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https://tex.stackexchange.com/questions/167650/is-there-a-more-recent-bibliography-style-file-bst-for-pnas/167802	# Is there a more recent bibliography style file (.bst) for PNAS? As per the PNAS author guidelines: References should be cited in numerical order as they appear in text. Because tables and figures will be inserted in the text where first cited, references in these sections should be numbered accordingly. Include the full title for each cited article. All authors (unless there are more than five) should be named in the citation. If there are more than five, list the first author's name followed by et al. Provide volume and issue numbers for journal articles as applicable; provide DOI numbers if volume/issue numbers are not available. Provide inclusive page ranges for journal articles and book chapters. Provide date of access for online sources. Cite databases in the text or as footnotes. Journal articles are cited as follows: 1. Neuhaus J-M, Sitcher L, Meins F, Jr, Boller T (1991) A short C-terminal sequence is necessary and sufficient for the targeting of chitinases to the plant vacuole. Proc Natl Acad Sci USA 88(22):10362-10366. Because the journal only accepts .bbl files embedded within the main .tex file, they do not provide a bibliography style. But their bibliography style has changed over the years (use of colon to separate number/volume and page numbers, volume number not in bold etc.), and there doesn't seem to be any up-to-date bibliography style. Google led me to this and this packages (first one was obsoleted by the second one), but they do not match the PNAS bibliography style. Obviously, this is not the end of the world as I could fix the .bbl file manually, but I was wondering whether there is a more recent version. bibtex or biblatex solutions are both okay, as the .bbl will be added in the main .tex file anyway (correct me if I'm wrong on this one please). Worst-case scenario, how much work would be required to modify the bibliography styles, given that I have never done any work on bibliography styles? • If there are not many modifications with respect to the numeric style, it would not be not very long — but that depends on your degree of familiarity with biblatex. Anyway it's much easier than modifying a .bst file. You could try to choose style = numeric as an option to biblatex, compile a bibliography and give a list of what's going wrong wrt PNAS requirements. – Bernard Mar 25 '14 at 19:08 • @Bernard It seems like modifying .bst files and playing with biblatex settings won't be necessary. There are tools, like Bibulous (used in the accepted answer) that generate the .bst files for you. Thanks for your comment! – sudosensei Mar 28 '14 at 19:22 • it seems to be an interesting project. Thanks for having pointed to it. – Bernard Mar 28 '14 at 19:44 • A yet another hack to obtain a journal-tailored style from a third party is to use pandoc, pandoc-citeproc, and feed it the CSL styles Zotero hat for virtually any journal. To prevent online HTTP queries, check if the style is not dependant (as they call it) or use the "root" style instead. – Oleg Lobachev Mar 15 '18 at 19:46 I'm not aware of a BibTeX style file for PNAS, but the Bibulous project does provide an easy way of customizing styles. For the style suggestions linked to by the OP, it took me only a few minutes to put together a complete style template to follow PNAS' requirements. Using the following main.bib database file @ARTICLE{Neuhaus, author = {Jean-Marc Neuhaus and Liliane Sitcher and Meins, Jr, Frederick and Thomas Boller}, year = {1991}, title = {A short C-terminal sequence is necessary and sufficient for the targeting of chitinases to the plant vacuole}, journal = {Proc Natl Acad Sci USA}, volume = {88}, number = {22}, pages = {10362-10366} } @INCOLLECTION{Hill, author = {Adrian V. S. Hill}, year = {1991}, title = {HLA associations with malaria in Africa: some implications for MHC evolution}, booktitle = {Molecular Evolution of the Major Histocompatibility Complex}, editor = {Jan Klein and Dagmar Klein}, publisher = {Springer}, pages = {403-420} } and the style template file main.bst (the lines below show the complete file) TEMPLATES: article = <au> (<year>) <title>. \textit{<journal>} <volume>(<number>): [<startpage>--<endpage>\|<startpage>\|<eid>\|].[ <note>] incollection = <au> (<year>) <title>. \textit{<booktitle>}[, vol.~<volume>, ][, <edition_ordinal>~ed.][, <null.if_singular(editorlist, edmsg1, edmsg2)>~<ed>][, <series>][, Chap.~<chapter>] (<publisher>, <address>)[, pp~<startpage>--<endpage>\|p~<startpage>\|<eid>\|].[ <note>] SPECIAL-TEMPLATES: authorlist = <author.to_namelist()> editorlist = <editor.to_namelist()> authorname.n = [<authorlist.n.prefix> ]<authorlist.n.last>[ <authorlist.n.first.initial()>][<authorlist.n.middle.initial().compress()>][, <authorlist.n.suffix>] au = <authorname.0>, ..., <authorname.9> editorname.n = [<editorlist.n.prefix> ]<editorlist.n.last>[ <editorlist.n.first.initial()>][<editorlist.n.middle.initial().compress()>][, <editorlist.n.suffix>] ed = <editorname.0>, ..., <editorname.9> null = {} OPTIONS: edmsg1 = ed edmsg2 = eds compiling the main.tex file \documentclass{article} \usepackage[paper=letterpaper, text={6.5in,9in},centering]{geometry} \makeatletter % \renewcommand{\@biblabel}[1]{#1.} \makeatother \begin{document} \nocite{Neuhaus,Hill} \bibliography{temp} \bibliographystyle{temp} \end{document} produces the following formatted result: This provides templates for only journal articles and articles/chapters in books, but the PNAS website provides guidelines for only these two. Templates for other entry types are easily derived from the two shown here. (For example, a book entry type template can be defined by adding another line book = <au> (<year>) <title>. ... in the lines below TEMPLATE: in the style template file.) • Nice one! I wasn't aware of this tool. Thanks a lot! – sudosensei Mar 28 '14 at 19:23 Also check out https://github.com/jburon/pnas2011.bst for PNAS bibtex style updated to match the current fashion in reference layout as of 4 September 2011. I found this in the PNAS site, so I guess it's the official bibliography style file. http://www.pnas.org/site/misc/pnas2011.bst The link to pnas2011.bst posted above is broken. However, PNAS offers pnas-new.bst as part of their official latex templates at: http://www.pnas.org/page/authors/latex. It's their template file, so I don't expect them to complain about the result :). Note that URLs must be cited via a note to conform to the suggested PNAS style (cf. http://www.citethisforme.com/guides/pnas/how-to-cite-a-website). E.g.: @misc{NIST_air_refractive_index, Author = {Stone Jr., Jack A. and Zimmerman, Jay H.}, Title = {Index of Refraction of Air, NIST publication}, Publisher = {National Institute of Science and Technology}, note = {Available at: \url{https://www.nist.gov/publications/index-refraction-air} [Accessed June 7, 2017]}, Year = {2001} }	2019-06-26 14:12:45	{"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": 0.33923932909965515, "perplexity": 4505.675083989234}, "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-2019-26/segments/1560628000353.82/warc/CC-MAIN-20190626134339-20190626160339-00131.warc.gz"}
http://crypto.stackexchange.com/questions?page=2&sort=newest	# All Questions 66 views ### counting points on elliptic curve Given an elliptic curve with equation $y^2=x^3+ax+b$, and i want to find the number of points $(a,b)\in E(\mathbb{F}_p)$ where the polynomial has repeated roots, how do i do it? I have an intuition it ... 50 views ### Regex searchable word list for space-less monoalphabetic substitution I am working on teaching myself basic cryptography, in the process I created a simple substitution cipher based on a single alphabet, and a statically defined offset (no key for now). Now that ... 147 views ### Can insecure algorithms be combined to form a secure algorithm? This is hypothetical as I can't think of any reason to do this, but out of curiosity.. Could I, for example, take the MD5 digest of a message and concatenate it with the SHA-1 digest (not quite ... 86 views ### Is a simple stream cipher “partially homomorphic” if no integrity check is applied? 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Is it the size of message, calculating the modular inverse, exponentiation? Say I have a fixed value of n to encrypt and decrypt ... 59 views ### Is the composition of collision resistant hash with non collision resistant hash a collision resistant hash? Suppose $h_1$ is collision resistant, and $h_2$ isn't. Will $h(x)=h_2(h_1(x))$ always be collision resistant? [Edited:] the output of h2 must be uniformly distributed 85 views ### Graphically representing points on Elliptic Curve over finite field I have taken elliptic curve $E\colon y^2=x^3-4x+20$, defined over $\mathbb{F}_{29}$. The number of points on the curve, $\left\|E(\mathbb{F}_{29})\right\|=37$. I took base point $P=(1,5)$, and got ... 159 views ### Using the same RSA keypair to sign and encrypt The RSA signature operation is basically the same as encrypting with the private key. In particular, both operations use the same kind of keys. 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I'm designing a function f that should be moderately hard to invert and very fast to evaluate in a modern CPU. The function will be used in a proof-of-work function. I've read that the middle-bits of ... 66 views ### Implementation of cryptography [closed] I want to know how we implement the research topic/thesis in cryptography area? I mean is it through C/C++ programmming or any other way? 79 views ### IV's from Crypto++ AutoSeededRNG always ends the same: Problem with the PRNG? I am using a AutoSeededRNG from Crypto++ to generate IVs for use in CBC encryption. I am encoding the IVs using Base64 for future use (decryption). I notice that ... 79 views ### What does this step in this protocol for the Yao's Millionaire Problem mean? I was reading “An Efficient Protocol for Yao’s Millionaires’ Problem” (Ioannidis and Grama 2003). 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I'm reading Multivariate Cryptography and would like to know why we use two affine transformations to scramble the central map? Why not use only one? 65 views ### Differences between working of Tor and onion routing [migrated] I am learning about Tor and Onion routing and I'm trying to compare both services. From the wiki link, I can see that Diffie Hellman (DH) handshake is used and when look into the “tor-design.pdf” for ... 112 views ### Can AES decrypt with a wrong key? The title is asking the question in layman terms, but what I really mean to ask is this: Let $x \in \lbrace 0,1 \rbrace^{128}$ (an arbitrary input block) and let $K \in \lbrace 0, 1 \rbrace^{128}$ ... 42 views ### On the structure of the UOV (Unbalanced Oil and Vinegar) signature scheme [closed] I'm study Multivariate Quadratic Cryptography. 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Why have AH and ESP security protocol when we could just encrypt / decrypt packets with a symmetrical key and with some block cipher using that symmetrical key? 102 views ### Publicly exposed hash of private key Would exposing a cryptographic hash function's digest (e.g. SHA-3) of RSA private key data compromise the key? If so, what are the possible (cryptanalysis-) vectors for attacking the key if an ... 31 views ### Can you clarify the proof that secure LR-two party protocols do not exist? In the paper “Multiparty Computation Secure Against Continual Memory Leakage” on pg. 1237, the footnote #1 discuss why it is not possible to construct a leakage resilient two-party protocol. But I'm ... 55 views ### Why is Lamport-Diffie secure? Why is Lamport-Diffie secure? I note that there is a demonstration based on onewayness (in the book postquantum cryptography). 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Is it possible to deduce the algorithm used, key length, etc just by looking at the minimum ... 203 views ### Is this encryption scheme perfectly secure? Let $m = 6$, and let $\mathbb{Z}_m$ denote the set $\{0,…,m-1\}$. Let $X \mod m$ denote the remainder obtained when dividing $X$ by $m$. (a) Consider the symmetric encryption scheme in ... 57 views ### How good is the RandomPool generator in Crypto++ library? How good is the RandomPool generator in Crypto++ library? I never heard of it before… I wonder if there is a Fortuna C implementation that I could use in an Objective-C project. Also: How secure is ... 49 views ### How can user privacy be preserved in certificateless cryptography? In certificateless cryptography, how can a user preserve his privacy from Key Generation Center? If the KGC becomes malicious, how can the user's privacy be preserved? I read "Privacy-Preserving ... 145 views ### What would make it impossible to deny that decryption of a package has taken place? I'm part of a team designing a system where a user have to be able to read a certain package, if he has preformed an action or series of actions that makes it legal for him to do so. The tricky part ... 46 views ### How do “sign” and “verify” algorithms work with certificateless crypto? I want to know in a practical scenario how could the CL-PKE works. For complete understanding of Certificateless Signature Scheme, I want to implement "Setup,Partial-secret key extract,user-key ... 111 views ### Can the XOR of two non-collision-resistant hashes be collision resistant? Suppose I have two hash functions, of which neither (or only one) is collision resistant, and I want to create a new hash function by taking the bitwise exclusive or (XOR) of the results of those two ... 49 views ### What are the key lengths used in (EC)DHE cipher suites in SSL? What are the key lengths used in (EC)DHE cipher suites in SSL? Browsers only display key lengths for public key encryption… 17 views ### TCR hash functions from MD5 I'm reading Collision-Resistant Hashing: Towards Making UOWHFs Practical (pag. 10 and 11) and here say The most direct way to construct a TCR hash function is to key a function like MD5 or SHA-1. ... 102 views ### Can homomorphic encryption filter? Often in articles homomorphic encryption is praised as the holy grail of encryption for cloud storage. This is done by suggesting that it can do any computation, and as such could be used for ... 58 views ### Authentication using certificateless signature With traditional PKI approach two parties can communicate with each other in following way Now, How the above approach will be applied using Certificateless Cryptography.? How two parties in two ...	2013-12-10 07:34:01	{"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": 0.7367919683456421, "perplexity": 2526.75078689939}, "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-2013-48/segments/1386164012753/warc/CC-MAIN-20131204133332-00070-ip-10-33-133-15.ec2.internal.warc.gz"}
http://mathoverflow.net/questions/63158/in-knot-theory-benefits-of-working-in-s3-instead-of-mathbbr3?answertab=active	In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$? In several textbooks on knot theory (e.g. Lickorish's, Rolfsen's) knots are considered in $\mathbb{R}^3$ or $S^3$. The reason for working in $S^3$ is sometimes given at the beginning of a text as that $S^3=\mathbb{R}^3\cup \{\infty\}$ is compact, so working there is "more convenient". What are some specific benefits or conveniences for working in $S^3$ instead of $\mathbb{R}^3$? I could think of one example: In knot theory we work in piecewise-linear category, so working in $S^3$ allows for considering finite triangulations of knot complements. Another example is in studying achirality, where Smith theory for fixed points of periodic maps on the sphere is used. Furthermore, some results in these textbooks are stated for knots in either $\mathbb{R}^3$ or $S^3$, but some other results are stated only for knots in $S^3$. Are there results that are correct for knots in $S^3$ but not for knots in $\mathbb{R}^3$? - I suspect one of the main motivations is pedagogical rather than theoretical. It's less of a stretch of the imagination to visualize knots in $\mathbb R^3$. Visualization in $S^3$, especially when it comes to symmetry, takes more time to digest. – Ryan Budney Apr 27 '11 at 17:28 Another question might be, why $S^3$ or $\mathbb{R}^3$? Why not some other 3-manifold? – Kevin H. Lin May 1 '11 at 6:53 @Kevin: $S^3$ is universal -- given an embedding in $S^3$, and given a manifold $M$, the connect sum of the embedding in $S^3$ with $M$ is an embedding in $M$. – Ryan Budney May 7 '11 at 18:56 @Kevin: For example, some authors have studied knots in the real projective space $\mathbb{R}P^3$: scholar.google.com/scholar?q=drobotukhina – hsp May 8 '11 at 12:01 I think it is much more convenient to work in $S^3$ when you want to compute the fundamental group of the complement of a toric knot , than in $R^3.$ In $S^3$ you use the torus containing the knot to divide $S^3$ into two $S^1 \times D^2$, and then apply Van Kampen theorem. In $R^3$ the corresponding divison of the complement of the knot is less natural, it is harder to describe and visualize. And ofcourse the fundamental group is the same if you take the knot-complement in $R^3$ or in $S^3.$ - You asked about the advantages of links in $S^3$ over links in $\mathbb R^3$ (or $B^3$, which I prefer). Here's an advantage of $B^3$ over $S^3$: Khovanov homology is a (functorial) invariant of links in $B^3$, but so far as I know there is no known proof that it is an invariant of links in $S^3$. More specifically, the standard literature on Khovanov homology constructs a functor from this category • Objects: links in $B^3$ • Morphisms: Isotopy classes of isotopies in $B^3\times I$ (or more generally isotopy classes of cobordisms in $B^3\times I\;$). to the category of bigraded chain complexes and homotopy classes of chain maps. The proof involves defining a chain map for each Reidemeister move and elementary cobordism, and then checking that these generators satisfy various "movie move" relations. If you want to prove an analogous theorem for links in $S^3$, then there is an additional "global movie move" to check. It corresponds to the case where a 1-parameter family of isotopies in $S^3\times I$ (i.e. a second order isotopy) transversely intersects {south pole of $S^3$} $\times I$. I don't think anyone knows how to prove invariance under this global movie move. Perhaps it's not even true. (Scott Morrison and I announced a proof a couple of years ago, but we later discovered a sign issue. We're currently writing up a $\mathbb Z/2$ version with Chris Douglas. If anyone knows how to prove/disprove the relation with $\mathbb Z$ coefficients, please speak up!) Morally, the reason for this difficulty is that there is no nice projection of links in $S^3$ to link diagrams in $S^2$. The obvious attempt at a map from $S^3$ to $S^2$ is not well-defined at the north and south pole of $S^3$, and this ambiguity actually matters for categorical/functorial link invariants. In summary, if you are studying categorical link invariants (like Khovanov homology) in terms of planar projections, you are dealing in links in $B^3$, not $S^3$. - A very down-to-earth but nice example (suggested by Kauffman himself during a seminar) of why working in $S^3$ can be useful. Let $T_{a,b}$, with $a,b$ coprime integers, be the the torus knot with $a$ horizontal windings and $b$ vertical windings. $T_{a,b}$ is equivalent to $T_{b,a}$ and this can be proved by constructing a homeomorphism $S^{3} \to S^{3}$ which sends one knot into the other. I'll leave the explicit construction for your entertainment. - There is also a homeomorphism of $R^3$ sending $T_{a,b}$ to $T_{b,a}$. – Sam Nead May 20 '11 at 21:46 I would definitely like to know it. It is not obvious to me... – user14548 May 26 '11 at 8:14 @sebastiano: Since all points of $S^3$ are equivalent, one can construct the desired homeomorphism $R^3 \to R^3$ by simply restricting the homeomorphism $S^3 \to S^3$ to the complement of a point. That said, your example can be salvaged as follows. There does not exist a homeomorphism $R^3 \to R^3$ which maps the torus containing $T_{a,b}$ to the torus containing $T_{b,a}$, and interchanges the knots. (Proof: cutting $R^3$ along the torus produces a compact and a non-compact piece.) – Dave Futer May 26 '11 at 19:58 In the same spirit as Bruno's answer (which focuses on hyperbolic geometry), let me mention a purely topological result. By the Lickorish-Wallace theorem, every closed, orientable 3-manifold can be obtained using Dehn surgery on a link in $S^3$. This result is central in the study of 3-manifolds, and simply doesn't hold true if one only considers knots in $R^3$. - One may freely pass from $S^3$ to $\mathbb R^3$ just by adding/removing a point; however $S^3$ is nicer when using 3-dimensional topology techniques, in particular geometrization. For instance, the complement of a "generic" (in some sense) knot in $S^3$ should admit a complete hyperbolic metric of finite volume, in which case it is called a hyperbolic knot. Such a metric is unique by (a version of) Mostow rigidity theorem and is hence a rigid nice geometric object assigned to the knot. The topology of the knot complement determines the knot by Gordon-Luecke theorem, hence such a rigid object is actually all you need to determine the knot. The Epstein-Penner decomposition is a canonical combinatorial decomposition of the complement of a hyperbolic knot (into ideal polyhedra) which may be used to determine algorithmically whether two hyperbolic knots are isotopic or not. To use all these very powerful geometric theorems and techniques you need to look at the knot complement in $S^3$, not in $\mathbb R^3$. A knot complement in $\mathbb R^3$ cannot have any complete hyperbolic metric of finite volume. - On the converse, given a knot in $\mathbb R^3$ if you apply geometrization to the complement the first step is to split off a ball summand. This gives you the corresponding knot complement in $S^3$ connect sum a ball, and then you're in the case you describe above. – Ryan Budney Apr 27 '11 at 19:54 Recall that the space of knots $\mathcal K(M)$ in a 3-manifold $M$ is the space of