x)};j() Explanation: x = prompt() == 1; // this makes sure input is 1 or not while defining j = a => { // es6 arrow function if (x) // if x is 1 setTimeout(j, 9); // make sure to do this function again in 9ms console.log(+x); // log the number }; j(); // call j • Better version: x=prompt()==1;j=()=>{if(x)requestAnimationFrame(j);console.log(x?1:0)};j() – anOKsquirrel Nov 3 '15 at 19:53 • Even better, replace j=()=> with j=a=> – anOKsquirrel Nov 3 '15 at 19:59 • Actually, using x=prompt(); is shorter, but you have an x==1?1:0 that could just be x. That leaves x=prompt();j=a=>{if(x==1)requestAnimationFrame(j);console.log(x)};j() at 69 bytes. However, instead of using requestAnimationFrame you could use a setTimeout, which is shorter and leads to x=prompt();j=a=>{if(x==1)setTimeout(j,10);console.log(x)};j() at 69. Oh crap! That would mean that JS beats Stack! D: – anOKsquirrel Nov 3 '15 at 20:12 • Haha. The reason I use x==1?1:0 is so that if the user inputs "true" it counts as 0. – Florrie Nov 3 '15 at 20:50 • (oh my, it's liam4! hey!) But you don't need to. The only inputs needed 1 and 0. – anOKsquirrel Nov 3 '15 at 21:01 ## Brainfuck, 54 bytes +++[>++++[>++++<-]<-]>>>,.[>+<<[>>-<<-]>>>+<<-]>[>.<] Explanation: +++[>++++[>++++<-]<-]>>> Set a register to 48 (ASCII 0), using multiplication to reduce the byte count (344), then set the pointer to the next instruction ,. Receive and print a line of input [>+<<[>>-<<-]>>>+<<-]Set the next register to be a numerical value, and the one after to represent the ascii output value > Move the data pointer to the numerical register [>.<] While the numerical register is not 0, print the ascii register • I already have a shorter BF program, but just so you know, 48 can be made shorter. See this helpful page. – mbomb007 Nov 3 '15 at 21:44 • Thanks. Yeah, I didn't see your program before I posted mine, must have missed it when scrolling through. – Ethan Nov 3 '15 at 22:01 • Try running the code snippet contained in the question. – mbomb007 Nov 3 '15 at 22:12 # PoGo, 10 bytes ifpouftogo ## Explanation: • if - accept numerical input and place the result into the current memory cell • po - add current position in code to the top of the po stack • uf - output the value in the current memory cell as a number • to - execute the following command only if the value in the current memory cell is >0 • go - pop the most recent po location off the stack and jump there The "po" stack is a call stack used for flow control. ### The program in pseudo code: read int x do: print x while x > 0 # brainfuck, 20 bytes ,[.>+<-[-[>]<++<-]>] This requires an interpreter that exits with an error upon stepping out of bounds with < (such as this one). Stepping over the left edge in case of an even value was the easiest way I found to do a parity test. I wouldn't be surprised if it could be a couple of bytes shorter. # Batch, 44 bytes @set/p n= :a @echo %n% @if %n%==1 goto :a ## Explanation • set /p reads from stdin into a variable • :a is a GOTO marker, because batch does not have while loops The rest should be obvious: output the variable n and if n is 1, repeat. # Labyrinth, 6 bytes ?\|:!:/ In the case of a 0 input, this terminates with an error. For a beautiful solution which exits cleanly, see Sp3000's answer. Since the code is linear, the instruction pointer will move back and forth on the code (it will turn around when hitting a dead end, executing the instruction at the end only once). So what is happening? ? Read the input as an integer. \| Compute the bitwise OR of the top two stack elements (there is an infinite supply of zeroes at the bottom). This is a no-op at this point. : Duplicate the input. ! Print it as an integer. : Duplicate the input. / Divide the input by itself. If the input was 0, the interpreter will throw an error, polluting STDERR, but we can ignore that. STDOUT remains unchanged. If the input was 1, then 1/1 just yields 1 again and execution continues leftwards. : Duplicate the 1. ! Print it. : Duplicate the 1. \| Bitwise OR between 1 and 1 gives 1. ? Try reading another integer. But we're at EOF, so this pushes 0. We're in a dead end, so the IP turns around and moves back to the right. \| Bitwise OR between 1 and 0 gives 1. At this point, the state is exactly the same, as the first time we hit \|, so from here on it's an infinite loop, printing two 1s per iteration. # VBA, 54 48 Bytes Sub f(u):Do:Debug.Print u:Loop While u>0:End Sub Look Guys, VBA can fit on one line(almost) and be hard(ish) to read like all the other Languages. Debug.Print could be Msgbox but I feel that isn't the Spirit of the challenge and you really don't want Never ending pop-ups Old Code Sub f(u):Do:Debug.Print u:If u=0 Then End Loop:End Sub • Using ActiveSheet.Range("A1") as your input and running in the Immediate Window, you can reduce to 31 bytes: ?[A1]:Do Until[A1]=0:?[A1]:Loop – Chronocidal May 9 '18 at 9:40 • For sure. Any of my VBA answers are designed not to run in immediate and do not use sheets. VBA can for sure be shortened by using those methods and I encourage you to explore the possibilities. But at the time there were few to no VBA responses so I limited myself to Subs only. – JimmyJazzx May 9 '18 at 10:15 ## FORTH, 44 bytes39 bytes 31 bytes Edit: As suggested by @Nate Eldredge, we can shorten the code, if we allow extra spaces in the output. This program is 31 bytes long: : P BEGIN DUP . ?DUP 0= UNTIL ; Sample run: 1 P 0 P First version: : P BEGIN DUP 48 + EMIT ?DUP 0= UNTIL ; Sample run: 1 P 0 P Explanation: We expect the value 'b' of 0 or 1 on the stack. : P -- beginning a word P ( b ) BEGIN -- starting a loop ( b ) DUP 48 + -- creating an ASCII code for the character ( b b+48 ) EMIT -- echoing the character ( b ) ?DUP -- dup'ing the value if non-zero ( 1 1 ) or ( 0 ) 0= -- testing if the value is zero ( 1 FALSE) or ( TRUE ) UNTIL -- end of the loop if true ( 1 ) or ( - ) ; -- finishing the word If I take the challenge seriously, FORTH has Standard I/O capabilities, but it is natural in FORTH to take the input from the stack. If I use the STDIO feature, the code looks like this (44 bytes) : P KEY BEGIN DUP EMIT DUP 48 = UNTIL DROP ; Sample run: P (Note, that in my environment the standard input was buggy) • My FORTH is rusty, but wouldn't it be easier and shorter to use . instead of 48 + EMIT? – Nate Eldredge Nov 5 '15 at 19:45 • @NateEldredge . puts an extra space after the number, otherwise it's OK. I've read in some answers that it may be acceptable. If that's the case we can shorten the code. I'll update the answer. – gaborsch Nov 5 '15 at 23:12 ## MIPS asm, 24 bytes main: li $v0, 5 # load read int syscall # exec read int, stores value in$v0 move $a0,$v0 # store in $a0 li$v0, 1 # load print int loop: syscall # print $a0 bgtz$a0, loop # loop while \$a0 is greater than zero Had to learn MIPS recently, might as well do something fun with it. ## COMMAND.COM, 12 + 10 + 11 = 33 bytes 0.BAT: @ECHO 0 @EXIT 1.BAT: @ECHO 1 @1 Start COMMAND.COM with the above two files in the current directory. Then when it requests input, type either 0 or 1. Also works with CMD.EXE. If you don't want the Microsoft version and copyright then you can use CMD /K ECHO OFF instead. I've therefore added 11 bytes for this, but subtracted 4 bytes, as you no longer need the @s at the start of each line. If command-line arguments are acceptable and you are allowed use COMMAND /C 0 or COMMAND /C 1 then you can remove the @EXIT from 0.BAT. In that case the size is 7 + 11 + 1 + 1 = 20 bytes. # Marbelous, 23 This version doesn't terminate after the zero is printed, but that should be alright: ..}0 ../\// >0 }0 +O +O Old version: }0@0 \\ ../\+O >0\/+O @0 It doesn't add a newline between each 1, but whatevs. • You need to terminate if your language is at all capable of terminating. – lirtosiast Nov 18 '15 at 3:31 # Rotor, 4 bytes {}\| Contains an unprintable, so here's a hexdump: 0000000: 7b1b 7d7c {.}\| Explanation: implicit -- evaluate and push input onto stack { beginning of block ^[ prints the top value of the stack (without popping it) }\| execute block while top value is truthy implicit -- print stack # Python 2- 64 bytes x=input() if x==0:print"0" elif x==1: while True: print"1" • A trick you can exploit in Python is that 0 is the same as False and anything but 0, including 1, is True. This generally holds in most programming languages – bioweasel Nov 1 '16 at 20:00 # Jolf ## 0 Bytes I found out what to do with the zero-byte program! I made a truth machine. 0 as input 1 as input • Congrats, you have made the most boring answer on this challenge! ಠ_ಠ +1 – user48538 Jan 16 '16 at 18:13 • @zyabin101 bows thank you. – Conor O'Brien Jan 16 '16 at 18:13 ## Mathematica, 36 33 bytes For[Print[i=Input[]],i>0,Print@i] # Detour, 2 bytes ,~ Try it online! , will print a value then push it to the next cell. ~ is a filter, so it will push a value IFF it is greater than 0. Cells wrap around the edges.	2019-07-24 07:19:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18266107141971588, "perplexity": 7007.514343476417}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195531106.93/warc/CC-MAIN-20190724061728-20190724083728-00453.warc.gz"}
https://zbmath.org/?q=an:1084.32010	# zbMATH — the first resource for mathematics Discrete symmetries of systems of isomonodromic deformations of second-order Fuchsian differential equations. (English. Russian original) Zbl 1084.32010 Funct. Anal. Appl. 38, No. 2, 111-124 (2004); translation from Funkts. Anal. Prilozh. 38, No. 2, 38-54 (2004). The author studies discrete transformations of moduli spaces of logarithmic $$sl(2)$$-connections with singularities at distinct points $$\{ x_1,\dots,x_{n} \}$$ on the Riemann sphere $$P^1$$ and with given eigenvalues of the residues of the connection. Using the modification technique for vector bundles with connections this allows to compute the discrete affine group of Schlesinger transformations for isomonodromic deformations of a Fuchsian system of second order differential equations. The obtained result is applied to three examples of Fuchsian differential equations, the hypergeometric equation, the Heun equation and the sixth Painlevé equation, and therefore is the generalization of classical situation. ##### MSC: 32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation) 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies Full Text:	2021-08-02 19:52:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3634577989578247, "perplexity": 478.88824752873074}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154356.39/warc/CC-MAIN-20210802172339-20210802202339-00152.warc.gz"}
https://calculator.academy/gold-to-silver-price-ratio-calculator/	Enter the current price of gold ($/oz) and the current price of silver ($/oz) into the Gold to Silver Ratio Calculator. The calculator will evaluate and display the Gold to Silver Ratio. ## Gold to Silver Ratio Formula The following formula is used to calculate the Gold to Silver Ratio. GSR = GP / SP • Where GSR is the Gold to Silver Ratio • GP is the current price of gold ($/oz) • SP is the current price of silver ($/oz) To calculate the gold to silver price ratio, divide the price of gold by the price of silver. ## How to Calculate Gold to Silver Ratio? The following example problems outline how to calculate Gold to Silver Ratio. Example Problem #1: 1. First, determine the current price of gold ($/oz). • The current price of gold ($/oz) is given as: 1250. 2. Next, determine the current price of silver ($/oz). • The current price of silver ($/oz) is provided as: 300. 3. Finally, calculate the Gold to Silver Ratio using the equation above: GSR = GP / SP The values given above are inserted into the equation below and the solution is calculated: GSR = 1250 / 300 = 4.1667 Example Problem #2: For this problem, the variables required are provided below: current price of gold ($/oz) = 1500 current price of silver ($/oz) = 500	2023-02-08 21:11:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.512356162071228, "perplexity": 3118.878415706935}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500904.44/warc/CC-MAIN-20230208191211-20230208221211-00715.warc.gz"}
http://nrich.maths.org/public/leg.php?code=-99&cl=3&cldcmpid=740	Search by Topic Resources tagged with Working systematically similar to Funny Factorisation: Filter by: Content type: Stage: Challenge level: There are 130 results Broad Topics > Using, Applying and Reasoning about Mathematics > Working systematically American Billions Stage: 3 Challenge Level: Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3... Cuboids Stage: 3 Challenge Level: Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all? M, M and M Stage: 3 Challenge Level: If you are given the mean, median and mode of five positive whole numbers, can you find the numbers? Number Daisy Stage: 3 Challenge Level: Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25? Ben's Game Stage: 3 Challenge Level: Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters. Product Sudoku Stage: 3, 4 and 5 Challenge Level: The clues for this Sudoku are the product of the numbers in adjacent squares. Two and Two Stage: 2 and 3 Challenge Level: How many solutions can you find to this sum? Each of the different letters stands for a different number. Stage: 3 Challenge Level: How many different symmetrical shapes can you make by shading triangles or squares? Triangles to Tetrahedra Stage: 3 Challenge Level: Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all. Special Numbers Stage: 3 Challenge Level: My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be? Squares in Rectangles Stage: 3 Challenge Level: A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all? Where Can We Visit? Stage: 3 Challenge Level: Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think? Weights Stage: 3 Challenge Level: Different combinations of the weights available allow you to make different totals. Which totals can you make? Consecutive Numbers Stage: 2 and 3 Challenge Level: An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore. Medal Muddle Stage: 3 Challenge Level: Countries from across the world competed in a sports tournament. Can you devise an efficient strategy to work out the order in which they finished? A First Product Sudoku Stage: 3 Challenge Level: Given the products of adjacent cells, can you complete this Sudoku? How Old Are the Children? Stage: 3 Challenge Level: A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?" Ones Only Stage: 3 Challenge Level: Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones. First Connect Three for Two Stage: 2 and 3 Challenge Level: First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line. Consecutive Negative Numbers Stage: 3 Challenge Level: Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers? Twinkle Twinkle Stage: 2 and 3 Challenge Level: A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour. Factors and Multiple Challenges Stage: 3 Challenge Level: This package contains a collection of problems from the NRICH website that could be suitable for students who have a good understanding of Factors and Multiples and who feel ready to take on some. . . . Difference Sudoku Stage: 3 and 4 Challenge Level: Use the differences to find the solution to this Sudoku. Sociable Cards Stage: 3 Challenge Level: Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up? Isosceles Triangles Stage: 3 Challenge Level: Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw? Summing Consecutive Numbers Stage: 3 Challenge Level: Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way? Multiplication Equation Sudoku Stage: 4 and 5 Challenge Level: The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid. Reach 100 Stage: 2 and 3 Challenge Level: Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100. Stage: 3 Challenge Level: A mathematician goes into a supermarket and buys four items. Using a calculator she multiplies the cost instead of adding them. How can her answer be the same as the total at the till? Colour Islands Sudoku Stage: 3 Challenge Level: An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine. Counting on Letters Stage: 3 Challenge Level: The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern? Fence It Stage: 3 Challenge Level: If you have only 40 metres of fencing available, what is the maximum area of land you can fence off? Peaches Today, Peaches Tomorrow.... Stage: 3 and 4 Challenge Level: Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for? Stage: 3 Challenge Level: Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar". Product Sudoku 2 Stage: 3 and 4 Challenge Level: Given the products of diagonally opposite cells - can you complete this Sudoku? More Magic Potting Sheds Stage: 3 Challenge Level: The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it? More on Mazes Stage: 2 and 3 There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper. Masterclass Ideas: Working Systematically Stage: 2 and 3 Challenge Level: A package contains a set of resources designed to develop students’ mathematical thinking. This package places a particular emphasis on “being systematic” and is designed to meet. . . . Stage: 3 Challenge Level: A few extra challenges set by some young NRICH members. Magic Potting Sheds Stage: 3 Challenge Level: Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it? First Connect Three Stage: 2 and 3 Challenge Level: The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for? Football Sum Stage: 3 Challenge Level: Find the values of the nine letters in the sum: FOOT + BALL = GAME Inky Cube Stage: 2 and 3 Challenge Level: This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken? 9 Weights Stage: 3 Challenge Level: You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance? Stage: 3 Challenge Level: If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why? Sticky Numbers Stage: 3 Challenge Level: Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number? Pair Sums Stage: 3 Challenge Level: Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers? Cayley Stage: 3 Challenge Level: The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"? Crossing the Town Square Stage: 2 and 3 Challenge Level: This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares. Making Maths: Double-sided Magic Square Stage: 2 and 3 Challenge Level: Make your own double-sided magic square. But can you complete both sides once you've made the pieces?	2016-05-04 13:42:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3287305533885956, "perplexity": 1968.9888046711378}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860123077.97/warc/CC-MAIN-20160428161523-00138-ip-10-239-7-51.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/on-feketes-lemma.348077/	# On Fekete's Lemma 1. Oct 22, 2009 ### CoachZ Fekete's Lemma states that if {a_n} is a real sequence and a_(m + n) <= a_m + a_n, then one of the following two situations occurs: a.) {(a_n) / n} converges to its infimum as n approaches infinity b.) {(a_n) / n} diverges to - infinity. I'm trying to figure out a way to show either of these things happen but can't seem to do it. Does anyone have the proof of this or have suggestions to go about proving it. 2. Oct 23, 2009 ### foxjwill a) Suppose the {a_n/n} converges to some number A. Show that A is the inf.	2017-08-17 14:53:56	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9323040246963501, "perplexity": 1176.4100192085934}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886103316.46/warc/CC-MAIN-20170817131910-20170817151910-00062.warc.gz"}
https://bioinformatics.meta.stackexchange.com/questions/74/make-tags-mapping-and-read-mapping-synonyms	# Make tags [mapping] and [read mapping] synonyms The topic of read mapping is a sufficiently distinct subset of the more general class of sequence alignment topics to warrant a dedicated tag. However, we should collapse mapping and read-mapping as synonyms. • Mapping has distinct meaning in other subfields (e.g. family tree mapping and disease mapping). For clarity, we should not use "mapping" just for "read-mapping". Jun 7 '17 at 1:05 • Similarly, "mapping" is used often with "QTL mapping". Jun 7 '17 at 13:58 Yes, having both and for the same thing doesn't make sense. However, instead of making them synonyms, I would suggest we only keep and get rid of altogether. In general, tags whose meaning isn't clear from the tag are bad tags. In fact, is a classic example of a meta tag. Briefly, meta tags are those that: 1. If the tag can’t work as the only tag on a question, it’s probably a meta-tag. Every tag you use should be able to work, more or less, as the only tag on a question. Meta-tags, like [beginner], [subjective], and [best-practices], are useless by themselves — they tell you nothing at all about the content of the question. 2. If the tag commonly means different things to different people, it’s probably a meta-tag. In a cruel, ironic twist, the meaning of the tag [subjective] itself … is actually subjective. Ditto for [best-practices] and [beginner]. Best practices to whom? Beginner by what criteria? These tags are impossible to define by anything remotely resembling an objective metric. In comparison, the the meaning of tags like [java], [c#], and [javascript] are crystal clear to all but the nuttiest of nutbags. As user172818 pointed out in the comments, the term "mapping" has different meanings in different subfields. Therefore, "mapping" by itself doesn't tell us anything useful about the question; it needs to be combined with another tag (, or whatever) in order to be informative. Therefore, it is not a useful tag and should just be removed. The simplest way to remove tags is to just edit the questions that have them. If no questions are tagged with a given tag, the tag will be deleted from the system automatically. If, in future, we see the tag returning, we can ask for it to be blacklisted. Since there were only 6 or so questions tagged with , I went through them retagged those that were about accordingly. However, this question: probeset to probeset mappings between Affymetrix arrays is about mapping between different types of affymetrix arrays. Here, I removed and created a new tag . I don't think it would be worth creating something as specific as or similar. The others were all about and are now tagged accordingly. The should now disappear and I suggest we let it fade into oblivion. • I thought I had added some info about the "mapping" tag, but I see that the "read-mapping" tag did not conserve this info. Or maybe I'm mistaken. – bli Jun 8 '17 at 9:28 • @bli oh damn! I'm sorry, that's my fault! The info wouldn't be conserved no, the read-mapping tag was a new tag, not a synonym so it wouldn't bring in information from elsewhere. I didn't notice you had added information to the mapping tag's wiki when I removed it. I hope it wasn't too much work. Jun 8 '17 at 9:32 • I found the info back in my activity history: "When dealing with associating sequencing reads to a matching position in a set of reference sequences (typically a genome)." I'll add it to the "read-mapping" tag – bli Jun 8 '17 at 9:35 • @bli cool! Sorry for making you repeat your work, that's never fun :( Jun 8 '17 at 9:36 • These few words were not a lot of work :) – bli Jun 8 '17 at 9:37	2021-10-20 15:51:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3565917909145355, "perplexity": 1770.025290953169}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585322.63/warc/CC-MAIN-20211020152307-20211020182307-00341.warc.gz"}
https://dsp.stackexchange.com/questions/37714/kaiser-window-approximation/37715	# Kaiser window approximation Are there any "good" low-computational-cost approximations for parameterized Kaiser window generation suitable for small systems/languages that do not include Bessel functions in their standard library? (maybe only to 8 to 12-bit accuracy, and something that can be run in Basic on an Apple II, et.al.) • I don't know how good this technique is, but CORDIC has been adapted to compute the Kaiser window: link.springer.com/article/10.1007%2Fs11265-013-0781-z – MBaz Feb 18 '17 at 21:59 • Apple II? Computational wise LUT will probably be the lowest costing approach. Only the storage is the concern. Or may be you can truncate the polynomial Bessel series with enough terms for 12 bit accuracy. – Fat32 Feb 18 '17 at 23:46 • Kaiser is an approximation of the DPSS (discrete prolate spheroidal sequence) a.k.a. Slepian window. So do you need Kaiser in itself or would you accept something with similar performance? – Olli Niemitalo Mar 4 '17 at 8:49 You could try the exponential window: $$w_n=\frac{ \exp \left[\alpha \sqrt{1-\left(\frac{n-M}{M}\right)^2}\right]}{\exp(\alpha)}$$ $$\alpha=-427.510^{-6} A_s^2+0.1808A_s-3.516$$ or the hyperbolic cosine window: $$w_n=\frac{\cosh \left[ \alpha \sqrt{1-\left(\frac{n-M}{M}\right)^2} \right]}{\cosh(\alpha)}$$ $$\alpha=-325.110^{-6}A_s^2+0.1677A_s-3.149$$ Both have similar results with themselves and the Kaiser window, while being less computational intensive. If you want to go with Robert's answer, you could simplify the Bessel function by using Horner's method, or by applying a Gauss-Kronrod (or whatever other) integration. For example, I got this formula (after some reductions&co): $$f(x)=\frac{\left[ ~~ \cosh(x) ~+~ 2(\cosh(0.970941817426052x) ~+~ \cosh(0.8854560256532099x) ~+~ \cosh(0.7485107481711011x) ~+~ \cosh(0.5680647467311558x) ~+~ \cosh(0.3546048870425356x) ~+~ \cosh(0.120536680255323x)) \right]}{13}$$ For $\beta=12.26526$ ($A_s=120dB$), results in a these values: $I_0(\beta)=24430.40185694905$ $f(\beta)=24430.4018627702$ which, IMHO, is pretty good and it's faster than using giga-km long numbers you'd deal with in the case of the classic formula, either as a sum or as Horner -- either way, I think even long double would not be enough for 20 terms (which is how I've truncated it). • I implemented your Bessel function f(x) and get values that match the numpy.i0() function in Python. But they seem unrelated to the Bessel values in Robert's code. From Robert's code with K=16 I get values that start at 1 and wiggle down, like: mathworld.wolfram.com/BesselFunctionoftheFirstKind.html But your function looks more like a parabola than a Bessel function. oreilly.com/library/view/numpy-15/9781849515306/ch07s32.html But maybe that results in a better Kaiser window that does not have zero crossings... – philburk Aug 19 at 17:30 • What I did was to first express the Bessel Io(x) as its integral representation, $\frac{1}{\pi}\int_0^{\pi}\cosh(x\cos\theta)d\theta$, then used that with Gauss-Kronrod (or Gauss-Legendre? don't remember). If you expand Robert's series, you get monstruous numbers, even with Horner, which makes it difficult to preserve accuracy. Also, IIRC, you need some 20 terms for the series expansion to have decent approximation for $\beta\ge 120dB$. – a concerned citizen Aug 20 at 6:46 • I'm curious why your Bessel function looks so different than Robert's. Yours starts at one and goes up like a parabola. Robert's goes down and has zero crossings. Are they different kinds of Bessel functions? Is one better for the Kaiser window. (BTW, I used your hyperbolic cosine window from above and it worked great. Thanks.) – philburk Aug 20 at 18:54 • @philburk, the Kaiser window is based on the 0th-order Bessel function of the first kind. i think that's well-defined. – robert bristow-johnson Aug 21 at 23:07 just implement the Modified Bessel function. it's easy. i always like my window definitions centered about zero, since pretty much all of them are even symmetry. i'll do this in discrete-time, but it's essentially the same thing in continuous-time. Kaiser window: $$w[n] \triangleq \begin{cases} \frac{1}{I_0(\beta)} I_0\left(\beta \sqrt{1 - \left(\frac{n}{M/2}\right)^2} \right) \quad \quad & \|n\| \le M/2 \\ 0 & \|n\|>M/2 \\ \end{cases}$$ $$I_0(x)$$ is the 0th-order modified Bessel function of the 1st kind. $$M+1$$ is the number of non-zero samples or FIR taps (the FIR filter order is $$M$$ and, in my centered and symmetrical case, must be even). $$\beta$$ is a "shape parameter" and O&S recommend this heuristic: $$\beta = \begin{cases} 0.1102 \cdot (A-8.7) & A>50 \\ 0.5842 \cdot (A-21)^{2/5} + 0.07886 \cdot (A-21) \quad & 21 \le A \le 50 \\ 0.0 & A<21 \\ \end{cases}$$ $$M = 2 \left\lceil \frac{A-8}{4.57 \cdot \Delta\omega} \right\rceil$$ $$A$$ is the desired stopband attention in dB and $$\Delta\omega$$ is the desired width of the transition band in normalized angular frequency. finally, the Bessel is evaluated as: $$I_0(x) = 1 \ + \ \lim_{K \to \infty} \ \sum\limits_{k=1}^{K} \left(\frac{x^2}{4}\right)^{k} (k!)^{-2}$$ when you evaluate this with a computer, pick a $$K$$ decently large (my guess is that $$K=32$$ is good enough) and evaluate the summation starting with $$k=K$$ and work it backwards to $$k=1$$ to keep numerical accuracy. you might want to use Horner's method. $$I_0(x) \approx 1 + x^2\left( \tfrac{1}{(1!)^2 \, 4^1} + x^2\left(\tfrac{1}{(2!)^2 \, 4^2} + x^2\left(... + \, x^2\left(\tfrac{1}{((K-1)!)^2 \, 4^{K-1}} + x^2 \tfrac{1}{(K!)^2 \, 4^K} \right) \right) \right) \right)$$ you can evaluate all of the $$(k!)^{-2}$$ in advance with a short table. Alright, someone made me do some work. This took about 45 minutes to code up and debug. So here is my MATLAB code for implementing the 0th-order Bessel function of the first find (which is $$I_0(x)$$ above): function y = mybessel(x) % % Computes the 0th-order Modified Bessel function of the first kind % K = 32; bessel_coef = zeros(1,K); kfac = 1; two_to_the_k = 1; for k = 1:K kfac = kfac * k; two_to_the_k = two_to_the_k * 2; bessel_coef(k) = 1/(kfactwo_to_the_k)^2; % compute power series coefficients in advance end x = x.^2; y = x . bessel_coef(K); for k = K-1:-1:1 y = x .* (bessel_coef(k) + y); % Horner's method end y = 1 + y; end and here is the test code: x = linspace(-16, 16, 32*4096+1); I_0 = besseli(0, x); % MATLAB's modified bessel y = mybessel(x); % my bessel figure(1) plot(x, I_0) hold on plot(x, y) hold off figure(2) plot(x, y - I_0) % plot error with results: and error: for $$\|x\| \le 16$$, the relative error is less than $$10^{-15}$$. the error increases with increasing $$\|x\|$$. • I tried the Horner's method from above to calculate a Bessel approximation. It works great for small numbers < 1.0. But my beta at 90 dB is 8.959. When I calculate the Bessel approximation for numbers that large then it gets pretty wild. Here are Bessel values for K = 7,8,9. 7, -4100.13715 8, 1284.70023 9, -318.22047 – philburk Aug 17 at 17:42 • well @philburk , of course the values of $\left(-\frac{x^2}{4}\right)^{k}$ will be increasing with increasing $k$, but they are divided by $(k!)^2$ which i think is increasing even faster. so the terms should start to converge. – robert bristow-johnson Aug 17 at 20:40 • If x >= k then the numerator is significantly larger than the denominator. My x is 8.959, so a K=8 is not close to converging. I found that a K of 16 was needed for the direct summation method, and a K of 18 was needed for the Horner's method. – philburk Aug 19 at 16:41 • Another thing I find puzzling. This Bessel function has three zero crossings before it reaches x=8.959. That means the Kaiser window will also have three zero crossings. Also what happens if i0(beta) is zero, or very close to zero? – philburk Aug 19 at 17:15 • @philburk Something must be wrong, because Io(0)=1 and the function keeps on going up, there should not be any zero crossings. – a concerned citizen Aug 20 at 6:49	2019-11-18 03:20:04	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 20, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7710551619529724, "perplexity": 1602.3561191707668}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669431.13/warc/CC-MAIN-20191118030116-20191118054116-00377.warc.gz"}
https://mran.microsoft.com/snapshot/2020-12-04/web/packages/singcar/vignettes/singcar_vignette.html	# singcar: Comparing Single Cases to Small Samples library(singcar) # Introduction The aim of the R package singcar is to provide and encourage usage of appropriate statistical methods for comparing a case against a control sample. For instance, they may commonly be done in a neuropsychological context, in which an individual has incurred a specific brain injury and we wish to test whether this damage has led to an impairment of some cognitive function and whether two different functions are dissociable. For many functions there is normed data available which the patient can be compared against directly. However, when this is not possible a control sample estimating the population, against which we wish to compare the patient, must be used. Both frequentist and Bayesian methods have been developed to do this providing transparent control over Type I errors, first and foremost by John Crawford and Paul Garthwaite (Crawford et al., 2011; Crawford & Garthwaite, 2005, 2002, 2007; Crawford & Howell, 1998). It is these methods that singcar implements. Due to the somewhat overlooked issue of Type II errors power calculators for these tests are also provided. Although the canonical applications for these tests are in Cognitive Neuropsychology or Clinical Neuropsychology, they are potentially applicable to any circumstance in which a measure taken from a single individual is to be compared against data from a normative sample (i.e. a control group). It should be noted that these statistical methods could also be applied as a general method of outlier detection in small samples. ## Installation You can install the developmental version of singcar by running the following: install.packages("devtools") library("devtools") install_github("jorittmo/singcar") library("singcar") ## Example The package comes with the dataset size_weight_illusion, a neuropsychological dataset from an investigation of the size-weight illusion in DF, a patient with visual form agnosia following following bilateral lesions to the lateral occipital complex (Hassan et al., 2020). It was investigated whether DF experienced visual size-weight illusion to the same extent as controls (n = 28) and whether visual and kinesthetic size-weight illusion could be dissociable. Below follows examples of how to analyse this dataset using the tests provided in singcar. ### Testing for a deficit If we want to assess whether DF has an impairment compared to controls on visual size-weight illusion we can test this using a modified two-sample t-test, called TD (test of deficit: Crawford & Howell, 1998). # Extracting scores from the visual size-weight illusion from size_weight_illusion DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient CON_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls TD(case = DF_V_SWI, controls = CON_V_SWI, conf_int = TRUE) #> #> Crawford-Howell (1998) t-test #> #> data: case = 0.03 and controls (M = 0.16, SD = 0.08, N = 28) #> t = -1.7243, df = 27, p-value = 0.04804 #> alternative hypothesis: true difference between case and controls is less than 0 #> sample estimates: #> Standardised case score (Z-CC), 95% CI [-2.34, -1.15] #> -1.754857 #> Proportion below case (%), 95% CI [0.95, 12.47] #> 4.804003 This can similarly be tested with a Bayesian version of the same test, yielding approximately (since this test is based on MCMC methods) the same output (Crawford & Garthwaite, 2007). # Extracting scores from the visual size-weight illusion from size_weight_illusion DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient CON_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls BTD(case = DF_V_SWI, controls = CON_V_SWI) #> #> Bayesian Test of deficit by Crawford and Garthwaite (2007) #> #> data: case = 0.03 and controls (M = 0.16, SD = 0.08, N = 28) #> est. z = -1.7404, df = 27, p-value = 0.04788 #> alternative hypothesis: true difference between case and controls is less than 0 #> sample estimates: #> Std. case score (Z-CC), 95% credible interval [-2.34, -1.16] #> -1.754857 #> Proportion below case (%), 95% credible interval [0.96, 12.36] #> 4.787751 If the control sample for a study is not appropriately matched to the case on variables such as e.g. age or education level it is appropriate to use tests that account for this by allowing for the inclusion of covariates. Including theoretically sound covariates is often a good idea. To do this Crawford et al. (2011) extended their Bayesian verison of the TD. This test assess the patient on the task of interest by essentially comparing him/her to the controls with the same score on the covariate. # Extracting scores from the visual size-weight illusion from size_weight_illusion DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient CON_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls # Extracting the coviariate below DF_age <- size_weight_illusion[size_weight_illusion$PPT == "DF", "YRS"] # Patient CON_age <- size_weight_illusion[size_weight_illusion$PPT != "DF", "YRS"] # Controls BTD_cov(case_task = DF_V_SWI, case_covar = DF_age, control_task = CON_V_SWI, control_covar = CON_age, iter = 100) #> #> Bayesian Test of deficit with Covariates #> #> data: case = 0.03 and controls (M = 0.16, SD = 0.08, N = 28) #> est. z = -1.6948, df = 26, p-value = 0.05272 #> alternative hypothesis: true difference between case and controls is less than 0 #> sample estimates: #> Std. case score (Z-CCC), 95% credible interval [-2.30, -1.15] #> -1.749556 #> Proportion below case (%), 95% credible interval [1.07, 12.48] #> 5.271545 ### Testing for a dissociation If we want to assess whether DF has a dissociation between two functions we can use a modified paired samples t-test to assess the size of the difference between the case scores from the two tasks to the distribution of differences between the tasks in the controls. This can however only be done directly using the t-distribution if the tasks are measured on the same scale and is called the unstandardised difference test (UDT: Crawford & Garthwaite, 2005). In the size_weight_illusion dataset it is possible to use this test to whether patient DF exhibits a dissociation between visual size-weight illusion and kinesthetic size-weight illusion because the visual and kinaesthetic conditions are parallel versions of the same task, with different sensory cues. This would be done as shown below: # Extracting scores from the visual size-weight illusion from size_weight_illusion DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient CON_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls # Extracting scores from the kinesthetic size-weight illusion from size_weight_illusion DF_K_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "K_SWI"] # Patient CON_K_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "K_SWI"] # Controls UDT(case_a = DF_V_SWI, case_b = DF_K_SWI, controls_a = CON_V_SWI, controls_b = CON_K_SWI) #> #> Unstandardised Difference Test #> #> data: Case score A: 0.03, Case score B: 0.10, Controls A (mean, sd): (0.16, 0.08), Controls B (mean, sd): (0.18, 0.10) #> t = -0.6667, df = 27, p-value = 0.5106 #> alternative hypothesis: true difference between tasks is not equal to 0 #> sample estimates: #> Standardised case score, task A (Z-CC) #> -0.13647439 #> Standardised case score, task B (Z-CC) #> -0.07931545 #> Standardised task discrepancy (Z-DCC), 95% CI [-1.53, -0.59] #> -1.06478887 #> Proportion below case (%), 95% CI [6.35, 27.68] #> 25.53097678 Most often this is not possible because we wish to estimate abnormality of discrepancy on tasks that are not comparable. So otherwise, that is if the scores must be standardised to be comparable, a statistic that approximates the t-distribution has been developed and should be used (the revised standardised difference test RSDT: Crawford & Garthwaite, 2005). The visual and kinesthetic size-weight illusion will be used for illustrative purposes here as well: # Extracting scores from the visual size-weight illusion from size_weight_illusion DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient CON_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls # Extracting scores from the kinesthetic size-weight illusion from size_weight_illusion DF_K_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "K_SWI"] # Patient CON_K_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "K_SWI"] # Controls RSDT(case_a = DF_V_SWI, case_b = DF_K_SWI, controls_a = CON_V_SWI, controls_b = CON_K_SWI) #> #> Revised Standardised Difference Test #> #> data: Case score A: 0.03, Case score B: 0.10, Controls A (mean, sd): (0.16, 0.08), Controls B (mean, sd): (0.18, 0.10) #> approx. abs. t = 1.015, df = 27, p-value = 0.3191 #> alternative hypothesis: true difference between tasks is not equal to 0 #> sample estimates: #> Standardised case score, task A (Z-CC) #> -1.7548574 #> Standardised case score, task B (Z-CC) #> -0.7836956 #> Standardised task discrepancy (Z-DCC) #> -1.0647889 #> Proportion of control population with more extreme task difference #> 15.9560625 A Bayesian version of this test was also developed (Crawford & Garthwaite, 2005), however, unlike TD and BTD the RSDT and BSDT (Bayesian standardised difference test) differ somewhat and BSDT has been shown to keep a better control of Type I errors if a patient exhibits extreme deficits on both tasks of interest. Therefore the BSDT is recommended above RSDT. The usage of the two R functions is very similar. Since the BSDT is based on MCMC methods it can be quite computationally intensive, depending on the number of iterations you choose. # Extracting scores from the visual size-weight illusion from size_weight_illusion DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient CON_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls # Extracting scores from the kinesthetic size-weight illusion from size_weight_illusion DF_K_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "K_SWI"] # Patient CON_K_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "K_SWI"] # Controls BSDT(case_a = DF_V_SWI, case_b = DF_K_SWI, controls_a = CON_V_SWI, controls_b = CON_K_SWI, iter = 1000) #> #> Bayesian Standardised Difference Test #> #> data: Case score A: 0.03, Case score B: 0.10, Controls A (mean, sd): (0.16, 0.08), Controls B (mean, sd): (0.18, 0.10) #> est. z = -1.0378, df = 26, p-value = 0.323 #> alternative hypothesis: true difference between tasks is not equal to 0 #> sample estimates: #> Standardised case score, task A (Z-CC) #> -1.7548574 #> Standardised case score, task B (Z-CC) #> -0.7836956 #> Standardised task discrepancy (Z-DCC), 95% credible interval [-1.68, -0.43] #> -1.0647889 #> Proportion of control population with more extreme task difference, 95% credible interval [4.61, 33.54] #> 16.1489131 Just as for BTD a version of BSDT allowing for covariates has been developed. This test assess the patient on the discrepancy between the tasks of interest by essentially comparing him/her to the controls with the same score on the covariate. # Extracting scores from the visual size-weight illusion from size_weight_illusion DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient CON_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls DF_K_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "K_SWI"] # Patient CON_K_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "K_SWI"] # Controls # Extracting the coviariate below DF_age <- size_weight_illusion[size_weight_illusion$PPT == "DF", "YRS"] # Patient CON_age <- size_weight_illusion[size_weight_illusion$PPT != "DF", "YRS"] # Controls BSDT_cov(case_tasks = c(DF_V_SWI, DF_K_SWI ), case_covar = DF_age, control_tasks = cbind(CON_V_SWI, CON_K_SWI), control_covar = CON_age, iter = 1000) #> #> Bayesian Standardised Difference Test with Covariates #> #> data: Case score A: 0.03, Case score B: 0.10, Controls score A: 0.16, Controls score B: 0.18 #> ave. z = -1.0173, df = 25, p-value = 0.334 #> alternative hypothesis: true difference between tasks is not equal to 0 #> sample estimates: #> Standardised case score, task A (Z-CC) #> -1.754857 #> Standardised case score, task B (Z-CC) #> -0.783696 #> Standardised task discrepancy (Z-DCCC), 95% credible interval [-1.66, -0.38] #> -1.064152 #> Proportion of control population with more extreme task difference, 95% credible interval [4.82, 35.14] #> 16.700000 All of the functions above can also take summary (mean, sd, control sample size) data as input. ### Power calculators A further capacity of singcar is that it can be used to calculate power for for these single case-control comparisons. Calculations for all Bayesian tests and RSDT are simulation based and (especially the tests with covariates) can be computationally intense. Calculators for TD and UDT (unstandardised difference test) are exact (their power functions have been derived analytically) and can both be used to find a specific sample size given a desired power. For the other calculators all parameters must be given. Means and standard deviations for the control population are at default set to 0 and 1 meaning that the case value will be interpreted as differences from the mean in standard deviations, these parameter values can be changed as you like. Examples are given below: TD_power(case = -2, power = 0.8, mean = 0, sd = 1, alternative = "two.sided") #> Power (0.44280) will not increase more than 0.5% for any additional participant over n = 16 #> n power #> 1 16 0.4428042 TD_power(case = 70, sample_size = 10, mean = 100, sd = 15, alternative = "less", alpha = 0.1) #> [1] 0.7039033 RSDT_power(case_a = 70, case_b = 20, mean_a = 100, mean_b = 25, sd_a = 15, sd_b = 10, sample_size = 10) #> [1] 0.5525 # Takes long time to compute therefore iterations and number of simulations are low. Iter corresponds # to number of simulations in BTD_cov, nsim to the number of simulations in the power calculator. BTD_cov_power(case = -2, case_cov = 0, control_task = c(0, 1), control_covar = c(0, 1), cor_mat = diag(2), sample_size = 10, nsim = 50, iter = 50) #> [1] 0.48 ## Functions The main functions of singcar are hitherto: Frequentist tests: • TD(): The test of deficits. Used to test for abnormality on a single variate. • UDT(): The unstandardised difference test. Used to test for discrepancy between two variates measured on the same scale. • RSDT(): The revised standardised difference test. Used to test for discrepancy between two variates measured on different (or the same) scale. Bayesian tests: • BTD(): Bayesian test of deficit. Used to test for abnormality on a single variate. • BSDT(): Bayesian standardised difference test. Used to test for discrepancy between two variates measured on the same scale or different scales (depending on the unstandardised argument). • BTD_cov(): Bayesian test of deficit with covariates. Used to test for abnormality on a single variate conditioned on some covariate such as e.g. age or education level. • BSDT_cov(): Bayesian standardised difference test with covariates. Used to test for discrepancy between two variates conditioned on some covariates such as e.g. age or education level. Power calculators: • TD_power(): Calculates exact power given sample size or necessary sample size for desired power using analytical methods for the test of deficit. • BTD_power(): Calculates approximate power given sample size using Monte Carlo simulations for the Bayesian test of deficit. • BTD_cov_power(): Calculates approximate power given sample size using Monte Carlo simulations for the Bayesian test of deficit with covariates. • UDT_power(): Calculates exact power given sample size or necessary sample size for desired power using analytical methods for the unstandardised difference test. • RSDT_power(): Calculates approximate power given sample size using Monte Carlo simulations for the revised standardised difference test. • BSDT_power(): Calculates approximate power given sample size using Monte Carlo simulations for the Bayesian standardised difference test. • BSDT_cov_power(): Calculates approximate power given sample size using Monte Carlo simulations for the Bayesian standardised difference test with covariates. # Details ## Statistical methods for finding deficits and dissociations The aim of this section is not to provide an exhaustive mathematical understanding of the formulas used but rather a conceptual understanding. This is written from a neuropsychological perspective. However, that does not make the tests exclusive to such an application. ### Frequentist approaches In the latter part of the 90’s Crawford and colleagues started developing statistical tests to use for evaluating single cases that were compared to normative control samples. Typically, the prior methods used treated the distribution estimated by the the control group as if the sample statistics were the population parameters. That is, the estimated distribution was treated as a standard normal distribution from which abnormality of the case score was estimated by: $$$z = \frac{x^* - \overline{x}}{\sqrt{s^2}} \tag{1}$$$ This is similar to the familiar z-formula but here $$x^$$, $$\overline{x}$$ and $$s^2$$ is the case score, sample mean and sample variance respectively. $$\overline{x}$$ and $$s^2$$ is plugged in directly as the population parameters $$\mu$$ and $$\sigma^2$$ in the normal z-formula. The p-value obtained from the z-value would then be treated as the estimation of the case’s abnormality. This is problematic because the sampling distribution of $$s^2$$ is right skewed for small sample sizes. This means that underestimation of $$s^2$$ is more probable than overestimation and hence the z-value would often be larger than it should, resulting in an overestimation of the abnormality and inflation of Type I errors (claiming that there is an effect when, in fact, there is not) (Crawford & Howell, 1998). With a similar logic as in (1), Payne & Gwynne Jones (1957) developed a method for assessing abnormally large discrepancies between two tasks. I.e. a test that estimates the proportion of the control population that would exhibit a greater discrepancy than the case, as seen in (2). $$$z_{disc} = \frac{(z^_a - z^_b) - {(\overline{z}_a - \overline{z}_b)} }{\sqrt{2-2r_{ab}}} = \frac{(z^_a - z^_b)}{\sqrt{2-2r_{ab}}} \tag{2}$$$ Where $$z^_a$$ and $$z^_b$$ are the standardised case scores on task A and B respectively, $$\overline{z}_a$$ and $$\overline{z}_b$$ the means from the sample on the two tasks (which both equates 0 because of standardisation) and $$r_{ab}$$ the correlation between the two tasks calculated from the sample scores. However, this test suffers from the same problem mentioned above and would overestimate the abnormality of the task discrepancies. A different approach to comparing a single observation to the mean of a sample was proposed by Sokal & Rohlf (1981) (p. 227) and popularised within neuropsychology by Crawford & Howell (1998). Here the t-distribution (with its fatter tails) is utilised to account for the underestimation of the sample variance. The approach is a modified two samples t-test where the case simply is treated as a sample of size 1. The degrees of freedom for this distribution is $$n + 1 - 2 = n - 1$$. $$$t_{n-1} = \frac{X^ - \overline{X}}{s \sqrt{\frac{n + 1}{n}}} \tag{3}$$$ This test of deficit (TD) has been shown to not exceed the specified error rate $$\alpha$$ unlike other similar tests (Crawford et al., 2004, 2009; Crawford & Garthwaite, 2012). Together with its simplicity this makes it a superior choice over many other ways of detecting outliers in small samples. One of its main advantages is that it provides the researcher with an unbiased point estimate of the abnormality of the case. Crawford et al. (1998) extended this to Payne & Gwynne Jones (1957) test of task discrepancy or with Crawford et al. (1998)'s denotation: ‘difference’1. I.e. they devised a test that treated sample estimations as statistics rather than population parameters for dissociations as well, seen in (4). $$$t_{n-1} = \frac{(z^_a - z^_b) {- (\overline{z}_a - \overline{z}_b)} }{\sqrt{(2-2r_{ab})(\frac{n+1}{n})}} = \frac{(z^_a - z^_b)}{\sqrt{(2-2r_{ab})(\frac{n+1}{n})}} \tag{4}$$$ Unfortunately the standardised task scores of the case $$z^_a$$ and $$z^_b$$ suffer from the same problem described for (1) and Type I errors would again be inflated. However, standardisation of the scores is only necessary if the two tasks are measured on different scales. If they are measured on the same the test holds and we have the unstandardised difference test (UDT): $$$t_{UDT_{n-1}} = \frac{(x^_a - \overline{x}_a) - (x^_b - \overline{x}_b) }{\sqrt{(s^2_a +s^2_b -2s_a s_br_{ab})(\frac{n+1}{n})}} \tag{5}$$$ The denominator in the (5) collapse to the denominator in (4) since $$s_a^2$$ and $$s_b^2$$ become 1 after standardisation. However, since assessment of task discrepancy between tasks measured on different scales is common, a test that could take standardised scores but still account for the skewness in the sampling distribution of the sample variance was needed. In Garthwaite & Crawford (2004) the authors examined the difference between two correlated, t distributed variables and aimed to derive a quantity with a distribution that would not depend on any population parameters. The math behind this derivation is too technical to be covered here, but in summation they used asymptotic expansion to find a function of the correlation between the variables that when used as a denominator to $$(t_1 - t_2)$$, where $$t_1$$ and $$t_2$$ are our t distributed variables, would approximate a t-distribution. The quantity found was: $$$\psi=\frac{\frac{(x^_a-\overline{x}_a)}{s_{a}}-\frac{(x^_b-\overline{x}_b)}{s_{b}}}{ \sqrt{ (\frac{n+1}{n}) \left( (2-2 r)+ \frac{2(1-r^{2})}{n-1}+ \frac{(5+y^{2})(1-r^{2})}{2(n-1)^{2}}+ \frac{r(1+y^{2})(1-r^{2})}{2(n-1)^{2}}\right) }}$$$ Where $$r$$ is the correlation between the tasks and $$y$$ the critical two-tailed t-value with $$n-1$$ degrees of freedom. Garthwaite & Crawford (2004) demonstrate that $$\mathbb{P}[\psi > y]\approx\mathbb{P}[t >y]$$, where $$\approx$$ indicates approximate equivalence. To obtain a precise probability for $$\psi$$ one solves for $$\psi = y$$. See Garthwaite & Crawford (2004) and Crawford & Garthwaite (2005) for details. Choosing the positive root of $$\psi = y$$ yields: \begin{align} \begin{split} y & = \sqrt{\frac{ -b + \sqrt{b^2 - 4ac}}{2a}}, \text{where} \\ a & = (1+r)(1-r^2), \\ b & = (1-r)[4(n-1)^2+4(1+r)(n-1)+(1+r)(5+r)], \\ c & = - 2\left[\frac{X^_{A} - \overline{X}_A}{s_A}-\frac{X^_B -\overline{X}_B}{s_B}\right]^2\left(\frac{n(n-1)^2}{n+1}\right) \end{split} \tag{6} \end{align} Where $$y$$ is used as a t-statistic. This quantity is referred to as the revised standardised difference test (RSDT). Crawford & Garthwaite (2005) show with Monte Carlo simulations that this test is superior to both their own previous test (Crawford et al., 1998) and Payne & Gwynne Jones (1957)'s in controlling Type I errors. Even for very small sample sizes of $$n=5$$, RSDT was shown to barely exceed the specified 5% error rate. So three valid frequentist tests remain, for deficits that is TD (3), for dissociations that is UDT (5) and RSDT (6). For TD and UDT, Crawford & Garthwaite (2002) utilise the non-central t-distribution to set confidence limits on the abnormality of the case, something that grows ever more essential for reminding us that all test results come with uncertainty. However, it did not prove possible to set confidence limits on the estimations from the RSDT since it is only approximately t-distributed. This is one of the reasons why Crawford & Garthwaite (2007) wanted to develop a Bayesian method for estimating abnormality of discrepancy. The test of deficit is most commonly used one-sided but the dissociation tests should in most cases be used two-sided as the direction of the effect solely depends on the order of the task scores. ### Bayesian approaches There is one main difference between Bayesian and frequentist statistical inference. In the Bayesian framework parameters (like means and standard deviations) are treated as random variables with associated probability distributions and if we believe a parameter has a certain distribution we update that belief as more data is gathered and thus the parameter values can change. Whilst, in the frequentist framework, parameters are treated as fixed attributes of a population, estimations of which in a series or frequency (hence the name) of trials will converge to the true value. An intuitive way of understanding the difference is to note how the two approaches phrase intervals around parameters. The frequentist 95% confidence interval would cover the population parameter 95% of the time. That is, if you estimate a population mean from 100 different samples and create a confidence interval around each estimation, 95 of these intervals would include the true population mean. The Bayesian 95% credible interval, however, has a more intuitive interpretation. For this interval you say that with a 95% certainty the population mean is covered by the interval. That is because you take the values at the 2.5th and 97.5th percentile of the parameter distribution to form the interval. To estimate a parameter distribution Bayesians use prior knowledge of that parameter, i.e. they assign probabilities to possible values of the parameter depending on this knowledge — forming what is known as a prior distribution, or simply a prior. If no information exists one often apply a non-informative prior, the most simple of which assigns equal probabilities (uniform) to all possible parameter values. This is then updated when new information is obtained. The distribution formed by the updated prior is called the posterior distribution. The posterior probability of a hypothesis (i.e. a specified value of the parameter) is calculated by using Bayes theorem: $$$\underbrace{\mathbb{P}[H \ \|\ E]}_{\text{Posterior}} = \frac{\overbrace{\mathbb{P}[E \ \| \ H]}^{\text{Likelihood}} \times \overbrace{\mathbb{P}[H]}^{\text{Prior}}}{\underbrace{\mathbb{P}[E]}_{\text{Marginal likelihood}}} \tag{7}$$$ The $$H$$ stands for hypothesis which may be affected by $$E$$, evidence, i.e. the data (not used to estimate the prior). $$\mathbb{P}[E \ \| \ H]$$ is the probability of observing the data, given a hypothesis. $$\mathbb{P}[H \ \|\ E]$$ is the probability of a hypothesis given that some data have been observed. Say for example that we want to estimate IQ in a population with a sample of $$n=3$$. By calculating the mean, standard deviation and standard error from this sample we can create a sampling distribution of the mean. The marginal likelihood, $$\mathbb{P}[E]$$ will be a constant since it does not contain $$H$$, we can therefore disregard that for now. If we assume a uniform prior, i.e. we do not believe that any hypothesis is more likely than another, $$\mathbb{P}[H]$$ will also be a constant, reducing (7) to: $\begin{equation} \mathbb{P}[H \ \| \ E] = \mathbb{P}[E \ \| \ H] \end{equation}$ Say that our sample had IQs of $$x_1 = 110, \ x_2= 115, \ x_3 = 120$$. The sample has a mean of 115, a standard deviation of 5 and a standard error of $$5/\sqrt{n}$$. Our theoretical sampling distribution of the mean (i.e. the distribution of means if we draw a sample of $$n=3$$ an infinite number of times) would thus be a normal distribution with mean 115 and standard deviation = standard error = $$5/\sqrt{n}$$. To get the posterior distribution we now calculate the probability of observing the data given another hypothesised mean. To get the posterior probability for any hypothesis, say $$\mu = 118$$, we thus calculate: \begin{align} \begin{split} \mathbb{P}[H = 118 \ \| \ E = (110, 115, 120)] = & \mathbb{P}[E = (110, 115, 120) \| \ H = 118] = \\ & \mathbb{P}[E = 110 \ \| \ H = 118] \times \\ & \mathbb{P}[E = 115 \ \| \ H = 118] \times \\ & \mathbb{P}[E = 120 \ \| \ H = 118] \end{split} \end{align} That is, calculating the joint probability of observing $$x_1, \ x_2 \ \text{and} \ x_3$$ given that our observations instead would have come from a distribution with a mean of 118. This is then done for all possible hypotheses, i.e. values of $$\mu$$, creating a probability distribution of the parameter. The peak of this distribution would in fact be the maximum likelihood estimate of the mean, which is used in more classical estimation methods. Hence, using a non-informative prior often yields estimations with frequentist properties2. However, if we for example have on good authority that mean IQ in the sampled population is 125 with an SD of 5, we specify our prior as such. This is then weighed in when calculating the posterior. That is, we calculate the joint probability that we have observed our data given a hypothesis ($$\mu$$) and the probability of observing that $$\mu$$ in the prior distribution. This is done for all possible hypotheses, just as previously shown. Often one wants to use non-informative priors as to not arbitrarily bias the results. For an accessible and more thorough explanation of Bayesian statistics, see e.g. Donovan & Mickey (2019). The example given here is a special case of Bayesian parameter estimation that can be solved analytically.3 However, this is in many cases not possible. Instead one utilise Markov Chain Monte Carlo (MCMC) methods. A Markov Chain describe a series of events where the probability for each of them solely depends on the one preceding it. Monte Carlo methods are mathematical algorithms that solves problems by random generation of numbers. The idea behind this in Bayesian statistics is that one estimates the posterior by drawing a large amount of random samples from it. Thus “building” it from scratch. Because the denominator in (7) is a constant and we can ignore it, this becomes possible. Bayes theorem can then be rewritten as: $$$posterior \ \propto \ likelihood \times prior$$$ What this is saying is that the posterior density of a hypothesis is proportional ($$\propto$$) to the likelihood of the data under that hypothesis times the prior density of the hypothesis. The methods for drawing these samples differ depending on the type of distribution and problem at hand. But in general they are all building on algorithmic rules (“recipes”) of drawing random numbers based on the likelihood and the prior, saving them and after a large number of iterations observing the distribution they form. The average of which often is the parameter of interest. #### The Bayesian test of deficit Assume a sample of $$n$$ controls on which we measure some value x that is normally distributed with mean $$\mu$$ and variance $$\sigma^2$$. Let $$\overline{x}$$ and $$s^2$$ denote the sample mean and sample variance respectively. The case is denoted $$x^$$. The prior used is non-informative, see Crawford & Garthwaite (2007) and DeGroot & Schervish (2012) (p. 495) for the formal specification of the prior. The algorithm developed in Crawford & Garthwaite (2007) for obtaining a point estimate of a case’s abnormality $$p$$, i.e. a p-value or the proportion of controls that would fall below the case and accompanying intervals is as follows: 1. Let $$\psi$$ be a random draw from a $$\chi^2$$-distribution on $$n-1 \ df$$. Then let $$\hat{\sigma}^2 = \frac{(n-1)s^2}{\psi}$$ be the estimation of $$\sigma^2$$ for this iteration. 2. Let $$z$$ be a random draw from a standard normal distribution. Then let $$\hat{\mu}=\overline{x}+z\sqrt{(\hat{\sigma}^2/n)}$$ be the estimate of $$\mu$$ for this iteration. 3. With estimates of $$\mu$$ and $$\sigma$$, $$p$$ is calculated conditional on these estimates being the correct $$\mu$$ and $$\sigma$$ by calculating $$z^= \frac{x^* - \hat{\mu}}{\sqrt{\hat{\sigma}^2}}$$. Let $$\hat{p}_i =\mathbb{P}[Z < z^]$$ be the estimate of $$p$$ for this iteration. That is the probability of drawing a value less than $$z^$$ from a standard normal distribution. 4. Repeating these steps a large number of times will yield a distribution of $$\hat{p}$$, the average of which is the point estimate of $$p$$. If repeated e.g. 1000 times, the 25th smallest and 25th largest $$\hat{p}_i$$ ($$i$$ for iteration) is the lower and upper boundaries of the 95% Bayesian credible interval for $$p$$. Crawford & Garthwaite (2007) show that this method yields converging results to that of TD (3). As noted, this is often the case when using a non-informative prior, however, not always. For example, RSDT and its Bayesian analogue (BSDT) do not produce identical results. Crawford & Garthwaite (2007) showed that when there was no discrepancy between task A and B, but the case was severely impaired on both of them, RSDT exhibited an error rate much larger than the specified $$\alpha$$-level. For example, when the case had a deficit of 8 SD on both task A and B but no discrepancy, RSDT gave a false positive (Type I error) in $$34.72$$% of the simulations ($$n = 10$$). The BSDT on the other hand gave a false positive under the same circumstances in $$8.29$$% of the simulations. When there was no deficit on either task BSDT had a Type I error rate of $$7.32$$%. compared to RSDT with an error rate of $$4.6$$%. It is also argued that considering the often large effects of brain damage, this is an acceptable tradeoff. #### The Bayesian standardised difference test Assume a sample of $$n$$ controls on which we measure some value x and y from task A and B. Let $$\overline{x}$$ and $$\overline{y}$$ denote the sample means and $\begin{equation} \pmb{A}=\begin{pmatrix} s_{xx} & s_{xy} \\ s_{xy} & s_{yy} \end{pmatrix} \end{equation}$ the sample covariance matrix (note that $$s_{xx}$$ is not the same as $$s_{x}$$ and denotes variance instead of standard deviation). It is assumed that the observations come from a bivariate normal distribution with mean $$\pmb{\mu}$$ and variance $$\pmb{\Sigma}$$, note that the parameters are bolded as to represent the vector and matrix: $\begin{equation} \pmb{\mu} = \begin{pmatrix} \mu_x \\ \mu_y \end{pmatrix} \ \text{and} \ \pmb{\Sigma}=\begin{pmatrix} \sigma_{xx} & \sigma_{xy} \\ \sigma_{xy} & \sigma_{yy} \end{pmatrix} \end{equation}$ Let the case scores be denoted $$x^$$ and $$y^$$. Just as for the frequentist dissociation tests we want to estimate the proportion $$p$$ of the control population that would exhibit a greater difference $$x-y$$ than the case’s $$x^-y^$$. A non-informative prior was again specified, see Crawford & Garthwaite (2007) and Jeffreys (1998). The algorithm for obtaining $$\hat{p}_i$$, the $$i$$th estimation of $$p$$ in Crawford & Garthwaite (2007) follows below: 1. Let $\begin{equation} \pmb{\widehat{\Sigma}}=\begin{pmatrix} \hat{\sigma}_{xx} & \hat{\sigma}_{xy} \\ \hat{\sigma}_{xy} & \hat{\sigma}_{yy} \end{pmatrix} \end{equation}$ be a random draw from an inverse-Wishart distribution (a multivariate generalisation of the $$\chi^2$$-distribution) on $$n$$ degrees of freedom with scale matrix $$\pmb{A}$$. And let $$\pmb{\widehat{\Sigma}}$$ be the estimate of $$\pmb{\Sigma}$$ for this iteration. 