embeddings $S^1 \hookrightarrow M$ (one must think a little to topologize this space usefully). Usual know theory is concerned with studying $\pi_0(\mathcal K(M))$ — its set of connected components — because already that's interesting. But from a topologist's point of view, one should be also interested in the higher homotopy groups. Then, although $\pi_0(\mathcal K(\mathbb R^3)) = \pi_0 (\mathcal K(S^3))$, because any classes in $\pi_0$ and $\pi_1$ can be arranged to avoid the point $S^3 \smallsetminus \mathbb R^3$, it's certainly not true that the higher homotopy groups are the same. (You already know this: that $\pi_3(S^3) > 0$ means that $\pi_2(\mathcal K(S^3)) > \pi_2(\mathcal K(\mathbb R^3))$. It's not just the topologists who are interested in these higher homotopies <edit>, which are related to </edit> higher bordisms. Physicists, whose quantum field theory connects with knot theory in a number of ways, are also interested, in programs that go by names like "categorification" and "string theory". For example, not long ago Witten released his paper proposing a physical interpretation for Khovanov homology. Actually, that paper interestingly illustrates my point above. Namely, Witten must carefully switch between $S^3$ and $\mathbb R^3$ a few times. $S^3$, being compact, is much better suited for defining certain integrals. On the other hand, $\mathbb R^3$ has its translation-invariant framing, and since many quantum field theoretical invariants are framing dependent, it matters whether you use the translation-invariant framing (which does not extend to $S^3$) or some other one. - Regarding your 1st paragraph, the homotopy and homology groups of both spaces are quite closely related -- the homotopy-types of the two spaces are related via a pair of fibre bundles which are readily understood. So in that sense there's not much difference. – Ryan Budney Apr 27 '11 at 17:19 I'm confused about the phrase "higher homotopies = higher bordisms". There is certainly a relationship between the two concepts, but I'm confused about why you wrote an "=" sign. – Daniel Moskovich Apr 27 '11 at 17:30 To be more precise, let $Emb(S^1,S^3)$ be the space of smooth embeddings of the circle in $S^3$, $K_{3,1}$ the space of long smooth embeddings of $\mathbb R$ in $\mathbb R^3$ and $Emb(S^1,\mathbb R^3)$ the space of embeddings of the circle in euclidean space. Then there are homotopy-equivalence $Emb(S^1,S^3) \simeq SO_4 \times_{SO_2} K_{3,1}$ and $Emb(S^1,\mathbb R^3) \simeq SO_4 \times_{SO_2} (C \rtimes K_{3,1})$. $C \rtimes K_{3,1}$ is the space of pairs consisting of a point in $K_{3,1}$ together with a point in its complement. – Ryan Budney Apr 27 '11 at 17:43 That might be a bit unclear $C \rtimes K_{3,1}$ is the set of pairs $(p,f)$ where $f \in K_{3,1}$ and $p \in \mathbb R^3 \setminus img(f)$. So it's the tautological bundle over $K_{3,1}$ with fibre the corresponding knot complement. – Ryan Budney Apr 27 '11 at 17:51 @Daniel: sorry I ment something much less precise than I guess I wrote. I only meant that these are related. @Ryan: Cool! – Theo Johnson-Freyd Apr 27 '11 at 18:05 Here is a way to say it directly in the language of three-manifolds. By Alexander's theorem, both $R^3$ and $S^3$ are irreducible. However, as Mark indicates, a knot complement in $R^3$ is reducible. In fact, a knot complement in $R^3$ decomposes as the connect sum of a copy of $R^3$ and... the knot complement in $S^3, which is irreducible. - This is a nice question! Knot theory is in fact knot-complement theory, and a knot complement in S3 is a compact 3-manifold, while a knot complement in R3 is an open 3-manifold. Compact (or closed) 3-manifolds are technically easier to work with than open 3-manifolds, because various chain complexes which you care about are finitely generated, and various analytic tools become available. Ultimately, I guess that the fundamental reason that people consider knots in S3 rather than in R3 is that bordism theory is tractable for compact manifolds, but not for open manifolds. In particular, bordism groups and Wall groups for compact manifolds are of finite rank (this matters for example for Blanchfield pairings). Analytically, characteristic numbers, topological and analytic index, analytic torsion and the eta invariant are only defined in the compact case, because for open manifolds the relevant integrals can diverge, elliptic operators need not be Fredholm, the spectrum of self-adjoint operators need not be discrete, etc. Classical knot theory usually takes place in a more low-tech world than the previous paragraph might have suggested; nevertheless, all of the above technology is lurking in the background to algebraic topological invariants associated to covering spaces (the Alexander polynomial), in particular to questions related to knot concordance. Because knots in R3 are fairly special objects, maybe classical knot theory can somehow be made to work despite the limitations listed above. But, in the absence of more general tools, I would have no idea how to do it. I think it would be interesting to have analogues to the classical algebraic invariants in knot theory for knots in R3! - The higher homotopy groups of the complement of a knot in$S^3$are all trivial. This is not true for a knot in$\mathbb{R}^3$(consider a$2\$-sphere enclosing the image of the knot). -	2014-10-22 12:35:49	{"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8161309361457825, "perplexity": 280.825491438848}, "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-2014-42/segments/1413507446943.4/warc/CC-MAIN-20141017005726-00370-ip-10-16-133-185.ec2.internal.warc.gz"}
http://electricalacademia.com/category/basic-electrical/page/3/	Home / Basic Electrical (page 3) # Basic Electrical ## Voltage or Electric Potential Difference: Definition, Unit, Symbol, Examples As we know that no current flows in a conductor unless a device (such as a battery) imparts energy to the free electrons. We say that the battery is the source of an electron-moving force or electromotive force, usually abbreviated to EMF. Electromotive force is a property that distinguishes an … ## What is Electric Current? Definition, Unit, Formula & Examples An electric current is a flow of charged particles. In most circuits, these charged particles are free electrons. Although there is a free electron for each copper ion, Figure 1 shows only one of these free electrons so that we may trace its motion through the lattice. In Figure 1(a), … ## Basic Electrical Circuit: Theory, Components, Working, Diagram A basic electrical circuit consists of three main components, a source of voltage, a load, and conductors. In Figure 1, a basic circuit is illustrated. This circuit consists of a battery as the source of electrical energy, a lamp as the electrical load, and two wires as the conductors connecting … ## How Does Static Electricity Work \| Static Electricity Applications Definition: The word static means at rest. Electricity can be at rest. The generation of static electricity can be demonstrated in many ways. When stroking the fur of a cat, you will notice that its fur is attracted to your hand as you bring your hand back over the cat. … ## Transient Response of Capacitor \| RC Circuit: Time Constant & Transient Response Definition: The response of current and voltage in a circuit immediately after a change in applied voltage is called the transient response. Refer to Figure 1. A capacitor and a resistor are connected in series across a voltage source. A circuit that contains resistance and capacitance is called an RC … ## Electrical Formulas Excel Sheet This particular Electrical Formulas Excel Sheet will assist you to get a quick answer to the quantities related to the following topics: 1 – Basic Concepts of Electricity 2 – Electrical Quantities and Components 3 – OHM’s Law 4 – Series Circuits 5 – Parallel Circuits 6 – Series-Parallel Circuits … ## Inductors in AC and DC Circuits The main action of an inductor is to resist a change in current. However, since the current in a DC circuit is constant, there is no induced voltage developed instantaneously across the inductor. The inductor does resist the initial inrush of current based on the time constant of the circuit. … ## Capacitors in AC and DC Circuits Capacitors in DC Circuits When a capacitor is placed in a DC circuit that is closed (current is flowing) it begins to charge. Charging is when the voltage across the plates builds up quickly to equal the voltage source. Once a capacitor reaches its fully charged state, the current … ## Power in an AC Circuit In an alternating-current circuit, power is dissipated in a resistor, but not in a pure inductor or a capacitor. Because the current in an RL circuit lags the supply voltage by an angle ϕ, the amount of useful power supplied to the circuit is proportional to Cosϕ. Similarly, in an … ## Conductors and Insulators Definition and Examples Conductors carry electric current. Insulators protect conductors and protect people from conductors. Whether a material is a conductor or an insulator depends on its atoms and on the relationship of each atom to its surrounding atoms. Insulators may break down if subjected to excessive voltages. Similarly, conductors may be destroyed … ## Passive Components in AC Circuits with Equations There are three key passive elements used in many electrical and electronic circuits such as: Resistor, Inductor, and Capacitor. All these three elements limit the current flow but in a dissimilar manner. Since passive elements exhaust energy, they cannot amplify the electrical signals power employed to them and will always … ## Series Parallel Circuit \| Series Parallel Circuit Examples Series-Parallel Circuit Definition Not all circuits are simple series or parallel arrangements. Many are combinations of parallel resistors connected in series with other resistors or combined with other parallel groups. These can be described as a series-parallel circuit. The simplest approach to analyzing a series-parallel circuit is to resolve each … ## Parallel Circuit Definition \| Parallel Circuit Examples Parallel Circuit Definition Resistors are said to be connected in parallel when the same voltage appears across every component. With different resistance values, different currents flow through each resistor. The total current taken from the supply is the sum of all the individual resistor currents. The equivalent resistance of a … Consider the relationship between voltage and current for a resistor (Ohm’s Law). Suppose that c current I1 (the excitation or input) is applied to a resistor, R. then the resulting voltage V1 (the response or output) is ${{V}_{1}}={{I}_{1}}R$ Similarly, if I2 is applied to R, then V2=I2R results. But if …	2019-03-21 20:13:04	{"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": 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": 0.5683079957962036, "perplexity": 1013.4723073486181}, "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-2019-13/segments/1552912202572.29/warc/CC-MAIN-20190321193403-20190321215403-00439.warc.gz"}
https://mathsgee.com/8305/what-are-binary-trees	MathsGee is Zero-Rated (You do not need data to access) on: Telkom \|Dimension Data \| Rain \| MWEB 0 like 0 dislike 33 views What are Binary Trees? \| 33 views 0 like 0 dislike A binary tree is either 1. an empty tree $\{\epsilon\}$ 2. a tree with a fixed root vertex such that each vertex has a left branch and a right branch (either of which may be empty) NB: Note that a binary tree with a right branch is not equal to a binary tree with a left branch, even if they have the same number of vertices. by Diamond (75,934 points) 0 like 0 dislike	2021-06-24 05:24:44	{"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": 0.5289905071258545, "perplexity": 1933.3631058256656}, "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-2021-25/segments/1623488551052.94/warc/CC-MAIN-20210624045834-20210624075834-00172.warc.gz"}
https://www.aimsciences.org/article/doi/10.1186/s41546-019-0040-8	# American Institute of Mathematical Sciences January 2019, 4: 6 doi: 10.1186/s41546-019-0040-8 ## Correction to: “Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting” 1. University of Jyvaskyla, Department of Mathematics and Statistics, P. O. Box 35, FI-40014 University of Jyvaskyla, Jyvaskyla, Finland; 2. Department of Applied Mathematics and Information Technology, Montanuniversitaet Leoben, Peter Tunner-Straße 25/I, A-8700 Leoben, Austria Received July 04, 2019 Revised July 04, 2019 Citation: Christel Geiss, Alexander Steinicke. Correction to: “Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting”. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 6-. doi: 10.1186/s41546-019-0040-8 ##### References: [1] Dellacherie, C., Meyer, P.-A.:Probabilities and potential. B. North-Holland Publishing Co., Amsterdam(1982) Google Scholar [2] Geiss, C., Steinicke, A.:Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting. Probab. Uncertain. Quant. Risk. 3(9) (2018) Google Scholar [3] He, S., Wang, J., Yan, J.:Semimartingale Theory and Stochastic Calculus. Science Press, CRC Press, New York (1992) Google Scholar show all references ##### References: [1] Dellacherie, C., Meyer, P.-A.:Probabilities and potential. B. North-Holland Publishing Co., Amsterdam(1982) Google Scholar [2] Geiss, C., Steinicke, A.:Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting. Probab. Uncertain. Quant. Risk. 3(9) (2018) Google Scholar [3] He, S., Wang, J., Yan, J.:Semimartingale Theory and Stochastic Calculus. Science Press, CRC Press, New York (1992) Google Scholar [1] Christel Geiss, Alexander Steinicke. Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 9-. doi: 10.1186/s41546-018-0034-y [2] Justin Cyr, Phuong Nguyen, Sisi Tang, Roger Temam. Review of local and global existence results for stochastic pdes with Lévy noise. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5639-5710. doi: 10.3934/dcds.2020241 [3] Shangzhi Li, Shangjiang Guo. Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 5101-5134. doi: 10.3934/dcdsb.2020335 [4] Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2543-2557. doi: 10.3934/dcds.2020374 [5] Min Niu, Bin Xie. Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises. 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Journal of Industrial & Management Optimization, 2017, 13 (1) : 1-21. doi: 10.3934/jimo.2016001 Impact Factor:	2021-08-02 18:02:25	{"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": 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": 0.5546323657035828, "perplexity": 10300.367477904647}, "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/1627046154356.39/warc/CC-MAIN-20210802172339-20210802202339-00618.warc.gz"}
https://researchprofiles.herts.ac.uk/en/publications/galactic-population-synthesis-of-radioactive-nucleosynthesis-ejec	# Galactic Population Synthesis of Radioactive Nucleosynthesis Ejecta Thomas Siegert, Moritz M. M. Pleintinger, Roland Diehl, Martin G. H. Krause, Jochen Greiner, Christoph Weinberger Research output: Contribution to journalArticlepeer-review ## Abstract Diffuse gamma-ray line emission traces freshly produced radioisotopes in the interstellar gas, providing a unique perspective on the entire Galactic cycle of matter from nucleosynthesis in massive stars to their ejection and mixing in the interstellar medium. We aim at constructing a model of nucleosynthesis ejecta on galactic scale which is specifically tailored to complement the physically most important and empirically accessible features of gamma-ray measurements in the MeV range, in particular for decay gamma-rays such as $^{26}$Al, $^{60}$Fe or $^{44}$Ti. Based on properties of massive star groups, we developed a Population Synthesis Code which can instantiate galaxy models quickly and based on many different parameter configurations, such as the star formation rate, density profiles, or stellar evolution models. As a result, we obtain model maps of nucleosynthesis ejecta in the Galaxy which incorporate the population synthesis calculations of individual massive star groups. Based on a variety of stellar evolution models, supernova explodabilities, and density distributions, we find that the measured $^{26}$Al distribution from INTEGRAL/SPI can be explained by a Galaxy-wide population synthesis model with a star formation rate of $4$-$8\,\mathrm{M_{\odot}\,yr^{-1}}$ and a spiral-arm dominated density profile with a scale height of at least 700 pc. Our model requires that most massive stars indeed undergo a supernova explosion. This corresponds to a supernova rate in the Milky Way of $1.8$-$2.8$ per century, with quasi-persistent $^{26}$Al and $^{60}$Fe masses of $1.2$-$2.4\,\mathrm{M_{\odot}}$ and $1$-$6\,\mathrm{M_{\odot}}$, respectively. Comparing the simulated morphologies to SPI data suggests that a frequent merging of superbubbles may take place in the Galaxy, and that an unknown but strong foreground emission at 1.8 MeV could be present. Original language English Astronomy & Astrophysics Accepted/In press - 24 Jan 2023 • astro-ph.GA • astro-ph.HE ## Fingerprint Dive into the research topics of 'Galactic Population Synthesis of Radioactive Nucleosynthesis Ejecta'. Together they form a unique fingerprint.	2023-03-20 10:06:59	{"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": 0.620644748210907, "perplexity": 3173.0001017955387}, "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/1679296943471.24/warc/CC-MAIN-20230320083513-20230320113513-00144.warc.gz"}
https://proxieslive.com/tag/first/	## Demon of the First Circle limitations Demon of the First Circle or any of the Demon spells seems ripe for exploitation. I’m wondering how other storytellers have dealt with this issue. I’m worried because of the ease of summoning, and the length the demon will serve you. The way I read the rules there is nothing preventing a sorcerer from summoning one demon every night. The risk of failing Int + Occult vs the demons resolve is also very small. Spend willpower and a stunt, and you’re almost guaranteed to succeed. That is before taking into consideration that most sorcerers will have high int and occult, and possibly an excellency to boost the roll if they feel like it. Once summoned the demon is your slave for a year and a day. Given the system, it feels like most sorcerers should be surrounded by a small army of demons to do their bidding. How have other storytellers dealt with this issue? Is there anything from e3 or the other editions that can be used as a guide to reduce the exploitability of the summon spells? ## How to get your first 1d6 sneak attack without class levels? There are numerous ways to get additional d6’s worth of sneak attack, but almost all of them require that you have 1d6 (or more) sneak attack before you can take them. How does one get +1d6 sneak attack damage from non-level sources when one has no sneak attack? There is at least one way—Martial Stance (assassin’s stance) from Tome of Battle, which actually gives +2d6—but that isn’t available until 12th or (if the DM is generous) maybe 9th, and requires another feat besides. • Any Wizards of the Coast published 3.5e material, as well as any 3.5e content from Dragon and Dungeon magazines, is legitimate. • While stuff earlier than Martial Stance’s 12th is preferred, any content available pre-epic is acceptable. • Epic content is not allowed, even if you somehow cheese into it prior to epic levels. • Any kind of polymorph or shape changing is not allowed. • The sneak attack must be available on a continuous, permanent basis. A magic item is acceptable, as is a 24-hour daily effect (e.g. spell), but not anything less than that. • Persistent Spell, and any other means of turning a less-than-24-hour-duration spell into a 24-hours duration spell, are not allowed. • LA counts as “levels” for this purpose, as do RHD. No ECL can be consumed by a valid answer. • Bloodlines are not allowed. • Any form of “undoing” levels (e.g. level loss from energy drain or resurrection, curing lycanthropy, various rituals) is not allowed. • No assumptions about which classes are taken may be made, though you may leave achieving the necessary BAB/saves/skill ranks as an exercise for the reader so long as they’re non-epic values. Really, what I want is feats and/or magic items that just say “you gain +1d6 [or more] sneak attack,” not some kind of shenanigan. Assassin’s stance proves it exists, and I’m fairly confident there’s some more out there (some named magic weapons, IIRC?). If you’re inclined to suggest a shenanigan, you may be well served running it by me in a comment before you spend your time—I have made every effort to provide a thorough list of the things I’m against, but if an answer manages to prove me wrong, I’m still not going to be happy with the shenanigan. ## What was the first published map for an adventure? What is the first officially published dungeon map from dnd? I believe there was no dungeon map in the first published book, but I’d like to know what the first map was that was published. ## How do I deal with a player who gets offended when other players get to loot first? I’m directing a 5e campaign and I have a party of three: A Tiefling Warlock (eloquent, rational type), a tiefling monk (rash, impulsive and danger seeking) and a half-elf rogue (his personality is not that well defined). We’re all adults: I’m 26, the warlock and the rogue are 25, and the monk is 29. I had this devil trapped inside a magic circle, and a puzzle involving a talking severed head and riddles. As a reward for freeing him, the devil drops three rings (one blue steel, one red iron and one orange copper) to the floor and says “Pick one”, before disappearing. The three stare at the rings, and the monk (who solved the riddle) said: “I will take the blue one first, you can get the others if they are still there.” The warlock was silent, staring at the rings and thinking. The rogue asked the warlock to identify the rings before touching them (not possible). When the monk heard that the warlock could not identify the rings he said “I reach for the blue one and I take it!” The other players did not react and I rushed to say, “The other two rings dissapeared!” And then it all broke down. The player who plays the