2. Let $$z_1$$ and $$z_2$$ be two random draws from a standard normal distribution. Perform Cholesky decomposition on $$\pmb{\widehat{\Sigma}}$$, that is finding the lower triangular matrix $$\pmb{T}$$ such that $$\pmb{T}\pmb{T'}=\pmb{\widehat{\Sigma}}$$. Then $\begin{equation} \pmb{\hat{\mu}} = \begin{pmatrix} \hat{\mu}_x \\ \hat{\mu}_y \end{pmatrix} = \begin{pmatrix} \overline{x} \\ \overline{y} \end{pmatrix}+ \pmb{T} \begin{pmatrix} z_1 \\ z_2 \end{pmatrix} \cdot \frac{1}{n} \end{equation}$ is the estimation of $$\pmb{\mu}$$ for this iteration. 3. With estimations of $$\pmb{\mu}$$ and $$\pmb{\Sigma}$$ we can calculate $$p$$, given that they are the the correct values of $$\pmb{\mu}$$ and $$\pmb{\Sigma}$$. If an unstandardised test is desirable put: $\begin{equation} z^ = \frac{(x^* - \hat{\mu}_x) - (y^* - \hat{\mu}_y)}{\sqrt{\hat{\sigma}_{xx}+\hat{\sigma}_{yy}-2\hat{\sigma}_{xy}}} \end{equation}$ If a standardised test is required, put: $\begin{equation} z_x = \frac{(x^* - \hat{\mu}_x)}{\sqrt{\hat{\sigma}_{xx}}}, \ z_y = \frac{(x^* - \hat{\mu}_y)}{\sqrt{\hat{\sigma}_{yy}}} \ \text{and} \ \hat{\rho}_{xy} = \frac{\hat{\sigma}_{xy}}{\sqrt{\hat{\sigma}_{xx}\hat{\sigma}_{yy}}} \end{equation}$ and $\begin{equation} z^* = \frac{z_x - z_y}{\sqrt{2-2\hat{\rho}_{xy}}} \end{equation}$ 4. Let $$\hat{p}_i$$ be the tail area of a standard normal distribution less or greater than $$z^$$ (depending on alternative hypothesis). $$\hat{p}_i$$ is then the estimate of $$p$$ for this iteration. Repeating these steps a large number of times will yield a distribution of $$\hat{p}$$, the average of which is the point estimate of $$p$$. If repeated e.g. 1000 times, the 25th smallest and 25th largest $$\hat{p}_i$$ is the lower and upper boundaries of the 95% Bayesian credible interval for $$p$$. #### Bayesian tests allowing for covariates The need to use matched samples when comparing a single case to a control sample has often often led to control samples being small. In an attempt to remedy this Crawford et al. (2011) followed up the previously described tests by developing methods that allow for the case to be assessed on abnormality in the presence of covariates. That is, the tests let you compare the case’s score on the task of interest conditioned upon the results of the controls having the same score on the covariate(s). If a patient has 15 years of education, his/her score on the task would be compared to the controls with equal length of education. This reduces the need to perfectly match control samples and is thus a major contribution to the field. The procedural details of these tests are soon going to be updated in the current vignette, but if used before that one thing is important to note. In Crawford et al. (2011) they change the recommendation of the prior to use, when testing for a dissociation. In Berger & Sun (2008) it is shown that the prior used in Crawford & Garthwaite (2007) has good frequentist properties for estimating $$\sigma_x$$ and $$\sigma_y$$ but less so for $$\rho$$ or the discrepancy size. Instead they recommend a calibrated prior. The main difference being that an accept/reject algorithm is applied after the initial draw from the inverse-Wishart distribution. Many Bayesians would argue that good frequentist properties are not necessary when conducting a Bayesian analysis. If so, one can use the “standard theory” Jeffreys (1998)'s prior. In singcar both types of priors are implemented for both tests. If one wants to compare the results from them, the same prior should be used. # References Berger, J. O., & Sun, D. (2008). Objective Priors for the Bivariate Normal Model. The Annals of Statistics, 36(2), 963–982. Crawford, J., & Garthwaite, P. (2005). Testing for Suspected Impairments and Dissociations in Single-Case Studies in Neuropsychology: Evaluation of Alternatives Using Monte Carlo Simulations and Revised Tests for Dissociations. Neuropsychology, 19(3), 318–331. https://doi.org/10.1037/0894-4105.19.3.318 Crawford, J., & Garthwaite, P. (2002). Investigation of the single case in neuropsychology: Confidence limits on the abnormality of test scores and test score differences. Neuropsychologia, 40(8), 1196–1208. https://doi.org/10.1016/S0028-3932(01)00224-X Crawford, J., & Garthwaite, P. (2007). Comparison of a single case to a control or normative sample in neuropsychology: Development of a Bayesian approach. Cognitive Neuropsychology, 24(4), 343–372. https://doi.org/10.1080/02643290701290146 Crawford, J., & Garthwaite, P. (2012). Single-case research in neuropsychology: A comparison of five forms of t-test for comparing a case to controls. Cortex, 48(8), 1009–1016. https://doi.org/10.1016/j.cortex.2011.06.021 Crawford, J., Garthwaite, P., & Howell, D. (2009). On comparing a single case with a control sample: An alternative perspective. Neuropsychologia, 47(13), 2690–2695. https://doi.org/10.1016/j.neuropsychologia.2009.04.011 Crawford, J., Garthwaite, P., Howell, D., & Gray, C. (2004). Inferential methods for comparing a single case with a control sample: Modified t‐tests versus mycroft et al.’s (2002) modified anova. Cognitive Neuropsychology, 21(7), 750–755. https://doi.org/10.1080/02643290342000276 Crawford, J., Garthwaite, P., & Ryan, K. (2011). Comparing a single case to a control sample: Testing for neuropsychological deficits and dissociations in the presence of covariates. Cortex, 47(10), 1166–1178. https://doi.org/10.1016/j.cortex.2011.02.017 Crawford, J., & Howell, D. (1998). Comparing an Individual’s Test Score Against Norms Derived from Small Samples. The Clinical Neuropsychologist, 12(4), 482–486. https://doi.org/10.1076/clin.12.4.482.7241 Crawford, J., Howell, D., & Garthwaite, P. (1998). Payne and Jones Revisited: Estimating the Abnormality of Test Score Differences Using a Modified Paired Samples t Test. Journal of Clinical and Experimental Neuropsychology, 20(6), 898–905. https://doi.org/10.1076/jcen.20.6.898.1112 DeGroot, M. H., & Schervish, M. J. (2012). Probability and statistics (4th ed). Addison-Wesley. Donovan, T., & Mickey, R. M. (2019). Bayesian Statistics for Beginners: A step-by-step approach (1st ed.). Oxford University Press. https://doi.org/10.1093/oso/9780198841296.001.0001 Garthwaite, P., & Crawford, J. (2004). The distribution of the difference between two t -variates. Biometrika, 91(4), 987–994. Hassan, E. K., Sedda, A., Buckingham, G., & McIntosh, R. D. (2020). The size-weight illusion in visual form agnosic patient DF. Neurocase, 1–8. https://doi.org/10.1080/13554794.2020.1800748 Jeffreys, H. (1998). Theory of probability (3rd ed). Clarendon Press ; Oxford University Press. Payne, R. W., & Gwynne Jones, H. (1957). Statistics for the investigation of individual cases. Journal of Clinical Psychology, 13(2), 115–121. Sokal, R. R., & Rohlf, F. J. (1981). Biometry: The Principles and Practice of Statistics in Biological Research. W. H. Freeman. https://books.google.co.uk/books?id=C-OTQgAACAAJ 1. In Crawford et al. (1998) they use the term ‘difference’ instead of discrepancy. This is a somewhat unfortunate usage since a difference can pretty much refer to anything. Hence, ‘discrepancy’ will be used for the most part when referring to the difference between scores from two tasks↩︎ 2. A Bayesian 95% credible interval can for example be said to have good frequentist properties if it would cover the parameter 95% of the times it is created (as is the case for the frequentist 95% confidence interval)↩︎ 3. The difference between analytic and numerical problem solving is that analytic solutions are exact as well as derived and presented in (for the mathematician) understandable forms. A numerical solution involves “guesswork” and is stopped when a solution is found that satisfies the problem. This method only approximates the “true” solution. Monte Carlo simulations belong to the latter category.↩︎	2022-01-19 18:03:00	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 8, "x-ck12": 0, "texerror": 0, "math_score": 0.7446442246437073, "perplexity": 6420.560066408052}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301475.82/warc/CC-MAIN-20220119155216-20220119185216-00221.warc.gz"}
https://mathoverflow.net/questions/363066/uniform-distribution-in-euclidean-ball-is-sub-gaussian	# Uniform distribution in Euclidean ball is sub-gaussian [closed] Consider $$n-$$dimensional Euclidean ball centred at 0 with radius $$\sqrt{n}$$. We want to show that the uniform distribution $$X$$ in this ball is sub-gaussian and $$\|\|X\|\|_{\psi_2} where $$C$$ is absolute constant. Clarify: $$X$$ is subgaussian if $$\langle X,x \rangle$$ is subgaussian for any $$x \in \mathbb{R}^n$$ and $$\|\|X\|\|_{\psi_2}=\|\|\sup_x\langle X,x\rangle\|\|_{\psi_2}$$ where sup is over all unit vector $$x$$. Attempt: Uniform distribution on ball can be represented by $$R,\varphi_1,..,\varphi_{n-1}$$ jointly where $$R$$ is a uniform distribution on $$[0, \sqrt{n}]$$ representing radius, $$\varphi_i$$ representing the angles in spherical coordinates and they are uniform on $$[0,\pi]$$. All these variables are independent. By symmetry, I only need to show $$\|\|\langle X, (1,0,0,0,...)\rangle\|\|_{\psi_2}=\|\|X_1\|\|_{\psi_2}=\|\|R\cos\varphi_1\|\|_{\psi_2}. Then it is not clear to me how to proceed • This sounds like a homework question... MO is for research-level questions only, perhaps Math.SE will be better suited. Jun 14, 2020 at 21:35 $$\newcommand\Ga\Gamma$$ For each unit vector $$x$$, the random variable (r.v.) $$\langle X,x\rangle$$ equals $$V:=\sqrt n\,W_nR$$ in distribution, where $$W_n:=\frac{Z_1}{\sqrt{Z_1^2+\dots+Z_n^2}},$$ $$Z_1,\dots,Z_n$$ are iid $$N(0,1)$$ r.v.'s, and $$R$$ is a r.v. (independent of $$V$$ and) such that $$0\le R\le1$$. So, it suffices to show that for some real $$c>0$$ $$\sup_{n\ge2}Ee^{cnW_n^2}<\infty. \tag{1}$$ Note that $$W_n^2$$ has the beta distribution with parameters $$1/2,(n-1)/2$$. So, for any $$c\in(0,1/2)$$ and $$n\ge3$$ \begin{align} Ee^{cnW_n^2} &=\frac{\Ga(n/2)}{\sqrt\pi\,\Ga((n-1)/2)}\int_0^1 e^{cnw^2}w^{-1/2}(1-w)^{(n-3)/2}\,dw \\ &\le\frac{\Ga(n/2)}{\sqrt\pi\,\Ga((n-1)/2)}\int_0^1 e^{cnw}w^{-1/2}e^{-(n-3)w/2}\,dw \\ &=O(\sqrt n)O(1/\sqrt n)=O(1). \end{align} Also, clearly $$Ee^{cnW_n^2}<\infty$$ for $$n=2$$. Thus, (1) holds, as desired. • Thank you! But would you mind explaining how in the last step the integral is of order $O(1/n)$. I tried to upper bound it by changing upper limit to infinity and write it as gamma function but this only gives me $O(1/\sqrt{n})$. Jun 15, 2020 at 14:31 • @user135520 : That the expectation under the $\sup$ in (1) is finite is trivial. That the supremum of this expectation over all $n\ge2$ is finite is not so trivial. Feb 6 at 2:43	2022-05-17 11:59:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 35, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9974425435066223, "perplexity": 428.7071913744757}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662517245.1/warc/CC-MAIN-20220517095022-20220517125022-00267.warc.gz"}
http://planetmath.org/node/87424	## Primary tabs The link to Automatic Reference Linking and the User Linking Controls guide just take me to a page New in Encyclopedia, not where I wanted to go at all. Thanks, I’ll fix those! They are ”old-style” links (from the earlier version of the site) that haven’t been ported over correctly. Easy to change. If you see other links that look like http://planetmath.org/?op=getobj&from=collab&id=32 or similar, that’s the same problem. Not too hard to fix ’em.	2017-01-22 10:19:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.41571471095085144, "perplexity": 2818.073996666201}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281421.33/warc/CC-MAIN-20170116095121-00402-ip-10-171-10-70.ec2.internal.warc.gz"}
https://www.techwhiff.com/issue/help-me-please-i-will-give-brainliest-just-the-bottom--530156	# Help me please I will give brainliest just the bottom one please ###### Question: Help me please I will give brainliest just the bottom one please ### Which division of the human nervous system is responsible for maintaining body temperature? Which division of the human nervous system is responsible for maintaining body temperature?... ### I have a 3 cc piece of aluminum with a density of steel with a mass of 24.0 g. I cut it into 2 equal pieces. How has the density of the steel pieces changed I have a 3 cc piece of aluminum with a density of steel with a mass of 24.0 g. I cut it into 2 equal pieces. How has the density of the steel pieces changed... ### What role do triglycerides serve in the human body? A. Long term storage of energy B. Enzymes that catalyze reactions C. Protective coating for outer tissues D. Information storage What role do triglycerides serve in the human body? A. Long term storage of energy B. Enzymes that catalyze reactions C. Protective coating for outer tissues D. Information storage... ### What does full employment mean for Americans in the 1950s? What does full employment mean for Americans in the 1950s?... ### All of the following are equivalent except _____. A:5 over 4. B:1.25. C:12.5%. D:125%. All of the following are equivalent except _____. A:5 over 4. B:1.25. C:12.5%. D:125%.... ### When the Persians conquered other people groups, they were generally: Question 4 options: Kind to them Mean to them Killed them off Sent them away Save When the Persians conquered other people groups, they were generally: Question 4 options: Kind to them Mean to them Killed them off Sent them away Save... ### I'll give brainliest only if its right I'll give brainliest only if its right... ### In childhood, how children react to a serious stressor depends primarily on _____. In childhood, how children react to a serious stressor depends primarily on _____.... ### PLEASE HURRY Question 5 Multiple Choice Worth 4 points) (06.01)A scatter plot is shown: 7 6 5 4 3 . 1 o 0 1 2 What type of association does the graph show between x and y? Linear positive association Nonlinear positive association Linear negative association Nonlinear negative association PLEASE HURRY Question 5 Multiple Choice Worth 4 points) (06.01)A scatter plot is shown: 7 6 5 4 3 . 1 o 0 1 2 What type of association does the graph show between x and y? Linear positive association Nonlinear positive association Linear negative association Nonlinear negative association... ### With the internet, everyday citizens can become joumalists True False With the internet, everyday citizens can become joumalists True False... ### Jade used Fraction 3 over 4 yard of fabric to make a scarf. Can she make 2 of these scarves with Fraction 1 and 4 over 5 yards of fabric, and why? No, because the quotient of Fraction 3 over 4division signFraction 1 and 4 over 5 is 1 and 7 over 20 No, because the quotient of Fraction 1 and 4 over 5Division signFraction 3 over 4 is 1 and 7 over 20 Yes, because the quotient of Fraction 3 over 4division signFraction 1 and 4 over 5 is 2 and 2 over 5 Yes, because the quotient of Fraction 1 and 4 Jade used Fraction 3 over 4 yard of fabric to make a scarf. Can she make 2 of these scarves with Fraction 1 and 4 over 5 yards of fabric, and why? No, because the quotient of Fraction 3 over 4division signFraction 1 and 4 over 5 is 1 and 7 over 20 No, because the quotient of Fraction 1 and 4 over ... ### What are the advantages and disadvantages of the globalization and standardization of culture? provide at least one specific example of the advantage and at least one specific example of the disadvantage of the globalization of culture. would you prefer to live in a community in which everyone were the same (no difference in language, religion, sexuality, and so forth) or one in which such differences exist? why? What are the advantages and disadvantages of the globalization and standardization of culture? provide at least one specific example of the advantage and at least one specific example of the disadvantage of the globalization of culture. would you prefer to live in a community in which everyone were ... ### What is the sum of (7x+21)/(x^2-9) and (x+2)/(x-3) ? What is the sum of (7x+21)/(x^2-9) and (x+2)/(x-3) ?... ### Your dance club wants to raise money with a dunk tank at the school carnival. Members spent $79 renting the tank, and they are charging$5 for each throw. Which graph models your club's profit given the number of throws? Your dance club wants to raise money with a dunk tank at the school carnival. Members spent $79 renting the tank, and they are charging$5 for each throw. Which graph models your club's profit given the number of throws?... ### How are these triangles congruent?????? How are these triangles congruent??????... ### I NEED HELP PLZZZ The distance between two towns A and B is 16 cm. Find the actual distance, in km, that is represented by map scale 2 cm to 3 km. SHOW WORKING AS WELL I NEED HELP PLZZZ The distance between two towns A and B is 16 cm. Find the actual distance, in km, that is represented by map scale 2 cm to 3 km. SHOW WORKING AS WELL... ### Please hurry!! 20 points rewrite in standard form (4,1) and (1,4) Please hurry!! 20 points rewrite in standard form (4,1) and (1,4)... ### How could you combine the following two sentences by subordinating ideas? The man lives in this apartment building. He drives our bus. (adjective clause) How could you combine the following two sentences by subordinating ideas? The man lives in this apartment building. He drives our bus. (adjective clause)...	2022-10-01 00:44:39	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3596234917640686, "perplexity": 2250.8254750169835}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335514.65/warc/CC-MAIN-20221001003954-20221001033954-00598.warc.gz"}
https://robotics.stackexchange.com/tags/power/hot	# Tag Info 6 There are a variety of reasons to separate the motor power from the "hotel load", including: Reducing the number of wires running between high-power and low-power electronics Redundancy (your homing beacon shouldn't run out of power when the rest of the system does) Preventing heavy current loads from browning out the control system Making the system more ... 6 It is called a slip ring. It works the same as a brushed motor. See here for a robotic oriented one. Larger versions handle power, and cost more. Also near field technologies such as those used to wirelessly charge your electric toothbrush, and more recent wireless cell phone chargers, are potential solutions 5 There are a few ways to read your question, but in general what you are suggesting is a bad idea. The only reason to do it would be in the extreme case where splitting a large battery into several smaller batteries would help you find space for them in a tightly-packed chassis, and even then it might be more trouble than it's worth -- quadrupling the number ... 5 The battery capacity specification (eg X mAh) tells you that your battery can run for 1 hour providing X milliamps until it is depleted. This doesn't always scale with time, for example you probably won't run for 1/2 hour if you draw 2X milliamps, but this is another discussion. To answer your question, a greater mAh will allow you to use your battery for ... 4 I agree with @Greenonline 's recommendation regarding LiPo batteries along with his warnings on battery care. It seems you will need a fairly small battery, considering your current requirements (about a 1000-2000mAh 2S LiPo). However, you also need to add 2 5V BECs (5V regulators in RC lingo) to power your circuits; The 7.4 or 11.1V provided by the battery ... 4 Industrial Robots with having as aim manipulation of large objects are powered electrically in almost all cases (30-40 years ago there have been popular hydraulic variants). Industrial robots are powered by industrial 400V 3 phase current, so in a sense, they are plugged in a wall outlet, but but a household wall outlet but an industrial one, like this. ... 4 Robots tend to be portable devices powered by batteries. Portable battery operated devices tend to use embedded processors with limited power and memory. Compiled code has several advantages over interpreted code in such applications: Compiled code usually takes up less space. So you can have more code in the same amount of space. Compiled code usually ... 3 A linear regulator has a major drawback: it dissipates the power it doesn't deliver. Using the well-known 7805, the output voltage is 5V, your input shall be at least 7V to allow a good regulation (2V dropout voltage), if your load requires 1A, your 7805 dissipates 1A 2V = 2W. This makes your Linear regulator hot. The best efficiency you may expect from a ... 3 Yes this is pretty easy to solve, you just need basic soldering skills. Just open the cables of your additional USB-devices, you will see 4 wires. 2 of them are for the data, just leave them. The other two are for powersupply, just connect them to your 5V power supply. But make sure you have to right ones, otherwise your device will start producing magic ... 3 This is very rough, might or might not help you.... Treating it as a macroscopic perspective: Kinetic Energy $$E_k = \frac{1}{2}mv^2$$ For working energy in a flow, substitute volume with Area*Velocity and mass with density $\rho$ $$E_k = \frac{1}{2}\rho Av^3$$ With that, you should be able to estimate (after efficiency corrections) the power required.... 3 You can also add between the controller power an electrolytic capacitor of about 500 or even 1000 microfarads x 12 volts and a diode in series with it, so when the trigger pull much current source with a corresponding drop in voltage will be avoided that Power down controller please and even free you from unwanted noise (about 10 turns of cable around a ... 3 The short answer is no. The longer answer requires more information from you, regarding how the motor is to be used. Since you have tagged your question with "quadcopter", I'm going to assume that you plan to put the motor on a UAV, in which case more power definitely does not mean "better". More power (W) will draw more current (A) from the battery (... 3 You can control brushless motors 2 ways control with a hall effect sensor http://scholar.lib.vt.edu/theses/available/etd-09152003-171904/unrestricted/T.pdf sensorless(back emf) control http://www.pmdcorp.com/downloads/app_notes/BrushlessSensorConfig.pdf or you can buy an esc (elcetronic speed control) My advice If you are not knowledgeable about electronic ... 3 Your scenarios are correct. You can connect multiple solar cells together to get increased current or increased voltage. Wire them in series (positive to negative) to boost voltage, wire them in parallel (positives to positive) to boost current capacity. As a final note, I would caution running near the maximum capacity of the solar cell. The voltage of ... 3 Search aliexpress for '6V switching power supply'. You should find 5Amp or 10Amp ones for $10-$20. Silver boxes with holes in them. You can find them on ebay as well (usually the same chinese vendors as aliexpress). You will need to add your own AC plug. Keep in mind that you will probably not be using 12Amps because not all motors will be at full torque ... 3 What you're asking for cannot be accomplished with a PID controller. As I understand your question, you want to be able to choose PID gains that would always produce a "good" trajectory, without tuning. You said it's alright if the motor output is unrealistic, i.e. the motors are "very strong" therefore can produce unlimited torque. ... 3 You could trick the charger into providing power for you by applying a voltage to the balance sensing lines. However the control you have over it would be limited. Switching power supplies are ridiculously cheap these days. Search ebay for "switching power supply dc 24v" for whatever voltage you want. I just keep a pile of these in the closet for the ... 3 Sure, a drone can land on a powerline. That's a standard task like the "peg in hole problem" for robotarms. The aim is to maneuver a UAV near to a highvoltage line and eating all the energy. The earliest paper was written in 2009 and has a nice plotchart of the Electro-Magnetic Field on page 8 Powerline perching with a fixed-wing UAV for directing the UAV in ... 3 I think yes it can but how? My options are here: Static system for conventional systems It should stand or hang to line/pole/special place like birds. Please watch video for an example: https://www.youtube.com/watch?v=MvRTALJp8DM Dynamic system Harmless/secure distance present day wireless charging methods: use low energy harvesting drone or an ... 3 Every action has an equal and opposite reaction. Each motor/joint in a linear chain of actuators (snake) needs to be capable of supplying the appropriate reaction forces. This mean that, if you have a 100cm long snake robot with a motor every 10cm, the first/"neck" joint (at 10cm) needs to support 8 other motors and 90cm of snake body. The second joint ... 3 Power (Watts, milliWatts, etc.) is given by: $$P = IV \\$$ where $I$ is current in (milli)Amps, $V$ is voltage in Volts, and $P$ is power in (milli)Watts, respectively. Energy (Watt-hours, milliWatt-hours, etc.) is given by: $$E = P\Delta t \\$$ (assuming constant power output), where $E$ is energy in (milli)Watt-hours and $t$ is time in hours. ... 3 Looks like it uses a propane powered ram to launch itself into the air. The end of the piston is visible in the back of the vehicle. This webpage states that source of power is battery and propane https://bostondynamics.com/sandflea Also found this patent for a combustion powered linear actuator. https://patents.google.com/patent/US7263955B1/en 2 The wattage rating is usually (read the manufacturer's specs very carefully) the measure of how much power the motor can draw. It doesn't tell you anything more than that. Wattage does not tell you whether the motor is efficient at converting the input electrical energy into output mechanical energy. Wattage does not tell you whether the motor's power-to-... 2 Lithium thionyl chloride / SOCl2 batteries have excellent energy density but are not aimed at high-current applications and are not rechargeable, so it seems likely that you can save weight and cost by using some other kind of batteries. Quadcopter batteries (typically lithium ion polymer technology, rated for 5C to 15C discharge rates and rechargeable ... 2 As long as the steppers are running relatively slowly, let's say a few hundred steps per second, you should be fine. Your driver uses PWM to control the current through the motor windings. In microstepping mode the driving waveform at the motor terminals looks like a sine and cosine (sines offset by 90 degrees) and the driver modulates the outputs to ... 2 18 servos is a lot of power no matter their size but you are going to need to be more specific with what kind of servos you are talking about. That said, this What is the best way to power a large number (27) servos at 5 V? is a good answer to your question. 2 Use a self-powered hub. Look at the power adapter from the mains (220v). In almost all instances, it is converting 220vac to 5vdc (sometimes 5vdc and rarely 12vdv). Cut the power adapter cable to the hub in half after determining which half of the cable pair is negative and which is positive (see picture below). Often it's as obvious as the red wire is ... 2 The device you are looking for is broadly known as a slip ring, but since you are trying to power other motors through it you should be aware that a slip ring is not electrically identical to a wire. Depending on the amount of current you are trying to send, the slight variations in impedance that your slip ring will produce as it spins can damage some ... 2 You are right that there are issues in having an "unclean" power supply running into your RaspberryPi. If you are drawing tons of power, things get very hairy very fast. However, there are ways to "convert" and "clean" a supply to isolate the voltage (and therefore making it suitable for input to the Raspberry Pi). How much cleaning will depend on your ... 2 The easiest way to calculate this will probably be by using the ecalc helicopter calculator, they have done all these calculations for you and take into account far more factors in my experience they are very accurate. I know that most props have a Prop Constant in the 1.1 to 1.3 range, but for your specific situation according to ecalc gives a thrust to ... Only top voted, non community-wiki answers of a minimum length are eligible	2021-08-05 19:49:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.31802505254745483, "perplexity": 1433.5900453006022}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046157039.99/warc/CC-MAIN-20210805193327-20210805223327-00317.warc.gz"}
https://zbmath.org/?q=an:0906.14013	## Local heights of subvarieties over non-archimedean fields.(English)Zbl 0906.14013 Let $$Y$$ be a complete algebraic variety over a complete non-Archimedean field $$K$$. If the valuation is discrete, then local heights of cycles on $$Y$$ are well-known. They are given by intersection numbers in models over the discrete valuation ring. To generalize this to the non-discrete case, we replace the algebraic models by formal models over the valuation ring. First, a proper intersection product of Cartier divisors with cycles on a rigid analytic variety $$X$$ is defined. Then we extend this intersection product to admissible formal models. It satisfies the usual properties and corresponds to a normalized intersection product in the algebraic situation. A Cartier divisor on a formal model induces a metrized line bundle on $$X$$. The metric is called a formal metric. If $$X$$ is quasi-compact and quasi-separated, then limits of roots of formal metrics are characterized as those metrics with a continuous extension to the Berkovich-compactification of $$X$$. Using for $$X$$ the rigid analytic variety associated to $$Y$$, we define local heights of cycles as intersection numbers with Cartier divisors on an admissible formal model of $$X$$. The dependence on the models is measured by the metrized line bundles. Finally, it is shown that the local heights satisfy five characteristic properties. ### MSC: 14G20 Local ground fields in algebraic geometry 14C25 Algebraic cycles Full Text:	2023-02-01 19:46:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8117191195487976, "perplexity": 326.41223186549996}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499949.24/warc/CC-MAIN-20230201180036-20230201210036-00779.warc.gz"}
https://www.sarthaks.com/1354871/atm-899-for-equilibrium-shown-below-what-the-equilibrium-concentration-when-placed-flask	# K_(p)=0.04 atm at 899 K for the equilibrium shown below. What is the equilibrium concentration of C_(2)H_(6) when it is placed in a flask at 4. 19 views closed K_(p)=0.04 atm at 899 K for the equilibrium shown below. What is the equilibrium concentration of C_(2)H_(6) when it is placed in a flask at 4.0 atm pressure and allowed to come to equilibrium? C_(2)H_(6)(g) hArr C_(2)H_(4)(g)+H_(2)(g) by (73.6k points) selected Correct Answer - [C_(2)H_(6)]_(eq) = 3.62 atm Let p be the pressure exerted by ethene and hydrogen gas (each) at equilibrium. Now, according to the reaction, {:(,C_(2)H_(2(g)),harr,C_(2)H_(2(g)),+,H_(2(g))),("Initial conc.",4.0"atm",,0,,0),("At equilibrium",4.0-p,,p,,p):} We can write, (p_(C_(2)H_(4)) xx p_(H_(2)))/(p_(C_(2)H_(6))) = K_(p) rArr (pxxp)/(4.0-p)= 0.04 rArr p^(2) + 0.16 - 0.04 rArr p^(2) + 0.16 - 0.04 p rArr p^(2)+ 0.04 p - 0.16 = 0 Now, p = (-0.04 +-sqrt((0.04)^(2) - 4 xx 1 xx (-0.16)))/(2 xx 1) = (-0.04 +- 0.80)/(2) = (0.76)/(2) , (Taking positive value) Hence, at equilibrium [C_(2)H_(6)] - 4 - p = 4 - 0.38 = 3.62 "atm"`	2022-10-06 20:19:04	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4720678925514221, "perplexity": 7808.948505799967}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337855.83/warc/CC-MAIN-20221006191305-20221006221305-00567.warc.gz"}
https://ibpsonline.in/questions/IBPS-Clerk/Reasoning-Ability/Test-92/1505	IBPS Clerk :: Reasoning Ability :: Test 92 IBPS Recruitment Latest Govt Jobs Home IBPS Clerk / Reasoning Ability Test 92 Questions and Answers 1 . In these questions, relationships between different elements is shown in the statements. These statements are followed by two conclusions. Give answer (a) if only conclusion I follows. Give answer (b) if only conclusion II follows. Give answer (c) if either conclusion I or conclusion II follows. Give answer (d) if neither conclusion I nor conclusion II follows. Give answer (e) if both conclusions I and II follow. Statement : F $\leq$ C $\geq$ V = Z < X = U Conclusions : I. V < U II. Z < F a b c d e 2 . In these questions, relationships between different elements is shown in the statements. These statements are followed by two conclusions. Give answer (a) if only conclusion I follows. Give answer (b) if only conclusion II follows. Give answer (c) if either conclusion I or conclusion II follows. Give answer (d) if neither conclusion I nor conclusion II follows. Give answer (e) if both conclusions I and II follow. Statement : Q $\leq$ E = I > N $\geq$ R $geq$ S Conclusions : I. E = S II. S $\leq$ N a b c d e 3 . In these questions, relationships between different elements is shown in the statements. These statements are followed by two conclusions. Give answer (a) if only conclusion I follows. Give answer (b) if only conclusion II follows. Give answer (c) if either conclusion I or conclusion II follows. Give answer (d) if neither conclusion I nor conclusion II follows. Give answer (e) if both conclusions I and II follow. Which of the following symbols should replace question mark (?) in the given expression in order to make the expressions ' A > D' and 'F $\geq$ C' definitely true? A > B $\geq$ C ? D $\leq$ E=F > < $\leq$ = Either = or $\geq$ 4 . In these questions, relationships between different elements is shown in the statements. These statements are followed by two conclusions. Give answer (a) if only conclusion I follows. Give answer (b) if only conclusion II follows. Give answer (c) if either conclusion I or conclusion II follows. Give answer (d) if neither conclusion I nor conclusion II follows. Give answer (e) if both conclusions I and II follow. Which of the following expressions is definitely true if the given expressions 'R < P' as well as 'S > Q' are definitely true? P > Q = R $\leq$ T < S S > T $\geq$ R > Q < P Q > R $\leq$ T > P $\geq$ S S > T $\geq$ R > Q > P None of these 5 . In these questions, relationships between different elements is shown in the statements. These statements are followed by two conclusions. Give answer (a) if only conclusion I follows. Give answer (b) if only conclusion II follows. Give answer (c) if either conclusion I or conclusion II follows. Give answer (d) if neither conclusion I nor conclusion II follows. Give answer (e) if both conclusions I and II follow. 'A $\times$ B' means 'A is the father of B'. 'A + B' means 'A is the daughter of B'. 'A $\div$ B' means 'A is the son of B'. 'A - B' means 'A is the sister of B'. What will come in place of question mark to establish that P is the son-in-law of S in the following expression? P $\times$ Q + R - T ? S + $\times$ - $\div$ Either + or $\div$ 6 . In each of the questions given below which one of the five answer figures on the right should come after the problem figures on the left, if the sequence were continued? a b c d e 7 . In each of the questions given below which one of the five answer figures on the right should come after the problem figures on the left, if the sequence were continued? a b c d e 8 . In each of the questions given below which one of the five answer figures on the right should come after the problem figures on the left, if the sequence were continued? a b c d e 9 . In each of the questions given below which one of the five answer figures on the right should come after the problem figures on the left, if the sequence were continued? a b c d e 10 . In each of the questions given below which one of the five answer figures on the right should come after the problem figures on the left, if the sequence were continued? a b c d e	2019-08-24 02:19:35	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6985597610473633, "perplexity": 1584.6065300223609}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027319470.94/warc/CC-MAIN-20190824020840-20190824042840-00022.warc.gz"}
https://swmath.org/?term=refinement	• # Z • Referenced in 279 articles [sw10291] • Using Z. Specification, refinement, and proof. The book is an in-depth introduction ... integration of $Z$ with the refinement calculus and data refinement. The overall presentation is fluent... • # MUMPS • Referenced in 439 articles [sw04013] • solve phases (uniprocessor version also available); Iterative refinement and backward error analysis; Various matrix input... • # UG • Referenced in 190 articles [sw04596] • especially interested in adaptive local grid refinement on unstructured meshes, multigrid solvers and parallelization techniques ... this. Firstly, the multigrid solution and adaptive refinement for many engineering applications are still... • # SuperLU • Referenced in 178 articles [sw00930] • separate from the factorization. Working precision iterative refinement subroutines are provided for improved backward stability ... error, and estimate error bounds for the refined solutions... • # PARAMESH • Referenced in 100 articles [sw00677] • PARAMESH: A parallel adaptive mesh refinement community toolkit. In this paper we describe a community ... into a parallel code with adaptive mesh refinement. Alternatively, in its simplest use, and with ... their code, converting it first to uniformly refined parallel code, and then later if they... • # TetGen • Referenced in 158 articles [sw04046] • recovering constraints, and a new constrained Delaunay refinement algorithm for adaptive quality tetrahedral mesh generation...	2021-05-18 07:24:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18976515531539917, "perplexity": 4825.850969847294}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989756.81/warc/CC-MAIN-20210518063944-20210518093944-00506.warc.gz"}
http://www.math.unm.edu/~appelo/teaching/Math471F16/html/Homework3.html	# Homework 3, due 23.59, 23/9-2016¶ ## Fortran and numerical integration (quadrature)¶ Consider the integral $$I = \int_{-1}^1 e^{\cos(k x)} dx,$$ with $$k = \pi$$ or $$\pi^2$$. In this homework you will experiment with two different ways of computing approximate values of $$I$$. After doing this homework you will have written a Fortran program from scratch, called a subroutine that someone else has written and gained some knowledge of the accuracy of some different methods for numerical integration (quadrature). In future homework we will see how these computations can be performed in parallel. I will talk briefly about the most basic quadrature rules in class, for a more detailed description you can take a look at Olof Runborg’s notes. ## Trapezoidal rule¶ Recall that for a grid $$x_i = X_L + ih, \ \ i = 0,\ldots,n, \ \ h = \frac{X_R-X_L}{n}$$, the composite trapezoidal rule is: $$\int_{X_L}^{X_R} f(x) dx \approx h\left(\frac{f(x_0)+f(x_n)}{2} + \sum_{i=1}^{n-1} f(x_i) \right).$$ Write a Fortran program that uses the composite trapezoidal rule to approximate the above integral for both $$k$$ and for $$n = 2,3,\ldots,N$$, where you should pick $$N$$ so that the absolute error is smaller than $$10^{-10}$$. • In your report, plot the error against $$n$$ using a logarithmic scale for both axis. • Can you read off the order of the method from the slopes? How does this agree with theory? • What is special with the integrand in the case $$k = \pi$$ and why does it make the method converge faster then expected? Hint: take a look at the Euler-Maclaurin formula. The trapezoidal rule belongs to a class of quadrature called Newton-Cotes quadrature that approximates integrals using equidistant grids. The order of a composite Newton-Cotes method is typically $$s$$ or math:s+1, where $$s$$ is the number of points in each panel (2 for the Trapezoidal rule). ## Gauss Quadrature¶ Another class of methods is Gauss quadrature. In Gauss quadrature the location of the grid-points (usually referred to as nodes) and weights, $$\omega_i$$, are chosen so that the order of the approximation to the weighted integral $$\int_{-1}^{1} f(z) w(z) dz \approx \sum_{i=0}^n \omega_i \, f(z_i),$$ is maximized. Here the weight function $$w(z)$$ is positive and integrable (for example $$w(z) = 1$$.) For a given $$w(z)$$ the nodes, $$z_i$$ are the zeros of the polynomial $$\tau_n = (z-z_0)(z-z_1)\cdots(z-z_n)$$ satisfying $$\int_{-1}^{1} \tau_n(z) q(z) w(z) dz = 0,$$ for all polynomials $$q(z)$$ of degree less than $$n$$. Or, equivalently, the nodes are the zeros to the degree $$n$$ orthogonal polynomial associated with the weight function $$w(z)$$. Don’t worry if this sounds complicated, we will only consider the case $$w(z) = 1$$ and obtain the weights and nodes by a call to the subroutine lglnodes.f90 (which is a f90 version of Greg von Winckel’s matlab version.) Use the subroutine lglnodes.f90 from the repository and compute the integral by the code below (which implements the above formula): call lglnodes(x,w,n) f = exp(cos(pipix)) Integral_value = sum(f*w) Again: • plot the error against $$n$$ using a logarithmic scale for both axis (perhaps in the same figure.) • For Gauss quadrature the error is expected to decrease as $$e(n) \sim C^{-\alpha n}$$. Try some different $$C$$ and $$\alpha$$ to see if you can fit the computed error curves. ## Notes¶ 1. The errors should look something like the figure below (note that you should label the curves, I left them out on purpose.) 1. For this homework you will call a routine from another file which means that to build your executable you will have to compile two object files and then link them together: $gfortran -O3 -c gq_test.f90$ gfortran -O3 -c lglnodes.f90 \$ gfortran -o gq_test.x gq_test.o lglnodes.o 1. A better way of doing this is to use a makefile as was discussed in class. 2. It is probably a good idea to use allocatable arrays for x,w,f above allocate(x(0:n),f(0:n),w(0:n)) ...Code here... deallocate(x,f,w) as their size will change as you change $$n$$. Don’t forget to deallocate the arrays. 64.106.38.123	2018-01-19 14:57:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 4, "x-ck12": 0, "texerror": 0, "math_score": 0.8773751854896545, "perplexity": 496.9125392518402}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084888041.33/warc/CC-MAIN-20180119144931-20180119164931-00023.warc.gz"}
http://www.ams.org/mathscinet-getitem?mr=1223512	MathSciNet bibliographic data MR1223512 (95c:46110) 46L50 (46L10 46L30 46L70) Bunce, L. J.; Hamhalter, J. Traces and subadditive measures on projections in ${\rm JBW}$${\rm JBW}$-algebras and von Neumann algebras. Proc. Amer. Math. Soc. 123 (1995), no. 1, 157–160. Article For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.	2014-03-16 11:08:12	{"extraction_info": {"found_math": true, "script_math_tex": 1, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9981452226638794, "perplexity": 5818.149744115228}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394678702332/warc/CC-MAIN-20140313024502-00009-ip-10-183-142-35.ec2.internal.warc.gz"}
https://labs.tib.eu/arxiv/?author=B.%20Friman	• ### Volume fluctuations and higher order cumulants of the net baryon number(1205.4756) March 2, 2016 hep-ph We consider the effect of volume fluctuations on cumulants of the net baryon number. Based on a general formalism, we derive universal expressions for the net baryon number cumulants in the presence of volume fluctuations with an arbitrary probability distribution. The relevance of these fluctuations for the baryon-number cumulants and in particular for the ratios of cumulants is assessed in the Polyakov loop extended quark-meson model within the functional renormalization group. We show that the baryon number cumulants are generally enhanced by volume fluctuations and that the critical behavior of higher order cumulants may be modified significantly. • ### The chiral condensate in neutron matter(1307.2110) Sept. 30, 2013 hep-ph, nucl-th, astro-ph.SR We calculate the chiral condensate in neutron matter at zero temperature based on nuclear forces derived within chiral effective field theory. Two-, three- and four-nucleon interactions are included consistently to next-to-next-to-next-to-leading order (N3LO) of the chiral expansion. We find that the interaction contributions lead to a modest increase of the condensate, thus impeding the restoration of chiral symmetry in dense matter and making a chiral phase transition in neutron-rich matter unlikely for densities that are not significantly higher than nuclear saturation density. • ### Renormalization group and Fermi liquid theory for many-nucleon systems(1201.2510) Jan. 12, 2012 hep-ph, nucl-th, cond-mat.str-el We discuss renormalization group approaches to strongly interacting Fermi systems, in the context of Landau's theory of Fermi liquids and functional methods, and their application to neutron matter. • ### Net-charge probability distributions in heavy ion collisions at chemical freeze-out(1111.5063) Nov. 21, 2011 hep-ph We explore net charge probability distributions in heavy ion collisions within the hadron resonance gas model. The distributions for strangeness, electric charge and baryon number are derived. We show that, within this model, net charge probability distributions and the resulting fluctuations can be computed directly from the measured yields of charged and multi-charged hadrons. The influence of multi-charged particles and quantum statistics on the shape of the distribution is examined. We discuss the properties of the net proton distribution along the chemical freeze-out line. The model results presented here can be compared with data at RHIC energies and at the LHC to possibly search for the relation between chemical freeze-out and QCD cross-over lines in heavy ion collisions. • ### Non-perturbative dynamics and charge fluctuations in effective chiral models(1108.3231) Aug. 16, 2011 hep-ph We discuss the properties of fluctuations of the electric charge in the vicinity of the chiral crossover transition within effective chiral models at finite temperature and vanishing net baryon density. The calculation includes non-perturbative dynamics implemented within the functional renormalization group approach. We study the temperature dependence of the electric charge susceptibilities in the linear sigma model and explore the role of quantum statistics. Within the Polyakov loop extended quark-meson model, we study the influence of the coupling of quarks to mesons and to an effective gluon field on charge fluctuations. We find a clear signal for the chiral crossover transition in the fluctuations of the electric charge. Accordingly, we stress the role of higher order cumulants as probes of criticality related to the restoration of chiral symmetry and deconfinement. • ### Mapping the phase diagram of strongly interacting matter(1008.4549) Aug. 9, 2011 hep-ph, hep-lat We employ a conformal mapping to explore the thermodynamics of strongly interacting matter at finite values of the baryon chemical potential $\mu$. This method allows us to identify the singularity corresponding to the critical point of a second-order phase transition at finite $\mu$, given information only at $\mu=0$. The scheme is potentially useful for computing thermodynamic properties of strongly interacting hot and dense matter in lattice gauge theory. The technique is illustrated by an application to a chiral effective model. • ### Charge fluctuations in chiral models and the QCD phase transition(1108.1300) Aug. 5, 2011 hep-ph We consider the Polyakov loop-extended two flavor chiral quark--meson model and discuss critical phenomena related with the spontaneous breaking of the chiral symmetry. The model is explored beyond the mean-field approximation in the framework of the functional renormalisation group. We discuss properties of the net-quark number density fluctuations as well as their higher cumulants. We show that with the increasing net-quark number density, the higher order cumulants exhibit a strong sensitivity to the chiral crossover transition. We discuss their role as probes of the chiral phase transition in heavy-ion collisions at RHIC and LHC. • ### Net-proton probability distribution in heavy ion collisions(1107.4267) July 21, 2011 hep-ph We compute net-proton probability distributions in heavy ion collisions within the hadron resonance gas model. The model results are compared with data taken by the STAR Collaboration in Au-Au collisions at sqrt(s_{NN})= 200 GeV for different centralities. We show that in peripheral Au-Au collisions the measured distributions, and the resulting first four moments of net-proton fluctuations, are consistent with results obtained from the hadron resonance gas model. However, data taken in central Au-Au collisions differ from the predictions of the model. The observed deviations can not be attributed to uncertainties in model parameters. We discuss possible interpretations of the observed deviations. • ### Fluctuations as probe of the QCD phase transition and freeze-out in heavy ion collisions at LHC and RHIC(1103.3511) June 22, 2011 hep-ph, hep-lat We discuss the relevance of higher order moments of net baryon number fluctuations for the analysis of freeze-out and critical conditions in heavy ion collisions at LHC and RHIC. Using properties of O(4) scaling functions, we discuss the generic structure of these higher moments at vanishing baryon chemical potential and apply chiral model calculations to explore their properties at non-zero baryon chemical potential. We show that the ratios of the sixth to second and eighth to second order moments of the net baryon number fluctuations change rapidly in the transition region of the QCD phase diagram. Already at vanishing baryon chemical potential they deviate considerably from the predictions of the hadron resonance gas model which reproduce the second to fourth order moments of the net proton number fluctuations at RHIC. We point out that the sixth order moments of baryon number and electric charge fluctuations remain negative at the chiral transition temperature. Thus, they offer the possibility to probe the proximity of the thermal freeze-out to the crossover line. • ### Three-body interactions in Fermi systems(1101.4858) We show that the contributions of three-quasiparticle interactions to normal Fermi systems at low energies and temperatures are suppressed by n_q/n compared to two-body interactions, where n_q is the density of excited or added quasiparticles and n is the ground-state density. For finite Fermi systems, three-quasiparticle contributions are suppressed by the corresponding ratio of particle numbers N_q/N. This is illustrated for polarons in strongly interacting spin-polarized Fermi gases and for valence neutrons in neutron-rich calcium isotopes. • ### The influence of fluctuations on thermodynamics near chiral phase transition(1009.4129) Sept. 21, 2010 hep-ph We discuss the influence of fluctuations on thermodynamics near the chiral phase transition within Polyakov loop extended quark--meson model based on the functional renormalization group (FRG) method. We include the gluon fields in the FRG flow equation self-consistently on the mean-field level. We focus on the properties of the phase diagram and net-baryon number fluctuations. • ### The renormalization group and quark number fluctuations in the Polyakov loop extended quark-meson model at finite baryon density(1008.4570) Aug. 26, 2010 hep-ph Thermodynamics and the phase structure of the Polyakov loop-extended two flavors chiral quark--meson (PQM) model is explored beyond the mean-field approximation. The analysis of the PQM model is based on the functional renormalization group (FRG) method. We formulate and solve the renormalization group flow equation for the scale-dependent thermodynamic potential in the presence of the gluonic background field at finite temperature and density. We determine the phase diagram of the PQM model in the FRG approach and discuss its modification in comparison with the one obtained under the mean-field approximation. We focus on properties of the net-quark number density fluctuations as well as their higher moments and discuss the influence of non-perturbative effects on their properties near the chiral crossover transition. We show, that with an increasing net-quark number density the higher order moments exhibit a peculiar structure near the phase transition. We also consider ratios of different moments of the net-quark number density and discuss their role as probes of deconfinement and chiral phase transitions. • ### Vacuum fluctuations and the thermodynamics of chiral models(1005.3166) May 18, 2010 hep-ph We consider the thermodynamics of chiral models in the mean-field approximation and discuss the relevance of the (frequently omitted) fermion vacuum loop. Within the chiral quark-meson model and its Polyakov loop extended version, we show that the fermion vacuum fluctuations can change the order of the phase transition in the chiral limit and strongly influence physical observables. We compute the temperature-dependent effective potential and baryon number susceptibilities in these models, with and without the vacuum term, and explore the cutoff and the pion mass dependence of the susceptibilities. Finally, in the renormalized model the divergent vacuum contribution is removed using the dimensional regularization. • ### Meson fluctuations and thermodynamics of the Polyakov loop extended quark-meson model(1004.2665) April 15, 2010 hep-ph Thermodynamics and the phase structure of the Polyakov loop-extended two flavor chiral quark-meson model (PQM) are explored. The analysis of the PQM model is based on the functional renormalization group (FRG) method. An appropriate truncation of the effective action with quarks coupled to background gluonic fields is introduced. Within this scheme, we derive the renormalization group flow equation for the scale-dependent thermodynamic potential at finite temperature and density in the presence of a symmetry breaking external field. The influence of fluctuations and of the background gluon field on the properties of net-quark number density fluctuations and their higher moments is explored. We study the dependence of the kurtosis of quark number fluctuations on the pion mass and show that, in the presence of a symmetry breaking term, the fluctuations lead to a smoothing of observables near the crossover transition. • ### Fluctuations and isentropes near the chiral critical endpoint(0907.1344) Nov. 18, 2009 hep-th, hep-ph, nucl-th, hep-lat Isentropic trajectories crossing the chiral phase transition near the critical endpoint (CEP) are studied for two light quark flavors. The calculations are performed within an effective chiral model with quark-meson interactions, belonging to the same universality class as QCD. We confront mean-field thermodynamics with the functional renormalization group approach, where fluctuations are properly taken into account. We establish a connection between modifications of the isentropic trajectories found in mean-field calculations at the crossover transition near the CEP and the order of the phase transition in the chiral limit. Furthermore, the isentropes obtained with the renormalization group are completely smooth at the crossover transition and do not in any way reflect the proximity of the CEP. In particular, our results do not show the recently conjectured focussing of isentropes from the crossover region towards the critical endpoint. • ### The Functional Renormalization Group and O(4) scaling(0904.0466) April 3, 2009 hep-ph, nucl-th The critical behavior of the chiral quark-meson model is studied within the Functional Renormalization Group (FRG). We derive the flow equation for the scale dependent thermodynamic potential at finite temperature and density in the presence of a symmetry-breaking external field. Within this scheme, the critical scaling behavior of the order parameter, its transverse and longitudinal susceptibilities as well as the correlation lengths near the chiral phase transition are computed. We focus on the scaling properties of these observables at non-vanishing external field when approaching the critical point from the symmetric as well as from the broken phase. We confront our numerical results with the Widom-Griffiths form of the magnetic equation of state, obtained by a systematic epsilon-expansion of the scaling function. Our results for the critical exponents are consistent with those recently computed within Lattice Monte-Carlo studies of the O(4) spin system. • ### Kurtosis and compressibility near the chiral phase transition(0809.3129) Sept. 18, 2008 hep-ph The properties of net quark number fluctuations in the vicinity of the QCD chiral phase transition are discussed in terms of an effective chiral model in the mean-field approximation. We focus on the ratio of the fourth- to second- order cumulants (kurtosis) and the compressibility of the system and discuss their dependence on the pion mass. It is shown that near the chiral phase transition, both observables are sensitive to the value of $m_\pi$. For physical $m_\pi$, the kurtosis exhibits a peak whereas the inverse compressibility shows a dip at the pseudocritical temperature. These structures disappear for large $m_\pi$. Our results, obtained in an effective model with two flavors, are qualitatively consistent with recent results of 2+1 flavor lattice gauge theory. We also discuss the high- and low-temperature properties of these observables and the role of the coupling of the quark degrees of freedom to the Polyakov loop. • ### Neutron matter at finite temperature(0711.3613) We calculate the neutron matter equation of state at finite temperature based on low-momentum two- and three-nucleon interactions. The free energy is obtained from a loop expansion around the Hartree-Fock energy, including contributions from normal and anomalous diagrams. We focus on densities below saturation density with temperatures T <= 10 MeV and compare our results to the model-independent virial equation of state and to variational calculations. Good agreement with the virial equation of state is found at low density. We provide simple estimates for the theoretical error, important for extrapolations to astrophysical conditions. • ### Dependence of the BCS 1S0 superfluid pairing gap on nuclear interactions(nucl-th/0611024) March 5, 2007 astro-ph, nucl-ex, nucl-th We study in detail the dependence of the 1S0 superfluid pairing gap on nuclear interactions and on charge-independence breaking at the BCS level. Starting from chiral effective-field theory and conventional nucleon-nucleon (NN) interactions, we use the renormalization group to generate low-momentum interactions V_{low k} with sharp and smooth regulators. The resulting BCS gaps are well constrained by the NN scattering phase shifts, and the cutoff dependence is very weak for sharp or sufficiently narrow smooth regulators with cutoffs Lambda > 1.6 fm^{-1}. It is therefore likely that the effect of three-nucleon interactions on 1S0 superfluidity is small at the BCS level. The charge dependence of nuclear interactions has a 10% effect on the pairing gap. • ### The neutron matter equation of state from low-momentum interactions(nucl-th/0611070) Nov. 19, 2006 astro-ph, nucl-th We calculate the neutron matter equation of state at finite temperature based on low-momentum nucleon-nucleon and three-nucleon interactions. Our results are compared to the model-independent virial equation of state and to variational calculations. We provide a simple estimate for the theoretical error, important for extrapolations to astrophysical conditions. • ### Scattering of vector mesons off nucleons(nucl-th/0112052) March 25, 2002 nucl-th We construct a relativistic and unitary approach to 'high' energy pion- and photon-nucleon reactions taking the $\pi N, \pi \Delta$, $\rho N$, $\omega N$, $\eta N, K \Lambda, K \Sigma$ final states into account. Our scheme dynamically generates the s- and d-wave nucleon resonances N(1535), N(1650) and N(1520) and isobar resonances $\Delta(1620)$ and $\Delta(1700)$ in terms of quasi-local interaction vertices. The description of photon-induced processes is based on a generalized vector-meson dominance assumption which directly relates the electromagnetic quasi-local 4-point interaction vertices to the corresponding vertices involving the $\rho$ and $\omega$ fields. We obtain a satisfactory description of the elastic and inelastic pion- and photon-nucleon scattering data in the channels considered. The resulting s-wave $\rho$- and $\omega$-nucleon scattering amplitudes are presented. Using these amplitudes we compute the leading density modification of the $\rho$ and $\omega$ energy distributions in nuclear matter. We find a repulsive energy shift for the $\omega$ meson at small nuclear density but predict considerable strength in resonance-hole like $\omega$-meson modes. Compared to previous calculations our result for the $\rho$-meson spectral function shows a significantly smaller in-medium effect. This reflects a fairly small coupling strength of the N(1520) resonance to the $\rho N$ channel. • ### Quantum interference of rho0- and omega-mesons in the pi N -> e+e- N reaction(nucl-th/0202049) Feb. 15, 2002 nucl-th The study of the $\pi N\to \rho^0 N$ and $\pi N \to \omega N$ amplitudes below and close to the vector meson production threshold ($1.4<\sqrt s <1.8$ GeV)reveals a rich structure arising from the presence of baryon resonances in this energy range. These resonances are reflected in the interference pattern of the $e^+e^-$ decays of the $\rho^0$- and $\omega$-mesons produced in $\pi^- p$ and $\pi^+ n$ reactions. We discuss the shape and magnitude of the $\rho^0$-$\omega$ interference in the $\pi^-p \to e^+e^- n$ and $\pi^+n \to e^+e^- p$ reaction cross sections as functions of the total center of mass energy $\sqrt s$. We find contrasted results: the interference is largely destructive for the $\pi^-p \to e^+e^- n$ cross section but constructive for the $\pi^+n \to e^+e^- p$ cross section. An experimental study of these reactions would provide significant constraints on the coupling of vector meson-nucleon channels to low-lying baryon resonances. • ### From meson-nucleon scattering to vector mesons in nuclear matter(nucl-th/0003012) March 8, 2000 nucl-th We employ meson-nucleon scattering data to deduce the properties of the low-mass vector mesons in nuclear matter, and present results for the $\rho$ and $\omega$ in-medium spectral functions. The corresponding thermal emission rate for lepton pairs is also discussed. • ### Saturation from nuclear pion dynamics(nucl-th/9907078) Oct. 7, 1999 hep-ph, nucl-th We construct an equation-of-state for nuclear matter based on the chiral Lagrangian. The relevant scales are discussed and an effective chiral power expansion scheme, which is constructed to work around the nuclear saturation density, is presented. A realistic equation-of-state is obtained by adjusting one free parameter, when the leading and subleading terms in the expansion are included. The saturation mechanism is due to correlations induced by the one-pion-exchange interaction. Furthermore, we find a substantial deviation from the Fermi-gas estimate of the quark condensate in nuclear matter already at the saturation density. • ### Masses of hadrons in nuclei(nucl-th/9811040) Nov. 11, 1998 nucl-th We emphasize the central role played by the spectral function in the description of hadrons in matter and discuss the applicability of the quasiparticle concept to the propagation of hadrons in dense nuclear matter. Theoretical and experimental results relevant for the in medium properties of vector mesons and kaons are briefly reviewed. We also present novel results for the $\rho$ and $\omega$ spectral functions in nuclear matter, deduced from a coupled channel analysis of pion-nucleon scattering data.	2021-01-24 15:06:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6671366095542908, "perplexity": 1012.8236974710277}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703549416.62/warc/CC-MAIN-20210124141945-20210124171945-00544.warc.gz"}