rogue wants to stop the monk from touching the rings. I make them do a dex throw and the rogue won. But I had already said the rings disappeared, so in order to keep the narrative going and not allow them to meta-game with the knowledge, I ruled that the rogue was able to stop the monk from taking the ring, but he touched it with a finger and the other two disappeared. The rogue starts complaining, claiming its not fair, and gets mad (the player). He takes it personally and starts to put away his things, really angry. I told him I already said the rings disappeared, and I’m not getting them back, and that I already gave him the chance to fight for the remaining ring, even though he has been getting all the magic items lately, and the monk has nothing yet. He says he wants me to pause before narrating the consequences of the players actions and ask everybody if they want to do something, so that everyone can react to everything. He starts making threats, saying he cant play with us if we can’t play like he wants. And when I explain to him that he has to roleplay that anger in-character, against the monk, he says “from now on my character is going to do the opposite of what the monk does and wants. And if he does something like this again, I’m killing him, and it’s not my fault because I warned him”. It took me a while to get him to calm down a little, but the session ended a bit on the cold side. For the record, the other two players side with me, and agree that events should flow naturally, even when it’s not totally fair to the players. What do you think? I don’t like this behavior, and being called unfair touches a sensitive string, because its been multiple times now that I’ve done things to keep this player happy, including talking to other players individually and telling them to tone down their arguments with him and giving him a good amount of the loot and letting him get away with some power-gamey stunts. ## ReplaceAll doesn’t replace all, factor out negative sign first I have this matrix:$$\left( \begin{array}{ccccc} \{\{0\},\{0,0,0\}\} & \{\{1\},\{0,0,0\}\} & \{\{0\},\{0,0,0\}\} & \{\{0\},\{0,0,0\}\} & \{\{0\},\{0,0,0\}\} \ \{\{-1\},\{0,0,0\}\} & \{\{0\},\{0,0,0\}\} & \{\{0\},\{0,0,0\}\} & \{\{0\},\{0,0,0\}\} & \{\{0\},\{0,0,0\}\} \ \{\{0\},\{0,0,0\}\} & \{\{0\},\{0,0,0\}\} & \{\{0\},\{0,0,0\}\} & \{\{0\},\{0,z,-y\}\} & \{\{0\},\{-z,0,x\}\} \ \{\{0\},\{0,0,0\}\} & \{\{0\},\{0,0,0\}\} & \{\{0\},\{0,-z,y\}\} & \{\{0\},\{0,0,0\}\} & \{\{0\},\{y,-x,0\}\} \ \{\{0\},\{0,0,0\}\} & \{\{0\},\{0,0,0\}\} & \{\{0\},\{z,0,-x\}\} & \{\{0\},\{-y,x,0\}\} & \{\{0\},\{0,0,0\}\} \ \end{array} \right)$$ When I run ReplaceAll (/.) on it using this$$\left\{\{\{1\},\{0,0,0\}\}\to X_1,\left\{\{t\},\left\{\frac{2 x}{3},\frac{2 y}{3},\frac{2 z}{3}\right\}\right\}\to X_2,\{\{0\},\{y,-x,0\}\}\to X_3,\{\{0\},\{z,0,-x\}\}\to X_4,\{\{0\},\{0,z,-y\}\}\to X_5,\{\{0\},\{0,0,0\}\}\to 0\right\}$$ it doesn’t replace everything: $$\left( \begin{array}{ccccc} 0 & X_1 & 0 & 0 & 0 \ \{\{-1\},\{0,0,0\}\} & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & X_5 & \{\{0\},\{-z,0,x\}\} \ 0 & 0 & \{\{0\},\{0,-z,y\}\} & 0 & X_3 \ 0 & 0 & X_4 & \{\{0\},\{-y,x,0\}\} & 0 \ \end{array} \right)$$ I expect:$$\left( \begin{array}{ccccc} 0 & X_1 & 0 & 0 & 0 \ -X_1 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & X_5 & -X_4 \ 0 & 0 & -X_5 & 0 & X_3 \ 0 & 0 & X_4 & -X_3 & 0 \ \end{array} \right)$$ Is there an automated way of doing these replacements? ## Sidebar widget not partial refreshing for first widget on add and remove action When you add the first widget in the sidebar it’s a refreshing customizer completely and not partial refreshing. • Go to Customizer • Go to the Widgets panel. • Remove all the widgets one by one. • At the last widget when you remove it, it refreshes the customizer. • When adding the first widget it again refreshes the customizer. • First onward, it partially refreshing the customizer as expected. Please find the attached video for further reference: https://a.cl.ly/WnulD9r6 Is it the right behavior? ## Add a word to the first // wp_get_attachment_url // as the subdomain name We use this code to display the link … $mp3Link = wp_get_attachment_url($ mp3_file_id); <a class="w3-black" href="'.\$ mp3Link.'" rel="nofollow" >mp3</a> It looks like this now : I want it to be like this : I mean, I just want to do this by editing the above code ## What happens with lethargy when Haste is cast again on a creature before first cast runs out? Haste states When the spell ends, the target can’t move or take actions until after its next turn, as a wave of lethargy sweeps over it. However, let’s say Character A cast Haste on Character B. In the 10th round (final round of haste if concentration not broken), Character B casts Haste on himself. What happens during the next round? 1. Do they lose a round of the “new” haste while they are lethargic and lose movement and actions? 2. Does the new haste override the lethargy until it ends? 3. Is it some combination? Retain AC/Dex modifications but lose actions/movement? 4. None of the above? ## Does it matter which weapon I attack with first when two-weapon fighting? The PHB on p. 195 (or the corresponding section of the basic rules) says: When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative. The rules do not make a distinction between “main-hand” and “off-hand” weapons. My character dual wields a shortsword with her right hand and a dagger with her left hand. Could my character attack with her left hand dagger first, doing 1d4 + STR/DEX mod damage, then attack with her right hand shortsword, doing 1d6 damage? Does it matter which weapon I attack with first when two-weapon fighting? ## A first timers question on hosting I created a website on wordpress and hosted it on google cloud. I followed a tutorial and got everything working on that part really nicely apart from a contact form. Heres where all started dawn on me that something is not right. I had my Gsuite account working with my domain host default nameservers but when I changed them to Google clouds name server my Email stopped working and when I changed the servers back I got my email working again. I think that here is many very knowledgeable people that knows what this noob did wrong. Hopefully somebody has time to point me in the right direction and if Im not in a completely wrong place asking this.	2021-01-27 03:18:22	{"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": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.29187828302383423, "perplexity": 2827.291288021123}, "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/1610704820894.84/warc/CC-MAIN-20210127024104-20210127054104-00392.warc.gz"}
https://socratic.org/questions/why-are-d-and-f-block-elements-called-transition-elements	# Why are d and f block elements called transition elements? Nov 23, 2017 They lie between the $s$-block (metals with low electronegativity) and the $p$-block (non-metals with high electronegativity) - representing the transition between the two. They typically are considered to be the titanium family through the copper family. #### Explanation: Transition metals typically have a medium electronegativity, and as such a unique set of behaviours that vary from the $s$-block metals. For example, gold and some of the other "noble metals" are highly resistant to corrosion and tend to remain in their elemental state rather than becoming oxidized, but the $s$-block metals often favor the oxidized state (like sodium for example). [This is due to the high electronegativity values (gold is $2.4$), whereas $s$-block metals would have values ranging from just below, to just above $1.0$).] Not all transition metals exhibit this property, but as electronegativity increases during the transition from metal to non-metal, the properties & behaviours begin to vary. From an electronic point of view, the transition metal have valence electrons that include their $n s$ electrons, while additional electrons are available in the partially-occupied $\left(n - 1\right) d$ orbitals. From left to right, the $d$ electrons become less and less available for bonding.	2019-10-14 20:45:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 10, "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": 0.7238897681236267, "perplexity": 2598.7803171413498}, "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-2019-43/segments/1570986655310.17/warc/CC-MAIN-20191014200522-20191014224022-00558.warc.gz"}
https://cs.stackexchange.com/questions/116639/addition-multiplication-and-apostrophe-used-to-represent-boolean-algebra-expre	# Addition, multiplication, and apostrophe used to represent boolean algebra expressions? I'm looking at a worksheet that expresses boolean logic expressions using multiplication, addition, and apostrophes; something I've never seen before. I can make a guess that the apostrophe is equivalent to ¬ (except it's suffixed instead of prefixed). But I'm not sure what the addition and multiplication of the variables/propositional atoms would mean. Furthermore, I don't know how a boolean logic formula can "output" something other than just a truth value... I can't seem to piece together with certainty the meaning of this representation. Could anyone take a look at the below example and maybe make a guess as to a translation of this representation to the more traditional ^, v, and ¬ symbols? This arose in the context of digital logic in terms of logic gates and such on a CPU, if that makes a difference. An alleged truth table of the above two expressions (the first row is filled in as an example): • @ShyPerson Thank you very much! I guessed that but didn't think it worked, but looking back now it would make sense! Do you know why they have a specific term written in the box? Is that just the term in the expression that turns out to be true, do you think? – James Ronald Nov 4 '19 at 12:52 • @JamesRonald: Yes, it looks like that's the term that makes the whole expression true. – ShyPerson Nov 4 '19 at 19:16 • @YuvalFilmus: Thanks for the encouragement to create an answer. I guess I'm still unsure about when to create a comment versus an answer. Is there some documentation somewhere that would clarify this? Many thanks – ShyPerson Nov 4 '19 at 19:17 • @ShyPerson Generally speaking, it’s better for questions to have an answer. – Yuval Filmus Nov 4 '19 at 19:18 • Do not post pictures of formulas, please. You can use LaTeX to type them, it's a lot more polite and generally friendlier. – Andrej Bauer Nov 4 '19 at 19:25 Usually $$+$$ means $$\lor$$, "multiplication" means $$\land$$, and $$'$$ means $$\lnot$$.	2020-08-05 17:07:38	{"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": 5, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6439087986946106, "perplexity": 651.7421617211089}, "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/1596439735963.64/warc/CC-MAIN-20200805153603-20200805183603-00109.warc.gz"}
http://math.stackexchange.com/questions/97056/uniform-continuity-and-differentiation?answertab=active	# Uniform Continuity and Differentiation Is the following true or false?: Let $f\colon [0,1) \to \mathbb{R}$ be a function differentiable in $[0,1)$ (where the derivative at zero means "right derivative") such that both $f$ and $f'$ are uniformly continuous in $(0,1)$. Then $f'$ is continuous. Note that the mistery lies at $x=0$. So the question is: can we say with these hypotheses that $f'(0)=\lim_{x\to 0^{+}}f'(x)$ (which exists thanks to the uniform continuity of $f'_{\mid (0,1)}$). Note also that the uniform continuity of $f'_{\mid (0,1)}$ makes redundant the analogous requirement for $f$ (which will even more become a Lipschitz function). - It's true. Since you know how to prove existence of the limit of $f'$ I'll focus on the fact that $f'(0)$ exists and is equal to the limit - it's not a big thing in fact. Note that Lagrange mean value theorem (following from Roll theorem) applies here. So we have $\frac{f(x)-f(0)}{x} = f'(\theta_x \cdot x)$ for some $\theta_x \in (0,1)$. Consequently: $$f'(0)=\lim_{x\to 0^+}\frac{f(x)-f(0)}{x} = \lim_{x\to 0^+} f'(\theta_x \cdot x)=\lim_{y\to 0+}f'(y).$$ - Note that the assumption of the existence of $f'(0)$ is not needed in the original post. Nice! – David Mitra Jan 6 '12 at 23:06 Right: the MVT needs differentiability in the open set only, and the first equality is the definition of the derivative at zero (which is shown to exist via the second and third equalities). – David Jan 6 '12 at 23:13 For $x>0$, consider $${f(x)-f(0)\over x}={1\over x}\int^x_0 f^\prime(y)\, dy.$$ As $x\downarrow 0$, the left hand side converges to $f^\prime(0)$, while the right hand side converges to $\lim_{x\to 0^+} f^\prime(x).$ This limit exists because $f^\prime$ has a continuous extension to $[0,1)$, by uniform continuity on $(0,1)$. - As you stated, thanks to the uniform continuity of $f'$, $L=\lim\limits_{x\rightarrow0^+} f'(x)$ exists (uniformly continuous functions map Cauchy sequences to Cauchy sequences). If $L\ne f'(0)$, then a contradiction to Darboux's Theorem can be obtained. Darboux's Theorem: If $f$ is differentiable on $I=[a,b]$ and if $k$ is a number between $f'(a)$ and $f'(b)$, then there is at least one point $c\in(a,b)$ such that $f'(c)=k$. -	2016-05-05 05:00:50	{"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": 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": 0.9918534159660339, "perplexity": 126.50901820830069}, "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-2016-18/segments/1461860125897.19/warc/CC-MAIN-20160428161525-00034-ip-10-239-7-51.ec2.internal.warc.gz"}
https://physics.stackexchange.com/questions/426231/how-does-a-wires-magnetic-field-appear-as-an-electric-field-when-the-wire-is-n	# How does a wire's magnetic field appear as an electric field, when the wire is neutral? [duplicate] It is well known that electromagnetic force depends on frame. I was reading a book, it says If a charge is moving parallel to a current carrying wire then a magnetic force will be exerted on charge. But if we start moving with the charge at same velocity then it is in rest for the moving frame but it will experience the force again and since both frames( stationary one and the moving one) has no acceleration w.r.t each other so acceleration on the charge will be same in both frames but reason of this acceleration in moving frame as there is no magnetic field must be an electric field My question is that it is well known that current carrying wire is neutral so how can there be an electric field in moving frame and if it is there then what is the origin of this field? As it was marked as a possible duplicate, I want to clarify that I want an intuitive answer. There is written that a moving current carrying wire will appear as charge. How it is possible when wire is neutral? I am new to electrodynamics so sorry if i ask bad questions. The short answer is length contraction. In the rest frame of the current-carrying wire, it appears neutral, but that is no longer the case if the wire is observed from a reference frame that is moving along the direction of the current, because • the wire contains components of different charges moving at different velocities, • each of those components will experience a different amount of length contraction, because the relative velocities to the new frame are different, • those length contractions will impact the apparent charge density as observed by the new frame of reference, • and therefore the total charge density will be nonzero, as observed in the new frame of reference. That nonzero charge density will then generate an electric field, which will attract or repel the (formerly moving, now stationary) charge. For a detailed exposition of this transformation, the go-to place is Ed Purcell's Electricity and Magnetism; for a condensed take, this video by Veritasium and MinutePhysics is a good introduction. For previous takes on this topic here on this site, see the search results here, and particularly How Special Relativity causes magnetism, current in wire + special relativity = magnetism, Special relativity and electromagnetism, Need clarity about relativity/magnetism explanation, How Special Relativity causes magnetism, Is magnetic field due to an electric current a relativistic effect?, and the many questions in their Linked and Related sidebars to the right. • Thanks for answering.... It has really cleared all my doubts....thanks – Himanshu Tyagi Sep 3 '18 at 7:22 The important thing to realize is that the wire is only neutral in one reference frame. In other frames it is charged. This is easy to see the other way: if you have some charge density $\rho$ in a frame where it is at rest, then in a frame where it is moving there is clearly a current density $J$ also. It turns out that it works the other way too although it isn’t as obvious. $\rho$ and $J$ have the same relationship as $t$ and $x$ in relativity. There is $\rho$ “dilation” and $J$ “contraction”. Importantly, the quantity $\rho^2-J^2$ is invariant (in units where c=1). So as $J$ changes in different frames, $\rho$ must also change to keep it invariant. This leads to the neutral current-carrying wire in one frame being charged in any other frame.	2020-02-25 00:57:30	{"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": 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": 0.6535652875900269, "perplexity": 290.75476371083494}, "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-2020-10/segments/1581875145989.45/warc/CC-MAIN-20200224224431-20200225014431-00545.warc.gz"}
http://gmatclub.com/forum/hand-shakes-102740.html?fl=similar	Find all School-related info fast with the new School-Specific MBA Forum It is currently 01 Sep 2015, 06:37 ### 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 # Events & Promotions ###### Events & Promotions in June Open Detailed Calendar # Hand shakes Author Message TAGS: Senior Manager Joined: 13 Aug 2010 Posts: 314 Followers: 1 Kudos [?]: 13 [0], given: 1 Hand shakes [#permalink] 12 Oct 2010, 20:27 00:00 Difficulty: (N/A) Question Stats: 100% (00:00) correct 0% (00:00) wrong based on 2 sessions Everyone shakes hands with everyone else in a room. Total number of handshakes is 66. Number of persons=? a.14 b.12 c.11 d.15 e.16 [Reveal] Spoiler: OA Retired Moderator Joined: 02 Sep 2010 Posts: 805 Location: London Followers: 87 Kudos [?]: 639 [0], given: 25 Re: Hand shakes [#permalink] 12 Oct 2010, 21:06 prab wrote: Everyone shakes hands with everyone else in a room. Total number of handshakes is 66. Number of persons=? a.14 b.12 c.11 d.15 e.16 In a room of n people, the number of possible handshakes is C(n,2) or n(n-1)/2 So n(n-1)/2 = 66 OR n(n-1)=132 OR n=12 _________________ Senior Manager Joined: 13 Aug 2010 Posts: 314 Followers: 1 Kudos [?]: 13 [0], given: 1 Re: Hand shakes [#permalink] 12 Oct 2010, 21:19 can you please explain why are we using n(n-1)/2, m not able to grab the concept. Retired Moderator Joined: 02 Sep 2010 Posts: 805 Location: London Followers: 87 Kudos [?]: 639 [0], given: 25 Re: Hand shakes [#permalink] 12 Oct 2010, 21:34 Number of handshakes will be the number of ways to choose 2 people out of n people. For every choice of two people, there is a handshake. This number is C(n,2) = $$\frac{n!}{(n-2)!2!}$$ _________________ Re: Hand shakes [#permalink] 12 Oct 2010, 21:34 Similar topics Replies Last post Similar Topics: 6 In Smithtown, the ratio of right-handed people to left-handed people i 6 09 Jun 2015, 02:06 1 If 11 persons meet at a reunion and each person shakes hands exactly o 4 08 Dec 2014, 05:26 2 If 10 persons meet at a reunion and each person shakes hands 2 09 Mar 2011, 14:12 Probability of hand with four aces 2 09 Feb 2011, 04:11 6 The minute hand of a clock 2 24 Oct 2010, 10:57 Display posts from previous: Sort by	2015-09-01 14:37:20	{"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": 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": 0.3012143075466156, "perplexity": 8243.034697478037}, "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/1440645191214.61/warc/CC-MAIN-20150827031311-00165-ip-10-171-96-226.ec2.internal.warc.gz"}