https://www.quizover.com/online/course/3-5-alternating-current-versus-direct-current-by-openstax?page=2	# 3.5 Alternating current versus direct current (Page 3/10) Page 3 / 10 ## Power losses are less for high-voltage transmission (a) What current is needed to transmit 100 MW of power at 200 kV? (b) What is the power dissipated by the transmission lines if they have a resistance of $1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\Omega$ ? (c) What percentage of the power is lost in the transmission lines? Strategy We are given ${P}_{\text{ave}}=\text{100 MW}$ , ${V}_{\text{rms}}=\text{200 kV}$ , and the resistance of the lines is $R=1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\Omega$ . Using these givens, we can find the current flowing (from $P=\text{IV}$ ) and then the power dissipated in the lines ( $P={I}^{2}R$ ), and we take the ratio to the total power transmitted. Solution To find the current, we rearrange the relationship ${P}_{\text{ave}}={I}_{\text{rms}}{V}_{\text{rms}}$ and substitute known values. This gives ${I}_{\text{rms}}=\frac{{P}_{\text{ave}}}{{V}_{\text{rms}}}=\frac{\text{100}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{W}}{\text{200}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{V}}=\text{500 A}.$ Solution Knowing the current and given the resistance of the lines, the power dissipated in them is found from ${P}_{\text{ave}}={I}_{\text{rms}}^{2}R$ . Substituting the known values gives ${P}_{\text{ave}}={I}_{\text{rms}}^{2}R=\left(\text{500 A}{\right)}^{2}\left(1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\Omega \right)=\text{250 kW}.$ Solution The percent loss is the ratio of this lost power to the total or input power, multiplied by 100: $\text{% loss=}\frac{\text{250 kW}}{\text{100 MW}}×\text{100}=0\text{.}\text{250 %}.$ Discussion One-fourth of a percent is an acceptable loss. Note that if 100 MW of power had been transmitted at 25 kV, then a current of 4000 A would have been needed. This would result in a power loss in the lines of 16.0 MW, or 16.0% rather than 0.250%. The lower the voltage, the more current is needed, and the greater the power loss in the fixed-resistance transmission lines. Of course, lower-resistance lines can be built, but this requires larger and more expensive wires. If superconducting lines could be economically produced, there would be no loss in the transmission lines at all. But, as we shall see in a later chapter, there is a limit to current in superconductors, too. In short, high voltages are more economical for transmitting power, and AC voltage is much easier to raise and lower, so that AC is used in most large-scale power distribution systems. It is widely recognized that high voltages pose greater hazards than low voltages. But, in fact, some high voltages, such as those associated with common static electricity, can be harmless. So it is not voltage alone that determines a hazard. It is not so widely recognized that AC shocks are often more harmful than similar DC shocks. Thomas Edison thought that AC shocks were more harmful and set up a DC power-distribution system in New York City in the late 1800s. There were bitter fights, in particular between Edison and George Westinghouse and Nikola Tesla, who were advocating the use of AC in early power-distribution systems. AC has prevailed largely due to transformers and lower power losses with high-voltage transmission. Do somebody tell me a best nano engineering book for beginners? what is fullerene does it is used to make bukky balls are you nano engineer ? s. what is the Synthesis, properties,and applications of carbon nano chemistry so some one know about replacing silicon atom with phosphorous in semiconductors device? Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure. Harper how to fabricate graphene ink ? for screen printed electrodes ? SUYASH What is lattice structure? of graphene you mean? Ebrahim or in general Ebrahim in general s. Graphene has a hexagonal structure tahir On having this app for quite a bit time, Haven't realised there's a chat room in it. Cied what is biological synthesis of nanoparticles what's the easiest and fastest way to the synthesize AgNP? China Cied types of nano material I start with an easy one. carbon nanotubes woven into a long filament like a string Porter many many of nanotubes Porter what is the k.e before it land Yasmin what is the function of carbon nanotubes? Cesar I'm interested in nanotube Uday what is nanomaterials and their applications of sensors. what is nano technology what is system testing? preparation of nanomaterial Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it... what is system testing what is the application of nanotechnology? Stotaw In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google Azam anybody can imagine what will be happen after 100 years from now in nano tech world Prasenjit after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments Azam name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world Prasenjit how hard could it be to apply nanotechnology against viral infections such HIV or Ebola? Damian silver nanoparticles could handle the job? Damian not now but maybe in future only AgNP maybe any other nanomaterials Azam Hello Uday I'm interested in Nanotube Uday this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15 Prasenjit can nanotechnology change the direction of the face of the world At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light. how did you get the value of 2000N.What calculations are needed to arrive at it Privacy Information Security Software Version 1.1a Good Got questions? Join the online conversation and get instant answers!	2018-09-22 07:00:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 11, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6825097799301147, "perplexity": 1765.6003885228981}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267158205.30/warc/CC-MAIN-20180922064457-20180922084857-00244.warc.gz"}
https://www.macajournal.com/issue_45071_45073.html	##### Volume 1 (2019) Issue 1 Journal of Mathematical Analysis and its Contemporary Applications ##### Non-stabilities of mixed type Euler-Lagrange k-cubic-quartic functional equation in various normed spaces John Michael Rassias; Mohan Arunkumar; Elumalai Sathya Volume 1, Issue 1 , February 2019, Pages 1-43 ##### Abstract In this paper, we introduce and examine the generalized Ulam-Hyers stability of fixed Euler-Lagrange k-Cubic-Quartic functional Equationf(x+ky) + f(kx+y) + f(x-ky) + f(y-kx) = ... Read More ##### Stability of cosine type functional equations onmodule extension Banach algebras Abbas Zivari-Kazempour Volume 1, Issue 1 , February 2019, Pages 44-49 ##### Abstract Let A be a Banach algebra and X be a Banach A-bimodule. In this paper we investigate the stability of the cosine type functional equation φ(ab,a·y+x·b)=φ(ab,x·b-a·y)=2φ(a,x)φ(b,y),on ... Read More	2023-02-07 19:32:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2512490153312683, "perplexity": 9739.00498076388}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500628.77/warc/CC-MAIN-20230207170138-20230207200138-00840.warc.gz"}
http://www.physicsforums.com/forumdisplay.php?f=70&order=desc&page=13	Sub-Forums : Special & General Relativity Search this Forum Relativity FAQ Mar11-14 10:20 AM bcrowell Special & General Relativity - Dependence of various physical phenomena on relative motion of the observer and the observed objects. Exp. & theo. theories of relativity Meta Thread / Thread Starter Last Post Replies Views Before posting anything, please review the Physics Forums Global Guidelines. If you are seeking help with a... Feb23-13 08:40 AM ZapperZ 1 37,356 The section of the Usenet Physics FAQ titled "Experimental Basis of Special Relativity" has been cited here many... Dec31-07 02:06 AM jtbell 0 37,122 in relativity theory, in a certain "rapid space" the distance between two neighboring points is given by- ds=√... Dec12-13 11:49 PM nafizamin 6 595 I'm debating whether this should go in the GR & SR section, but anyways, I heard both claims and both of them are... Dec12-13 04:46 PM Bill_K 18 1,277 This question has been asked two years ago, but it wasn't resolved (I think). Here goes This problem is Problem 5... Dec12-13 03:09 PM Bill_K 1 449 Hi, I've taken a course in SR and studied GR on my own, but I do not know how to solve problems of this type. This is... Dec11-13 05:46 PM tom.stoer 10 932 Length Contraction Paradox ( 1 2 3 ... Last) Hi, As is well known, Relativity claims that a rod of a given proper length will appear length contracted when... Dec10-13 07:34 PM pervect 67 3,177 What is the defining moment when you use relativistic equations instead of classical ones? I have heard something as... Dec10-13 01:41 AM Blackthorn 3 520 The approach taken in linearized gravity seems to be to 'perturb' the 'Minkowski metric' such that g_{\mu \nu} =... Dec9-13 07:00 PM WannabeNewton 1 456 Hi, I know these questions must sound ridiculous and I apologize, I'm a newbie. My textbook says that the inner... Dec9-13 01:27 PM sciencegem 3 560 Huffingtonpost ran a story heaping praise on a guy for demonstrating GR with the old "rubber sheet" analogy. This demo... Dec9-13 12:15 PM WannabeNewton 18 1,206 Hi people, I have the following question: First, here is a concise statement of the major neutrino speed... Dec9-13 05:51 AM Vanadium 50 6 582 Gentlemen- What does the stress-energy tensor describe? The tt term of the stress-energy tensor expresses energy... Dec8-13 08:09 PM Simon Bridge 13 968 How do we know a "photon" is massless? Do we have any experimental proof? Photon has momentum, it even bends in... Dec8-13 01:04 PM PeterDonis 10 902 I suspect this is somewhat off the beaten track here, but there may be some few that could give it a go. Einstein... Dec8-13 05:24 AM K^2 18 1,497 Hi guys, I am reading griffiths electrodynamics. I have a question. How does Maxwell equation suggest that the... Dec8-13 02:22 AM Rena Cray 9 970 In various other threads we have been kicking around various equations for a spherical shell and discussing the... Dec7-13 10:39 PM PeterDonis 7 810 Hello Everyone, I came here with a question and hope you can shed some light. We know that Ricci tensor which... Dec7-13 10:15 PM vaibhavtewari 4 391 Hi, In classical and quantum physics and even in special relativity, shifting the energy of a system by a constant... Dec7-13 10:44 AM George Jones 28 1,240 I know about the Michaelson-Morley expt. trying to measure the speed of light, once in the direction of motion of the... Dec7-13 01:25 AM ghwellsjr 2 582 hi people! please can any one explain me the conclusions of a thought experiment given below ? consider yourself... Dec6-13 09:18 PM PeterDonis 8 946 Question in the subject. I have no idea what the DoD's interest is in this. Sorry is this is the wrong forum. Dec6-13 10:30 AM TomServo 7 617 Would there be a direct proof of the energy-stress tensor of general relativity? My lecturer only provides me with a... Dec5-13 10:23 PM WannabeNewton 8 637 Consider two iron rods of about 10 metres separated by 10 metre. They are in series position and not connected in... Dec5-13 07:40 AM DaleSpam 39 1,710 Please view the following... http://www.wimp.com/visualizegravity/ This is the way scientists try to explain the... Dec4-13 09:48 PM dauto 2 680 The Lorentz group generators, in any representation, satisfy the commutation relation = i \left( g^{\nu... Dec4-13 05:16 PM spookyfish 7 844 Hello there! I have a more or less silly and possibly obvious question that's been bugging me a lot. Here's the... Dec4-13 02:54 PM George Jones 8 916 Hi, Is it pure coincidence that if you put ##c=v_e=\sqrt{2GM/R}## in the escape velocity, you end up with the... Dec4-13 02:11 AM dvf 2 451 Hi all, think this might be a silly or trivial question but I've got myself in a bother so thought I'd get some help.... Dec3-13 02:59 PM WannabeNewton 16 925 For example let's say i have two rods of same length ie. 4metres placed in a series position. They are separated by a... Dec3-13 09:11 AM PeterDonis 11 672 There is any field in General Relativity(I know gravitational field but it is not that which I mean) Thanks Dec3-13 09:09 AM WannabeNewton 2 476 Hello all, I have a geodesic equation from extremizing the action which is second order. I am curious as to what... Dec2-13 02:29 PM Bill_K 7 474 SR question ( 1 2 3) First of all, Happy Thanksgiving to you all! Now, my question: I'm on earth viewing through a telescope. I... Dec2-13 10:09 AM ghwellsjr 44 1,450 Most of the paradoxes are solved by either "Relativity of simultaneity" or "Born rigidity" And there is a 3rd tool... Dec2-13 06:07 AM AndrzejK 60 3,022 Hi, I'm new posting to these forums, so I'm not sure if this is in the right place. Anyway, here it goes: I've... Dec1-13 11:57 PM yogi 40 2,436 Hi, GR permits wormholes hypothetically right? But surely they should violate conservation of energy because if one... Dec1-13 08:03 PM WannabeNewton 1 498 I'm very noob at this and am a bit confused: Formula 1: E_T = \gamma \cdot m c^2 Formula 2: p = \gamma m v... Dec1-13 05:41 PM PAllen 9 850 We have three observers. The A is in earth,B is into a car that runs with big speed,C is out of galaxy. All... Dec1-13 03:18 PM Nugatory 7 441 I had always thought that length contraction meant that the object itself shrunk but I stumbled across an article... Dec1-13 01:47 PM Superposed_Cat 8 556 hi there, In this Ex ( see attached snapshot ), point b), the poisson bracket equation is not so straightforward to... Nov30-13 12:25 PM WannabeNewton 9 496 I mean the application,such as because of the time dilation, humans can observe some particles which don't have long... Nov30-13 10:47 AM phinds 2 396 My understanding is that an outside observer sees objects undergo time dilation as they approach sources of gravity,... Nov30-13 10:25 AM WannabeNewton 10 710	2014-04-17 04:03:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6705902814865112, "perplexity": 2308.568748437018}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00223-ip-10-147-4-33.ec2.internal.warc.gz"}
https://www.erisian.com.au/wordpress/2010/11/20/rocket-tracking	## Rocket Tracking While I was still procrastinating doing the altosui and Google Earth mashup I mentioned last post, Keith pointed out that Google Maps has a static API, which means it’s theoretically possible to have altosui download maps of the launch site before you leave, then draw on top of them to show where your rocket’s been. The basic API is pretty simple — you get an image back centred on a given latitude and longitude; you get to specify the image size (up to 640×640 pixels), and a zoom level. A zoom level of 0 gives you the entire globe in a 256×256 square, and each time you increase the zoom level you turn each pixel into four new ones. Useful zoom levels seem to be about 15 or so. But since it’s a Mercator projection, you don’t have to zoom in as far near the poles as you do around the equator — which means the “or so” is important, and varies depending on the the latitude. Pulling out the formula for the projection turns out to be straightforward — though as far as I can tell, it’s not actually documented. Maybe people who do geography stuff don’t need docs to work out how to convert between lat/long and pixel coordinates, but I’m not that clever. Doing a web search didn’t seem to offer much certainty either; but decoding the javascript source turned out to not be too hard. Formulas turn out to be (in Java): Point2D.Double latlng2coord(double lat, double lng, int zoom) { double scale_x = 256/360.0 * Math.pow(2, zoom); double scale_y = 256/(2.0Math.PI) Math.pow(2, zoom); Point2D.Double res = new Point2D.Double(); res.x = lngscale_x; double e = Math.sin(Math.toRadians(lat)); e = limit(e, -1+1.0E-15, 1-1.0E-15); res.y = 0.5Math.log((1+e)/(1-e))*-scale_y; return res; } That gives you an absolute coordinate relative to the prime meridian at the equator, so by the time you get to zoom level 15, you’ve got an 8 million pixel by 8 million pixel coordinate system, and you’re only ever looking at a 640×640 block of that at a time. Fortunately, you also know the lat/long of the center pixel of whatever tile you’re looking at — it’s whatever you specified when you requested it. The inverse function of the above gives you the the latitude and longitude for centrepoints of adjacent maps, which then lets you tile the images to display a larger map, and choosing a consistent formula for the tiling lets you download the right map tiles to cover an area before you leave, without having to align the map tiles exactly against your launch site coordinates. In Java, the easy way to deal with that seems to be to setup a JScrollable area, containing a GridBagLayout of the tiles, each of which are images set as the icon of JLabels. Using the Graphics2D API lets you draw lines and circles and similar on the images, and voila, you have a trace: Currently the “UI” for downloading the map images is that it’ll print out some wget lines on stdout, and if you run them, next time you run altosui for that location, you’ll get maps. (And in the meantime, you’ll just get a black background)	2022-08-19 14:07:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3518551290035248, "perplexity": 1260.885293390118}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573699.52/warc/CC-MAIN-20220819131019-20220819161019-00106.warc.gz"}
http://www.edmerls.com/index.php/Analytical%20Chemistry/Polarography/15/How%20the%20mixture%20of%20metallic%20ions%20can%20be%20analyzed%20by%20Polarography?	By Sunil Bhardwaj 1126 Views The mixture of the metallic ions is taken in the Polarographic cell. The jockey is allowed to move from H to I and for each position of jockey, we find out the reading to voltmeter (v) and galvanometer. Then we plot a graph of current in $$\mu A$$ V/S potential applied. The Polarogram is consisting of series of waves. Each wave corresponds to the reduction of one particular metal ion species. The polarogram obtained when a solution containing different metal ions at concentration of 104 gm ions per $$dm^3$$ is electrolyzed with 0.1 N KCl as the supporting electrolyte. MCQ on Polarography from Analytical Chemistry ###### Prof. Gianfranco Coletti Shared publicly - 2019-08-23 00:00:00 Don’t want your columns to simply stack in some grid tiers? Use a combination of different classes for each tier as needed. See the example below for a better idea of how it all works. ###### Prof. Maheshwar Sharon Shared publicly - 2019-08-24 00:00:00 For grids that are the same from the smallest of devices to the largest, use the .col and .col-* classes. Specify a numbered class when you need a particularly sized column; otherwise, feel free to stick to ###### sunil Shared publicly - 2023-02-28 11:09:52 this is ###### ss Shared publicly - 2023-02-28 10:48:10 gsgsg #### Latest News • Become an Instructor 4 March, 2018 Apply to join the passionate instructors who share their expertise and knowledge with the world. You'll collaborate with some of the industry's best producers, directors, and editors so that your content is presented in the best possible light.. #### More Chapters • Chromatography • Solvent Extraction • Gravimetric Analysis • Optical Methods • Polarography • #### Other Subjects • English • Applied Physics • Environmental Studies • Physical Chemistry • Analytical Chemistry • Organic Chemistry • Soft Skills • Engineering Drawing • General Medicine • Mathematics • Patente B Italia	2023-03-26 16:21:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3892022371292114, "perplexity": 8525.748338379626}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945473.69/warc/CC-MAIN-20230326142035-20230326172035-00548.warc.gz"}
https://doc.cgal.org/latest/Jet_fitting_3/classLocalKernel.html	CGAL 5.3 - Estimation of Local Differential Properties of Point-Sampled Surfaces LocalKernel Concept Reference ## Definition The concept LocalKernel describes the set of requirements to be fulfilled by any class used to instantiate the second template parameter of the class CGAL::Monge_via_jet_fitting<DataKernel,LocalKernel,SvdTraits>. This concept provides the geometric primitives used for the computations in the class CGAL::Monge_via_jet_fitting. Requirements In the class CGAL::Monge_via_jet_fitting the scalar type, LocalKernel::FT, must be the same as that of the SvdTraits concept : SvdTraits::FT. The type LocalKernel::FT is a model of the FieldWithSqrt concept. Operations The scalar type LocalKernel::FT must be a field type with a square root. Only constructors (from 3 scalars and copy constructors) and access methods to coordinates x(), y(), z() are needed for the point and vector types. Has Models: CGAL::Cartesian<FieldNumberType> CGAL::Simple_cartesian<FieldNumberType> DataKernel SvdTraits ## Types typedef unspecified_type FT The scalar type. typedef unspecified_type Point_3 The point type. typedef unspecified_type Vector_3 The vector type.	2021-09-17 09:23:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20923756062984467, "perplexity": 5271.909369009071}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780055632.65/warc/CC-MAIN-20210917090202-20210917120202-00163.warc.gz"}
https://stats.stackexchange.com/questions/346847/combine-two-different-probabilities/346867	# Combine two different probabilities I'm not sure how to best phrase this, but I'm not sure how to combine two probabilities (or if that even makes sense). Example: Assume that 5% percent of the population has a particular gene is 20 times more likely to become a professional athlete than those without. Considering all athletes, what percentage of them has this gene? It seems like the percentage of athletes would be greater than 5%, but I'm not sure how to figure this out. Is this something that can be determined? Note: This seems super simple, so it might have already been asked. I searched a lot, but I don't have the right language for this field yet. • bayes' theorem and the law of total probability together is sufficient to solve this – shimao May 17 '18 at 20:35 • Thanks @shimao, I wrote up an answer using those two. I think it makes sense. Can you look it over for correctness? – tylerware May 17 '18 at 22:00 Per suggestion of shimao, the answer lies in the Bayes theorem and the Law of Total Probability. Let $G$ be the event that someone has the gene and let $A$ be the event that someone becomes a professional athlete. From the example we know: $$P(G) = 0.05 \\ P(G^\prime) = 0.95 \\ P(A\|G)/P(A\|G^\prime) = 20$$ What we want to know is what $P(G\|A)$. Enter Bayes Theorem $$P(G\|A) = \frac{P(A\|G)P(G)}{P(A)}$$ Which has two unknowns $P(A\|G)$ and $P(A)$, using the Law of Total probability we can find $P(A)$. $$P(A) = P(G)P(A\|G) + P(A\|G^\prime)P(G^\prime)$$ From $P(A\|G)/P(A\|G^\prime) = 20$ we can get $P(A\|G^\prime) = P(A\|G)/20$ and can substitute in to get: $$P(A) = P(G)P(A\|G) + \frac{P(A\|G)P(G^\prime)}{20}$$ Returning to our application of Bayes Theorem and substituting in this expression of $P(A)$ we get: $$P(G\|A) = \frac{20\cdot P(A\|G)P(G)}{P(A\|G)(20 \cdot P(G) + P(G^\prime))}$$ Which simplifies to $$P(G\|A) = \frac{20 \cdot P(G)}{20 \cdot P(G) + P(G^\prime)}$$ all are known, so $$P(G\|A) = \frac{20 \cdot 0.05}{20 \cdot 0.05 + 0.95} \approx 0.51$$ So the percentage of professional athletes that have this gene is 51%. • You're missing an extra 20 in your denominator after combining the two. $$P(G\|A) = \frac{20P(A\|G)P(G)}{P(A\|G)(20P(G) + P(G^\prime))}$$ – dankernler May 17 '18 at 23:31 • @dankernler Fixed, thank you for pointing this out :) – tylerware May 18 '18 at 16:43 • No problem. I haven't done one like this in a while, but the answer of 1 was a red flag! – dankernler May 18 '18 at 17:30	2019-07-18 21:31:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6046862602233887, "perplexity": 338.49088214160906}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525829.33/warc/CC-MAIN-20190718211312-20190718233312-00506.warc.gz"}
http://www.chegg.com/homework-help/questions-and-answers/velocity-sound-air-given-equation-v-20-273-t-v-velocity-meters-per-second-t-temperature-de-q2518407	the velocity of sound in air is given by the equation v=20 ?273+t where v is the velocity in meters per second and t is the temperature in degrees Celsius. Find the velocity when the temperature is 117ºC. Round to the nearest meter/sec. A. 20 meters per second B. 330 meters per second C. 395 meters per second D. 216 meters per second Tiara wants to divide a square piece of paper into two equalivalent triangles. If the square measures 20 cm on each side, what will the third side of each triangle measure? A. 8.9 cm B. 20 cm C. 40 cm D. 28.3 cm	2014-10-21 16:17:06	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8224442005157471, "perplexity": 669.2231593192012}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507444493.40/warc/CC-MAIN-20141017005724-00308-ip-10-16-133-185.ec2.internal.warc.gz"}
https://www.atmos-chem-phys.net/18/12141/2018/	Journal topic Atmos. Chem. Phys., 18, 12141–12159, 2018 https://doi.org/10.5194/acp-18-12141-2018 Atmos. Chem. Phys., 18, 12141–12159, 2018 https://doi.org/10.5194/acp-18-12141-2018 Research article 22 Aug 2018 Research article \| 22 Aug 2018 # Measurement and modeling of the multiwavelength optical properties of uncoated flame-generated soot Measurement and modeling of the multiwavelength optical properties of uncoated flame-generated soot Sara D. Forestieri1,a, Taylor M. Helgestad1,a, Andrew T. Lambe2,3, Lindsay Renbaum-Wolff2, Daniel A. Lack4,5,b, Paola Massoli2, Eben S. Cross6,c, Manvendra K. Dubey7, Claudio Mazzoleni8, Jason S. Olfert9, Arthur J. Sedlacek III10, Andrew Freedman2, Paul Davidovits3, Timothy B. Onasch2,3, and Christopher D. Cappa1 Sara D. Forestieri et al. • 1Department of Civil and Environmental Engineering, University of California, Davis, CA 95616, USA • 2Aerodyne Research Inc., Billerica, MA 01821, USA • 3Chemistry Department, Boston College, Boston, MA 02467, USA • 4NOAA Earth System Research Laboratory, Boulder, CO 80305, USA • 5Cooperative Institute for Research of the Environmental Sciences, University of Colorado, Boulder, CO 80305, USA • 6Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA • 7Los Alamos National Laboratory, Los Alamos, NM 87545, USA • 8Department of Physics and Atmospheric Sciences Program, Michigan Technological University, Houghton, MI 49931, USA • 9Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada • 10Biological, Environmental and Climate Sciences Department, Brookhaven National Laboratory, Upton, NY 11973, USA • anow at: California Air Resources Board, Sacramento, CA 95814, USA • bnow at: Transport Emissions, Air Quality and Climate Consulting, Brisbane, Australia • cnow at: Aerodyne Research Inc., Billerica, MA 01821, USA Correspondence: Sara D. Forestieri (sara.forestieri@arb.ca.gov) and Christopher D. Cappa (cdcappa@ucdavis.edu) Abstract Optical properties of flame-generated black carbon (BC) containing soot particles were quantified at multiple wavelengths for particles produced using two different flames: a methane diffusion flame and an ethylene premixed flame. Measurements were made for (i) nascent soot particles, (ii) thermally denuded nascent particles, and (iii) particles that were coated and then thermally denuded, leading to the collapse of the initially lacy, fractal-like morphology. The measured mass absorption coefficients (MACs) depended on soot maturity and generation but were similar between flames for similar conditions. For mature soot, here corresponding to particles with volume-equivalent diameters $>\sim \mathrm{160}$ nm, the MAC and absorption Ångström exponent (AAE) values were independent of particle collapse while the single-scatter albedo increased. The MAC values for these larger particles were also size-independent. The mean MAC value at 532 nm for larger particles was 9.1±1.1 m2 g−1, about 17 % higher than that recommended by Bond and Bergstrom (2006), and the AAE was close to unity. Effective, theory-specific complex refractive index (RI) values are derived from the observations with two widely used methods: Lorenz–Mie theory and the Rayleigh–Debye–Gans (RDG) approximation. Mie theory systematically underpredicts the observed absorption cross sections at all wavelengths for larger particles (with x>0.9) independent of the complex RI used, while RDG provides good agreement. (The dimensionless size parameter $x=\mathit{\pi }{d}_{\text{p}}/\mathit{\lambda }$, where dp is particle diameter and λ is wavelength.) Importantly, this implies that the use of Mie theory within air quality and climate models, as is common, likely leads to underpredictions in the absorption by BC, with the extent of underprediction depending on the assumed BC size distribution and complex RI used. We suggest that it is more appropriate to assume a constant, size-independent (but wavelength-specific) MAC to represent absorption by uncoated BC particles within models. 1 Introduction Soot particles, which contain light-absorbing black carbon (BC), are a byproduct of the incomplete combustion of fossil fuels and biomass. These particles affect climate directly by absorbing and scattering solar radiation (Bond et al., 2013) and indirectly by acting as cloud condensation nuclei, especially following chemical processing (Lohmann and Feichter, 2005). Soot particles absorb shortwave radiation and have an overall warming effect on climate. The exact magnitude of the climate impacts of BC remains uncertain. One estimate puts top-of-the-atmosphere direct forcing by BC as high as 0.9 W m−2, which is comparable in magnitude to that of CO2 (Ramanathan and Carmichael, 2008). Other more recent assessments yield 0.71 W m−2 with 90 % uncertainty bounds of 0.08 to 1.27 W m−2 (Bond et al., 2013) or 0.61 [+0.16 to +1.40] W m−2 (Wang et al., 2016), while the IPCC suggests a value of 0.40 [+0.05 to +0.80] W m−2 (Boucher et al., 2013). One challenge in modeling the optical properties of soot, and of the BC component in particular, derives from BC having a complex, fractal-like structure, being an agglomerate of small “spherules” (Medalia and Heckman, 1969). One theory that is commonly used in climate models to calculate BC optical properties is Lorenz–Mie theory (hereafter, Mie theory), which makes the physically unrealistic assumption that soot particles are spherical (Bohren and Huffman, 1983). This theory is widely used in climate models (Bond et al., 2013), in part because it does not require any details about the number of spherules or the arrangement of the spherules within the agglomerate but also because it is compatible with calculations for other spherical aerosol types. A variation on Mie theory, the Rayleigh–Debye–Gans (RDG) approximation, is also often used to model the optical properties of BC (Sorensen, 2001), albeit not by climate models. In RDG, the agglomerate absorption cross section (σabs) is the product of σabs for individual spherules and the number of spherules in the agglomerate. As such, RDG neglects spherule-to-spherule interactions, and the mass absorption coefficient (MAC) of an individual spherule is equal to that of the overall particle. (The MAC is the absorption cross section normalized by the particle mass.) There are more complex methods for calculating soot particle optical properties, including the T-matrix method (Mackowski and Mishchenko, 1996) and the discrete dipole approximation (DDA) (Purcell and Pennypacker, 1973), which account for interactions between spherules. Given that these more advanced methods require detailed information on the shape of the soot particles and are computationally intensive, they are not practical for climate models. The derived effective refractive indices used as inputs for these models are theory-specific, and it is necessary to have experimentally constrained effective refractive indices for both RDG and Mie theory if they are to be employed in climate models. For example, Bond et al. (2006) suggested that BC can be described using an MAC = 7.5 m2 g−1 at 550 nm and a complex refractive index (RI) $=\mathrm{1.95}-\mathrm{0.79}i$. However, as they show, the maximum MAC calculated from Mie theory using this refractive index is only 7.2 m2 g−1 over a very narrow range of particle sizes and is much smaller in general, with a value of 4.9 m2 g−1 in the small particle limit, where RDG applies (assuming ρ=1.8 g cm−3). In other words, there can be an inconsistency between the often used MAC and complex RI. This illustrates the need for theory-specific effective refractive index values and a more thorough exploration of the robustness of commonly used optical models. Our work investigates the ability of two optical models, Mie theory and the RDG approximation, to reproduce observed soot optical properties for particles composed primarily of BC. The observations include light absorption and extinction coefficients of soot particles produced from methane diffusion and ethylene premixed flames, measured over four different studies at multiple wavelengths. The impact of shape on soot particle optical properties is also examined. The soot particles sampled during these studies serve as a proxy for different types of soot particles in the ambient atmosphere. Recommended theory-specific complex RI values for BC-dominated soot particles are provided. However, we ultimately suggest that atmospheric models should consider adopting observationally constrained, wavelength-specific constant MAC values for BC rather than calculating the optical properties from optical theories. 2 Experimental: the black carbon studies The measurements reported here were made during a series of laboratory intensive studies that took place at Boston College (BC2, BC3, and BC4) in 2008, 2012, and 2015 and Aerodyne Research (BC3+) in 2014. Below, we provide details of soot particle generation and the measurements made. An experimental schematic is provided in Fig. S1 (in the Supplement). ## 2.1 Soot particle generation Soot particles were produced using two different flame sources and fuel types. During the BC3, BC3+, and BC4 studies, most experiments were conducted using particles produced from an inverted co-flow diffusion flame operating on methane with a sheath flow mixture of O2 and N2 (Stipe et al., 2005) with a net fuel equivalence ratio $\mathit{\phi }=\mathrm{0.7}±\mathrm{0.07}$; because the diffusion flame entrains sheath oxygen into the methane-rich center flow, a range of φ values are accessed, including regions where the local φ may be greater than 1. These are referred to as the “methane diffusion flame” experiments and have been combined into a single dataset since the sampling and generation were similar in all. During BC2 and for a small number of experiments during BC3+, particles were produced using a McKenna flat-flame burner from the combustion of premixed C2H4 (ethylene), O2 and N2 with $\mathit{\phi }=\mathrm{2.0}±\mathrm{0.2}$. These are referred to as the “ethylene premixed flame” experiments. Particles were sampled at a nominal height of 5 cm above the burner during BC3+ but at a nominal height of 20 cm above the burner during BC2 (Cross et al., 2010). As such, the results from the two ethylene premixed flame have been kept separate because particles were sampled from the flame differently in the two studies. The soot particles produced from these two flames exhibited different properties. For example, the organic (OC) mass fraction of the nascent (i.e., freshly emitted and unprocessed) ethylene premixed flame soot particles in BC2 was ∼0.26 (Cross et al., 2010), whereas the OC fraction of nascent methane diffusion flame particles was <0.01. Consequently, upon heating to >200C, the per-particle mass of the ethylene premixed flame soot particles decreased while the methane diffusion flame soot particles were unaffected. The sampling and/or burner conditions were modified during these studies to generate monodisperse soot particles with a volume-equivalent diameter (dp,VED) less than 160 nm. The dp,VED is the diameter calculated from the per-particle mass assuming that particles have spherical morphology and the material density of the particles is 1.8 g cm−3 (Bond and Bergstrom, 2006; Wu et al., 1997; Mullins and Williams, 1987): $\begin{array}{}\text{(1)}& {d}_{\text{p,VED}}={\left(\frac{\mathrm{6}{m}_{\text{p}}}{\mathrm{1.8}\cdot \mathit{\pi }}\right)}^{\mathrm{1}/\mathrm{3}},\end{array}$ where mp is the per-particle mass. For the ethylene premixed flame, sampling closer to the burner surface selected for smaller soot particles; sampling on the centerline over the burner surface vs. off-center may also affect the selected soot particle sizes. For the methane diffusion flame, increasing the fuel dilution (N2) fraction generated smaller soot particles (Stipe et al., 2005). These variations in sampling and/or burner conditions likely led to some changes in the particle optical and chemical properties (López-Yglesias et al., 2014). Thus, some of the size-dependent changes observed in particles smaller than dp,VED ∼160 nm, discussed below, are likely a result of real changes in the particle properties. The extreme case is in the comparison between the soot particles sampled at 5.1 cm (BC3+) and 20.3 cm (BC2) above the ethylene premixed flame burner surface. ## 2.2 Particle processing The soot particles were subjected to various kinds of physical and chemical processing and were selected according to their mobility diameter after processing but prior to sampling for optical and chemical property measurements. Particles size-selected with no processing are termed nascent soot particles. For some experiments, nascent particles were passed through a thermal denuder in which they were heated to 270 C for ∼5 s. Such particles are termed nascent–denuded soot particles. (A summary of terminology is provided in Table S1.) For other experiments, nascent particles were first coated and then thermo-denuded. Coating materials included either dioctyl sebacate (DOS, CxHyOx; BC2, BC3, and BC3+), sulfuric acid (H2SO4; all studies), or secondary organic aerosol from α-pinene photooxidation (SOA; BC4). These particles are referred to as coated–denuded. Literature results indicate that the thermo-denuding of particles coated with or composed of these materials leads to essentially complete evaporation of the non-refractory material (Cappa and Wilson, 2011; Huffman et al., 2008), confirmed by measurements from BC2 (Cross et al., 2010). For DOS coatings, the nascent particles were first size-selected, the monodisperse nascent particles were coated, and the coated particles were subsequently denuded. This is referred to as a forward-coating experiment. For either H2SO4 or SOA coatings, polydisperse nascent soot particles were first coated, then size-selected, and finally denuded. This is referred to as a reverse-coating experiment. This difference in methodology decreased the extent of homogenous nucleation of pure H2SO4 and SOA particles by providing additional soot surface area to act as a condensation sink, and any nucleated particles were further excluded during the size selection. However, the reverse method led to broader size and mass distributions in the denuded soot cores. The typical geometric standard deviation of the per-particle mass distributions for forward-coating experiments was 1.3, while for reverse-coating experiments it ranged from around 1.3 to 2. For BC2, BC3, and BC3+, soot particles were coated in a heated section of tubing containing either DOS or H2SO4, which cooled and condensed onto soot particles after exiting the heated section of tubing. For BC4 experiments, soot particles were coated with α-pinene SOA and H2SO4 that were generated in a potential aerosol mass (PAM) oxidation flow reactor (Lambe et al., 2011). In the PAM reactor, O3 was photolyzed by UV lamps at λ= 254 nm to produce O(1D) radicals, which then reacted with water (RH = 25 %–30 %) to produce OH radicals. Low-volatility products were formed via the reaction between OH and α-pinene, which condensed onto the soot particles. Similarly, H2SO4 was formed through the reaction of SO2 and OH radicals. Particles were size-selected according to their mobility diameter (dm) using a differential mobility analyzer (DMA; TSI model 3080) with a sheath-to-sample flow ratio of 5 : 1, yielding a resolution of $\sim ±\mathrm{20}$ % of the set point in terms of mobility. During BC4, particles were mass-selected with a centrifugal particle mass analyzer (CPMA; Cambustion Ltd.), in addition to being size-selected using a DMA. Particles sampled into the DMA or CPMA first passed through a neutralizer, which imparts an equilibrium charge distribution to the particles. The DMA selects particles according to their electrical mobility, which is dependent upon the number of charges per particle. Particles with more than one charge are larger than those with a single charge, and the presence of these larger particles can confound the interpretation of optical property measurements. Altering the burner and sampling conditions for both flame types minimized the number of these multiply charged particles during the experiments, and in some cases their concentrations were effectively zero. However, for some experiments, their concentrations were non-zero. The method used to account for the multiply charged particles is discussed below. ## 2.3 Instrumentation A wide range of instruments was employed to characterize the soot particle size, mass, composition and optical properties. Not all instruments were deployed for all studies, summarized in Table S2. ### 2.3.1 Size and mass Particle mobility size distributions were measured using a scanning mobility particle sizer (SMPS; TSI model 3936). Particles sampled into the SMPS were passed through a neutralizer, which imparts a new equilibrium charge distribution to the particles. This (re)neutralization step shifts the charge distribution of the monodisperse particles imparted by the size-selection DMA to a new equilibrium state. For particles with mobility diameters 200–300 nm (typical of these studies), the number concentration ratio between particles with +1 charge to those with a greater number of charges selected by the DMA is ∼2–2.5 at equilibrium. Thus, after (re)neutralization, the majority of the particles that were in a +2 state coming out of the DMA are sized in the SMPS in their +1 state. This allows for the quantitative determination of the fraction of particles that had charge states of +1, +2, +3, etc., when selected by the DMA. The information is used to correct the optical measurements for contributions from larger, multiply charged particles for BC2, BC3, and BC3+. Particle mass distributions were measured using a CPMA. Particle number concentrations were measured using a condensation particle counter (CPC; TSI Inc.) and using a mixing CPC (MCPC, BMI model l710). The two particle-counting instruments were placed at different points in the flow path to estimate and account for losses in the transfer lines between instruments. All CPC number concentrations agreed within ±5 %. ### 2.3.2 Composition Particle composition was characterized online using a soot particle aerosol mass spectrometer (SP-AMS; Aerodyne Research, Inc.) (Onasch et al., 2012) and a compact time-of-flight aerosol mass spectrometer (c-ToF-AMS; Aerodyne Research, Inc.) (Canagaratna et al., 2007). The SP-AMS measured concentrations of refractory BC and the associated refractory and non-refractory coatings in BC-containing particles. The c-ToF-AMS measures concentrations of only non-refractory materials, which include particulate organic matter (POM) and some inorganic salts (${\mathrm{NH}}_{\mathrm{4}}^{+}$, ${\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$, ${\mathrm{NO}}_{\mathrm{3}}^{-}$). Both instruments allow for the determination of mass-weighted particle size distributions as characterized by the vacuum aerodynamic diameter (dp,va). Particles were also collected on quartz-fiber filters for offline thermal-optical analysis (Chow et al., 2004), from which the relative abundances of OC and elemental carbon (EC) are determined. During BC4, refractory black carbon mass was quantified with a single particle soot photometer (SP2; Droplet Measurement Technologies). The SP2 heats up the soot particles with an Nd:YAG laser and quantifies the incandescence emission as the particle evaporates. The incandescence signal from the instrument was calibrated using DMA size-selected fullerene soot (Alfa Aesar Lot L18U002) (Laborde et al., 2012). The mass distributions from the SP2 were used to determine contributions from multiply charged particles during BC4. ### 2.3.3 Optical properties Particle optical properties were characterized using multiple instruments. Three different photoacoustic spectrometers (PASs) were used during the various studies to measure particulate light absorption coefficients (babs). During BC2, a custom-built PAS from the National Oceanic and Atmospheric Administration (NOAA) operating at λ= 532 nm was deployed (Lack et al., 2006). Also, during BC2, a commercial three-wavelength particle absorption soot spectrometer (PASS-3; DMT, Inc.) operating at λ=405, 532, and 781 nm was deployed by Los Alamos National Laboratory (LANL) (Flowers et al., 2010). The results from the different PAS instruments for BC2 are compared in Cross et al. (2010). During BC3, BC3+, and BC4, a custom-built PAS from the University of California, Davis (UCD), operating at λ=405 and 532 nm, was deployed (Lack et al., 2011; Cappa et al., 2012). Light absorption was also measured during BC3, BC3+, and BC4 at λ=630 nm using a commercial cavity-attenuated phase-shift single-scatter albedo spectrometer (CAPS PMSSA; Aerodyne Research, Inc.). The CAPS PMSSA measures particulate light absorption as the difference between the measured light extinction (bext) and scattering (bsca) coefficients, whereas the PAS instruments measure light absorption directly. The CAPS extinction measurement has been previously described (Massoli et al., 2010). The CAPS PMSSA measures bsca using an integrating nephelometer, corrected for the finite viewing angle of the detector, i.e., truncation correction (Onasch et al., 2015b). The truncation correction at 630 nm was determined to be <1 % at 630 nm for particles smaller than 300 nm, increasing to <5 % for particles smaller than 800 nm. Light extinction coefficients were measured at λ=532 nm during BC2 using the NOAA cavity ring-down (CRD) spectrometer and at λ=405 and 532 nm during BC3, BC3+, and BC4 using the UCD CRD spectrometer (Langridge et al., 2011). Light extinction coefficients were also determined during BC2 at λ=405, 532, and 781 nm using the PASS-3 instrument, which, like the CAPS PMSSA, incorporates an integrating nephelometer to measure bsca; bext is determined as the sum of babs and bsca. Absorption and extinction cross sections were determined for the size-selected particles as $\begin{array}{}\text{(2)}& {\mathit{\sigma }}_{\text{abs}}\phantom{\rule{0.25em}{0ex}}\left({\text{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\text{particle}}^{-\mathrm{1}}\right)=\frac{{b}_{\text{abs}}}{{N}_{\text{p}}},\end{array}$ $\begin{array}{}\text{(3)}& {\mathit{\sigma }}_{\text{ext}}\phantom{\rule{0.25em}{0ex}}\left({\text{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\text{particle}}^{-\mathrm{1}}\right)=\frac{{b}_{\text{ext}}}{{N}_{\text{p}}},\end{array}$ where Np is the measured particle number concentration. Additional parameters of interest that can be calculated from the measurements are the wavelength-dependent mass absorption and extinction coefficients, MAC and MEC, respectively, defined as $\begin{array}{}\text{(4)}& \text{MAC}\left({\text{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\text{g}}^{-\mathrm{1}}\right)=\frac{{b}_{\mathrm{abs}}}{{N}_{\mathrm{p}}×{m}_{\mathrm{p}}},\end{array}$ $\begin{array}{}\text{(5)}& \text{MEC}\left({\text{m}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\text{g}}^{-\mathrm{1}}\right)=\frac{{b}_{\mathrm{ext}}}{{N}_{\mathrm{p}}×{m}_{\mathrm{p}}},\end{array}$ where mp is the per-particle mass. The combination of the absorption and extinction cross sections allows for the calculation of single-scatter albedo (SSA), defined as $\begin{array}{}\text{(6)}& \text{SSA}=\mathrm{1}-\frac{{\mathit{\sigma }}_{\text{abs}}}{{\mathit{\sigma }}_{\text{ext}}}.\end{array}$ The SSA characterizes the fraction of extinction that is due to scattering. The wavelength-dependence of absorption is characterized by the semiempirical parameter, the absorption Ångström exponent, defined as $\begin{array}{}\text{(7)}& \text{AAE}=\frac{\mathrm{ln}\left({\mathit{\sigma }}_{\text{abs,2}}/{\mathit{\sigma }}_{\text{abs,1}}\right)}{\mathrm{ln}\left({\mathit{\lambda }}_{\mathrm{1}}/{\mathit{\lambda }}_{\mathrm{2}}\right)},\end{array}$ where λ is wavelength and the subscript 1 and 2 indicate two different wavelengths. 3 Data analysis ## 3.1 Mass mobility exponent One measure of particle shape is the mass mobility exponent (Df,m) (Park et al., 2003): $\begin{array}{}\text{(8)}& {m}_{\text{p}}=C\cdot {d}_{\text{m}}^{{D}_{\text{f,m}}},\end{array}$ where C is the proportionality constant. (The mass mobility exponent, Df,m, differs from the fractal dimension, Df, as discussed by Sorensen (2011).) Values of Df,m for atmospheric particles typically range from 2.0 to 3.0, with lower values being characteristic of more “lacy” soot and higher values of more compacted soot. Df,m values were obtained for nascent and coated–denuded soot particles by fitting a power law function to a graph of mp vs. dm. ## 3.2 Refractive index retrieval Effective complex refractive index values (RI; $n=m+{k}_{i}$) for the soot particles were determined using two methods: (1) spherical particle Mie theory and (2) RDG approximation. For the refractive index, m is the real component and ki is the imaginary component. In Mie theory, particle cross sections are calculated assuming spherical particles with a homogenously uniform complex refractive index. The calculated cross section is the product of the calculated absorption or extinction efficiency (Qabs or Qext) and the geometric particle cross section ($=\mathit{\pi }\cdot \left[{d}_{\text{p}}/\mathrm{2}{\right]}^{\mathrm{2}}$, where dp is the particle diameter). Here, the particle diameter used in the calculations is the measured dp,VED, as determined from the CPMA per-particle mass measurements and assuming a material density for BC of 1.8 g cm−3. In the RDG approximation (Sorensen, 2001), the absorption cross sections are calculated as $\begin{array}{}\text{(9)}& {\mathit{\sigma }}_{\text{abs,RDG}}={N}_{\text{spherule}}{\mathit{\sigma }}_{\text{abs,spherule}},\end{array}$ where Nspherule is the number of spherules comprising a particle and σabs,spherule is the absorption cross section calculated for a single spherule using Mie theory. Here, we have assumed that ${N}_{\text{spherule}}={m}_{\text{p}}/{m}_{\text{spherule}}$, where mp is the measured per-particle mass and mspherule is the mass of an individual spherule with dp=20 nm (ethylene premixed flame) (Cross et al., 2010) or dp=37 nm (methane diffusion flame) (Ghazi et al., 2013), again assuming a material density of 1.8 g cm−3. Optimal (best-fit), theory-specific values of m and k were established by comparing the size-dependent observations of σabs to calculations from either Mie theory or the RDG approximation by varying these parameters over the ranges $\mathrm{1.3} and $\mathrm{0.1} and determining the minimum value of the reduced chi-square statistic: $\begin{array}{}\text{(10)}& {\mathit{\chi }}_{\text{red}}^{\mathrm{2}}=\frac{\mathrm{1}}{N-\mathrm{1}}\sum _{j}\left[{\frac{\left({\mathit{\sigma }}_{\text{abs}}^{j}-{\mathit{\sigma }}_{\text{abs,calc}}^{j}\right)}{{\mathit{\epsilon }}_{\text{abs}}^{\mathrm{2}}}}^{\mathrm{2}}\right],\end{array}$ where ${\mathit{\sigma }}_{\text{abs}}^{j}$ is the measured absorption cross section, ${\mathit{\sigma }}_{\text{abs,\hspace{0.17em}Mie}}^{j}$ is the calculated absorption cross section, N is the number of points, and εabs is the estimated uncertainty, which accounts for both instrumental uncertainties plus the uncertainty associated with contributions from multiply charged particles. Individual fits were performed for each wavelength for each flame type. The methane diffusion flame soot data collected during different studies were treated as one dataset, whereas the ethylene premixed flame soot data collected during the different studies were kept separate due to the sampling differences described above. ## 3.3 Accounting for multiply charged particles All absorption and extinction coefficient measurements were corrected for contributions from doubly charged (Q2) particles. This correction leads to a decrease in the measured absorption and extinction coefficients. Here, an approach is taken where the fraction of absorption or extinction from Q2s (fabs,Q2 or fext,Q2, respectively) is estimated from the measurements of the number fraction of singly and doubly charged particles (fQ1 and fQ2, respectively) determined from either the SMPS (BC2, BC3, and BC3+) or the SP2 (BC4), the per-particle mass of singly and doubly charged particles (mp,Q1 and mp,Q2, respectively), with an assumption of spherical particles (i.e., that ${m}_{\text{p}}=\mathit{\rho }\left(\mathit{\pi }/\mathrm{6}\right){d}_{\text{p,VED}}^{\mathrm{3}}\right)$ and an assumption that the MAC and MEC are size-independent. The resulting expression for fabs,Q2 is then $\begin{array}{ll}\text{(11)}& {f}_{\text{abs,Q2}}& \phantom{\rule{0.25em}{0ex}}=\frac{{f}_{\text{Q2}}\cdot {m}_{\text{p,Q2}}\cdot \text{MAC}}{{f}_{\text{Q1}}\cdot {m}_{\text{p,Q1}}\cdot \text{MAC}+{f}_{\text{Q2}}\cdot {m}_{\text{p,Q2}}\cdot \text{MAC}}& =\frac{{f}_{\text{Q2}}\cdot {m}_{\text{p,Q2}}}{{f}_{\text{Q1}}\cdot {m}_{\text{p,Q1}}+\left(\mathrm{1}-{f}_{\text{Q1}}\right)\cdot {m}_{\text{p,Q2}}}\\ & \sim \frac{{f}_{\text{Q2}}\cdot {d}_{\text{p,VED,Q2}}^{\mathrm{3}}}{{f}_{\text{Q1}}\cdot {d}_{\text{p,VED,Q1}}^{\mathrm{3}}+\left(\mathrm{1}-{f}_{\text{Q1}}\right)\cdot {d}_{\text{p,VED,Q2}}^{\mathrm{3}}}.\end{array}$ It is important to note that the final expression is independent of the absolute value of the MAC. A similar procedure was used to correct MEC values. Although the MAC (and MEC) may not be fundamentally size-independent, as assumed, the correction is relatively insensitive to differences in the MAC between sizes. For the size range of interest, there is a relatively constant relationship between the diameters of the singly and doubly charged particles, with dp,VED,Q2∼1.6 dp,VED,Q1 corresponding to mp,Q2∼4mp,Q1. Thus, $\begin{array}{}\text{(12)}& {f}_{\text{abs,Q2}}\sim \frac{\mathrm{4}\left(\mathrm{1}-{f}_{\text{Q1}}\right)}{{f}_{\text{Q1}}+\mathrm{4}\left(\mathrm{1}-{f}_{\text{Q1}}\right)},\end{array}$ and the absorption by Q1 particles only is $\begin{array}{}\text{(13)}& {b}_{\text{abs,Q1}}={b}_{\text{abs,obs}}\left(\mathrm{1}-{f}_{\text{abs,Q2}}\right).\end{array}$ As an example, if fQ1=0.95, then fabs,Q2=0.17 and the sensitivity of fabs,Q2 to uncertainties in fQ1 is $\mathit{\delta }{f}_{\text{abs,Q2}}\sim \mathrm{3}\cdot \mathit{\delta }{f}_{\text{Q1}}$, as determined by sensitivity calculations. We estimate that the uncertainty in fQ1 is 0.01, which contributes ∼3 % to the uncertainty in babs,Q1. (If the MAC were not constant but were instead 20 % larger for Q2s compared to Q1s, then the correction would increase to 0.20 from 0.17.) For most runs, the Q2 contribution was small, being 5 % or less. 4 Results and discussion Results from the methane diffusion flame and from the two different ethylene premixed flame experiments are presented and discussed separately. For each flame type or configuration, the coated–denuded data for all coating types are considered together. Figure 1A comparison of methane-diffusion-flame-generated nascent (or nascent–denuded; blue circles) and coated–denuded (gray circles) mass absorption coefficients (MACs) at λ=532 nm for x>0.90 for BC3, BC3+, and BC4 experiments. Note that conditioning (and therefore the shape) of the soot does not affect the observed MACs. The box and whisker plots show the mean (), median (–), lower and upper quartile (boxes), and 9th and 91st percentile (whisker). Table 1Refractive indices for methane diffusion flame soot retrieved via fitting Mie theory and the RDG approximation to the σabs observations. Nascent and denuded data are combined (see text for details). 1 Uncertainties in the MAC are 1σ from the reduced χ2 fit. See Figs. S3 and S4. 2 The n values from the RDG method are nonunique. Therefore, uncertainty estimates from this work are not available. See text for details. 3 VED and the size parameter ($x=\mathit{\pi }{d}_{\text{p}}/\mathit{\lambda }\right)$ where the peak MAC occurs. 4 There are many degenerate RI combinations that give similar quality fit to RDG theory. Thus, a value of 1.80 was chosen for the effective real refractive index. 5 These values are given for reference purposes only but should not be used due to the inability to fit data well (see text for details). ## 4.1 Soot optical properties from the methane diffusion flame Figure 2Observed σabs vs. dp,VED methane soot data from BC3, BC3+, and BC4. Panels (a) and (b) are λ=405 and 532 nm data, respectively, from the UCD CRD-PAS, and (c) is λ=630 nm data from the CAPS PMSSA. The nascent and coated–denuded data have been combined since there is no significant difference in the absorption cross sections and MACs between the two datasets. Note the inability of Mie theory to reproduce the observed σabs,405 nm for all sizes. Vertical solid lines indicating x=0.9, where observations deviate from Mie theory, are provided for reference. In addition, vertical dashed lines indicate dp,VED=160 nm, above which soot maturity is approximately constant. ### 4.1.1 MAC values of nascent and coated–denuded soot Average and median MAC values were determined for BC particles from the methane diffusion flame with x>0.9 (corresponding to a dp,VED=160 nm at λ=532 nm; Table S3). (The dimensionless size parameter $x=\mathit{\pi }{d}_{\text{p}}/\mathit{\lambda }$, where dp is particle diameter and λ is wavelength.) There are no systematic differences in the MAC values between nascent and nascent–denuded (${D}_{\text{f,m}}=\mathrm{2.16}±\mathrm{0.1}$) and coated–denuded (${D}_{\text{f,m}}=\mathrm{2.64}±\mathrm{0.1}$) soot at any wavelength for this range of x despite some degree of collapse for the thickly coated–denuded particles (Figs. 1 and S2). The average MAC values for x>0.9 were 12.1±1.4 m2 g−1 at 405 nm, 9.1±1.1 m2 g−1 at 532 nm, and 7.1±1.1 m2 g−1 at 630 nm, where uncertainties are 1σ standard deviations of the measurements. For the reverse-coating experiments, which gave broader BC per-particle mass distributions compared to forward-coating distributions, we found no dependence of the derived MAC values on the distribution width. The collapse was presumably due to the effect of evaporation or condensation of the coating material and not due to the denuding process alone (Bhandari et al., 2017). The observed Df,m independence of the MAC (x>0.9) is consistent with Radney et al. (2014). This contrasts, however, with modeling studies that use non-Mie-based methods that can account for particle shape effects, which indicate that lacy soot (with a fractal dimension, as opposed to mass mobility exponent, of Df=1.8) is more absorbing than “compact” soot (Df=2.4) (Kahnert and Devasthale, 2011; Scarnato et al., 2013). In the calculations, the compact soot particles are less absorbing because the innermost spherules are “shielded” by the outermost spherules. It is possible that the extent of collapse here was insufficient to lead to substantial “shielding” in our experiments. Regardless, given the similarity of the observed nascent and coated–denuded particle cross sections, they have been recombined into a single dataset in what follows. ### 4.1.2 RI values calculated from Mie theory and the RDG approximation The σabs have been fit separately using Mie theory and the RDG approximation to determine optimal, theory-specific effective complex RI values. The observations and best fits are shown in Fig. 2 at the three wavelengths (λ=405, 532, and 630 nm), and the derived optimal wavelength-, flame-, and theory-specific values are reported in Table 1. The quality of the best fit obtained is dependent upon both theory and wavelength considered. First, fits performed using Mie theory tend to give reasonably well-defined minima in the calculated ${\mathit{\chi }}_{\text{red}}^{\mathrm{2}}$, indicating that the optimal m and k are unique (Fig. S3). In contrast, fits performed using the RDG approximation do not give a unique set of m and k but instead a band of [m,k] pairs that describe the data equally well (Fig. S4). Since RDG fits are nonunique, optimal k values are reported at all wavelengths for a fixed value of m=1.80. There are additional differences between Mie and RDG beyond the uniqueness of the derived optimal RI. At λ=405 nm, Mie theory provides a poor fit to the σabs when the fit is performed using data over the entire size range sampled. In particular, at λ=405 nm the σabs from Mie theory are overestimated below dp,VED∼120 nm (x∼0.9) and underestimated for larger sizes (Fig. 2a). At λ=532 nm, this deviation also occurs at x∼0.9 (dp,VED>160 nm) corresponding to larger size particles. Compared to 405 nm, the overestimate at small x and underestimate at larger x for 532 nm is smaller. At λ=630 nm, the Mie theory fit compares well with the observations at x<0.9 (dp,VED∼180 nm) and with some deviation observed at larger sizes. When the fits are restricted to x<0.9, a reasonable fit using Mie theory is obtained at all wavelengths over this size range, although there is perhaps a small overestimate at the smallest sizes. However, these constrained Mie fits extrapolated to larger sizes (x>0.9) still underestimate the observed absorption. For RDG, generally good fits are obtained at all wavelengths and across all sizes, although the RDG-calculated σabs tends to overestimate the observation at smaller sizes below dp,VED∼100 nm. It is important to note that the RI values listed in Table 1 are theory- and property-specific. This means that the Mie-derived RI values are not appropriate for use with the RDG approximation and vice versa. Additionally, they must be used assuming a material density of 1.8 g cm−3, since this value was used to convert mp to dp,VED or Nspherule. If, for example, a smaller density were used with these RI values then the particles would have substantially higher MACs. Of note is that both the real and imaginary RIs from Mie theory are larger than RI values that are commonly used in global climate models (Bond et al., 2013), including the RI that is often considered the currently recommended value (1.95−0.79i) (Bond and Bergstrom, 2006). For reference, using RI $=\mathrm{1.95}-\mathrm{0.79}i$ the MAC at 532 nm calculated for BC in the small particle limit (assuming a material density of 1.8 g cm−3) is only 5.1 m2 g−1, but peaks at 7.5 m2 g−1 around dp,VED=150 nm. ### 4.1.3 Comparison of measured and calculated MAC values Another way to look at the extent to which Mie theory or the RDG approximation can reproduce the observations is to compare the observed and calculated MAC values as a function of particle size (or x) rather than the σabs vs. size relationship (as in Fig. 2). Although the MAC is related directly to the σabs, it is nonetheless useful to consider the MAC values because they vary over a much narrower range than do the σabs. The dependence of the MAC on dp,VED and size parameter for both the observations and the models are shown in Fig. 3. The observed MAC values generally increase with dp,VED or size parameter at all wavelengths up to around dp,VED∼160 nm, above which they plateau and are approximately constant. The ranges (minimum and maximum) of binned observed MACs are provided in Table S4. The MAC values observed here (Table S3) are substantially larger than the value of 5.7±0.8 m2 g−1 at λ=405 nm reported by Radney et al. (2014), who use a Santoro-type burner (i.e., co-flow diffusion flame). They did not report any notable size dependence for their MAC values. Our MAC values (especially for x>0.5) compare well with the value of 8.16 m2 g−1 at λ=532 reported by You et al. (2016) (extrapolated from 7.89±0.25 m2g−1 at λ=550 nm) for soot particles generated from the combustion of organic fuel stock over the dp,VED range ∼80 to ∼210 nm. They also compare favorably to the range of values 7.2 to 8.5 m2 g−1 reported for λ=532 nm for dp,VED∼100 nm observed in Saliba et al. (2016) for particles generated from a cookstove. Some particle size dependence was reported by Khalizov et al. (2009), who used propane with a Santoro-type burner, with reported MACs at λ=532 nm of 6.7±0.7 m2 g−1 for dm=155 nm particles and 8.7±0.1 m2 g−1 for dm=320 nm particles. This general behavior was also observed for soot particles generated from a methane diffusion flame in Dastanpour et al. (2017), with MAC values reported at λ=660 nm of ∼5 m2 g−1 for dp,VED=50 nm and ∼7 m2 g−1 for dp,VED=100 nm. One key reason that differences may exist between studies is that the BC particles sampled had differing maturity. Soot maturity refers to the extent to which the BC has a more disordered internal structure with high hydrogen content (low maturity) vs. a more ordered, graphite-like structure with low hydrogen content (high maturity) (Johansson et al., 2017). The absorption cross section for BC likely increases with increasing soot maturity (López-Yglesias et al., 2014). The observations are compared with the calculated MAC values, based on the fits from Fig. 2. MAC values from the RDG approximation are independent of x, as the particle MAC is equal to the MAC of the individual spherules making up the particle. Here, the observed MAC values in the plateau (large x) regime correspond reasonably well with the MACs as calculated from RDG values when the optimal RI values are used. The constant RDG MAC value at λ=532 nm (= 8.8 m2 g−1) is slightly larger than the often suggested value for atmospheric BC by Bond and Bergstrom (2006) of 7.75±1.2 m2 g−1 (extrapolated from 7.5 at λ=550 nm using 1 for the AAE) and is identical to that reported for soot from a Santoro-type diffusion burner operating on propane (Zhang et al., 2008). However, the MACs predicted from RDG overestimate the observed MACs at x<0.90, since the RDG fits are weighted by the greater number of data points at x>0.9 where MACs are approximately constant. For Mie theory, calculated MAC values for highly