https://www.borealisai.com/research-blogs/explainability-i-local-post-hoc-explanations/	Machine learning systems are increasingly deployed to make decisions that have significant real-world impact. For example, they are used for credit scoring, to produce insurance quotes, and in various healthcare applications. Consequently, it’s vital that these systems are trustworthy. One aspect of trustworthiness is explainability. Ideally, a non-specialist should be able to understand the model itself, or at the very least why the model made a particular decision. Unfortunately, as machine learning models have become more powerful, they have also become larger and more inscrutable (figure 1). At the time of writing, the largest deep learning models have trillions of parameters. Although their performance is remarkable, it’s clearly not possible to understand how they work by just examining the parameters. This trend has led to the field of explainable AI or XAI for short. Figure 1. Explainability vs. performance. There has been a trend for models to get less explainable as they get more powerful. This is partly just because of the increasing number of model parameters (potentially trillions for large models in the top-left corner). The decision tree is a rare example of a non-linear model which is both easy to understand, and provides reasonable performance. It is unknown whether it is possible to build models in the desirable top-right corner of this chart. Adapted from AAAI XAI Tutorial 2020. Explainable AI methods are useful for three different groups of stakeholders. For machine learning scientists, they are useful for debugging the model itself and for developing new insights as to what information can be exploited to improve performance. For business owners, they help manage business risk related to AI systems. They can provide insight into whether the decisions can be trusted and how to answer customer complaints. Finally, for the customers themselves, XAI methods can reassure them that a decision was rational and fair. Indeed, in some jurisdictions, a customer has the right to demand an explanation (e.g., the under the GDPR regulations in Europe). ### Two considerations There are two related and interesting philosophical points. Firstly, it is well known that humans make systematically biased decisions, use heuristics, and cannot explain their reasoning processes reliably. Hence, we might ask whether it is fair to demand that our machine learning systems are explicable. This an interesting question, because at the current time, it is not even known whether it is possible to find models that are explicable to humans and still have good performance. However, regardless of the flaws in human decision making, explainable AI is a worthy goal even if we do not currently know to what extent it is possible. Secondly, Rudin 2019 has argued that the whole notion of model explainability is flawed, and that explanations of a model must be wrong. If they were completely faithful to the original model, then we would not need the original model, just the explanation. This is true, but even if it is not possible to explain how the whole model works, this does not mean that we cannot get insight into how a particular decision is made. There may be an extremely large number of local explanations pertaining to different inputs, each of which can be understood individually, even if we cannot collectively understand the whole model. However, as we shall see, there is also a strand of XAI that attempts to build novel transparent models which have high performance, but that are inherently easy to understand. ### What makes a good explanation? In this section, we consider the properties that a good explanation should have (or alternatively that a transparent model should have). First, it should be easily understandable to a non-technical user. Models or explanations based on linear models have this property; it’s clear that the output increases when certain inputs increase and decreases when other inputs increase, and the magnitude of these changes differs according to the regression coefficient. Decision trees are also easy to understand as they can be described as a series of rules. Second, the explanation should be succinct. Models or explanations based on small decision trees are reasonably comprehensible, but become less so as the size of the tree grows. Third, the explanation should be accurate and have high-fidelity. In other words, it should predict unseen data correctly, and in the same way as the original model. Finally, an explanation should be complete; it should be applicable in all situations. Adherence to these criteria is extremely important. Indeed, Gilpin et al. (2019) argue that it is fundamentally unethical to present a simplified description of a complex system to increase trust, if the limits of that approximation are not apparent. ### Taxonomy of XAI approaches There are a wide range of XAI approaches that are designed for different scenarios (figure 2). The most common case is that we have already trained a complex model like a deep neural network or random forest and do not necessarily even have access to its internal structure. In this context, we refer to this as a black box model, and we seek insight into how it makes decisions. Explanations of existing models are referred to as post-hoc explanations. Figure 2. Taxonomy of XAI methods. If we do not already have a model that we need to explain, we can develop a model that is inherently interpretable. If we already have a model, then we must use a post-hoc method. One approach is to distill this into a simpler and more interpretable model. However, if we only use this for explanations, then the explanations are unreliable to the extent that the results differ. If we replace the original model entirely, then we may sacrifice performance. If we decide to work with just the existing model, then there are two main families of methods. Local models explain a single decision at a time, whereas global models attempt to explain the entire model behaviour. See also Singh (2019). There are three main types of post-hoc explanation. First, we can distill the black box model into a surrogate machine learning model that is intrinsically interpretable. We could then either substitute this model (probably sacrificing some performance) or use it to interpret the original model (which will differ, making the explanations potentially unreliable). Second, we can try to summarize, interrogate, or explain the full model in other ways (i.e., a global interpretation). Third, we can attempt to explain a single prediction (a local interpretation). If we do not already have an existing model, then we have the option of training an intrinsically interpretable model from scratch. This might be standard ML model that is easy to understand like a linear model or tree. However, this may have the disadvantage of sacrificing the performance of more modern complex techniques. Consequently, recent work has investigated training models with high performance but which are still inherently interpretable. In part I of this blog, we consider local post-hoc methods for analyzing black box models. In part II, we consider methods approximate the entire model with a surrogate, and models that provide global explanations at the level of the dataset. We also consider families of model that are designed to be inherently interpretable. ### Local post-hoc explanations Local post-hoc explanations sidestep the problem of trying to convey the entire in an interpretable way by focusing on just explaining a particular decision. The original model consists of a very complex function mapping inputs to outputs. Many local post-hoc models attempt to just describe the part of the function that is close to the point under consideration rather than the entire function. In this section we’ll describe the most common local post-hoc methods for a simple model (figure 3) with two inputs and one output. Obviously, we don’t really need to ‘explain’ this model, since the whole function relating inputs to outputs can easily be visualized. However, it will suffice to illustrate methods that can explain much more complex models. Figure 3. Model used to describe XAI techniques. The model has two inputs $x_{1}$ and $x_{2}$ and returns a probability $Pr(y=1)$ that indicates the likelihood that the input belongs to class 1 (brighter means higher probability). The red points are positive training points and the green points are negative training points. The green line represents the decision boundary. Obviously, we do not need to “explain” this model as we can visualize it easily. Nonetheless, it can be used to elucidate local post-hoc XAI methods. ### Individual conditional expectation (ICE) An individual conditional expectation or ICE plot (Goldstein et al. 2015) takes an individual prediction and shows how it would change as we vary a single feature. Essentially, it answers the question: what if feature $x_{j}$ had taken another value? In terms of the input output function, it takes a slice through a single dimension for a given data point (figure 4). Figure 4. Individual conditional expectation. a) We wish to understand why the purple point is assigned to the negative class. We can do this by considering what would happen if we changed either feature $x_{1}$ (cyan line) or $x_{2}$ (blue line). b) The effect of changing feature $x_{1}$. We see that the point might have been classified as positive (so $Pr(y)>0.5)$ if $x_{1}$ had a higher value. c) The effect of changing feature $x_{2}$. We see that the point might have been classified as positive if $x_{2}$ had a lower value. ICE plots have the obvious disadvantage that they can only interrogate a single feature at a time and they do not take into account the relationships between features. It’s also possible that some combinations of input features never occur in real-life, yet ICE plots display these and do not make this clear. ### Counterfactual explanations ICE plots create insight into the behaviour of the model by visualizing the effect of manipulating one of the model inputs by an arbitrary amount. In contrast, counterfactual explanations manipulate multiple features, but only consider the behaviour within the vicinity of the particular input that we wish to explain. Counterfactual explanations are usually used in the context of classification. From the point of view of the end user, they pose the question “what changes would I have to make for the model decision to be different?”. An oft-cited scenario is that of someone whose loan application has been declined by a machine learning model whose inputs include income, debt levels, credit history, savings and number of credit cards. A counterfactual explanation might indicate that the loan decision would have been different if the applicant had three fewer credit cards and an extra $5000 annual income. From an algorithmic point of view, counterfactual explanations are input data points$\mathbf{x}^{}$that trade off (i) the distance${dist1}[{f}[\mathbf{x}^], y^]$a between the actual function output${f}[\mathbf{x}^]$and the desired output$y^$and (ii) the proximity${dist2}\left[\mathbf{x}, \mathbf{x}^\right]$to the original point$\mathbf{x}$(figure 5). To find these points we define a cost function of the form: $$\hat{\mathbf{x}}^,\hat{\lambda} = \mathop{\rm argmax}_{\lambda}\left[\mathop{\rm argmin}_{\mathbf{x}^}\left[{dist1}\left[{f}[\mathbf{x}], y^\right] + \lambda\cdot {dist2}\left[\mathbf{x}, \mathbf{x}^*\right]\right]\right] \tag{1}$$ where the positive constant$\lambda$controls the relative contribution of the two terms. We want$\lambda$to be as large as possible so that the counterfactual examples is as close as possible to the original example. Figure 5. Counterfactual explanations. a) We want to explain a data point (purple point 1) which was classified negatively. One way to do this is to ask, how we would have to change the input so that it is classified positively. In practice, this means finding and returning the closest point on the decision boundary (cyan point 2). In a real-life situation, a customer might be able to take remedial action to move the inputs to this position. b) This remedial action may be impractical if there are many input features, and so usually we seek sparse counterfactual examples where we have only changed a few features (here just feature$x_{1}$). c) One problem with counterfactual examples is that there be many potential ways to modify the input (brown point 1) to change the classification (points 2, 3 and 4). It’s not clear which one should be presented to the end user. This formulation was introduced by Wachter et al. (2017) who used the squared distance for${dist1}[\bullet]$and the Manhattan distance weighted with the inverse median absolute deviation of each feature for${dist2}[\bullet]$. In practice, they solved this optimization problem by finding a solution for the counterfactual point for a range of different values of$\lambda$and then choosing the largest$\lambda$for which the proximity to the desired point was acceptable. This method is only practical if the model output is a continuous function of the input and we can calculate the derivatives of the model output with respect to the input efficiently. The above formulation has two main drawbacks that were addressed by Dandl et al. (2018). First, we would ideally like counterfactual examples where only a small number of the input features have changed (figure 5); this is both easier to understand and more practical in terms of taking remedial action. To this end, we can modify the function${dist2}$to encourage sparsity in the changes. Second, we want to ensure that the counterfactual example falls in a plausible region of input space. Returning to the loan example, the decision could be partly made based on two different credit ratings, but these might be highly correlated. Consequently, suggesting a change where one remains low, but the other increases is not helpful as this is not realizable in practice. To this end, Dandl et al. (2018) proposed adding a second term that penalizes the counterfactual example if it is far from the training points. A further important modification was made by McGragh et al. (2018) who allow the user to specify a weight for each input dimension that effectively penalizes changes more or less. This can be used to discourage finding counter-factual explanations where the change to the input are not realisable. For example, proposing a change in an input variable that encodes an individuals age is not helpful as this cannot be changed. A drawback of counterfactual explanations is that there may be many possible ways to modify the model output by perturbing the features locally and it’s not clear which is most useful. Moreover, since most approaches are based on optimization of a non-linear function, it’s not possible to ensure that we have find the local minimum; even if we fail to find any counterfactual examples within a predefined distance from the original, this does not mean that they do not exist. ### LIME Local interpretable model-agnostic explanations or LIME (Ribeiro et al. 2016) approximate the main machine learning model locally around a given input using a simpler model that is easier to understand. In some cases, we may trade off the quality of the local explanation against its complexity. Figure 6 illustrates the LIME algorithm with a linear model. Samples are drawn randomly and passed through through the original model. They are then weighted based on their distance to the example that we are trying to explain. Then a linear model is trained using these weighted samples to predict the original model outputs. The linear model is interpretable and is accurate in the vicinity of the example under consideration. Figure 6. Local interpretable model-agnostic explanations (LIME). a) We wish to explain the purple point. Samples are drawn randomly and weighted by their proximity to the point of interest (red and green points). b) We use these samples to train a simpler, interpretable model like a linear model. c) The region around point of interest on original function is closely approximated by d) the interpretable linear model. This method works well for continuous tabular data, and can be modified for other data types. For text data, we might perturb the input by removing or replacing different words rather than sampling and we might use a bag of words model as the approximating model. So, we could understand the output of a spam detector based on BERT by passing multiple perturbed sentences through the BERT model, retrieving the output probability of spam and then building a sparse bag of words model that approximates these results. For image data, we might explain the output of a classifier based on a convolutional network by approximating it locally with a weighted sum of the contributions of different superpixels. We first divide the image into superpixels, and then perturb the image multiple times by setting different combinations of these superpixels to be uniform and gray. These images are passed through the classifier and we store the output probability of the top-rated class. Then we build a sparse linear model that predicts this probability from the presence or absence of each superpixel (figure 7). Figure 7. LIME explanations for image classification. a) Original image to be explained which was classified as `tree frog’. b) Grouping input features by dividing into superpixels. c-e) Replace randomly selected subsets of superpixels with blank gray regions, run through model, and store model probability. f) Build sparse linear model explaining model probability in terms of presence or absence of superpixels. It appears that the head region is mainly responsible for the classification as tree-frog. g) Repeating this process for the class “billiards” which was also assigned a relatively high probability by the model. Adapted from Ribeiro et al. (2016) ### Anchors Ribeiro et al. (2018) proposed anchors which are local rules defined on the input values around a given point that we are trying to explain (figure 8). In the simplest case, each rule is a hard threshold on an input value of the form$x_{1} < \tau$. The rules are added in a greedy way with the aim being to construct rules where the precision is very high (i.e., when the rule is true, the output is almost always the same as the original point). We cannot exhaustively evaluate every point in the rule region and so this is done by considering the choice of rules as a multi-armed bandit problem. In practice, rules are extended in a greedy manner by adding more constraints on them to increase the precision for a given rule complexity. In a more sophisticated solution, beam search is used and the overall rule is chosen to maximize the coverage. This is the area of the data space that the rule is true over. Figure 8. Anchors. One way to explain the purple point is to search for simple set of rules that explain the local region. In this case, we can tell the user that 100% of points the where$x_{1}$is between 0.51 and 0.9 and$x_{2}$is between 0.52 and 0.76 are classified as positive. Ribeiro et al. (2018) applied this message to an LSTM model that predicted the sentiment of reviews. They perturbed the inputs by replacing individual tokens (words) with other random words with the same part of speech tag, with a probability that is proportional to their similarity in the embedding space and measure the response of the LSTM to each. In this case their rules just consist of the presence of words, and for the sentence This movie is not bad, they retrieve the rule that when the both the words “not” and “bad” are present, the sentence has positive sentiment 95% of the time. ### Shapley additive explanations Shapley additive explanations describe the model output${f}[\mathbf{x}_i]$for a particular input$\mathbf{x}_{i}$as an additive sum: $$\label{eq:explain_add_shap} {f}[\mathbf{x}_i] = \psi_{0} + \sum_{d=1}^{D} \psi_d \tag{2}$$ of$D$contributing factors$\psi_{D}$associated with the$D$dimensions of the input. In other words, the change in performance from a baseline$\psi_0$is attributed to a sum of changes$\psi_{d}$associated with the input dimensions. Consider the case where there are only two input variables (figure 9), choosing a particular ordering of the input variables and a constructing the explanation piece by piece. So, we might set: \begin{eqnarray} \psi_{0} &=& \mathbb{E}_{\mathbf{x}}\left[{f}[\mathbf{x}]\right]\nonumber \\ \psi_{1} &=& \mathbb{E}_{\mathbf{x}}\left[{f}[\mathbf{x}]\|x_{1}\right] – \left(\mathbb{E}_{\mathbf{x}}\left[{f}[\mathbf{x}]\right]\right) \nonumber \\ \psi_{2} &=& \mathbb{E}_{\mathbf{x}}\left[{f}[\mathbf{x}]\|x_{1},x_{2}\right]-\left(\mathbb{E}_{\mathbf{x}}\left[{f}[\mathbf{x}]\right]-\mathbb{E}_{\mathbf{x}}\left[{f}[\mathbf{x}]\|x_{1}\right]\right). \tag{3}\end{eqnarray} The first term contains the expected output without observing the input. The second term is the change to the expected output given that we have only observed the first dimension$x_{1}$. The third term gives the additional change to expected output now that we have observed dimensions$x_{1}$and$x_{2}$. In each line, the term in brackets is the right hand side from the previous line, which is why these represent changes. Figure 9. Shapley additive explanations. a) Consider explaining the model output$\mbox{f}[\mathbf{x}]$for point$\mathbf{x}$. b) We construct the explanation as a linear combination of three terms. c) The first term is what we know before considering the data at all. This is the average model output (bottom left corner of panel a). The second term is the change due to what we know from observing feature$x_{1}$. This is calculated by marginalizing over feature$x_{2}$and reading off the prediction (bottom of panel a). We then subtract the first term in the sum to measure the change that was induced by feature$x_{1}$. The third term consists of the remaining change that is needed to get the true model output and is attributable to$x_{2}$. d) We can visualize the cumulative changes due to each feature. e-g) If we repeat this procedure, but consider the features in a different order, then we get a different results. Shapley additive explanations take a weighted average of all possible orderings and return the additive terms$\psi_{\bullet}$that explain the positive or negative contribution of each feature. Assuming we could calculate these terms, they would obviously have the form of equation 2. However, the input order was arbitrary. If we took a different ordering of the variables so that$x_{2}$is before$x_{1}$, then we would get different values \begin{eqnarray} \psi_{0} &=& \mathbb{E}_{\mathbf{x}}\left[{f}[\mathbf{x}]\right]\nonumber \\ \psi_{2} &=& \mathbb{E}_{\mathbf{x}}\left[{f}[\mathbf{x}]\|x_{2}\right] – \left(\mathbb{E}_{\mathbf{x}}\left[{f}[\mathbf{x}]\right]\right) \nonumber \\ \psi_{1} &=& \mathbb{E}_{\mathbf{x}}\left[{f}[\mathbf{x}]\|x_{1},x_{2}\right]-\left(\mathbb{E}_{\mathbf{x}}\left[{f}[\mathbf{x}]\right]-\mathbb{E}_{\mathbf{x}}\left[{f}[\mathbf{x}]\|x_{2}\right]\right). \tag{4}\end{eqnarray} The idea of Shapley additive explanations is to compute the values$\psi_{d}$in equation 2 by taking a weighted average of the$\psi_{i}$over all possible orderings. If the set of indices is given by$\mathcal{D}=\{1,2,\ldots, D\}$, then the final Shapley values are $$\label{eq:explain_shap_expect} \psi_{d}[{f}[\mathbf{x}]] = \sum_{\mathcal{S}\subseteq \mathcal{D}} \frac{\|\mathcal{S}\|!(D-\|\mathcal{S}\|-1)!}{D!}\left(\mathbb{E}\left[{f}[\mathbf{x}]\|\mathcal{S}\right] – \mathbb{E}\left[{f}[\mathbf{x}]\|\mathcal{S}_{\setminus d}\right]\right). \tag{5}$$ This computation takes every subset of variables that contains$x_{d}$and computes the expected value of the function given this subset takes the particular values for that data point with and without$x_{d}$itself. This result is weighted and contributes to the final value$\psi_{d}$. The particular weighting (i.e., the first term after the sum) can be proven to be the only one that satisfies the properties of (i) local accuracy (the Shapley values sum to the true function output) (ii) missingness (an absent feature has a Shapley value/ attribution of zero, and (iii) consistency (if the marginal contribution of a feature increases or stays the same, then the Shapley value should increase or stay the same). The eventual output of this process is a single value associated with each feature that represents how much it increased or decreased the model output. Sometimes this is presented as a force diagram (figure 10) which is a compact representation of all of these values. Figure 10. Force diagram for Shapley additive explanations. The final result is explained by an additive sum of a scalars associated with each input feature (PTRATIO, LSTAT, RM etc.). The red features increase the output and the blue feature decrease it. The horizontal size associated with each feature represents the magnitude of change. Via Lundberg and Lee (2017). ### Computing SHAP values Computing the SHAP values using (equation 5) is challenging, although it can be done exactly for certain models like trees (see Lundberg et al., 2019). The first problem is that there are a very great number of subsets to test. This can be resolved by approximating the sum with a subset of samples. The second problem is how to compute the expectation terms. For each term, we consider the data$\mathbf{x}$being split into two parts$\mathbf{x}_{\mathcal{S}}$and$\mathbf{x}_{\overline{\mathcal{S}}}$. Then the expectation can be written as: $$\mathbb{E}[{f}[\mathbf{x}]\|\mathcal{S}]=\mathbb{E}_{x_{\overline{\mathcal{S}}}\|x_{\mathcal{S}}}\left[{f}[\mathbf{x}]\right]. \tag{6}$$ It is then possible to make some further assumptions that can ease computation. We might first assume feature independence: $$\mathbb{E}_{x_{\overline{\mathcal{S}}}\|x_{\mathcal{S}}}\left[{f}[\mathbf{x}]\right] \approx \mathbb{E}_{x_{\overline{\mathcal{S}}}}\left[{f}[\mathbf{x}]\right], \tag{7}$$ and then further assume model linearity: $$\mathbb{E}_{x_{\overline{\mathcal{S}}}}\left[{f}[\mathbf{x}]\right] \approx {f}\left[\mathbf{x}_{\mathcal{S}}, \mathbb{E}_[\mathbf{x}_{\overline{\mathcal{S}}}]\right]. \tag{8}$$ When we use this latter assumption, the model can replicate LIME and this is known as kernel SHAP. Recall that LIME fits a linear model based on weighted samples, where the weights are based on the proximity of the sample to the point that we are trying to explain. However, these weights are chosen heuristically. Shapley additive explanations with feature independence and linearity also fit local linear model from points$\mathbf{x}'[\mathcal{S}] = [\mathbf{x}_{\mathcal{S}}, \mathbb{E}_{x_{\overline{\mathcal{S}}}}]$where some subset of$\overline{\mathcal{S}}\$ of the features have been replaced by their expected values. In practice the expectations are computed by substituting other random values from the training set. The weights for KernelShap are given (non-obviously) by: $$\omega[\mathcal{S}] = \frac{D-1}{(D \:{choose}\: \|\overline{\mathcal{S}}\|)\|\overline{\mathcal{S}}\|(D-\|\overline{\mathcal{S}}\|)}. \tag{9}$$ ### Conclusion In part I of this blog, we have discussed the importance of explainability for AI systems. We presented a taxonomy of such methods and described a number of methods for creating local post-hoc explanations of a black box model. These help the user understand why a particular decision was made but do not try to explain the whole model in detail. In part II of this blog, we will consider global explanations and models that are interpretable by design.	2023-02-07 20:54:47	{"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": 7, "x-ck12": 0, "texerror": 0, "math_score": 0.4634045660495758, "perplexity": 594.927059392111}, "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/1674764500641.25/warc/CC-MAIN-20230207201702-20230207231702-00207.warc.gz"}
http://gmatclub.com/forum/as-a-bicycle-salesperson-norman-earns-a-fixed-salary-of-111038.html?oldest=1	Find all School-related info fast with the new School-Specific MBA Forum It is currently 04 May 2015, 19:14 ### 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 # Events & Promotions ###### Events & Promotions in June Open Detailed Calendar # As a bicycle salesperson, Norman earns a fixed salary of $20 Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: Manager Joined: 11 Feb 2011 Posts: 141 Followers: 3 Kudos [?]: 57 [1] , given: 21 As a bicycle salesperson, Norman earns a fixed salary of$20 [#permalink] 17 Mar 2011, 05:39 1 KUDOS 5 This post was BOOKMARKED 00:00 Difficulty: 75% (hard) Question Stats: 51% (02:34) correct 49% (01:24) wrong based on 251 sessions As a bicycle salesperson, Norman earns a fixed salary of $20 per week plus$6 per bicycle for the first 6 bicycles he sells, $12 per bicycle for the next 6 bicycles he sells, and$18 per bicycle for every bicycle sold after first 12. This week, he earned more than twice as much as he did last week. If he sold x bicycles last week and y bicycles this week, which of the following statements must be true? I. y>2x II. y>x III. y>3 A. I only B. II only C. I and II D. II and III E. I, II, III [Reveal] Spoiler: OA _________________ target:-810 out of 800! Last edited by Bunuel on 31 Mar 2012, 16:39, edited 2 times in total. Edited the OA Director Status: Matriculating Affiliations: Chicago Booth Class of 2015 Joined: 03 Feb 2011 Posts: 926 Followers: 13 Kudos [?]: 219 [3] , given: 123 Re: word problem [#permalink] 17 Mar 2011, 06:07 3 KUDOS 2 This post was BOOKMARKED I will throw in the values of x and y since its going to be tough using algebra. For first 6 bicycles - he gets $6 / bicycle For next 6 bicycles - he gets$12 / bicycle For > 12 bicycles - he gets $18 / bicycle Constraint : This week he earned more than twice as much as he did last week. Paraphrase I: Did he double the quantity of bicycles sold to earn more than double the revenue from last week? I dont think so. Reasons - Lets say last week he sold 13 bicycles. Last week revenue = 20 + 66 + 612 + 118 = 146 1462 + 1 = 292 + 1 = 293. To make this revenue he could sell (293 - 128)/18 = 165/18 i.e. 10 more than 12 bicycles Total bicycles sold this week = 12 + 10 = 22 (which is less than twice the bicyles sold last week) Hence I is ruled out. That leaves the options - B and D. B Vs D. I have to verify statement III Paraphrase III: Did he double the revenue from last week by selling minimum of 4 bicycles this week? Lets assume the contradiction is true. He sold 3 bicycles this week and 1 bicycle last week. Last week revenue = 20 + 61 = 26 This week revenue = 20 + 63 = 38 38 is less than twice 26. So the contradiction fails. Hence III is true. Answer D. Intern Joined: 29 Dec 2011 Posts: 7 Concentration: Finance Schools: Yale '14 GMAT 1: 710 Q48 V38 GMAT 2: 720 Q49 V40 GPA: 3.3 Followers: 0 Kudos [?]: 14 [0], given: 6 Re: word problem [#permalink] 31 Mar 2012, 16:30 Can someone verify the OA? I get D as well. Math Expert Joined: 02 Sep 2009 Posts: 27215 Followers: 4228 Kudos [?]: 41022 [6] , given: 5654 Re: word problem [#permalink] 31 Mar 2012, 17:03 6 This post received KUDOS Expert's post jj97cornell wrote: Can someone verify the OA? I get D as well. Correct answer is D. OA edited. As a bicycle salesperson, Norman earns a fixed salary of$20 per week plus $6 per bicycle for the first 6 bicycles he sells,$12 per bicycle for the next 6 bicycles he sells, and $18 per bicycle for every bicycle sold after first 12. This week, he earned more than twice as much as he did last week. If he sold x bicycles last week and y bicycles this week, which of the following statements must be true? I. y>2x II. y>x III. y>3 A. I only B. II only C. I and II D. II and III E. I, II, III II and III are obviously always true: II. y>x --> since this week, Norman earned more than he did last week and the total salary is in direct relationship with the # of bicycle sold, then y (# of bicycle sold this week) must be more than x (# of bicycle sold last week); III. y>3 --> if Norman sold 3 bicycles this week then this week he earned 20+36=$38, which cannot be more than twice as much as he earned the last week, since the minimum salary is fixed to $20. So y must be more than 3; I. y>2x --> if y=12 and x= 6 then this week Norman earned 20+66+612=$128, and the last week he earned 20+66=$56.$128 is more than twice as much as $56, so the condition in the stem holds but y=2x, which means that III is not always true. Answer: D. _________________ Manager Joined: 21 Feb 2012 Posts: 115 Location: India Concentration: Finance, General Management GMAT 1: 600 Q49 V23 GPA: 3.8 WE: Information Technology (Computer Software) Followers: 0 Kudos [?]: 40 [0], given: 15 MANHATTAN PS3 [#permalink] 11 May 2012, 02:17 As a bicycle salesperson, Norman earns a fixed salary of$20 per week plus $6 per bicycle for the first six bicycles he sells,$12 per bicycle for the next six bicycles he sells, and $18 per bicycle for every bicycle sold after the first 12. This week, Norman earned more than twice as much as he did last week. If he sold x bicycles last week and y bicycles this week, which of the following statements must be true? I. y > 2x II. y > x III. y > 3 A. I only B. II only C. I and II D. II and III E. I, II, and III Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 5453 Location: Pune, India Followers: 1335 Kudos [?]: 6784 [1] , given: 177 Re: MANHATTAN PS3 [#permalink] 11 May 2012, 10:37 1 This post received KUDOS Expert's post 1 This post was BOOKMARKED piyushksharma wrote: As a bicycle salesperson, Norman earns a fixed salary of$20 per week plus $6 per bicycle for the first six bicycles he sells,$12 per bicycle for the next six bicycles he sells, and $18 per bicycle for every bicycle sold after the first 12. This week, Norman earned more than twice as much as he did last week. If he sold x bicycles last week and y bicycles this week, which of the following statements must be true? I. y > 2x II. y > x III. y > 3 A. I only B. II only C. I and II D. II and III E. I, II, and III I think II and III are pretty straight forward and I am assuming you have no problem deciding about those. Let me add here what I thought about I. One way is that you can try to find a case where he earns twice as much but doesn't sell twice as many bikes. Another is a more intuitive approach. You know that initially, he has to sell more bikes to make some money (he earns only$6 from first 6 bikes and $12 from next 6 bikes. First$20 is too small an amount). Later on, he gets $18 per bike which means he makes money at a much faster rate. Hence, later on, he can double the amount he made previously very quickly and by selling far fewer bikes. Hence it is not essential that he needs to sell twice as many bikes to make twice as much money. Hence y may not be greater than 2x. _________________ Karishma Veritas Prep \| GMAT Instructor My Blog Veritas Prep GMAT course is coming to India. Enroll in our weeklong Immersion Course that starts March 29! Veritas Prep Reviews Senior Manager Joined: 13 Aug 2012 Posts: 464 Concentration: Marketing, Finance GMAT 1: Q V0 GPA: 3.23 Followers: 17 Kudos [?]: 257 [0], given: 11 Re: As a bicycle salesperson, Norman earns a fixed salary of$20 [#permalink] 01 Dec 2012, 22:00 Test the inequalities: I. $$y>2x$$ Let x = 1 bicycle; Earnings: 26 dollars Let y = 3 bicylce; Earnings: 38 dollars Is 38 more than twice of 26? NO! II. $$y > x$$ Surely, there must be more bicycles sold in the second week. Always true! YES! III. y>3 Testing I, we found that when y = 3 and x = 1, we still couldn't achieve the condition that the second week's earning is more than twice the first. Therefore, y must be greater than 3. YES! _________________ Impossible is nothing to God. Senior Manager Joined: 17 Dec 2012 Posts: 395 Location: India Followers: 19 Kudos [?]: 243 [0], given: 10 Re: As a bicycle salesperson, Norman earns a fixed salary of $20 [#permalink] 29 Mar 2013, 00:19 AnkitK wrote: As a bicycle salesperson, Norman earns a fixed salary of$20 per week plus $6 per bicycle for the first 6 bicycles he sells,$12 per bicycle for the next 6 bicycles he sells, and $18 per bicycle for every bicycle sold after first 12. This week, he earned more than twice as much as he did last week. If he sold x bicycles last week and y bicycles this week, which of the following statements must be true? I. y>2x II. y>x III. y>3 A. I only B. II only C. I and II D. II and III E. I, II, III Given: 1. The number of bicycles sold last week = x 2. The number of bicycles sold this week = y 3. Let earnings of last week and this week be s1 and s2 resp. s2> 2s1 Question: 1. Is y > 2x 2. Is y > x 3. Is y > 3 Basically the question asks us to relate number of bicycles sold in each of 2 weeks based on the relation between the earnings in those 2 weeks. 1. Earnings in the current week can be higher than that of the last week only when the number of bicycles sold is higher in the current week. i.e., only when y>x 2. If the number of bicycles sold during the current week <4, then the earnings in the current week cannot be more than double that of the previous week. 3. Now let us assume y=2x. Since we are assuming twice the bicycles are sold this week over that of the previous week , if we take x=18, then y=36. 4. Let us calculate s1 and s2. s1= earnings from the first 12 bicycles + earnings from the next 6 bicycles = 128+ 108= 236 s2= earnings from the 12 bicycles+ earnings from the next 24 bicycles= 128+ 432= 560 5. s2>2s1 even when y=2x We see from (1) above statement II is true, from (2) above statement III is true, from (5) above statement I need not be true. The answer is choice D. _________________ Srinivasan Vaidyaraman Sravna Test Prep http://www.sravna.com Classroom Courses in Chennai Free Online Material SVP Joined: 06 Sep 2013 Posts: 2039 Concentration: Finance GMAT 1: 770 Q0 V Followers: 24 Kudos [?]: 294 [0], given: 354 Re: word problem [#permalink] 17 Jan 2014, 04:10 Bunuel wrote: jj97cornell wrote: Can someone verify the OA? I get D as well. Correct answer is D. OA edited. As a bicycle salesperson, Norman earns a fixed salary of$20 per week plus $6 per bicycle for the first 6 bicycles he sells,$12 per bicycle for the next 6 bicycles he sells, and $18 per bicycle for every bicycle sold after first 12. This week, he earned more than twice as much as he did last week. If he sold x bicycles last week and y bicycles this week, which of the following statements must be true? I. y>2x II. y>x III. y>3 A. I only B. II only C. I and II D. II and III E. I, II, III II and III are obviously always true: II. y>x --> since this week, Norman earned more than he did last week and the total salary is in direct relationship with the # of bicycle sold, then y (# of bicycle sold this week) must be more than x (# of bicycle sold last week); III. y>3 --> if Norman sold 3 bicycles this week then this week he earned 20+36=$38, which cannot be more than twice as much as he earned the last week, since the minimum salary is fixed to $20. So y must be more than 3; I. y>2x --> if y=12 and x= 6 then this week Norman earned 20+66+612=$128, and the last week he earned 20+66=$56.$128 is more than twice as much as $56, so the condition in the stem holds but y=2x, which means that III is not always true. Answer: D. Bunuel, nice approach +1 On I though, I'm having some issues picking the correct numbers, how can I decide which numbers to use to prove this case not necessarily true? Cheers! J Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 5453 Location: Pune, India Followers: 1335 Kudos [?]: 6784 [0], given: 177 Re: word problem [#permalink] 20 Jan 2014, 21:50 Expert's post jlgdr wrote: I'm having some issues picking the correct numbers, how can I decide which numbers to use to prove this case not necessarily true? Cheers! J There are no correct/incorrect numbers. You can just try to understand the logic using numbers. 6 bikes -$6 each i.e. total $36 next 6 bikes -$12 each i.e. total $72 So 12 bikes for a total sum of$108 But for every subsequent bike, he gets $18 so the next$108 he will be able to make by selling just 6 bikes. So even if he earns twice as much as before, he doesn't need to sell twice as many bikes. _________________ Karishma Veritas Prep \| GMAT Instructor My Blog Veritas Prep GMAT course is coming to India. Enroll in our weeklong Immersion Course that starts March 29! Veritas Prep Reviews Senior Manager Joined: 15 Aug 2013 Posts: 331 Followers: 0 Kudos [?]: 20 [0], given: 23 Re: As a bicycle salesperson, Norman earns a fixed salary of $20 [#permalink] 03 Aug 2014, 15:19 VeritasPrepKarishma wrote: piyushksharma wrote: As a bicycle salesperson, Norman earns a fixed salary of$20 per week plus $6 per bicycle for the first six bicycles he sells,$12 per bicycle for the next six bicycles he sells, and $18 per bicycle for every bicycle sold after the first 12. This week, Norman earned more than twice as much as he did last week. If he sold x bicycles last week and y bicycles this week, which of the following statements must be true? I. y > 2x II. y > x III. y > 3 A. I only B. II only C. I and II D. II and III E. I, II, and III I think II and III are pretty straight forward and I am assuming you have no problem deciding about those. Let me add here what I thought about I. One way is that you can try to find a case where he earns twice as much but doesn't sell twice as many bikes. Another is a more intuitive approach. You know that initially, he has to sell more bikes to make some money (he earns only$6 from first 6 bikes and $12 from next 6 bikes. First$20 is too small an amount). Later on, he gets $18 per bike which means he makes money at a much faster rate. Hence, later on, he can double the amount he made previously very quickly and by selling far fewer bikes. Hence it is not essential that he needs to sell twice as many bikes to make twice as much money. Hence y may not be greater than 2x. Hi Karishma, I'm intrigued by your intuitive approach. To backtrack a little -- word problems as a whole seem to be the biggest time suck for me. I spent 4 minutes on this problem, and although I got it right, I can't seem to figure out how to speed things up when it comes to word problems as such. Is there a strategy you recommend to tackle word problems in general? I know that this is a vague question but any help would be appreciated. Can you recommend other word problems to do to help with practice? Regarding what you said, to me, 2 seemed very straight forward but I still went and checked statement 3. Yes, in hindsight, all of this looks very simple after reading your explanation but I'm not as certain during the test. Any thoughts would be appreciated. Thanks Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 5453 Location: Pune, India Followers: 1335 Kudos [?]: 6784 [1] , given: 177 Re: As a bicycle salesperson, Norman earns a fixed salary of$20 [#permalink] 04 Aug 2014, 20:55 1 KUDOS Expert's post russ9 wrote: VeritasPrepKarishma wrote: piyushksharma wrote: As a bicycle salesperson, Norman earns a fixed salary of $20 per week plus$6 per bicycle for the first six bicycles he sells, $12 per bicycle for the next six bicycles he sells, and$18 per bicycle for every bicycle sold after the first 12. This week, Norman earned more than twice as much as he did last week. If he sold x bicycles last week and y bicycles this week, which of the following statements must be true? I. y > 2x II. y > x III. y > 3 A. I only B. II only C. I and II D. II and III E. I, II, and III I think II and III are pretty straight forward and I am assuming you have no problem deciding about those. Let me add here what I thought about I. One way is that you can try to find a case where he earns twice as much but doesn't sell twice as many bikes. Another is a more intuitive approach. You know that initially, he has to sell more bikes to make some money (he earns only $6 from first 6 bikes and$12 from next 6 bikes. First $20 is too small an amount). Later on, he gets$18 per bike which means he makes money at a much faster rate. Hence, later on, he can double the amount he made previously very quickly and by selling far fewer bikes. Hence it is not essential that he needs to sell twice as many bikes to make twice as much money. Hence y may not be greater than 2x. Hi Karishma, I'm intrigued by your intuitive approach. To backtrack a little -- word problems as a whole seem to be the biggest time suck for me. I spent 4 minutes on this problem, and although I got it right, I can't seem to figure out how to speed things up when it comes to word problems as such. Is there a strategy you recommend to tackle word problems in general? I know that this is a vague question but any help would be appreciated. Can you recommend other word problems to do to help with practice? Regarding what you said, to me, 2 seemed very straight forward but I still went and checked statement 3. Yes, in hindsight, all of this looks very simple after reading your explanation but I'm not as certain during the test. Any thoughts would be appreciated. Thanks Hey Russ, Familiarity creates intuition. When you see a lot of word problems, you are often able to see what is going to work and usually it is correct. Till a few years back, I use to rely on algebra (equations) to solve all word problems. Then, a mentor forced me to see the big picture, the reason behind every step and how the steps are meant for machines only - how we are quite capable of using reason and logic to solve most questions in a reasoning based test such as GMAT. Now the problem is that when you need to give a solution to someone, just saying that use intuition is not helpful. You can barely explain it in a face-to-face situation. Also, confidence comes with practice. You will start feeling confident in your inferences from the given data once you see that you are getting most of them right on practice questions. I will suggest you to start every word problem by trying to infer whatever you can from the given data. Try to minimize your use of equations (you can't let them go completely). Look for alternative solutions for every problem. Soon. you will start coming up with your own intuitive solutions. _________________ Karishma Veritas Prep \| GMAT Instructor My Blog Veritas Prep GMAT course is coming to India. Enroll in our weeklong Immersion Course that starts March 29! Veritas Prep Reviews Intern Joined: 10 Nov 2013 Posts: 4 Followers: 0 Kudos [?]: 0 [0], given: 20 Re: As a bicycle salesperson, Norman earns a fixed salary of $20 [#permalink] 13 Sep 2014, 01:49 i still feel option (B) is correct because it says y>3 and you guys tested the condition with y=3 and x=1 but what about when y=4 and x=1 or 2 then the earning last week add up to 26 or 32 and the earnings this week is merely 44 and either ways the earnings last week is more than half of the earnings this week Can you guys please clarify on this approach? Math Expert Joined: 02 Sep 2009 Posts: 27215 Followers: 4228 Kudos [?]: 41022 [0], given: 5654 Re: As a bicycle salesperson, Norman earns a fixed salary of$20 [#permalink] 13 Sep 2014, 06:22 Expert's post swaydzlycan wrote: i still feel option (B) is correct because it says y>3 and you guys tested the condition with y=3 and x=1 but what about when y=4 and x=1 or 2 then the earning last week add up to 26 or 32 and the earnings this week is merely 44 and either ways the earnings last week is more than half of the earnings this week Can you guys please clarify on this approach? If Norman sold 3 bicycles this week then this week he earned 20+36=$38, which cannot be more than twice as much as he earned last week, since the minimum salary is fixed to$20. So y must be more than 3. _________________ Intern Joined: 17 Dec 2014 Posts: 11 Followers: 0 Kudos [?]: 1 [0], given: 0 Re: As a bicycle salesperson, Norman earns a fixed salary of $20 [#permalink] 17 Dec 2014, 15:46 As a bicycle salesperson, Norman earns a fixed salary of$20 per week plus $6 per bicycle for the first 6 bicycles he sells,$12 per bicycle for the next 6 bicycles he sells, and $18 per bicycle for every bicycle sold after first 12. This week, he earned more than twice as much as he did last week. If he sold x bicycles last week and y bicycles this week, which of the following statements must be true? I. y>2x II. y>x III. y>3 A. I only B. II only C. I and II D. II and III E. I, II, III What if you just do: x=1 so y>21=2 so let's say y=3 20+(16)=26 = earnings last week 20+(36)=38 = earnings this week He earned this week more than twice as much as last week so 38 must be bigger than 262. 