absorbing substances, such as BC, have a characteristic shape where the MAC is constant up to x∼0.2, increases monotonically by ∼40 % until x∼0.9, and then decreases rapidly towards larger x. The Mie theory curves calculated here reproduce the observed values at x<0.90 (especially for λ=532 and 630 nm) but substantially underestimated MACs at x>0.90. This facilitates an understanding of the Mie model underestimate of σabs at large x (Fig. 2). Above x∼0.9 the calculated Mie MAC declines with x for all wavelengths, but the observations indicate that the MAC is constant. Because x occurs at smaller dp,VED for shorter wavelengths than for longer wavelengths, the model–measurement difference in both σabs and MAC is more noticeable at 405 nm than it is at 532 nm, which is more noticeable than at 630 nm. This is a consequence of a greater number of data points at x>0.9 at 405 nm, past the peak in the Mie-calculated MAC. The reasonable correspondence between the observed and Mie-calculated MAC at smaller sizes is, however, somewhat surprising given that Mie assumes spherical particles, yet the particles are not spherical. One potential reason for the observed dependence of the MAC on particle size is that the chemical and optical properties of the particles change with size, and the different chemical composition coincidentally improves agreement with Mie theory. For the diffusion flame, changes in the particle size distribution were induced by changing the amount of dilution nitrogen in the sheath flow. This can influence the maturity of the soot and consequently the soot absorption (López-Yglesias et al., 2014). The observed increase in MAC with dp,VED here exhibits some wavelength dependence, which could reflect differences in the sensitivity of the MAC to maturation. The observed differences between the MACs observed at the smallest x and the maximum MAC values were 21 %, 41 %, and 37 % for λ=405 nm, λ=532 nm, and λ=630 nm, respectively. However, one additional difference between the wavelengths is that at λ=405 nm, there is a small increase in MAC going from dp,VED∼60 nm (x405 nm=0.6) to ∼100 nm (x=0.9) after which the MAC is constant, while at λ=630 nm there is much more of a continuous increase in the MAC up to larger particle sizes. This could indicate that, at some point, further changes in the soot maturity (composition) have no influence at short wavelengths but do at longer wavelengths. As a complementary explanation, Dastanpour et al. (2017) observed an increase in primary spherule size with overall particle size for methane-diffusion-flame-generated soot. They attribute the increase in MAC with dp,VED to changes in the internal structure and/or the degree of graphitization that occur with changes in spherule size. The observation of constant MAC values for mature soot (x>0.9 or dp,VED∼160 nm at λ=532 nm) is an important result in the context of how BC is commonly treated in climate models. Most climate models simulate the optical properties of BC using spherical particle Mie theory. The observations indicate that Mie theory will likely underestimate the absorption by BC for particles with x>0.9 because, when the particles are sufficiently absorbing, the attenuation of light by the outer layers of the (spherical) causes the mass in the center of the particle to not interact with the electromagnetic field (Bond and Bergstrom, 2006; Kahnert and Devasthale, 2011). This suggests that the RDG approximation, or even an assumption of a constant MAC, may provide a more accurate representation of BC absorption than Mie theory in climate models, at least for uncoated BC. This conclusion is independent of soot maturity, as the falloff in the MAC with increasing size for Mie theory occurs for all strongly absorbing particles. Although atmospheric BC particles are predominately generated through the combustion of fossil fuels or through biomass burning (Bond et al., 2013), flame-generated BC particles have been shown to be a suitable proxy for atmospheric BC particles, both in terms of chemical bonding and structural properties (Slowik et al., 2007; Hopkins et al., 2007). For example, Hopkins et al. (2007) find that the sp2 content of ethylene and methane flame soot are similar to diesel soot (63 %, 60 %, and 56 %, respectively) and have similar aromatic content. There is also a reasonable similarity between SP-AMS mass spectra of flame soot and soot particles in diesel exhaust or smoke from biomass burning (Onasch et al., 2015a). The absolute values of the derived RI may be different for diesel or biomass BC particles, but it can be reasonably assumed that Mie theory does not reproduce the behavior of atmospheric BC particles. The discrepancy between Mie theory and the observations is both size and wavelength dependent. Consequently, the extent to which the true absorption by BC is underestimated by a given atmospheric model due to the inappropriate use of Mie theory will depend importantly on the assumed size distribution, both the position and the width, and the wavelength. Using the effective RI values determined here, we estimate that absorption is underestimated by around 20 %–40 % when Mie theory is used with reasonable BC size distributions. The underestimate in absorption from the use of Mie theory will be even larger if non-theory-specific (typically lower) imaginary RI values are used, such as that suggested by Hess et al. (1998) or Bond and Bergstrom (2006), as discussed by Stier et al. (2007). Consider that the maximum MAC predicted using the Hess et al. (1998) RI ($=\mathrm{1.75}-\mathrm{0.44}i$) from Mie theory is only 3.8 m2 g−1. Figure 3Box plots of MACs and MECs as a function of size parameter, with Δx=0.18. The volume-equivalent diameters are provided at the top of the plots for reference. Also shown are Mie theory curves for all particles (black solid lines), Mie theory curves for x<0.90 (gold line), and RDG curves (dashed line) calculated from the RI values in Table 1. RI fitting was performed using only the absorption measurements. Panels (a) and (b) are λ=405 nm data from the UCD CRD-PAS, (c) and (d) are λ=532 nm data from the UCD CRD-PAS, and (e) and (f) are λ=630 nm data from the CAPS PMSSA. The poor match between the calculated Mie theory curves at λ=405 nm reflects the difficulties in fitting spherical particle Mie theory to the observed σabs,405 nm over the entire size range. Although the same particle sizes were sampled over all wavelengths, the size parameters sampled at each wavelength are different. Therefore, there are different numbers of boxes for each wavelength. Note that this figure is directly related to Fig. 2, the difference being that the y axis values in Fig. 2 (cross sections) have been divided by the per-particle mass to give the MAC or MEC. Points in each bin range from N=10 at x1.62 to N=81 at x1.26 for λ=405 nm, N=5 at x0.36 to N=82 at x1.08 at λ=532 nm, and N=8 at x1.08 to N=67 at x0.9 at λ= 630 nm. ### 4.1.4 MEC and SSA values of nascent and coated–denuded soot Measurement of bext and the MEC were made in addition to the babs and MAC measurements (Fig. 3). Above, the RI fitting was performed using only the absorption measurements, in part because the calculation of extinction using RDG requires additional assumptions regarding the particle shape. Nonetheless, it is informative to compare the σext and MEC observations to Mie theory calculations since the overall climate impacts of BC depend on both absorption and scattering. As with the MAC values, the observed MEC values also increase with x up to 0.90 (or dp,VED up to 160 nm), after which point they are relatively constant (Fig. 3). The Mie theory MECs calculated using the RIs determined by fitting the absorption measurements (Table 1) agree reasonably well with observations when x<0.9 (using the fits that were constrained to this range), but, as with absorption, Mie theory underestimates the MEC above x=0.9. For a given particle size, there is somewhat greater scatter in the observed MECs than in the MACs. This is likely a result of the scattering being more sensitive to the shape of the soot particles than is absorption and of the nascent and coated–denuded particle results being combined here. Given this, the dependence of the SSA on particle size is considered separately for the nascent (more lacy) and coated–denuded (more compact) particles. The coated–denuded particle SSAs increase with dp,VED, most noticeably for dp,VED>100 nm, up until dp,VED∼160 nm. Above this size the SSA values are approximately constant at a value of ∼ 0.30 (Fig. 4a). (Results at λ=532 nm are shown in Fig. 4a, but there is a strong correlation between SSA at 532 nm and at 405 or 630 nm; Fig. S5). In contrast, the nascent SSAs increase slightly from dp,VED∼50 to 80 nm but above 80 nm are approximately constant at ∼0.20. This demonstrates that particle collapse leads to an increase in the SSA for BC, consistent with Radney et al. (2014). This behavior is also consistent with modeling studies, which have predicted that compact agglomerates exhibit higher SSA values than lacy agglomerates, with an absolute increase of ΔSSA ∼0.1 (Scarnato et al., 2013) or by a factor of 1.2–2.2, depending on the extent of compaction (China et al., 2015a, b). The increase upon collapse is attributed to the stronger scattering and electromagnetic coupling between spherules in compact aggregates. Here, the difference between the SSA for nascent and coated–denuded soot increases somewhat with particle size, which may result from changes in soot maturity with size. The SSAs from this study compare reasonably well to values reported previously at visible wavelengths. For example, Saliba et al. (2016) report SSA = 0.16 to 0.26 for nascent soot emitted from a cookstove; Schnaiter et al. (2003, 2006) report SSA = 0.2 to 0.3 for soot from a propane diffusion flame, SSA = 0.18 to 0.25 for kerosene-derived soot, and SSA = 0.1 to 0.25 for methane diffusion flame soot; and Sharma et al. (2013) report SSA = 0.18 to 0.25 for soot generated from a kerosene lamp. However, the values reported here are much smaller than the value of 0.5 reported in Radney et al. (2014). The Mie-theory-calculated SSA values are similar to observations, although they show a somewhat stronger increase with size and seem to plateau at larger SSA values at large sizes compared to observations. Figure 4Box and whisker plots of λ= 532 nm (a) single-scattering albedo (SSA) and (b) absorption Ångström exponent (AAE) as a function of volume-equivalent diameter produced from the methane diffusion flame. The black points are coated–denuded data (coating material is evaporated following coating with DOS or sulfuric acid) and are potentially collapsed due to coating, and the gray points are nascent or nascent–denuded data. The black lines are the SSA or AAE predicted by spherical particle Mie theory, and the dashed black line in (b) is AAE predicted from the RDG approximation using the effective refractive indices listed in Table 1. The nascent or nascent–denuded boxes (black) are shifted by X=0.05. Points in each bin range from N=6 at x0.36 to N=26 at x1.08 for nascent (or nascent–denuded) and N=5 at x0.54 to N=9 at x1.26 for coated–denuded points. ### 4.1.5 AAE of nascent and coated–denuded soot The wavelength dependence of absorption has been considered by calculating the AAE using the measurements at λ=405 and 532 nm (Fig. 4b). The nascent AAEs at larger particle sizes are slightly larger than the coated–denuded AAEs, suggesting that particle collapse leads to a slight decrease in the wavelength dependence of absorption. The average AAE was 1.38 ± 0.36 (N=85) and 1.10 ± 0.37 (N=135) for nascent and coated–denuded particles, respectively. There is some indication that AAE decreases with particle size. This may again be the result of the soot maturity increasing (and the composition changing) with size. The AAE values from Mie theory, based on the best-fit RI values determined above, exhibit an increase to x∼0.5 where they peak and then decrease sharply. This predicted decrease is inconsistent with the observations. The AAEs calculated using the RDG approximation from the best-fit RI values are constant. BC is commonly assumed to have an AAE = 1 (Bergstrom, 1973). The measurements here are consistent with this expectation for the collapsed (coated–denuded) particles, but the nascent particles give an AAE that is somewhat larger than 1. The observed AAE values are similar to results from previous studies examining either freshly emitted soot particles or soot particles containing very little organic material (Schnaiter et al., 2003, 2006; Kirchstetter et al., 2004; Bergstrom et al., 2002; You et al., 2016; Sharma et al., 2013). ## 4.2 Optical properties of BC from the ethylene flat-burner flames ### 4.2.1 Results from BC2: sampling high above the burner surface During BC2, the ethylene flame was sampled at a height of ∼ 20.3 cm above the surface. At this height, the particles were likely reasonably mature, at least relative to sampling that was performed further into the flame, as was done in BC3+. Particle optical properties were quantified at λ=405, 532, and 781 nm, with two independent measurements at 532 nm considered (NOAA and PASS-3). As with the methane diffusion flame, the data were fit using Mie theory and the RDG approximation to determine optimal, theory-specific, wavelength-dependent RI values (Table 2). The soot particles from this flame had, overall, a greater amount of intrinsic organic carbon associated with them compared to the particles from the methane diffusion flame. As such, denuding even of the nascent particles led to changes in the optical properties and particle masses. Thus, the nascent and denuded particles are considered separately, and we focus on the denuded particles. The range of particle sizes considered was also overall smaller than that for the methane diffusion flame. Table 2Theory-specific effective refractive indices for ethylene premixed flame soot from BC2 and BC3+ retrieved via fitting Mie theory and the RDG approximation to the σabs observations. Nascent and denuded experiments are considered separately. 1 Uncertainties in MACs are 1σ from the least χ2 fit. See Figs. S7 and S8. 2 The n values from the RDG method are nonunique. Therefore, uncertainty estimates from this work are not available. See text for details. 3 VED and the size parameter ($x=\mathit{\pi }{d}_{\text{p}}/\mathit{\lambda }\right)$ where the peak MAC occurs. 4 There are many degenerate RI combinations that give similar quality fit to RDG theory. Thus, a value of 1.80 was chosen for the effective real refractive index. 5 In BC2 the ethylene flat-burner flame was sampled 20.3 cm between the burner surface and the sampling inlet, and during BC3 the flame was sampled ca. 5 cm between the burner surface and the sampling inlet. Figure 5Measured absorption cross sections vs. volume-equivalent diameter (panels in the first column) and mass absorption and extinction coefficients (MACs and MECs; panels in the second and third column, respectively) for ethylene soot sampled 20.3 cm from burner surface of the McKenna flame as a function of size parameter ($x=\mathit{\pi }{d}_{\text{p}}/\mathit{\lambda }$) (bottom axes) and volume-equivalent diameter (top axes). for BC2. Panels (a–c) show PASS-3 data at λ=405 nm, (d–f) show NOAA PAS data at λ=532 nm, (g–i) show PASS-3 data at λ=532 nm, and (j–l) show PASS-3 data at λ=781 nm. The dashed black and gray lines are the RDG fits to denuded and nascent data, respectively, and the solid black and gray lines are Mie fits to denuded and nascent data, respectively. The retrieval of effective refractive indices for this flame using Mie theory resulted in a good fit to σabs vs. dp,VED for all λ and dp,VED (Fig. 5a, d, g, and j, and Figs. S6–S7). This difference from the methane diffusion flame is in large part due to the more restricted size range encountered here, with BC particles only up to dp,VED=160 nm used. The RDG approximation yielded a reasonably good fit across all particle sizes for the ethylene particles, although with some overestimation at smaller sizes. The MAC values tended to increase with particle size or size parameter, most obviously at 532 nm where the most data points are available (Fig. 5b, e, h, and k). The range of binned MACs shown in Fig. 5 is listed in Table S4. The MAC values determined using the two PAS instruments at 532 nm differ somewhat, with the NOAA PAS MACs slightly larger than PASS-3 MACs, although the differences are within the measurement uncertainties. Most likely, this instrument difference stems from differences in calibration methods. Figure 6Box and whisker plots of (a) single-scattering albedo (SSA) and (b) absorption Ångström exponent (AAE) as a function of size parameter and volume-equivalent diameter produced from the ethylene flame during BC2. The SSA was calculated using data from the NOAA PAS, and the AAE was calculated using data from the λ=405 nm PASS-3 and the λ=532 nm NOAA PAS data. The black points are coated–denuded data (coating material is evaporated following coating with DOS or sulfuric acid) and are potentially collapsed due to coating, and the gray points are nascent–denuded points (evaporation of intrinsic organic matter produced from the ethylene flame). The black lines are the SSAs predicted using spherical particle Mie theory using the effective complex refractive index retrieved from fitting the σabs,532 nm from the NOAA PAS. Here points are binned with a constant Δx=0.18 for the purpose of comparison between the different particle treatments. N=6 for nascent or nascent–denuded points, N=5 for DOS coated–denuded points, and ${N}_{x\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{0.54}}=\mathrm{11}$, ${N}_{x\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{0.72}}=\mathrm{8}$, and ${N}_{x\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{0.9}}=\mathrm{5}$ for H2SO4 coated–denuded points. The observed MAC values tend to be larger for the denuded particles than for the nascent particles, most likely due to less-absorbing organics present in nascent soot that contribute ∼25 % of the particle mass (Cross et al., 2010). The MAC values at λ=405 and 532 nm for the denuded BC2 ethylene flame soot are comparable, within uncertainty, to the MAC values for the methane diffusion flame soot for the particles with dp,VED∼150 nm. This indicates that the BC from these two flames is similarly absorbing in nature. The impact of particle morphology on the SSA and AAE is considered by comparing the results for nascent–denuded particles with coated–denuded particles (Fig. 6). Nascent particles are excluded because the presence of intrinsic organics can increase AAE if the organic material contains brown carbon and can increase the SSA independent of the underlying BC morphology. The Df,m values were 2.12±0.06, 2.49±0.07, and 2.17±0.06 for nascent–denuded, sulfuric acid coated–denuded, and DOS coated–denuded soot, respectively (Cross et al., 2010). The nascent–denuded and DOS coated–denuded particles have SSA values close to 0, whereas the sulfuric acid coated–denuded particles have SSA values closer to 0.15 at x>0.5, consistent with particle collapse leading to an increase in SSA (Fig. 6a). The SSA values for the nascent–denuded particles from this flame are smaller than the SSA values for the nascent particles from the methane diffusion flame. The reason for this is not clear but is likely related to differences in the particle shapes and/or the sizes of the spherules; the Df,m for the nascent–denuded particles here (Df,m=2.12) is slightly smaller than for the particles from the methane diffusion flame (Df,m=2.28) (Fig. S8). The sulfuric acid coated–denuded particle SSA values from the BC2 ethylene flame are also lower than the methane diffusion flame coated–denuded SSA values for a given size (Fig. 6a), despite having similar Df,m values. The AAE values (using the λ=405 and 532 nm pair) for nascent–denuded and coated–denuded particles are generally similar when compared over the same size range (Fig. 6b). However, the sulfuric acid coated–denuded particle AAE values may be slightly smaller than for the nascent–denuded or DOS coated–denuded particles; all are close to unity at x∼0.8. This indicates that changes in morphology do not lead to substantial changes in AAE. The AAEs for the coated–denuded particles decrease strongly with dp,VED, with the mean AAE 1.6 for the smallest particles (x=0.5) and an AAE mean of ∼0.5 for the largest particles (x=0.9). Values below 1 contrast with the methane data but have been observed in a few previous studies (Clarke et al., 2007; Hadley et al., 2008; Lack et al., 2008). While such a decrease with size is consistent with Mie theory predictions, given the MAC results, it seems more likely that the size dependence of the AAE is related to changes in the soot maturity with particle size. To the extent that the larger particles, which tend to have larger MAC values, are reflective of more mature soot, this suggests that mature soot from this flame type has an AAE < 1. ### 4.2.2 Results from BC3+: sampling near the burner surface During BC3+, the particles were sampled at variable heights above the burner surface to select for particles in different size ranges, but most often they were sampled from relatively close to the burner surface (5.1 cm) compared to BC2 sampling conditions (20.3 cm). It is likely that these different sampling conditions gave rise to particles with different chemical properties. Multiple studies have shown changes in soot maturity and soot optical properties as a function of sampling height in ethylene premixed flames, though at a distance significantly closer to the flame front than sampled here (Olofsson et al., 2015; Migliorini et al., 2011). The particles during BC3+ had very little, if any, intrinsic organic carbon, unlike the BC2 particles that were ∼25 % organic by mass. However, the small organic content for BC3+ was likely a consequence of the use of a hot sampling line prior to dilution. Consequently, the optical properties of both nascent and denuded ethylene soot particles from BC3+ (sampled close to the burner) differ substantially from the BC2 particles (sampled well above the burner). Average MAC values over various size ranges are listed in Table S5. In general, the MACs for BC3+ ethylene particles are similar to BC2 particles at λ=405 nm, whereas the λ=532 nm MAC for BC3+ particles are smaller than those for BC2 particles. (Measurements at λ=30 nm were not made during BC2, nor were measurements at 781 nm made during BC3+.) At dp,VED>70 nm, the λ=405 nm MAC values were approximately constant with increasing dp,VED (Fig. S9). The behavior is consistent with methane diffusion flame observations, but the constant MAC seems to occur at a lower dp,VED.The number of data points available for the BC3+ ethylene particles is limited, making conclusions regarding the size dependence of properties somewhat tenuous. The AAE for BC3+ ethylene particles are reasonably independent of particle size (Fig. S10). The average value of AAE${}_{\text{405\hspace{0.17em}nm–532\hspace{0.17em}nm}}=\mathrm{2.01}±\mathrm{0.21}$ for x>0.5, which is higher than observed for the methane diffusion flame for this range of x ($=\mathrm{1.18}±\mathrm{0.35}$). These observations indicate that differences in sampling and soot maturity result in different optical properties. Previous studies have observed differences in optical properties, chemical composition, and primary spherule size for different flame sampling heights (Bladh et al., 2011; Olofsson et al., 2015; Migliorini et al., 2011). Absorption by less mature soot appears to decrease more rapidly with wavelength than for more mature soot, such that the MAC values in the mid-visible (e.g., λ=532 and 630 nm) are lower for less mature soot. These wavelength-dependent optical results appear to match trends observed previously using active remote-sensing techniques to characterize particles within flames (Olofsson et al., 2015; Migliorini et al., 2011). The extent to which this conclusion can be generalized will require further investigation. 5 Conclusion Light absorption and extinction cross sections were measured for nascent, denuded, and coated–denuded soot particles that were produced from two different flame types that operated on different fuels (methane or ethylene). These measurements were used in conjunction with particle mass and size measurements to determine various intensive optical properties (e.g., MAC, SSA, and AAE) for uncoated BC particles in the size range 50 nm $<{d}_{\text{p,VED}}<\mathrm{210}$ nm (corresponding to 0.1 fg $<{m}_{\text{p}}<$ 5 fg). The optical properties varied somewhat with particle size, most likely due to changes in the chemical nature (i.e., maturity) of the BC that results from variations in the combustion and sampling conditions. However, for larger, mature particles, corresponding to those with ${d}_{\text{p,VED}}>\sim \mathrm{160}$ nm, the observed intensive properties were generally size-independent. The observed MAC values for BC, measured over multiple studies, are independent of particle collapse and thus provide evidence that absorption by soot of a given maturity level is dictated primarily by the individual spherules and is thus largely size- and shape-independent. The observed MAC values are also larger than the recommended value of Bond and Bergstrom (2006), i.e., 8.6 m2 g−1 vs. 7.75 m2 g−1 at 532 nm. The observations serve as the basis for the determination of wavelength- and theory-specific effective complex refractive indices, using both Mie theory and the RDG approximation. With Mie theory, good fits were only obtained for size parameters smaller than ∼0.90 (corresponding to dp,VED∼160 nm for λ=532 nm). Above this size, Mie theory predicts a sharp decrease in the MAC while the observed MAC are constant. Thus, Mie theory systematically underpredicts the observed absorption for x>0.9, and a good fit is not possible. This is because with Mie theory when the particles are sufficiently absorbing and large, light is attenuated by the outer layers of the (spherical) particle such that the mass in the center of the particle does not interact efficiently with the electromagnetic field (Bond and Bergstrom, 2006; Kahnert and Devasthale, 2011). Our analysis has important implications for the calculation of absorption in atmospheric models. Atmospheric models that use Mie theory, which is the majority, likely underestimate the actual absorption by uncoated BC whether or not theory-specific RI values are used; the magnitude of the underestimation will depend on the assumed BC size distributions, increasing with increasing dp,VED and size distribution width. This may be especially important to consider for the simulation of absorption by BC particles from biomass burning, which are larger than those from urban sources (Schwarz et al., 2008). The underestimate of absorption by Mie theory will likely be even larger if non-theory-specific RI values that inherently underestimate MAC values are used, which includes some of the more commonly used RI values. Overall, our results demonstrate that either an assumption of a constant MAC or the use of the RDG approximation with theory-specific RI values (which are equivalent) in atmospheric models are likely to provide for a more accurate representation of absolute absorption by uncoated BC than does Mie theory. Further work will be necessary to understand how these results for uncoated BC will impact calculations of absorption by coated BC, for which absorption can be enhanced. However, our results suggest that the absolute absorption by coated particles will be similarly underestimated if core-shell Mie theory is used, regardless of the accuracy of the absorption enhancement calculation. Data availability Data availability. The data associated with this paper are archived at the UC Davis DASH data repository and are available for download from https://doi.org/10.25338/B8JP4V (Forestieri and Cappa, 2018). Supplement Supplement. Author contributions Author contributions. SDF and CDC led the analysis. TBO and PD conceived the overall project, with contributions from CDC, ATL, and others on experimental design. All coauthors contributed to data collection and/or processing. SDF and CDC prepared the paper with particular input from TBO and contributions from all coauthors. Competing interests Competing interests. The authors declare that they have no conflict of interest. Acknowledgements Acknowledgements. The BC2 study was supported by the US Department of Energy (DOE) grant no. DE-FG02-05ER63995 and the Atmospheric Chemistry Program of the National Science Foundation grants no. ATM-0525355. The BC3 and BC3+ studies were supported by the DOE ASR program grant DE-SC0006980, the EPA STAR Program grant R835033, and the NSF grant ATM-0854916. The BC4 study was supported by DOE grant DE-SC0011935 and the NSF grants AGS-1244918 (Boston College) and AGS-1244999 (Aerodyne), and the participation of the SP2 was made possible by DOE. 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Cite this document Summary â¦ Albert Einstein After studying this Unit, you will be able to â¢â¢â¢ explain the terms : system and surroundings; â¢â¢â¢ discriminate between close, open and isolated systems; â¢â¢â¢â¢ explain internal energy, work and heat; â¢â¢â¢ â¦ (a) Specific heat capacity at constant volume ( c v ):- Specific heat capacity at constant volume is defined as the amount of heat required to raise the temperature of 1 g of the gas through 1º C keeping volume of the gas constant. 13) Explain the rule to find specific heat for aqueous solutions. A Notebook for Topics in Thermodynamics of Solutions and Liquid Mixtures . As the modern sciences developed and became increasingly specialized, physics came to denote that part of physical science not included in astronomy, chemistry, geology, and engineering.Physics â¦ Browse more Topics under Thermodynamics. Dear Readers, Welcome to Thermodynamics multiple choice questions and answers with explanation. Compressibility charts. Essay Writing Help. All properties are point â¦ Avogadro's hypothesis 10. Work and energy of closed systems. You also get idea about the type of questions and method to answer in your Class 11th examination. (a) one value of specific heat (ft) two values of specific heat (c) three values of specific heat (d) no value of specific heat (e) one value under some conditions and two values under other conditions. Introduction to Thermodynamics; Thermodynamic Processes; First Law of Thermodynamics; Second Law of Thermodynamics; Reversible and Irreversible Process; Carnot Engine; Heat Engines and Heat Pumps; Internal Energy . Steam Tables. This note covers the following topics: systems surroundings and thermodynamic variables work and equilibrium introduced, temperature and the zeroth law of thermodynamics, basic properties of basic systems, reversible processes, internal energy: heat capacities and the first law of thermodynamics, isothermal and adiabatic expansions, ideal gas and â¦ It states that when a system approaches absolute zero, the entropy of the system approaches a minimum value and all process get a stop. Yes there is some differences between the thermodynamic study in physics and chemistry. Heat transfer project topics for Mechanical Engineers. Browse more Topics under Thermodynamics Introduction to Thermodynamics Thermodynamic Processes First Law of Thermodynamics Second Law of Thermodynamics Reversible and Irreversible Process Carnot Engine Heat Engines and Heat Pumps History of Thermodynamics Thermodynamics has become part and parcel of our life. And more! This note covers the following topics: Partial Derivatives, Temperature, Thermal Conduction, Thermodynamic Processes, Properties of Gases, The First and Second Laws of Thermodynamics, Heat Capacity, and the Expansion of Gases, Enthalpy, The Joule and Joule-Thomson Experiments, Heat Engines, The Clausius-Clapeyron Equation, Adiabatic Demagnetization, Nernst's â¦ Enthalpy. Thermodynamics is the study of the relationship between heat, work, and the associated flow of energy. Let us imagine a column of air of cross-sectional â¦ Heat â¦ Q&A GPA Calculator Thesis Generator Essay Topic Generator Free Essay Topics Study Guides Donate Paper. Thermodynamic efficiencies. Mass flow rates. 57. Thermodynamics notes for mechanical engineering students. Measurable outcomes (assessment method) : 1) To be able to state the First Law and to define heat, work, thermal efficiency and the difference between various â¦ Entropy. Energy Transfer And Thermodynamics - Math Problem Example. A thermodynamic system is defined as a quantity of matter of fixed mass and identity. Question from very important topics are covered by NCERT Exemplar Class 11. the properties of each and every molecule and ways in which they interact are taken into consideration to characterize the behaviour of a group of molecules. Here we have a free online quiz which includes mcqs questions and answers related to the topic of Heat and Thermodynamics . Home; Subjects; Other; Energy Transfer And Thermodynamics; Nobody downloaded yet. THERMODYNAMICS It is the only physical theory of universal content concerning which I am convinced that, within the framework of the applicability of its basic concepts, it will never be overthrown. All the individuals who are currently preparing for any physics subject related exam or just want to improve their general knowledge related to this topic, should attempt these tests in order to complete their preparation in a short time period with ease. That formula is much like the formula for heat ( $$q$$ ) transfer during isothermal expansion and you can rightfully conclude that: $\Delta S = {q_{\rm rev}\over T}$ Now the change in entropy is calculated via the reversible heat flow in/out of the system divided by the temperature at which the transfer took place. Albert Einstein After studying this Unit, you will be able to â¢â¢â¢ explain the terms : system and surroundings; â¢â¢â¢ discriminate between close, open and isolated systems; â¢â¢â¢ explain internal energy, work and heat; â¢â¢â¢ â¦ Thermodynamics app almost covers important topics of Thermodynamics chapter wise 1. Equilibrium â¦ This energy stands for the total energy of the system. Ans: a. Rumfordâs observation of the proportionality between heat â¦ Path function: Their magnitudes depend on the path followed during a process as well as the end states. Home; Subjects; Chemistry; Energy Transfer And Thermodynamics; Nobody downloaded yet. Two Years â¦ Fundamentally, thermodynamics in physics and chemistry is the â¦ We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. In addition to thermodynamics, basic heat transfer and fluid mechanics relationships are discussed, with an emphasis on the effect on plant balance equations. It is a set of theories that correlate macroscopic properties that we can measure (such as temperature, volume, and pressure) to energy and its capability to deliver work. 11 th Class; 12 th Class; Complete Course (11 th & 12 th Class) Physical Chemistry; Organic Chemistry; Inorganic Chemistry; Tablet Learning Program; Test Series. For an aqueous solution of salts, the specific heat can be estimated by assuming the specific heat of the solution equal to that of the water alone. Here, heat, temperature, specific heat, and latent heat are concepts which are studied. Zeroth law of thermodynamics 7. definition of ideal gas in thermodynamics 8. Here you can get Class 11 Important Questions Chemistry based on NCERT Text book for Class XI. Chemistry Class 11 Important Questions are very helpful to score high marks in â¦ Heat was not formally recognized as a form of energy until about 1798, when Count Rumford (Sir Benjamin Thompson), a British military engineer, noticed that limitless amounts of heat could be generated in the boring of cannon barrels and that the amount of heat generated is proportional to the work done in turning a blunt boring tool. Specially developed for the Mechanical Engineering freshers and professionals, these model â¦ Click Here for Class 11 Chemistry Notes. Thermodynamics is branch of science that deals with the system in equilibrium states only,ie how much heat is transfered from one equilibrium state to another equilibrium state but heat trasfer tells us the rate with which heat is transfered,either in equilibrium or not.So this the reason unit of heat in thermo is joule but in heat transfer it is watt(j/s) Mechanical , Diploma Mechanical Students For Final year Submission . Chemical thermodynamics is the study of how work and heat relate to each other both in chemical reactions and in changes of states. This article contain list of projects for mechanical engineering students related to heat transfer projects , heat and mass transfer mechanical Projects . Over three hundred Topics in Thermodynamics (which can addressed individually) describe the thermodynamic properties of aqueous solutions and aqueous mixtures. This list contain projects which are helpful for B.E. Advanced Search. In physics we studies the laws of thermodynamics and used to study the anthropic of Universe and objects. Rate of flow of heat in conduction is given by If you looking For heat transfer projects for Engineering Diploma , B.E. Essay Writing Help. Newtonâs law of cooling is another important topic studied under the thermodynamics of physics. The properties c v and c p are referred to as specific heats (or heat capacities) because under certain special conditions they relate the temperature change of a system to the amount of energy added by heat transfer. Third Law of thermodynamics: You can get answers on law of thermodynamics assignment. Heat and Thermodynamics Formulas. This course covers all the topics needed to gain an understanding of the basics of thermodynamics. Two Years; One Year; View Complete List; NEET . The Gibbs energies of these systems are discussed leading through successive differential operations to enthalpies, volumes, heat capacities, compressibilities â¦ Absolute zero Activity (thermodynamics) Activity-based costing Adiabatic process Aerothermodynamics Arrow of time Black-hole thermodynamics and superstring theory Blackbody Boiling Bolometer Boyle's law Brayton cycle Calorimetry Carnot cycle Charles' law Chemical energy Chemical equilibrium Chemical â¦ Process A: W A = 10 kJ Process b: W B = 7 kJ; Point Function: They depend on the state only, and not on how a system reaches that state. 1. Q&A GPA Calculator Thesis Generator Essay Topic Generator Free Essay Topics Study Guides Donate Paper. According to the first law of thermodynamics, for â¦ The anal- ysis of thermal systems is achieved through the â¦ Thermodynamics is the study of energy transformations and the relationships among properties of substances. Second Law of Thermodynamics: It states that heat cannot spontaneously flow from a space having less temperature to the space having a high temperature. Heat and work â¦ Their SI units are J/kg K or J/mol K. Two specific heats are defined for gases, one for constant volume (c v) and one for constant pressure (c p). THERMODYNAMICS It is the only physical theory of universal content concerning which I am convinced that, within the framework of the applicability of its basic concepts, it will never be overthrown. Topics include: Select topic Subtopics: Acoustics; Atomic and molecular physics ... Thermodynamics and heat. Live Online Classes. ; Electricity and Magnetism ; Chemistry ; energy transfer and thermodynamics ; Waves and Oscillations ; and! Â¦ heat transfer project topics for mechanical Engineers if you looking for heat transfer projects, heat topics under heat and thermodynamics thermodynamics quantity... 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Study of energy transformations and the relationships among properties of substances Readers, Welcome to thermodynamics multiple choice questions method... The anthropic of Universe and objects the total energy of the engineering purposes, heat ( q ) are functions. Thermodynamics 7. definition of ideal topics under heat and thermodynamics in thermodynamics ( which can addressed individually ) describe the thermodynamic study physics. Physics and Chemistry Guides Donate Paper One to One ; Self study Courses anal- of. And identity choice questions and method to answer in your Class 11th examination for engineering Diploma,.. Study of how work and heat relate to each Other both in chemical reactions in. Â¦ Question from very important topics of thermodynamics chapter wise 1 looking for heat transfer project topics for Engineers! 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Heat relate to each Other both in chemical reactions and in changes of states introduced identify! Transformations and the relationships among properties of substances Years Program ; One Year Program One... Magnitudes depend on the path followed during a process as well as the end states questions Chemistry on... You can get Class 11 important questions Chemistry based on NCERT Text book for Class XI room or thermodynamics! And mass transfer mechanical projects to identify the variables of thermodynamics 7. definition of ideal gas in thermodynamics ( can. For â¦ Select topic Subtopics: Acoustics ; Atomic and molecular physics... thermodynamics and used study... Aqueous mixtures for most of the system heat values topic Generator Free Essay topics study Guides Donate Paper to Delete... And used to study the anthropic of Universe and objects many decades of experience with phenomena! Also get idea about the type of questions and method to answer in your Class 11th examination mechanical projects law... Project topics for mechanical Engineers room or â¦ thermodynamics notes for mechanical engineering students some differences between the thermodynamic of... Article contain list of projects for engineering Diploma, B.E questions Chemistry based on NCERT Text book Class. Identify the variables of thermodynamics: you can get answers on law thermodynamics. Study Guides Donate Paper â¦ Select topic Subtopics: Acoustics ; Atomic and molecular physics... and. Changes of states it is the study of how work and heat relate to each Other both chemical! Thermodynamics app almost covers important topics are covered by NCERT Exemplar Class 11 important questions Chemistry based NCERT...	2021-05-18 18:04:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3777703642845154, "perplexity": 1797.5128619581246}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991288.0/warc/CC-MAIN-20210518160705-20210518190705-00309.warc.gz"}
https://stats.stackexchange.com/questions/178408/a-possible-mistake-in-a-conditional-probability-derivation	# A possible mistake in a conditional probability derivation The following is a derivation of a density from a paper I am currently studying. Sorry for the bad quality, it is quite an old paper. I need to clarify that $R$ has the standard exponential density in $(0,\infty)$, $U$ is uniform on $(0,1)$ and they are independent. The population correlation coefficient $\rho$ is a constant of course. $X$ and $Y$ come from the standard bivariate normal distribution, hence the trigonometric representation, but this plays no role here, I believe. What I do not understand is how the author reaches these conclusions for positive or negative $t$. It seems to me that the division by a negative number and the nonnegativity of $R$ are not properly taken into account. I could be mistaken of course so I would appreciate some advice. Thank you. • @Xi'an Thank you for your comment. This representation is derived from the fact that $XY = \left[ (X+Y)^2 -(X-Y)^2 \right]/4$ with $X-Y$ and $X+Y$ independent. Since the sum has variance $2(1+\rho)$ and the difference $2(1-\rho)$, then $XY$ has the same distribution as $$\frac{1}{2}\left((1+\rho)Z_1^2 -(1-\rho) Z_2 ^2 \right)$$ where $Z_i$ now has the standard normal distribution. Then the result follows by putting $Z_1=\sqrt{-2\log(U_1})\cos(2\pi U_2)$ and $Z_2=\sqrt{-2\log(U_1)} \sin(2\pi U_2)$, the Box-Muller transform and uitlizing that $- \log(U)$ has the standard exponential distribution – JohnK Oct 23 '15 at 20:59 • @Xi'an No problem. Would you say that the subsequent steps are correct, then? – JohnK Oct 23 '15 at 21:08 When $t\ge 0$, \begin{align} P(R(\cos(\pi U)&+\varrho)\ge t,\cos(\pi U)+\varrho\ge 0)\\ &+P(R(\cos(\pi U)+\varrho)\ge t,\cos(\pi U)+\varrho\le 0)\\ &=P(R(\cos(\pi U)+\varrho)\ge t,\cos(\pi U)+\varrho\ge 0) \end{align} since the second term is zero, $R$ being multiplied there by a negative term. Thus $$P(XY\ge t) = P(R(\cos(\pi U)+\varrho)\ge t,\cos(\pi U)+\varrho\ge 0)\quad\qquad \\ \ \ \ = P(R(\cos(\pi U)+\varrho)\ge t,U\le\cos^{-1}(-\varrho)/\pi)\\ = \int_0^{\cos^{-1}(-\varrho)/\pi} P(R(\cos(\pi U)+\varrho)\ge t)\,\text{d}u$$ seems to be correct. When $t\le 0$, since $$R(\cos(\pi U)+\varrho)\ge t$$ is always true when $\cos(\pi U)+\varrho\ge 0$, \begin{align} P(R(\cos(\pi U)&+\varrho)\ge t,\cos(\pi U)+\varrho\ge 0)\\ &+P(R(\cos(\pi U)+\varrho)\ge t,\cos(\pi U)+\varrho\le 0)\\ =P(\cos(\pi U)&+\varrho\ge 0)\\ &+P(R(-\cos(\pi U)-\varrho)\le -t,\cos(\pi U)+\varrho\le 0)\\ =P(\cos(\pi U)&+\varrho\ge 0)\\ &+P\left\{R\le t\big/(\cos(\pi U)+\varrho),U\ge\cos^{-1}(-\varrho)/\pi\right\}\\ =P(\cos(\pi U)&+\varrho\ge 0)\\ &+\int_{\cos^{-1}(-\varrho)/\pi}^1 P\left\{R\le t\big/(\cos(\pi u)+\varrho\right\} \end{align} so this seems to be correct as well.	2019-10-17 02:48:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9466641545295715, "perplexity": 608.4735064521292}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986672548.33/warc/CC-MAIN-20191017022259-20191017045759-00398.warc.gz"}
https://www.snapsolve.com/solutions/Figureshows-a-capacitor-made-of-two-circular-plates-each-of-radius-12-cm-and-sep-1672370140277761	Home/Class 12/Physics/ QuestionPhysicsClass 12 Figure shows a capacitor made of two circular plates each of radius $$12\text{ cm}$$, and separated by $$5.0\;\text{cm}.$$ The capacitor is being charged by an external source (not shown in the figure). The charging current is constant and equal to $$0.15\;A$$. Is Kirchhoff’s first rule $$($$junction rule$$)$$ valid at each plate of the capacitor? Explain. Yes, Kirchhoff’s first junction rule is valid at each plate of the capacitor provided that we take the sum of conduction and displacement for current.	2022-07-06 01:49:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7700061798095703, "perplexity": 509.30958520072915}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104655865.86/warc/CC-MAIN-20220705235755-20220706025755-00440.warc.gz"}
https://proofwiki.org/wiki/Definition:Differential/Functional	Definition:Differential/Functional Definition Let $J \sqbrk y$ be a differentiable functional. Let $h$ be an increment of the independent variable $y$. Then the term linear with respect to $h$ is called the differential of the functional $J$, and is denoted by $\delta J \sqbrk {y; h}$. Notes For a differentiable functional is holds that: $\Delta J \sqbrk {y;h} = \phi \sqbrk {y; h} + \epsilon \size h$ where $\phi$ is linear with respect to $h$. Thus: $\delta J \sqbrk {y; h} = \phi \sqbrk {y; h}$ Also known as The differential $\delta J \sqbrk {y; h}$ is also known as the (first) variation.	2019-07-23 13:54:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9696615934371948, "perplexity": 249.6479504769018}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195529406.97/warc/CC-MAIN-20190723130306-20190723152306-00202.warc.gz"}
https://proxies123.com/tag/neumann/	## Equivalence of projections in smaller von Neumann algebra I came across the following assertion and I can’t understand why it’s true. We are given with two finite equivalent projection $$esim f$$ in some von Neumann algebra $$A$$ (with a unit of course). It’s known that the projection $$q=evee f$$ is also finite, so we infer the algebra $$qAq$$ is a finite algebra (that is, the unit element is a finite projection). Now suppose that $$q-esim q-f$$ in $$qAq$$, show that $$q-esim q-f$$ also in $$A$$. Comment: Technically it is true that $$q-esim q-f$$ in $$qAq$$ so we don’t really need to assume that. ## differential equations – Solving PDE with Neumann boundary conditions I want to solve a pde on a triangular domain, such that you can’t leave the domain (which I think would just be Neumann boundary conditions). My pde takes the following form: $$frac{partial u}{partial x}(A(x,y)u(x,y)) + frac{partial u}{partial y}(B(x,y)u(x,y)) + frac{partial^2 u}{partial x^2}(C(x,y)u(x,y)) + frac{partial^2 u}{partial x partial y}(E(x,y)u(x,y)) + frac{partial^2 u}{partial y^2}(F(x,y)u(x,y)) = 0.$$ However, I’m struggling to solve this numerically in mathematica. When I use the code below, mathematica doesn’t do anything, just returning exactly what I wrote. My code is as follows, where A,B,C,E,F are functions of x and y I have already defined: ``````domain = Triangle({{0, 0}, {1, 0}, {0, 1}}); eqn = D(A(x,y)u(x, y), x) + D(B(x, y)u(x, y), y) + D(C(x,y)u(x, y), x, x) + D((E(x,y))u(x, y), x, y) + D((F(x, y)u(x, y), y, y) == 0; s = NDSolve(eqn, NeumannValue(0, ...), u, {x, y} (Element) domain); `````` I also require that the integral of $$u(x,y)$$ over the domain be equal to 1 – is there a way of including this condition in the pde solver? ## Von Neumann Random Byte Debuting in Python I need to create a method for you should be biased `bytes` of the method `get_raw_bytes(length)` using the von Neumann deburring method. Here is my current code: ``````def get_debiased_bytes(length): arr_debiased_bytes = () debiased_byte = 0 bit_counter = 0 while len(arr_debiased_bytes) < length: raw_bytes = get_raw_bytes(length 5) for byte in raw_bytes: for k in range(0, 8, 2): bit1 = byte >> k & 1 bit2 = byte >> k + 1 & 1 if bit1 != bit2: debiased_byte = debiased_byte << 1 \| bit1 bit_counter += 1 if bit_counter == 8: arr_debiased_bytes.append(debiased_byte) debiased_byte = 0 bit_counter = 0 return bytes(arr_debiased_bytes(:length)) `````` I think my code can still be improved and sorted. Can someone help me with this? ## pde – Solving the Cauchy problem for the wave equation using Neumann boundary conditions I've been trying to solve this problem for a while, but I can't seem to figure out what to do with the Neumann Limit Condition. The problem is solving the following: $$begin {equation } u_ {tt} -u_ {xx} = 0 \ u (x, 0) = 0 \ u_t (x, 0) = begin {cases} pi cos ( pi (x-1)), y 1 For the next 1) $$x in (- infty, infty)$$ two) $$x in (0, infty) text {y} u_x (0, t) = 0$$ 3) $$x in (- infty, 3) text {y} u (3, t) = 0$$ I know how to do 1, just apply D & # 39; Alemebert's formula. But I am lost by 2 and 3. ## differential equations – Numerical solution for ODE 1-D with Neumann conditions only I tried to solve a simple ODE with only Neumann conditions like But obviously this does not work. I must add a useless Dirichlet condition to make it work I have verified that the solution is correct, but how can I get it without including the useless condition? The code is ``````Sol = NDSolveValue({Piecewise({{1, x < 0}, {2.25, x > 0}}, 0)u(x) + Derivative(2)(u)(x) == NeumannValue(I(2E^(Ix) - u(x)), x == -4Pi) + NeumannValue((0. - 1.5I)u(x), x == 4Pi), DirichletCondition(1, False)}, u, {x, -4Pi, 4Pi}) ReImPlot(Sol(x), {x, -4Pi, 4Pi}) `````` ## reference request – Estimation of the Dirichlet map to Neumann for mixed limit value problems To be specific, consider a model problem in $$Omega subset mathbb {R} ^ 2$$ with Lipschitz limit, $$begin {cases} – Delta u = 0 & text {in} Omega, \ u = g & mbox {on} Gamma_D, \ partial_n u = g_N & text {on} Gamma_N, end {cases}$$ where $$partial Omega = bar Gamma_D cup bar Gamma_N$$, $$Gamma_N cap Gamma_D = emptyset$$. I'm curious to know if there is literature on the DtN map (existence, estimates, explicit construction as layer potentials) in the Dirichlet boundary: $$mathcal {V}: H ^ {1/2} ( Gamma_D) a H ^ {1/2} ( Gamma_D),$$ such that $$\| mathcal {V} g \| _ {- 1/2, Gamma_D} ^ 2 leq langle mathcal {V} g, g rangle _ { Gamma_D} + ( text {terms in} Gamma_N)$$ ## Laplacian's smallest non-zero eigenvalue with Neumann contour conditions of a ball in a simply connected pinched negative curvature space In the document, Small values of multiple geometric finites, http://www.math.uni-bonn.de/people/ursula/eigenvalue.pdf, page 11, in the last paragraph, the author stated that "Now the Value own smaller than zero with Neumann contour conditions of a radio ball $$rho$$ in a variety of simply connected curvature contained in $$(- κ, −1)$$ is not smaller than $$(n – 1) ^ 2/4$$"I struggled to find a reference to support this claim. If we use the Dirichlet boundary condition, it follows from the comparison of the monotonicity of the domain and the curvature of the balls. However, for the Neumann condition it depends on the geometry of the multiple and the limit.I consulted (BCD) cited on the line and I still did not see the idea.Maybe I am missing something or Neumann's own value for a ball in pinched curvature has been explicitly calculated. I also have doubts about the following statement, "therefore, by domain monotonicity, the second proper value for each of the sets $$U_i$$ with mixed boundary conditions (i.e., Neumann boundary conditions in $$∂U_i ∩ Ω$$ and border conditions of Dirichlet in $$∂Ω$$) is not less than $$(n – 1) ^ 2/4$$ Also here $$U_i$$ is a Dirichlet domain as polygons within the radio ball $$rho$$ previously mentioned. $$Ω$$ It is an open set with compact closure and lower limit in the radius of injectivity $$rho$$. It is not clear if the author is using the monotonicity of Neumann or the monotonicity of Dirichelt. For the ball and $$U_i$$, since there are domains that do not cross $$∂Ω$$, the mixed boundary condition becomes the Neumann boundary condition. We cannot conclude Neumann's own value of the smaller sets. $$U_i$$ is greater than Neumann's own value of the ball it contains $$U_i$$ directly, as the usual Neumann monotonicity requires a partition and actually gives the inverse inequality instead of the desired one. Maybe I made a mistake in my argument or is there any reference to Neumann's own value / monotonicity that works this way. Thank you. ## or operational algebras or – Norms for calculating operator polynomials in the Hilbert space and generalized inequality of von Neumann Leave $$T$$ be an operator $$l ^ 2 ({ mathbb {Z} _ { geq 0}}) a l ^ 2 ({ mathbb {Z} _ { geq 0}})$$, $$e_n mapsto sqrt {1 – q ^ {2 (n + 1)}} e_ {n + 1}$$, where $$0 . I want to compute $$\| f (T, T ) \|$$ (operator standard) for any $$f in mathbb {C} (z, bar z)$$. Operator $$T $$, of course, is Hilbert's conjugate and $$T ^ e_0 = 0$$, $$T ^ {} e_n = sqrt {1-q ^ {2n}} e_ {n-1}$$ for $$n> 0$$. I'm not sure if that calculation is possible and would be happy to show that $$\| f (T, T ) \| to sup limits_ {\| z \| leq 1} f (z, bar z)$$ how $$q$$ It goes to 1 because it's really why I need to calculate the rules first. I think there is a generalization of von Neumann's inequality as $$q$$-analysis or something (there are many generalizations) because the usual inequality proves it for polynomials $$g in mathbb {C} (z)$$ (without $$bar z$$) Question: Does anyone know any useful facts or inequalities related to my question? Something that could help. ## 1D transport equation with Neumann conditions I am trying to solve the second order differential equation of the type of heat transfer in a finite system x<0,L>, in particular: ``````eq = D(T(x, t), t) - k*D(T(x, t), {x, 2}) == 0; `````` (For simplicity, I suppose that hereafter k = 1 and L = 1). The system is connected to a semi-infinite source of power 1 for x (-infty, 0), while the & # 39; temperature & # 39; initial for x <0.1) is 0. This provides the initial condition for the equation in parts form: ``````T(x, 0) == Piecewise({{1, x < 0}, {0, x >= 0}}) `````` I also assume that Neumann's boundary conditions: 1) the "heat" flow in the system is proportional to the temperature gradient between the source and the system: ``````Derivative(1, 0)(T)(0., t) == (T(0., T) - 1) `````` and 2) there is no loss of & # 39; heat & # 39; in the second system limit: ``````Derivative(1, 0)(T)(1, t) == 0} `````` Merging all the elements I obtained a code: ``````sol = NDSolve({eq, T(x, 0) == Piecewise({{1, x < 0}, {0, x >= 0}}), Derivative(1, 0)(T)(0., t) == (T(0., t) - 1), Derivative(1, 0)(T)(1, t) == 0}, T, {x, 0, 1}, {t, 0, 1}) Plot3D(Evaluate(T(x, t) /. sol), {x, 0, 1}, {t, 0, 1}) `````` When I ran the Mathematica code (v8.0) he sent me a warning about the inconsistency of the initial conditions / limit: I guess I confused something with the behavior at x = 0. However, I got a graph, not exactly what I want (T (x, t) = 0). Then I changed the initial conditions to ``````T(x, 0) == Piecewise({{1, x <= 0}, {0, x > 0}}) `````` to impose the heat flux through x = 0. Once again, I received the warning and a graph: this time I can see some distributions of & # 39; temperature & # 39; in the system, but the result seems to be bad, since the inconsistency of the initial conditions / limit and also the non-physical results (T> 1, while it cannot be higher than the source temperature T = 1). Can anyone help me overcome the problem with the initial limits / conditions? I am using Mathematica 8, so I cannot use the NeumannValue function. ## o.operator algebras – Trace extension in von Neumann subalgebra No Yes $$Gamma$$ is a group without torsion and with the property of infinite conjugation class then the algebra $$L Gamma$$ it's a factor of type $$II$$ and has a single normal state of affairs. Now leave $$x$$ in $$Gamma$$ Different from identity. The von Neumann algebra $$S$$ generated by $$x$$ is isomorphic to $$L ^ infty (S ^ 1)$$, the algebra of measurable functions in the circle. It has many normal trative states.	2020-09-26 15:09:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 60, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5845428109169006, "perplexity": 1428.535663052068}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400244231.61/warc/CC-MAIN-20200926134026-20200926164026-00280.warc.gz"}
https://masteranza.wordpress.com/2007/11/11/2-killer-identities/	Identities NO. 2 : Algebra, some Euler’s work 11Nov07 First identity comes from Lebesgue: $(a^2+b^2+c^2+d^2)^2 = (a^2+b^2-c^2-d^2)^2 +(2ac+2bd)^2+(2ad+2bc)^2$ and the second one from Euler: $(a_1 ^2 + a_2^2 + a_3^2+ a_4^2)(b_1 ^2 + b_2^2 + b_3^2+ b_4^2) = (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4)^2 + (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2 + (a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2)^2 + (a_1 b_4 + a_2 b_3 - a_3 b_2+ a_4 b_1)^2$ One Response to “Identities NO. 2 : Algebra, some Euler’s work” 1. 1 Astrid Denaya Lesa Thanks for your nice formula. I promise to visit this blog next time.	2018-03-23 03:27:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22546374797821045, "perplexity": 5723.317605902876}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257648177.88/warc/CC-MAIN-20180323024544-20180323044544-00427.warc.gz"}
https://www.gradesaver.com/textbooks/math/calculus/calculus-10th-edition/chapter-8-integration-techniques-l-hopital-s-rule-and-improper-integrals-8-3-exercises-page-530/30	## Calculus 10th Edition $\frac{sec^23x}{6}+ \frac{\ln{\|cos3x\|}}{3}+C$ Find the indefinite integral $\int tan^33xdx$ $\int tan^23xtan3xdx$ $\int (sec^23x -1)tan3x dx$, Convert a tangent to secant $\int sec3xsec3xtan3x - \int tan3xdx$ Use u-substitution For the first integration let $u=sec3x$, $du=3sec3xtan3xdx$ For the second let $u=3x$, and $du=3dx$ $\frac{1}{3}\int udu - \frac{1}{3} \int tanudu$, Integrate $\frac{1}{6}u^2 + \frac{1}{3} \ln{\|cosu\|} +C$ $\frac{sec^23x}{6}+ \frac{\ln{\|cos3x\|}}{3}+C$	2018-10-23 06:29:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9982613921165466, "perplexity": 625.8847616841265}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583516071.83/warc/CC-MAIN-20181023044407-20181023065907-00105.warc.gz"}
https://aas.org/archives/BAAS/v32n3/dps2000/299.htm	DPS Pasadena Meeting 2000, 23-27 October 2000 Session 53. Solar System Origin Posters Displayed, 1:00pm, Monday - 1:00pm, Friday, Highlighted Tuesday and Thursday, 3:30-6:30pm, C101-C105, C211 ## [53.12] Gas Accretion Flows onto Giant Protoplanets: High-Resolution Two-Dimensional Local Simulation T. Tanigawa, S. Watanabe (Dept. of Earth and Planetary Sciences, Nagoya Univ.) We investigate the gas accretion flows onto the giant protoplanets from protoplanetary disks in detail in order to clarify the gas capturing process of the giant planets after the onset of a gravitational instability of the proto-atmosphere, using high-resolution two-dimensional numerical simulations with local Cartesian coordinates. We use ZEUS-2D code, which is a kind of finite-difference time-marching methods, and obtain the steady gas-accretion flows depending on normalized sound speed \tilde{c} corresponding to the ratio of disk scale height to Hill radius. Then we find that; (1) Accretion flow patterns: There exist two types of steady shocks; a pair of bow shocks around the planetary gravitational sphere and a pair of spiral shocks in the sphere. And only gases in narrow bands are able to flow into the planetary gravitational sphere. The band location is about 2.5 Hill radius away from the planetary orbit and the width is about 0.1 Hill radius. The band width increases with decreasing \tilde{c} and the band location approaches the planetary orbit with decreasing \tilde{c}. (2) Gas accretion rates onto the giant protoplanets: When the gas temperature is given by the radiative equilibrium with central star and the ratio of specific heats \gamma is chosen to unity (isothermal), we get \dot{M} = 8.0 \times 10-3 M\rm E (a/{\rm 5.2AU})-1.5 (M\rm p/10M\rm E)1.3 (f\Sigma0/\Sigma\rm min) {\rm yr}-1, where f is the depletion factor of the surface density at the flow-in band caused by the formation of a gap, \Sigma0 is the original surface density, which has no gap, and \Sigma\rm min is the surface density of minimum mass disk model. The accretion rate decreases with increasing \gamma. (3) Migration of giant planets: The circum-planetary spiral-shock structures may strongly affect the torque about a central star exerted on the planet by the gas. The author(s) of this abstract have provided an email address for comments about the abstract: tanigawa@eps.nagoya-u.ac.jp [Previous] \| [Session 53] \| [Next]	2016-07-27 10:05:10	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.849101185798645, "perplexity": 3596.328668082467}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257826759.85/warc/CC-MAIN-20160723071026-00064-ip-10-185-27-174.ec2.internal.warc.gz"}
https://www.gamedev.net/forums/topic/404946-dealing-with-mousewheel-input/	# Dealing with MouseWheel input ## Recommended Posts Hi guys. I'm using C#, MDX, and DirectInput. I have a MouseInput class which works wonders. However, the MouseState.Z member which is supposed to return how much the mouse wheel has turned works in a mysterious way. Basically, it works but every time I turn the wheel and from one "click" to the next, it returns that as 120 instead of 1. Now of course you could say "Why don't you just divide by 120 and you've got the right number of "clicks" the mouse wheel has turned?" The problem is, how can I be sure that on someone else's computer one "click" will not return 200 or 500 or whatever? So is there a way to check how much each "click" is worth or to simply return the number of "clicks?" Thanks. ##### Share on other sites Well, the managed documentation stinks, but search for GetGranularity (it's part of the DeviceProperties class accessable from Device.Properties). For the mouse wheel, it will return the number you're looking for. I just haven't figured out what to pass the method to specify that I'm interested in the wheel. ##### Share on other sites Thanks a lot. I tried this: Console.WriteLine("Mouse Granularity: {0}", _mouse.Properties.GetGranularity(ParameterHow.ByUsage, _mouse.DeviceInformation.Usage)); But got an UnsupportedException when the code is run. [Edited by - yaroslavd on July 21, 2006 12:29:22 AM] ##### Share on other sites In C++ there is a defined value for it: /* Value for rolling one detent */ #define WHEEL_DELTA 120 Wouldn't there be something similar for managed code? ##### Share on other sites Maybe, that's what I'm trying to figure out. Can't find anything so far, though. ##### Share on other sites As DXnut stated, WHEEL_DELTA is what you are looking for. It is specified as 120 and since it is a const your program sets it. Your code should handle getting non-multiples of 120, but 120 would still be the conceptual "one tick" even if someone's mouse took two ticks to get to 120. Large quantities of code are being written unable to handle such a result though, so it's your choice. I am not familiar with what GetGranularity() does, but your "UnsupportedException" makes me guess it could be trying to get the information from the mouse itself. If this is the case, your device probably does not support this feature. Also keep in mind you should check the capabilities of a device prior to using functionality to prevent errors. You probably do this with GetCaps() for your video card. ##### Share on other sites Where is this WHEEL_DATA specified in Managed DirectInput? And I'm pretty sure that my device supports it because my mouse has a wheel. I just have no idea what to pass to the function. ##### Share on other sites SystemInformation.MouseWheelScrollDelta Property ##### Share on other sites Thank you. Rating+ all around!	2017-10-23 15:36:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18526969850063324, "perplexity": 1498.744986749736}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187826114.69/warc/CC-MAIN-20171023145244-20171023165244-00605.warc.gz"}