38<52 so this means y>2x does not have to be true. Is this correct or is this the wrong way? Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 5453 Location: Pune, India Followers: 1335 Kudos [?]: 6784 [0], given: 177 Re: As a bicycle salesperson, Norman earns a fixed salary of$20 [#permalink] 17 Dec 2014, 22:12 Expert's post Lars1988 wrote: As a bicycle salesperson, Norman earns a fixed salary of $20 per week plus$6 per bicycle for the first 6 bicycles he sells, $12 per bicycle for the next 6 bicycles he sells, and$18 per bicycle for every bicycle sold after first 12. This week, he earned more than twice as much as he did last week. If he sold x bicycles last week and y bicycles this week, which of the following statements must be true? I. y>2x II. y>x III. y>3 A. I only B. II only C. I and II D. II and III E. I, II, III What if you just do: x=1 so y>21=2 so let's say y=3 20+(16)=26 = earnings last week 20+(36)=38 = earnings this week He earned this week more than twice as much as last week so 38 must be bigger than 262. 38<52 so this means y>2x does not have to be true. Is this correct or is this the wrong way? To prove that (I) needn't hold, you need to find numbers where he earned more than twice but y was not greater than twice of x. You have done the opposite - you have taken a case where y is greater than twice of x and shown that he did not earn more than twice. This doesn't prove that (I) needn't hold. The numbers you need to consider would be say x = 12, y = 24 (y is NOT MORE than twice of x) Last week's earning = 20 + 66 + 126 = 128 This week's earning = 20 + 66 + 126 + 1212 = 128 + 144 (More than twice of last week's earning) So he could earn more than twice of last week's earning and still, y > 2x may not hold. _________________ Karishma Veritas Prep \| GMAT Instructor My Blog Veritas Prep GMAT course is coming to India. Enroll in our weeklong Immersion Course that starts March 29! Veritas Prep Reviews Intern Joined: 17 Dec 2014 Posts: 11 Followers: 0 Kudos [?]: 1 [0], given: 0 Re: As a bicycle salesperson, Norman earns a fixed salary of $20 [#permalink] 18 Dec 2014, 03:24 VeritasPrepKarishma wrote: Lars1988 wrote: As a bicycle salesperson, Norman earns a fixed salary of$20 per week plus $6 per bicycle for the first 6 bicycles he sells,$12 per bicycle for the next 6 bicycles he sells, and $18 per bicycle for every bicycle sold after first 12. This week, he earned more than twice as much as he did last week. If he sold x bicycles last week and y bicycles this week, which of the following statements must be true? I. y>2x II. y>x III. y>3 A. I only B. II only C. I and II D. II and III E. I, II, III What if you just do: x=1 so y>21=2 so let's say y=3 20+(16)=26 = earnings last week 20+(36)=38 = earnings this week He earned this week more than twice as much as last week so 38 must be bigger than 262. 38<52 so this means y>2x does not have to be true. Is this correct or is this the wrong way? To prove that (I) needn't hold, you need to find numbers where he earned more than twice but y was not greater than twice of x. You have done the opposite - you have taken a case where y is greater than twice of x and shown that he did not earn more than twice. This doesn't prove that (I) needn't hold. The numbers you need to consider would be say x = 12, y = 24 (y is NOT MORE than twice of x) Last week's earning = 20 + 66 + 126 = 128 This week's earning = 20 + 66 + 126 + 12*12 = 128 + 144 (More than twice of last week's earning) So he could earn more than twice of last week's earning and still, y > 2x may not hold. Oke I thought because y can be bigger than 2x and he can earn more than twice last week x=1 and y=6 for example than 56>52 but if y is 3, 4 or 5, which is bigger dan 2x, y<52 and he did not earn more than twice last week. So it can be true but also false. So that's why I thought y>2x does not stand at all time. I now understand that I have to find y=2x and that he earned this week more than twice last week but I thought the other way around could also solve the problem. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 5453 Location: Pune, India Followers: 1335 Kudos [?]: 6784 [0], given: 177 Re: As a bicycle salesperson, Norman earns a fixed salary of$20 [#permalink] 18 Dec 2014, 19:09 Expert's post Lars1988 wrote: Oke I thought because y can be bigger than 2x and he can earn more than twice last week x=1 and y=6 for example than 56>52 but if y is 3, 4 or 5, which is bigger dan 2x, y<52 and he did not earn more than twice last week. So it can be true but also false. So that's why I thought y>2x does not stand at all time. I now understand that I have to find y=2x and that he earned this week more than twice last week but I thought the other way around could also solve the problem. Think logically - Say if A is true implies B must be true, does it mean that if B is true then A must be true too? Not necessary, right? For example, you know that if it rains, the ground gets wet. Now if you see the ground wet, can you say that it must have rained? No necessary, right? Perhaps someone spilled water on the ground - we don't know. _________________ Karishma Veritas Prep \| GMAT Instructor My Blog Veritas Prep GMAT course is coming to India. Enroll in our weeklong Immersion Course that starts March 29! Veritas Prep Reviews Re: As a bicycle salesperson, Norman earns a fixed salary of $20 [#permalink] 18 Dec 2014, 19:09 Similar topics Replies Last post Similar Topics: 3 As a bicycle salesperson, Norman earns a fixed salary of$20 3 02 Nov 2009, 03:03 As a bicycle salesperson, Norman earns a fixed salary of $20 1 13 Sep 2008, 01:52 3 As a bicycle salesperson, Norman earns a fixed salary of$20 5 15 Jan 2008, 10:20 As a bicycle salesperson, Norman earns a fixed salary of $20 1 09 Dec 2007, 14:19 As a bicycle salesperson, Norman earns a fixed salary of$20 1 08 Dec 2007, 10:36 Display posts from previous: Sort by	2015-05-05 03:14:35	{"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": 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": 0.3530099093914032, "perplexity": 2558.8089249840987}, "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-2015-18/segments/1430455222810.45/warc/CC-MAIN-20150501044022-00012-ip-10-235-10-82.ec2.internal.warc.gz"}
https://www.semanticscholar.org/paper/Minimal-Model-Fusion-Rules-From-2-Groups-Akman-Feingold/a6bd3aad2d8362e87a61574330b6a6e3a01c5a21	# Minimal Model Fusion Rules From 2-Groups @article{Akman1996MinimalMF, title={Minimal Model Fusion Rules From 2-Groups}, author={Fusun Akman and Alex J. Feingold and Michael D. Weiner}, journal={Letters in Mathematical Physics}, year={1996}, volume={40}, pages={159-169} } • Published 5 January 1996 • Mathematics, Physics • Letters in Mathematical Physics AbstractThe fusion rules for the (p,q)-minimal model representations of the Virasoro algebra are shown to come from the group $$G = \mathbb{Z}_2^{p + q - 5}$$ in the following manner. There is a partition $$G = P_1 \cup \; \cdot \cdot \cdot \; \cup P_N$$ into disjoint subsets and a bijection between $$\{ P_1 ,\;...,\;P_N \}$$ and the sectors $$\{ S_1 ,\;...,\;S_N \}$$ of the (p,q)-minimal model such that the fusion rules S_i * \;S_j = \sum\nolimits_k {D(S_i ,S_j ,S_k )S_k… 4 Citations Type A fusion rules from elementary group theory • Mathematics, Physics • 2000 We show how the fusion rules for an affine Kac-Moody Lie algebra g of type A_{n-1}, n = 2 or 3, for all positive integral level k, can be obtained from elementary group theory. The orbits of the kth A new perspective on the Frenkel–Zhu fusion rule theorem • Mathematics, Physics • 2008 Abstract In this paper we prove a formula for fusion coefficients of affine Kac–Moody algebras first conjectured by Walton [M.A. Walton, Tensor products and fusion rules, Canad. J. Phys. 72 (1994) Fusion Rules for Affine Kac-Moody Algebras This is an expository introduction to fusion rules for affine Kac-Moody algebras, with major focus on the algorithmic aspects of their computation and the relationship with tensor product ## References SHOWING 1-10 OF 19 REFERENCES Spinor construction of the c = 1/2 minimal model • Physics, Mathematics • 1995 The representation theories of affine Kac-Moody Lie algebras and the Virasoro algebra have been found to have interesting connections with many other parts of mathematics and have played a vital role Vertex operator algebras associated to representations of affine and Virasoro Algebras • Mathematics • 1992 The first construction of the integrable highest-weight representations of affine Lie algebras or loop algebras by Kac i-K] was greatly inspired by the generalization of the Weyl denominator formula Bombay Lectures On Highest Weight Representations Of Infinite Dimensional Lie Algebras • Mathematics • 1983 This book is a collection of a series of lectures given by Prof. V Kac at Tata Institute, India in Dec '85 and Jan '86. These lectures focus on the idea of a highest weight representation, which goes Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory - Nucl. Phys. B241, 333 (1984) • Physics • 1984 We present an investigation of the massless, two-dimentional, interacting field theories. Their basic property is their invariance under an infinite-dimensional group of conformal (analytic) On Axiomatic Approaches to Vertex Operator Algebras and Modules • Mathematics • 1993 Introduction Vertex operator algebras Duality for vertex operator algebras Modules Duality for modules References. Yi-Zhi Huang • J. Lepowsky, On Axiomatic Approaches to Vertex Operator Algebras and Modules, Memoirs Amer. Math. Soc., Vol. 104, No. 594, Amer. Math. Soc., Providence, RI • 1993 Vertex operator algebras associated to representations of aane and Virasoro algebras, Duke Math • J • 1992	2022-01-25 01:14:50	{"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": 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": 0.5757152438163757, "perplexity": 1451.5432134815107}, "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/1642320304749.63/warc/CC-MAIN-20220125005757-20220125035757-00113.warc.gz"}
https://lexparency.org/eu/32013R0575/ART_325an/latest	Capital Requirements Regulation (CRR) ### Article 325an — Intra-bucket correlations for credit spread risk for securitisations not included in the ACTP 1. Between two sensitivities WSk and WSl within the same bucket, the correlation parameter ρkl shall be set as follows: • ρkl = ρkl(tranche) · ρkl(tenor) · ρkl(basis) where: • ρkl(thranche) shall be equal to 1 where the two names of sensitivities k and l are within the same bucket and are related to the same securitisation tranche (more than 80 % overlap in notional terms), otherwise it shall be equal to 40 %; • ρkl(tenor) shall be equal to 1 where the two vertices of the sensitivities k and l are identical, otherwise it shall be equal to 80 %; and • ρkl(basis) shall be equal to 1 where the two sensitivities are related to the same curves, otherwise it shall be equal to 99,90 %. 2. The correlation parameters referred to in paragraph 1 shall not apply to bucket 25 in Table 7 of Article 325am(1). The own funds requirement for the delta risk aggregation formula within bucket 25 shall be equal to the sum of the absolute values of the net weighted sensitivities allocated to that bucket: $${K _{b ^{(bucket 25)}}} = {\sum _{k} \|\mathrm{WS} _{k}\|}$$	2020-01-18 15:02:09	{"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": 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": 0.6437145471572876, "perplexity": 1886.6336594698691}, "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-2020-05/segments/1579250592636.25/warc/CC-MAIN-20200118135205-20200118163205-00341.warc.gz"}
https://en.m.wikipedia.org/wiki/Kerr%E2%80%93Newman_metric	# Kerr–Newman metric The Kerr–Newman metric is the most general asymptotically flat, stationary solution of the Einstein–Maxwell equations in general relativity that describes the spacetime geometry in the region surrounding an electrically charged, rotating mass. It generalizes the Kerr metric by taking into account the field energy of an electromagnetic field, in addition to describing rotation. It is one of a large number of various different electrovacuum solutions, that is, of solutions to the Einstein–Maxwell equations which account for the field energy of an electromagnetic field. Such solutions do not include any electric charges other than that associated with the gravitational field, and are thus termed vacuum solutions. This solution has not been especially useful for describing non-black-hole astrophysical phenomena, because observed astronomical objects do not possess an appreciable net electric charge,[citation needed] and the magnetic field of stars arise through other processes. As a model of realistic black holes, it omits any description of infalling baryonic matter, light (null dusts) or dark matter, and thus provides at best an incomplete description of stellar mass black holes and active galactic nuclei. The solution is of theoretical and mathematical interest as it does provide a fairly simple cornerstone for further exploration. The Kerr–Newman solution is a special case of more general exact solutions of the Einstein–Maxwell equations with non-zero cosmological constant.[1] ## History In Dec 1963 Kerr and Schild found the Kerr-Schild metrics that gave all Einstein spaces that are exact linear perturbations of Minkowski space. In early 1964 Roy Kerr looked for all Einstein-Maxwell spaces with this same property. By Feb 1964 the special case where the Kerr-Schild spaces were charged (this includes the Kerr-Newman solution) was known but the general case where the special directions were not geodesics of the underlying Minkowski space proved very difficult. The problem was given to George Debney to try to solve but was given up by March 1964. About this time Ezra T. Newman found the solution for charged Kerr by guesswork. In 1965, Ezra "Ted" Newman found the axisymmetric solution of Einstein's field equation for a black hole which is both rotating and electrically charged.[2][3] This formula for the metric tensor ${\displaystyle g_{\mu \nu }\!}$ is called the Kerr–Newman metric. It is a generalisation of the Kerr metric for an uncharged spinning point-mass, which had been discovered by Roy Kerr two years earlier.[4] Four related solutions may be summarized by the following table: Non-rotating (J = 0) Rotating (J ≠ 0) Uncharged (Q = 0) Schwarzschild Kerr Charged (Q ≠ 0) Reissner–Nordström Kerr–Newman where Q represents the body's electric charge and J represents its spin angular momentum. ## Overview of the solution Newman's result represents the simplest stationary, axisymmetric, asymptotically flat solution of Einstein's equations in the presence of an electromagnetic field in four dimensions. It is sometimes referred to as an "electrovacuum" solution of Einstein's equations. Any Kerr–Newman source has its rotation axis aligned with its magnetic axis.[5] Thus, a Kerr–Newman source is different from commonly observed astronomical bodies, for which there is a substantial angle between the rotation axis and the magnetic moment.[6] Specifically, neither the Sun, nor any of the planets in the Solar system have magnetic fields aligned with the spin axis. Thus, while the Kerr solution describes the gravitational field of the Sun and planets, the magnetic fields arise by a different process. If the Kerr–Newman potential is considered as a model for a classical electron, it predicts an electron having not just a magnetic dipole moment, but also other multipole moments, such as an electric quadrupole moment.[7] An electron quadrupole moment has not yet been experimentally detected; it appears to be zero.[7] In the G = 0 limit, the electromagnetic fields are those of a charged rotating disk inside a ring where the fields are infinite. The total field energy for this disk is infinite, and so this G=0 limit does not solve the problem of infinite self-energy.[8] Like the Kerr metric for an uncharged rotating mass, the Kerr–Newman interior solution exists mathematically but is probably not representative of the actual metric of a physically realistic rotating black hole due to issues with the stability of the Cauchy horizon, due to mass inflation driven by infalling matter. Although it represents a generalization of the Kerr metric, it is not considered as very important for astrophysical purposes, since one does not expect that realistic black holes have an significant electric charge (they are expected to have a miniscule positive charge, but only because the proton has a much larger momentum than the electron, and is thus more likely to overcome electrostatic repulsion and be carried by momentum across the horizon). The Kerr–Newman metric defines a black hole with an event horizon only when the combined charge and angular momentum are sufficiently small:[9] ${\displaystyle J^{2}/M^{2}+Q^{2}\leq M^{2}.}$ An electron's angular momentum J and charge Q (suitably specified in geometrized units) both exceed its mass M, in which case the metric has no event horizon and thus there can be no such thing as a black hole electron — only a naked spinning ring singularity.[10] Such a metric has several seemingly unphysical properties, such as the ring's violation of the cosmic censorship hypothesis, and also appearance of causality-violating closed timelike curves in the immediate vicinity of the ring.[11] A 2007 paper by Russian theorist Alexander Burinskii describes an electron as a gravitationally confined ring singularity without an event horizon. It has some, but not all of the predicted properties of a black hole.[12] As Burinskii described it: In this work we obtain an exact correspondence between the wave function of the Dirac equation and the spinor (twistorial) structure of the Kerr geometry. It allows us to assume that the Kerr–Newman geometry reflects the specific space-time structure of electron, and electron contains really the Kerr–Newman circular string of Compton size.[12] ## Limiting cases The Kerr–Newman metric can be seen to reduce to other exact solutions in general relativity in limiting cases. It reduces to: • The Kerr metric as the charge Q goes to zero. • The Reissner–Nordström metric as the angular momentum J (or a = JM ) goes to zero. • The Schwarzschild metric as both the charge Q and the angular momentum J (or a) are taken to zero. • Minkowski space if the mass M, the charge Q, and the rotational parameter a are all zero. Alternately, if gravity is intended to be removed, Minkowski space arises if the gravitational constant G is zero, without taking the mass and charge to zero. In this case, the electric and magnetic fields are more complicated than simply the fields of a charged magnetic dipole; the zero-gravity limit is not trivial. ## The metric The Kerr–Newman metric describes the geometry of spacetime for a rotating charged black hole with mass M, charge Q and angular momentum J. The formula for this metric depends upon what coordinates or coordinate conditions are selected. Two forms are given below: Boyer-Lindquist coordinates, and Kerr-Schild coordinates. The gravitational metric alone is not sufficient to determine a solution to the Einstein field equations; the electromagnetic stress tensor must be given as well. Both are provided in each section. ## Boyer-Lindquist coordinates One way to express this metric is by writing down its line element in a particular set of spherical coordinates,[13] also called Boyer–Lindquist coordinates: ${\displaystyle c^{2}d\tau ^{2}=-\left({\frac {dr^{2}}{\Delta }}+d\theta ^{2}\right)\rho ^{2}+\left(c\,dt-a\sin ^{2}\theta \,d\phi \right)^{2}{\frac {\Delta }{\rho ^{2}}}-\left(\left(r^{2}+a^{2}\right)d\phi -ac\,dt\right)^{2}{\frac {\sin ^{2}\theta }{\rho ^{2}}}}$ where the coordinates (r, θ, ϕ) are standard spherical coordinate system, and the length-scales: ${\displaystyle a={\frac {J}{Mc}}\,,}$ ${\displaystyle \ \rho ^{2}=r^{2}+a^{2}\cos ^{2}\theta \,,}$ ${\displaystyle \ \Delta =r^{2}-r_{s}r+a^{2}+r_{Q}^{2}\,,}$ have been introduced for brevity. Here rs is the Schwarzschild radius of the massive body, which is related to its total mass-equivalent M by ${\displaystyle r_{s}={\frac {2GM}{c^{2}}}}$ where G is the gravitational constant, and rQ is a length-scale corresponding to the electric charge Q of the mass ${\displaystyle r_{Q}^{2}={\frac {Q^{2}G}{4\pi \epsilon _{0}c^{4}}}}$ where 1/(4πε0) is Coulomb's force constant. ### Electromagnetic field tensor in Boyer-Lindquist form The electromagnetic potential in Boyer–Lindquist coordinates is[14][15] ${\displaystyle A_{\mu }=\left({\frac {r\ r_{Q}}{\rho ^{2}}},0,0,-{\frac {c^{2}\ a\ r\ r_{Q}\sin ^{2}\theta }{\rho ^{2}\ G\ M}}\right)}$ while the Maxwell-tensor is defined by ${\displaystyle F_{\mu \nu }={\frac {\partial A_{\nu }}{\partial x^{\mu }}}-{\frac {\partial A_{\mu }}{\partial x^{\nu }}}\ \to \ F^{\mu \nu }=g^{\mu \sigma }\ g^{\nu \kappa }\ F_{\sigma \kappa }}$ In combination with the Christoffel symbols the second order equations of motion can be derived with ${\displaystyle {{{\ddot {x}}^{i}=-\Gamma _{jk}^{i}\ {{\dot {x}}^{j}}\ {{\dot {x}}^{k}}+q\ {F^{ik}}\ {{\dot {x}}^{j}}}\ {g_{jk}}}}$ where ${\displaystyle q}$ is the charge per mass of the testparticle. ## Kerr–Schild coordinates The Kerr–Newman metric can be expressed in the Kerr–Schild form, using a particular set of Cartesian coordinates, proposed by Kerr and Schild in 1965. The metric is as follows.[16][17][18] ${\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+fk_{\mu }k_{\nu }\!}$ ${\displaystyle f={\frac {Gr^{2}}{r^{4}+a^{2}z^{2}}}\left[2Mr-Q^{2}\right]}$ ${\displaystyle \mathbf {k} =(k_{x},k_{y},k_{z})=\left({\frac {rx+ay}{r^{2}+a^{2}}},{\frac {ry-ax}{r^{2}+a^{2}}},{\frac {z}{r}}\right)}$ ${\displaystyle k_{0}=1.\!}$ Notice that k is a unit vector. Here M is the constant mass of the spinning object, Q is the constant charge of the spinning object, η is the Minkowski metric, and a=J/M is a constant rotational parameter of the spinning object. It is understood that the vector ${\displaystyle {\vec {a}}}$ is directed along the positive z-axis, i.e. ${\displaystyle {\vec {a}}=a{\hat {z}}}$ . The quantity r is not the radius, but rather is implicitly defined like this: ${\displaystyle 1={\frac {x^{2}+y^{2}}{r^{2}+a^{2}}}+{\frac {z^{2}}{r^{2}}}}$ Notice that the quantity r becomes the usual radius R ${\displaystyle r\to R={\sqrt {x^{2}+y^{2}+z^{2}}}}$ when the rotational parameter a approaches zero. In this form of solution, units are selected so that the speed of light is unity (c = 1). In order to provide a complete solution of the Einstein–Maxwell equations, the Kerr–Newman solution not only includes a formula for the metric tensor, but also a formula for the electromagnetic potential:[16][19] ${\displaystyle A_{\mu }={\frac {Qr^{3}}{r^{4}+a^{2}z^{2}}}k_{\mu }}$ At large distances from the source (R >> a), these equations reduce to the Reissner–Nordström metric with: ${\displaystyle A_{\mu }={\frac {Q}{R}}k_{\mu }}$ In the Kerr–Schild form of the Kerr–Newman metric, the determinant of the metric tensor is everywhere equal to negative one, even near the source.[1] ### Electromagnetic fields in Kerr-Schild form The electric and magnetic fields can be obtained in the usual way by differentiating the four-potential to obtain the electromagnetic field strength tensor. It will be convenient to switch over to three-dimensional vector notation. ${\displaystyle A_{\mu }=\left(-\phi ,A_{x},A_{y},A_{z}\right)\,}$ The static electric and magnetic fields are derived from the vector potential and the scalar potential like this: ${\displaystyle {\vec {E}}=-{\vec {\nabla }}\phi \,}$ ${\displaystyle {\vec {B}}={\vec {\nabla }}\times {\vec {A}}\,}$ Using the Kerr–Newman formula for the four-potential in the Kerr–Schild form yields the following concise complex formula for the fields:[20] ${\displaystyle {\vec {E}}+i{\vec {B}}=-{\vec {\nabla }}\Omega \,}$ ${\displaystyle \Omega ={\frac {Q}{\sqrt {({\vec {R}}-i{\vec {a}})^{2}}}}\,}$ The quantity omega (${\displaystyle \Omega }$ ) in this last equation is similar to the Coulomb potential, except that the radius vector is shifted by an imaginary amount. This complex potential was discussed as early as the nineteenth century, by the French mathematician Paul Émile Appell.[21] ## Irreducible mass The total mass-equivalent M, which contains the electric field-energy and the rotational energy, and the irreducible mass Mirr are related by[22][23] ${\displaystyle M_{\rm {irr}}={\frac {1}{2}}{\sqrt {2M^{2}-r_{Q}^{2}c^{4}/G^{2}+2M{\sqrt {M^{2}-(r_{Q}^{2}+a^{2})c^{4}/G^{2}}}}}}$ which can be inverted to obtain ${\displaystyle M={\frac {4M_{\rm {irr}}^{2}+r_{Q}^{2}c^{4}/G^{2}}{2{\sqrt {4M_{\rm {irr}}^{2}-a^{2}c^{4}/G^{2}}}}}}$ In order to electrically charge and/or spin a neutral and static body, energy has to be applied to the system. Due to the mass–energy equivalence, this energy also has a mass-equivalent; therefore M is always higher than Mirr. If for example the rotational energy of a black hole is extracted via the Penrose processes[24][25], the remaining mass-energy will always stay greater than or equal to Mirr. ## Important surfaces Event horizons and ergospheres of a charged and spinning black hole in pseudospherical r,θ,φ and cartesian x,y,z coordinates. Setting ${\displaystyle 1/g_{rr}}$ to 0 and solving for ${\displaystyle r}$ gives the inner and outer event horizon, which is located at the Boyer-Lindquist coordinate ${\displaystyle r_{\text{H}}^{\pm }={\frac {r_{\rm {s}}}{2}}\pm {\sqrt {{\frac {r_{\rm {s}}^{2}}{4}}-a^{2}-r_{Q}^{2}}}.}$ Repeating this step with ${\displaystyle g_{tt}}$ gives the inner and outer ergosphere ${\displaystyle r_{\text{E}}^{\pm }={\frac {r_{\rm {s}}}{2}}\pm {\sqrt {{\frac {r_{\rm {s}}^{2}}{4}}-a^{2}\cos ^{2}\theta -r_{Q}^{2}}}.}$ Testparticle in orbit around a spinning and charged black hole (a/M=0.9, Q/M=0.4) ## Equations of motion For brevity, we further use dimensionless natural units of ${\displaystyle G=M=c=K=1}$ , with Coulomb's constant ${\displaystyle K}$ , where ${\displaystyle a}$ reduces to ${\displaystyle Jc/G/M^{2}}$ and ${\displaystyle Q}$ to ${\displaystyle Q/M\ {\sqrt {K/G}}}$ , and the equations of motion for a testparticle of charge ${\displaystyle q}$ become[26][27] ${\displaystyle {\dot {t}}}$ ${\displaystyle ={\frac {\csc ^{2}\theta \ ({L_{z}}(a\ \Delta \sin ^{2}\theta -a\ (a^{2}+r^{2})\sin ^{2}\theta )-q\ Q\ r\ (a^{2}+r^{2})\sin ^{2}\theta +E((a^{2}+r^{2})^{2}\sin ^{2}\theta -a^{2}\Delta \sin ^{4}\theta ))}{\Delta \rho ^{2}}}}$ ${\displaystyle {\dot {r}}=\pm {\frac {\sqrt {((r^{2}+a^{2})\ E-a\ L_{z}-q\ Q\ r)^{2}-\Delta \ (C+r^{2})}}{\rho ^{2}}}}$ ${\displaystyle {\dot {\theta }}=\pm {\frac {\sqrt {C-(a\cos \theta )^{2}-(a\ \sin ^{2}\theta \ E-L_{z})/\sin \theta }}{\rho ^{2}}}}$ ${\displaystyle {\dot {\phi }}={\frac {E\ (a\ \sin ^{2}\theta \ (r^{2}+a^{2})-a\ \sin ^{2}\theta \ \Delta )+L_{z}\ (\Delta -a^{2}\ \sin ^{2}\theta )-q\ Q\ r\ a\ \sin ^{2}\theta }{\rho ^{2}\ \Delta \ \sin ^{2}\theta }}}$ with ${\displaystyle E}$ for the total energy and ${\displaystyle L_{z}}$ for the axial angular momentum. ${\displaystyle C}$ is the Carter constant: ${\displaystyle C=p_{\theta }^{2}+\cos ^{2}\theta \left(a^{2}(1-E^{2})+{\frac {L_{z}^{2}}{\sin ^{2}\theta }}\right)=a^{2}\ (1-E^{2})\ \sin ^{2}\delta +L_{z}^{2}\ \tan ^{2}\delta ={\rm {const.}}}$ where ${\displaystyle p_{\theta }={\dot {\theta }}\ \rho ^{2}}$ is the poloidial component of the testparticle's angular momentum, and ${\displaystyle \delta }$ the orbital inclination angle. Ray traced shadow of a spinning and charged black hole with the parameters a²+Q²=1M². The left side of the black hole is rotating towards the observer. ${\displaystyle L_{z}={\frac {v^{\phi }\ {\bar {R}}}{\sqrt {1-v^{2}}}}={\rm {const.}}}$ and ${\displaystyle E={\sqrt {\frac {\Delta \ \rho ^{2}}{(1-v^{2})\ \chi }}}+\Omega \ L_{z}={\rm {const.}}}$ are also conserved quantities. ${\displaystyle \Omega =-{\frac {g_{t\phi }}{g_{\phi \phi }}}={\frac {a\left(2r-Q^{2}\right)}{\chi }}}$ is the frame dragging induced angular velocity. The shorthand term ${\displaystyle \chi }$ is defined by ${\displaystyle \chi =\left(a^{2}+r^{2}\right)^{2}-a^{2}\ \sin ^{2}\theta \ \Delta }$ The relation between the coordinate derivatives ${\displaystyle {\dot {r}},\ {\dot {\theta }},\ {\dot {\phi }}}$ and the local 3-velocity ${\displaystyle v}$ is ${\displaystyle v^{r}={\dot {r}}\ {\sqrt {\frac {\rho ^{2}\ (1-v^{2})}{\Delta }}}}$ ${\displaystyle v^{\theta }={\dot {\theta }}\ {\sqrt {\rho ^{2}\ (1-v^{2})}}}$ for the poloidial, ${\displaystyle v^{\phi }={\frac {L_{z}{\sqrt {1-v^{2}}}}{\bar {R}}}}$ for the axial and ${\displaystyle v={\frac {\sqrt {{\dot {t}}^{2}-\varsigma ^{2}}}{\dot {t}}}={\sqrt {\frac {\chi \ (E-L_{z}\ \Omega )^{2}-\Delta \ \rho ^{2}}{\chi \ (E-L_{z}\ \Omega )^{2}}}}}$ for the total local velocity, where ${\displaystyle {\bar {R}}={\sqrt {-g_{\phi \phi }}}={\sqrt {\frac {\chi }{\rho ^{2}}}}\ \sin \theta }$ is the axial radius of gyration (local circumference divided by 2π), and ${\displaystyle \varsigma ={\sqrt {g^{tt}}}={\frac {\chi }{\Delta \ \rho ^{2}}}}$ the gravitational time dilation component. The local radial escape velocity for a neutral particle is therefore ${\displaystyle v_{\rm {esc}}={\frac {\sqrt {\varsigma ^{2}-1}}{\varsigma }}}$ . ## References 1. ^ a b Stephani, Hans et al. Exact Solutions of Einstein's Field Equations (Cambridge University Press 2003). See page 485 regarding determinant of metric tensor. See page 325 regarding generalizations. 2. ^ Newman, Ezra; Janis, Allen (1965). "Note on the Kerr Spinning-Particle Metric". Journal of Mathematical Physics. 6 (6): 915–917. Bibcode:1965JMP.....6..915N. doi:10.1063/1.1704350. 3. ^ Newman, Ezra; Couch, E.; Chinnapared, K.; Exton, A.; Prakash, A.; Torrence, R. (1965). "Metric of a Rotating, Charged Mass". Journal of Mathematical Physics. 6 (6): 918–919. Bibcode:1965JMP.....6..918N. doi:10.1063/1.1704351. 4. ^ Kerr, RP (1963). "Gravitational field of a spinning mass as an example of algebraically special metrics". Physical Review Letters. 11 (5): 237–238. Bibcode:1963PhRvL..11..237K. doi:10.1103/PhysRevLett.11.237. 5. ^ Punsly, Brian (10 May 1998). "High-energy gamma-ray emission from galactic Kerr–Newman black holes. I. The central engine". The Astrophysical Journal. 498 (2): 646. Bibcode:1998ApJ...498..640P. doi:10.1086/305561. All Kerr–Newman black holes have their rotation axis and magnetic axis aligned; they cannot pulse. 6. ^ Lang, Kenneth (2003). The Cambridge Guide to the Solar System. Cambridge University Press. p. 96. ISBN 9780521813068 – via Google Books. 7. ^ a b Rosquist, Kjell (2006). "Gravitationally induced electromagnetism at the Compton scale". Classical and Quantum Gravity. 23 (9): 3111–3122. arXiv:gr-qc/0412064. Bibcode:2006CQGra..23.3111R. doi:10.1088/0264-9381/23/9/021. 8. ^ Lynden-Bell, D. (2004). "Electromagnetic magic: The relativistically rotating disk". Physical Review D. 70 (10): 105017. arXiv:gr-qc/0410109. Bibcode:2004PhRvD..70j5017L. doi:10.1103/PhysRevD.70.105017. 9. ^ Meinel, Reinhard (29 October 2015). "A Physical Derivation of the Kerr–Newman Black Hole Solution". In Nicolini P., Kaminski M., Mureika J., Bleicher M. (eds.). 1st Karl Schwarzschild Meeting on Gravitational Physics. Springer Proceedings in Physics. 170. pp. 53–61. arXiv:1310.0640. doi:10.1007/978-3-319-20046-0_6. ISBN 978-3-319-20045-3.CS1 maint: uses editors parameter (link) 10. ^ Burinskii, Alexander (2008). "The Dirac-Kerr electron". Gravitation and Cosmology. 14: 109–122. arXiv:hep-th/0507109. doi:10.1134/S0202289308020011. 11. ^ Carter, Brandon (1968). "Global structure of the Kerr family of gravitational fields". Physical Review. 174 (5): 1559. doi:10.1103/PhysRev.174.1559. 12. ^ a b Burinskii, Alexander (2007). "Kerr geometry as space-time structure of the Dirac electron". arXiv:0712.0577 [hep-th]. 13. ^ Hajicek, Petr et al. An Introduction to the Relativistic Theory of Gravitation, page 243 (Springer 2008). 14. ^ Brandon Carter: Global structure of the Kerr family of gravitational fields (1968) 15. ^ Luongo, Orlando; Quevedo, Hernando (2014). "Characterizing repulsive gravity with curvature eigenvalues". Physical Review D. 90 (8): 084032. arXiv:1407.1530. Bibcode:2014PhRvD..90h4032L. doi:10.1103/PhysRevD.90.084032. 16. ^ a b Debney, G. C.; Kerr, R. P.; Schild, A. (1969). "Solutions of the Einstein and Einstein‐Maxwell Equations". Journal of Mathematical Physics. 10 (10): 1842–1854. doi:10.1063/1.1664769.. Especially see equations (7.10), (7.11) and (7.14). 17. ^ Balasin, Herbert; Nachbagauer, Herbert (1994). "Distributional energy--momentum tensor of the Kerr--Newman spacetime family". Classical and Quantum Gravity. 11 (6): 1453–1461. arXiv:gr-qc/9312028. Bibcode:1994CQGra..11.1453B. doi:10.1088/0264-9381/11/6/010. 18. ^ Berman, Marcelo. “Energy of Black Holes and Hawking’s Universe” in Trends in Black Hole Research, page 148 (Kreitler ed., Nova Publishers 2006). 19. ^ Burinskii, A. “Kerr Geometry Beyond the Quantum Theory” in Beyond the Quantum, page 321 (Theo Nieuwenhuizen ed., World Scientific 2007). The formula for the vector potential of Burinskii differs from that of Debney et al. merely by a gradient which does not affect the fields. 20. ^ Gair, Jonathan. "Boundstates in a Massless Kerr–Newman Potential" Archived 2011-09-26 at the Wayback Machine. 21. ^ Appell, Math. Ann. xxx (1887) pp. 155–156. Discussed by Whittaker, Edmund and Watson, George. A Course of Modern Analysis, page 400 (Cambridge University Press 1927). 22. ^ 23. ^ Eq. 57 in Pradhan, Parthapratim (2014). "Black hole interior mass formula". The European Physical Journal C. 74 (5): 2887. arXiv:1310.7126. Bibcode:2014EPJC...74.2887P. doi:10.1140/epjc/s10052-014-2887-2. 24. ^ Charles Misner, Kip S. Thorne, John. A. Wheeler: Gravitation, pages 877 & 908 25. ^ Bhat, Manjiri; Dhurandhar, Sanjeev; Dadhich, Naresh (1985). "Energetics of the Kerr-Newman black hole by the penrose process". Journal of Astrophysics and Astronomy. 6 (2): 85–100. doi:10.1007/BF02715080. 26. ^ 27. ^ Hackmann, Eva; Xu, Hongxiao (2013). "Charged particle motion in Kerr–Newmann space-times". Physical Review D. 87 (12): 4. arXiv:1304.2142. Bibcode:2013PhRvD..87l4030H. doi:10.1103/PhysRevD.87.124030.	2019-11-12 13:51:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 67, "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": 0.9055816531181335, "perplexity": 1425.0595017688565}, "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/1573496665573.50/warc/CC-MAIN-20191112124615-20191112152615-00082.warc.gz"}
https://ccssmathanswers.com/180-days-of-math-for-sixth-grade-day-56-answers-key/	# 180 Days of Math for Sixth Grade Day 56 Answers Key By accessing our 180 Days of Math for Sixth Grade Answers Key Day 56 regularly, students can get better problem-solving skills. ## 180 Days of Math for Sixth Grade Answers Key Day 56 Direcions Solve each problem. Question 1. 163+25 = 188. Explanation: By adding 163 and 25 which is 163+25 = 188. Question 2. Multiply 65 and 10. _______________________ 65×10 = 650. Explanation: The multiplication of 65 and 10 is 65×10 which is 650. Question 3. 35÷7 = 5. Explanation: By dividing 35 by 7 we will get the result as 35÷7 which is 5. Question 4. Is -5 a negative number? _______________________ Yes. Explanation: Yes, as there is a minus symbol. So -5 is a negative number. Question 5. Arrange the fractions in descending order. $$\frac{1}{3}$$, $$\frac{1}{2}$$, $$\frac{1}{4}$$ ___ ____ ___ $$\frac{1}{4}$$, $$\frac{1}{3}$$, $$\frac{1}{2}$$. Explanation: Given the fractions are $$\frac{1}{3}$$, $$\frac{1}{2}$$, $$\frac{1}{4}$$. Now we will change the fraction to decimals which is 0.33, 0.5, 0.25. So the descending order is 0.25 which is $$\frac{1}{4}$$ and 0.33 which is $$\frac{1}{3}$$, and 0.5 $$\frac{1}{2}$$. Question 6. -1 + 7 + (-3) = ___ -1 + 7 + (-3) = 3. Explanation: Given the expression is -1 + 7 + (-3), so -1 + 7 + (-3) = 6 – 3 = 3. Question 7. 19 24 = 43 19+24 = 23. Explanation: By adding 19 and 24 we will get 43. Question 8. Solve for k. $$\frac{k}{7}$$ = 5 k = _____ k = 35. Explanation: Given that $$\frac{k}{7}$$ = 5, k = 5×7 = 35. Question 9. What is the elapsed time from 11:34 A.M. to 12:06 P.M.? _______________________ The elapsed time will be 1 hour and 1 minute. Explanation: Given the time is 11:34 A.M. to 12:06 P.M. So the elapsed time will be 1 hour and 1 minute. Question 10. What is the sum of the interior angles of a triangle? _______________________ 180 degrees. Explanation: The sum of the interior angles of a triangle is 180 degrees. Question 11. Order the size of each group from largest to smallest. _______________________ _______________________ Women, girls, boys, and men. Explanation: The order of the size of each group from largest to smallest is women, girls, boys, and men. Question 12. Conner has $3 more than Chi and$2 more than Cher. Jan has $1 less than Cher. Together they have$28. How much does Conner have? ______________________ Conner had $11. Explanation: Given that Conner has$3 more than Chi and $2 more than Cher and Jan has$1 less than Cher. Let the money Cher had to be X, so Jan will have $(X-1), Chi will have$(X+2) and Conner will have $(X+3) and together they have$28. So (X-1) + (X+2) + (X+3) = $28 3X + 4 =$28, X = $$\frac{24}{3}$$ So Jan had $(X-1) =$(8-1) = $7, Chi had$(X+2) = $(8+1) =$9, Conner had $(X+3) =$(8+3) = \$11.	2022-05-19 01:32:44	{"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": 0.4981662333011627, "perplexity": 3062.186388957736}, "config": {"markdown_headings": true, "markdown_code": false, "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/1652662522741.25/warc/CC-MAIN-20220519010618-20220519040618-00272.warc.gz"}
https://codereview.stackexchange.com/questions/87580/hackerearth-find-array-index-with-cumulative-sum-till-there-matching-a-given-ta	# HackerEarth: Find array index with cumulative sum till there matching a given target The problem was like this: Given an array: [1, 2, 1, 3, 4] where each index value shows the production of that particular day. So, at index 0, production is 1, index 1 - production is 2, so on... Then we're given some targets (N numbers of them). For a given target T, we've to find the index till which the cumulative production matches the target. For e.g., a target of 4 will be achieved by index 2 (1 + 2 + 1 = 4 >= 4). Similarly, a target of 10 will be achieved by index 4 (1 + 2 + 1 + 3 + 4 = 11 >= 10). Basically, the target is achieved for index i, if cumulative sum till that index is greater than target. We've find the minimum index for each target. For the above array, sample input and output would be: 4, 2, 10 2, 1, 4 My idea for the problem is: While reading the array as input, I pre-process the array to store cumulative at each index. So, I'll convert the above array to: [1, 3, 4, 7, 11] and then for the given target, I'll find the minimum value in array greater than that target (can easily be done using binary search). Assuming the total number of targets input is Q and size of initial array is T, my algorithm would be O(Qlog(T)), which seems to be quite fine. But, it failed with TLE for input #5 on hackerearth. Then I thought of maintaining a target cache (Map<Long, Integer>), which I'll populate after reading each array element. That map will store the index of each possible target till the maximum value. For the above array, the map which I build would be like: {1=0, 2=1, 3=1, 4=2, 5=3, 6=3, 7=3, 8=4, 9=4, 10=4, 11=4} So, for each target, I can just do map.get(target), to get the index. For this solution, pre-processing numbers, and building map should be O(maxTarget), where maxTarget is the maximum value a target can get. And then fetching result is just O(1). But again, this also failed with TLE. I'm completely out of thought to modify this to make the last input pass. Here's my code for 2nd approach (map): public static void main(String[] args) throws Exception { int T = Integer.parseInt(line); StringTokenizer tokenizer = new StringTokenizer(input, " "); long[] numbers = new long[T]; Map<Long, Integer> targetCache = new HashMap<Long, Integer>(); int i = 0; while (tokenizer.hasMoreTokens()) { int nextTarget = Integer.parseInt(tokenizer.nextToken()); long previousTarget = i == 0 ? 0 : numbers[i - 1]; numbers[i] = previousTarget + nextTarget; long currentCumulativeTarget = numbers[i]; for (long j = previousTarget + 1; j <= currentCumulativeTarget; j++) { targetCache.put(j, i); } i++; } System.out.println(targetCache); for (int x = 0; x < Q; x++) { Integer index = targetCache.get(target); // If target can't be matched, print -1 System.out.println(index == null ? -1 : index); } } How can the above solution be optimized further? Any leads? BTW, I can't test the solution, as that was the part of a interview test, and I can't see that question again. With Java performance it is important to create methods other than the main method. Java compilation is method-based, and it often relates to how often the method is called. Since the main method is only called once, it is seldom optimized very well. In addition to that, Java println is also slow, as it requires access to the console, and it is synchronizes, and flushes it. As a result, many println calls are also slower than fewer larger printlns. So, batch up your output as much as you can. All in all, create methods, batch the output, and .... use a binary search. Then, use some useful Java native methods. Split is a good one, and also try-with-resources for the input streams. With the format of your input, a Scanner may be simpler too.... private static int[] readInts(Scanner scanner) { int size = scanner.nextInt(); int[] values = new int[size]; for (int i = 0; i < size; i++) { values[i] = scanner.nextInt(); } return values; } private static void accumulateValues(int[] values) { for (int i = 1; i < values.length; i++) { values[i] += values[i - 1]; } } public static void main(String[] args) { try (Scanner scanner = new Scanner(System.in)) { accumulateValues(values); int[] results = new int[searches.length]; for (int i = 0; i < searches.length; i++) { results[i] = Arrays.binarySearch(values, searches[i]); if (results[i] < 0) { results[i] = -results[i] - 1; } } StringBuilder sb = new StringBuilder(); for (int r : results) { sb.append(r).append("\n"); } System.out.println(sb.toString()); } } • Damn!! Never thought of such things. Unfortunately I couldn't test the solution now, but your answer does makes sense. – Rohit Jain Apr 22 '15 at 5:03	2020-02-17 08:18:48	{"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": 0.3233638405799866, "perplexity": 2564.715953519991}, "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-2020-10/segments/1581875141749.3/warc/CC-MAIN-20200217055517-20200217085517-00034.warc.gz"}
http://kullabs.com/classes/subjects/units/lessons/notes/note-detail/2316	Note on Cube Root • Note • Things to remember • Exercise • Quiz To find cube root, make triple of equal factors. The opposite of cubing a number is called finding the cube root. A cube root is a number, that is multiplied by itself three times in order to create a cubic value. A cube root of a number x is a number, such that a3= x. All real numbers (except zero) have exactly one real cube root. Cube of 6 = 6³ =216 Cube root of 216 = 6 Examples • The cube root of 64 is 4 ( because 4x4x4=64) • The cube root of 125 is 5 ( because 5x5x5=125) • The cube root of 512 is 8 ( because 8x8x8=512 ) The symbol, $$\sqrt [3]{}$$, means cube root, so $$\sqrt [3]{27}$$ means "cube root of 27" and $$\sqrt[3]{64}$$means "Cube root of 64" Thus $$\sqrt [3]{27}$$ = $$\sqrt [3]{3^3}$$ = 3 and $$\sqrt[3]{64}$$ = $$\sqrt[3]{4^3}$$ = 4 A natural number is known as a perfect cube or a cube number. Cube root of a perfect cube can be found by factorization method. • The number should be the factor of the prime number or should be expressed as the factor of the prime number. • Make triples of the factor and each triple should be equal. • Take one factor from each triple. • The product is the cube root of the given number. Examples 1. Find the cube root of 2×2×2×3×3×3 = 2 × 3 = 6 2. Find the cube root of 729. Solution: $$\sqrt[3]{729}$$ = $$\sqrt[3]{3×3×3×3×3×3}$$ = $$\sqrt[3]{3^3×3^3}$$ = 3×3 = 9 • A cube root is a number, that multiplied by itself three times in order to create a cubic value. • To find cube root, make triple of equal factors. • The opposite of cubing a number is called finding the cube root. . Very Short Questions Solution: cube of 6 = 63 = 6×6×6 = 216 Solution: Cube of 16 = 163 =16 ×16 ×16 = 4096 Solution: cube root of 125 =$$\sqrt[3]{125}$$ =$$\sqrt[3]{5×5×5}$$ =$$\sqrt[3]{5^3}$$ = 5 5 125 5 25 5 Solution: Cube of 20 = 203 = 20 × 20 × 20 = 8000 Soln: Cube of 35=(35)3 =35×35×35 =42875 Solution: Cube of 400 = 4003 = 400×400×400 = 64000000 Solution: 3 81 3 27 3 9 3 81 = 3×3×3 =33 ∴ The required number is 3. Solution: 2 128 2 64 2 32 2 16 2 8 2 4 2 128 = 23×23×2 ∴The required number is 2. Solution: 3 135 3 45 3 15 5 135 = 33× 5 ∴The required number is 5. Solution: Cube root of 1331 = $$\sqrt{1331}$$ = $$\sqrt[3]{11 ×11 ×11}$$ = 11 11 1331 11 121 11 Solution: 3 243 3 81 3 27 3 9 3 243 = 33× 32 ∴The required number is 3. Solution: 2 192 2 96 2 48 2 24 2 12 2 6 3 192 = 23×23×3 ∴The required number is 3. Solution: 5 625 5 125 5 25 5 325 = 53×5 ∴The required number is 5. Solution: 3 675 3 225 3 75 5 25 5 675 = 33×52 ∴The required number is 5. 0% 2000 9000 8000 8500 64000000 72000000 63000000 6400000 1 5 7 2 8 5 9 10 9 8 3 6 2 7 3 5 2 5 10 11 6 5 3 4 4 8 7 2 16 8 12 5 8 6 3 4 -243 -343 243 343 • Find the cube of:0.06 (frac{27}{125000}) (frac{25}{124000}) (frac{28}{125100}) (frac{27}{126000}) 1.9 2.8 3.9 1.8 44 33 22 11 DISCUSSIONS ABOUT THIS NOTE No discussion on this note yet. Be first to comment on this note	2018-11-16 17:05:58	{"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": 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": 0.6472732424736023, "perplexity": 1055.3936527785736}, "config": {"markdown_headings": false, "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-2018-47/segments/1542039743105.25/warc/CC-MAIN-20181116152954-20181116174954-00453.warc.gz"}
https://socratic.org/questions/what-is-the-arclength-of-f-t-t-sqrt-t-2-2-t-e-t-2-on-t-in-1-1-1	# What is the arclength of f(t) = (t/sqrt(t^2+2),t/e^(t-2)) on t in [-1,1]? May 26, 2018 $s = {\int}_{- 1}^{1} \sqrt{\frac{4}{{t}^{2} + 2} ^ 3 + {e}^{4 - 2 t} {\left(1 - t\right)}^{2}} \approx 22.9212$ #### Explanation: Using the Arc Length Formula, we have $\textcolor{b l u e}{s = {\int}_{- 1}^{1} \sqrt{{\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)}^{2} + {\left(\frac{\mathrm{dy}}{\mathrm{dt}}\right)}^{2}} \mathrm{dt}}$ Since $x \left(t\right) = \frac{t}{\sqrt{{t}^{2} + 2}}$, differentiating and simplifying, we have $\frac{\mathrm{dx}}{\mathrm{dt}} = \frac{2}{{t}^{2} + 2} ^ \left(\frac{3}{2}\right)$ Similarly, $\frac{\mathrm{dy}}{\mathrm{dt}} = \left(1 - t\right) {e}^{2 - t}$ Substituting into the formula, we have $s = {\int}_{- 1}^{1} \sqrt{{\left(\frac{2}{{t}^{2} + 2} ^ \left(\frac{3}{2}\right)\right)}^{2} + {\left(\left(1 - t\right) {e}^{2 - t}\right)}^{2}} \mathrm{dt}$ $= {\int}_{- 1}^{1} \sqrt{\frac{4}{{t}^{2} + 2} ^ 3 + {e}^{4 - 2 t} {\left(1 - t\right)}^{2}}$ This integral unfortunately cannot be expressed in terms of standard functions or constants, so the best you can get is either an infinite series or an approximation. In this case, the integral evaluates to $s \approx 22.9212$, which is our final answer. For arclength type questions, please take note that the vast majority of the resultant integrals will, like this one, be unsolvable in terms of standard functions. ### Explanation for Arc Length Formula To find the arc length $\Delta s$ over a time $\Delta t$, you can try to approximate it with a straight line, using the hypotenuse of the right-angled triangle with sides $\Delta x , \Delta y$, which is the change in the x- and y- coordinates over the time $t$. $\Delta s \approx \sqrt{\Delta {x}^{2} + \Delta {y}^{2}}$ Of course, this will often be wildly inaccurate since most curves aren't exactly a straight line. However, for a continuous function, the smaller time interval you choose, the more accurate the above approximation is (you can imagine zooming in on a function's graph: the more you zoom, the closer the function looks like a straight line) Thus, at the limit when you shorten the time interval to be really (infinitesimally) small, you get $\mathrm{ds} = \sqrt{{\mathrm{dx}}^{2} + {\mathrm{dy}}^{2}}$ With $\mathrm{ds} , \mathrm{dx} , \mathrm{dy}$ here just representing really small changes in $s , x , y$ To manipulate it to suit the given parametric, we can adjust the equation such that $\mathrm{ds}$ is in terms of $\mathrm{dt}$. $\mathrm{ds} = \sqrt{{\left(\frac{\mathrm{dx}}{\mathrm{dt}} \cdot \mathrm{dt}\right)}^{2} + {\left(\frac{\mathrm{dy}}{\mathrm{dt}} \cdot \mathrm{dt}\right)}^{2}}$ $= \sqrt{\left({\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)}^{2} + {\left(\frac{\mathrm{dy}}{\mathrm{dt}}\right)}^{2}\right) {\mathrm{dt}}^{2}}$ $= \sqrt{{\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)}^{2} + {\left(\frac{\mathrm{dy}}{\mathrm{dt}}\right)}^{2}} \mathrm{dt}$ So for the arc length $s$ from $t = a$ to $t = b$, we have, by integrating both sides, $s = {\int}_{a}^{b} \sqrt{{\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)}^{2} + {\left(\frac{\mathrm{dy}}{\mathrm{dt}}\right)}^{2}} \mathrm{dt}$	2022-08-20 02:55:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 25, "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": 0.9592089653015137, "perplexity": 261.59048184847757}, "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-2022-33/segments/1659882573876.92/warc/CC-MAIN-20220820012448-20220820042448-00376.warc.gz"}
http://codeforces.com/problemset/problem/453/D	D. Little Pony and Elements of Harmony time limit per test 6 seconds memory limit per test 256 megabytes input standard input output standard output The Elements of Harmony are six supernatural artifacts representing subjective aspects of harmony. They are arguably the most powerful force in Equestria. The inside of Elements of Harmony can be seen as a complete graph with n vertices labeled from 0 to n - 1, where n is a power of two, equal to 2m. The energy in Elements of Harmony is in constant movement. According to the ancient book, the energy of vertex u in time i (ei[u]) equals to: Here b[] is the transformation coefficient — an array of m + 1 integers and f(u, v) is the number of ones in the binary representation of number (u xor v). Given the transformation coefficient and the energy distribution at time 0 (e0[]). Help Twilight Sparkle predict the energy distribution at time t (et[]). The answer can be quite large, so output it modulo p. Input The first line contains three integers m, t and p (1 ≤ m ≤ 20; 0 ≤ t ≤ 1018; 2 ≤ p ≤ 109). The following line contains n (n = 2m) integers e0[i] (1 ≤ e0[i] ≤ 109; 0 ≤ i < n). The next line contains m + 1 integers b[i] (0 ≤ b[i] ≤ 109; 0 ≤ i ≤ m). Output Output n lines, the i-th line must contain a single integer et[i] modulo p. Examples Input 2 2 100004 1 2 30 1 0 Output 146614	2018-05-26 21:26:54	{"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": 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": 0.3573001027107239, "perplexity": 678.9606634966595}, "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-2018-22/segments/1526794867904.94/warc/CC-MAIN-20180526210057-20180526230057-00161.warc.gz"}
https://www.zbmath.org/?q=ut%3Aasymtotics+cc%3A35	# zbMATH — the first resource for mathematics Trace asymptotics for fractional Schrödinger operators. (English) Zbl 1290.35301 Summary: This paper proves an analogue of a result of R. Bañuelos and A. Sá Barreto [Commun. Partial Differ. Equations 20, No. 11–12, 2153–2164 (1995; Zbl 0843.35016)] on the asymptotic expansion for the trace of Schrödinger operators on $$\mathbb R^d$$ when the Laplacian $$-{\Delta}$$, which is the generator of the Brownian motion, is replaced by the non-local integral operator $$(-{\Delta})^{{\alpha}/2}$$, $$0 < {\alpha} < 2$$, which is the generator of the symmetric stable process of order $${\alpha}$$. These results also extend recent results of R. Bañuelos and S. Y. Yolcu [J. Lond. Math. Soc., II. Ser. 87, No. 1, 304–318 (2013; Zbl 1270.35142)] where the first two coefficients for $$(-{\Delta})^{{\alpha}/2}$$ are computed. Some extensions to Schrödinger operators arising from relativistic stable and mixed-stable processes are obtained. ##### MSC: 35R11 Fractional partial differential equations 35J10 Schrödinger operator, Schrödinger equation 35K08 Heat kernel 60G52 Stable stochastic processes Full Text: ##### References: [1] Applebaum, D., Lévy processes and stochastic calculus, (2009), Cambridge University Press · Zbl 1200.60001 [2] (Arendt, W.; Schleich, W. P., Mathematical Analysis of Evolution, Information, and Complexity, (2009), Wiley-VCH Verlag GmbH & Co. KGaA Weinheim, Germany) · Zbl 1159.94001 [3] Bañuelos, R.; Kulczycki, T., Trace estimates for stable processes, Probab. Theory Related Fields, 142, 313-338, (2008) · Zbl 1184.60012 [4] Bañuelos, R.; Kulczycki, T.; Siudeja, B., On the heat trace of symmetry stables processes on Lipschitz domains, J. Funct. Anal., 257, 3329-3352, (2009) · Zbl 1189.60100 [5] Bañuelos, R.; Sá Barreto, A., On the heat trace of Schrödinger operators, Comm. Partial Differential Equations, 20, 2153-2164, (1995) · Zbl 0843.35016 [6] Bañuelos, R.; Yildirim, S., Heat trace of non-local operators, J. Lond. Math. Soc., 87, 1, 304-318, (2013) · Zbl 1270.35142 [7] P. Billingsley, Probability and Measure, third edition Wiley Series in Probability and Mathematical Statistics, 1995, pp. 327-352 378-386. [8] Bogdan, K., Potential analysis of stable processes and its extensions, Lecture Notes in Math., vol. 1980, (2009), Springer-Verlag [9] Chen, Z. Q.; Kim, P.; Song, R., Dirichlet heat kernel estimates for $$\operatorname{\Delta}^{\alpha / 2} + \operatorname{\Delta}^{\beta / 2}$$, Illinois J. Math., 54, 1357-1392, (2010) · Zbl 1268.60101 [10] Chen, Z.-Q.; Kumagai, T., Heat kernel estimates for jump processes of mixed types on metric measure spaces, Probab. Theory Related Fields, 140, 277-317, (2008) · Zbl 1131.60076 [11] Colin de Verdière, Y., Une formule de trace pour lʼuperateur de Schrödinger dans $$\mathbb{R}^3$$, Ann. Sci. Ecole Norm. Sup., 14, 27-39, (1981) · Zbl 0482.35068 [12] Datchev, K.; Hezari, H., Inverse problems in spectral geometry · Zbl 1316.35001 [13] Davies, E. B., Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, (1989), Cambridge University Press · Zbl 0699.35006 [14] Donnelly, H., Compactness of isospectral potentials, Trans. Amer. Math. Soc., 357, 1717-1730, (2005) · Zbl 1062.58033 [15] Hiroshima, F.; Ichinose, T.; Lörinczi, J., Path integral representation for Schrödinger operators with Bernstein functions of the Laplacian, (2010) [16] Jakubowski, T.; Szczypkowski, K., Estimates of gradient perturbations series, (2011) [17] Khoshnevisan, D., Topics in probability: Lévy processes, lecture notes · Zbl 0910.60061 [18] Lieb, E. H., Calculation of exchange second viral coefficient of a hard-sphere gas by path integrals, J. Math. Phys., 8, 43-52, (1967) [19] McKean, H. P.; van Moerbeke, P., The spectrum of hillʼs equation, Invent. Math., 30, 217-274, (1975) · Zbl 0319.34024 [20] Penrose, M. D.; Penrose, O.; Stell, G., Sticky spheres in quantum mechanics, Rev. Math. Phys., 6, 947-975, (1994) · Zbl 0841.60094 [21] Sebah, P.; Gourdon, X., Introduction to the gamma function, (2002) [22] Simon, B., Schrödinger semigroups, Bull. Amer. Math. Soc., 7, 447-526, (1982) · Zbl 0524.35002 [23] van den Berg, M., On the trace of the difference of Schrödinger heat semigroups, Proc. Roy. Soc. Edinburgh Sect. A, 119, 169-175, (1991) · Zbl 0767.47022 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.	2021-06-20 11:05:51	{"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": 0.8690693378448486, "perplexity": 3558.8554023910465}, "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/1623487660269.75/warc/CC-MAIN-20210620084505-20210620114505-00539.warc.gz"}
https://includestdio.com/6807.html	# Can’t connect to local MySQL server through socket ‘/var/lib/mysql/mysql.sock’ (2) ## The Question : 272 people think this question is useful I am getting the following error when I try to connect to mysql: Can't connect to local MySQL server through socket '/var/lib/mysql/mysql.sock' (2) Is there a solution for this error? What might be the reason behind it? 227 people think this answer is useful Are you connecting to “localhost” or “127.0.0.1” ? I noticed that when you connect to “localhost” the socket connector is used, but when you connect to “127.0.0.1” the TCP/IP connector is used. You could try using “127.0.0.1” if the socket connector is not enabled/working. 193 people think this answer is useful Ensure that your mysql service is running service mysqld start Then, try the one of the following following: (if you have not set password for mysql) mysql -u root mysql -u root -p 29 people think this answer is useful If your file my.cnf (usually in the etc folder) is correctly configured with socket=/var/lib/mysql/mysql.sock you can check if mysql is running with the following command: mysqladmin -u root -p status try changing your permission to mysql folder. If you are working locally, you can try: sudo chmod -R 777 /var/lib/mysql/ that solved it for me 24 people think this answer is useful The MySQL server is not running, or that is not the location of its socket file (check my.cnf). 22 people think this answer is useful Most likely mysql.sock does not exist in /var/lib/mysql/. If you find the same file in another location then symlink it: For ex: I have it in /data/mysql_datadir/mysql.sock Switch user to mysql and execute as mentioned below: su mysql That solved my problem 20 people think this answer is useful If you are on a recent RHEL, you may need to start mariadb (an open source mysql db) instead of the mysql db: yum remove mysql You should then be able to access mysql in the usual fashion: mysql -u root -p 15 people think this answer is useful In my case I have moved socket file to another location inside /etc/my.cnf from /var/lib/mysql/mysql.sock to /tmp/mysql.sock Even after restarting the mysqld service, I still see the error message when I try to connect. ERROR 2002 (HY000): Can't connect to local MySQL server through socket '/var/lib/mysql/mysql.sock' (2) The problem is with the way that the client is configured. Running diagnostics will actually show the correct socket path. eg ps aux \| grep mysqld Works: mysql -uroot -p -h127.0.0.1 mysql -uroot -p --socket=/tmp/mysql.sock Does not Work: mysql -uroot -p mysql -uroot -p -hlocalhost You can fix this problem by adding the same socket line under [client] section inside mysql config. 14 people think this answer is useful Just edit /etc/my.cnf Add following lines to my.cnf [mysqld] socket=/var/lib/mysql/mysql.sock [client] socket=/var/lib/mysql/mysql.sock Restart mysql and connect again mysql -u user -p password database -h host; 12 people think this answer is useful Check if your mysqld service is running or not, if not run, start the service. If your problem isn’t solved, look for /etc/my.cnf and modify as following, where you see a line starting with socket. Take a backup of that file before doing this update. socket=/var/lib/mysql/mysql.sock Change to socket=/opt/lampp/var/mysql/mysql.sock -u root 7 people think this answer is useful MariaDB, a community developed fork of MySQL, has become the default implementation of MySQL in many distributions. So first you should start, $sudo systemctl start mariadb If this fails rather try, $ sudo systemctl start mysqld Then to start mysql, $mysql -u root -p As of today, in Fedora the package is named mariadb And in Ubuntu it is called mariadb-server. So you may have to install it if its not already installed in your system. ## The Answer 11 7 people think this answer is useful Make sure you have enough space left in /var. If Mysql demon is not able to write additional info to the drive the mysql server won’t start and it leads to the error Can't connect to local MySQL server through socket '/var/lib/mysql/mysql.sock' (2) Consider using expire_logs_days = 10 max_binlog_size = 100M This will help you keep disk usage down. ## The Answer 12 5 people think this answer is useful Here’s what worked for me: ln -s /var/lib/mysql/mysql.sock /tmp/mysql.sock service mysqld restart ## The Answer 13 4 people think this answer is useful Please check whether another mysql service is running. ## The Answer 14 4 people think this answer is useful Make sure you started the server: mysql.server start Then connect with root user: mysql -uroot ## The Answer 15 3 people think this answer is useful If your mysql was previously working and has stopped suddenly just “reboot” the server. Was facing this issue on my CentOS VPS.-> Was constantly getting Can't connect to local MySQL server through socket '/var/lib/mysql/mysql.sock'(2) Tried all techniques, finally restarting the server fixed the issues -> shutdown -r now Hope this helps !! ## The Answer 16 3 people think this answer is useful try echo 0 > /selinux/enforce ## The Answer 17 3 people think this answer is useful if you change files in /var/lib/mysql [ like copy or replace that ], you must set owner of files to mysql this is so important if mariadb.service restart has been faild chown -R mysql:mysql /var/lib/mysql/* chmod -R 700 /var/lib/mysql/* ## The Answer 18 2 people think this answer is useful First enter “service mysqld start” and login ## The Answer 19 2 people think this answer is useful Please ensure you have installed MySQL server correctly, I met this error many times and I think it’s complicated to debug from the socket, I mean it might be easier to reinstall it. If you are using CentOS 7, here is the correct way to install it: First of all, add the mysql community source yum install http://dev.mysql.com/get/mysql-community-release-el7-5.noarch.rpm Then you can install it by yum install mysql-community-server Start it with systemctl: systemctl start mysqld ## The Answer 20 1 people think this answer is useful My problem was that I installed mysql successfully and it worked fine. But one day, the same error occurred. Can’t connect to local MySQL server through socket ‘/var/lib/mysql/mysql.sock’ (2) And no mysql.sock file existed. This sollution solved my problem and mysql was up and running again: Log in as root: sudo su - Run: systemctl stop mysqld.service systemctl start mysqld.service systemctl enable mysqld.service Test as root: mysql -u root -p mysql should now be up and running. I hope this can help someone else as well. ## The Answer 21 1 people think this answer is useful Note that while mysql reads the info of the location of the socketfile from the my.cnf file, the mysql_secure_installation program seems to not do that correctly at times. So if you are like me and shuffle things around at installationtime you might get into the situation where you can connect to the database with mysql just fine, but the thing can not be secured (not using that script anyway). To fix this the suggestion from sreddy works well: make a softlink from where the script would expect the socket to where it actually is. Example: ln -s /tmp/mysql.sock /var/lib/mysql/mysql.sock (I use /tmp/ as a default location for sockets) ## The Answer 22 1 people think this answer is useful It worked for me with the following changes Whatever path for socket is mentioned in [mysqld] and same in [client] in my.cnf and restart mysql [mysqld] socket=/var/lib/mysql/mysql.sock [client] socket=/var/lib/mysql/mysql.sock ## The Answer 23 1 people think this answer is useful One way to reproduce this error: If you meant to connect to a foreign server but instead connect to the non existent local one: eric@dev ~$ mysql -u dev -p ERROR 2002 (HY000): Can't connect to local MySQL server through socket '/var/lib/mysql/mysql.sock' (2) eric@dev ~ $ So you have to specify the host like this: eric@dev ~$ mysql --host=yourdb.yourserver.com -u dev -p Welcome to the MySQL monitor. Commands end with ; or \g. Your MySQL connection id is 235 Server version: 5.6.19 MySQL Community Server (GPL) Type 'help;' or '\h' for help. Type '\c' to clear the current input statement. mysql> show databases; +-------------------------+ \| Database \| +-------------------------+ \| information_schema \| \| mysql \| \| performance_schema \| +-------------------------+ 3 rows in set (0.00 sec) mysql> exit Bye eric@dev ~ \$ 1 people think this answer is useful This might be a stupid suggestion but make 100% sure your DB is still hosted at localhost. For example, if a Network Admin chose (or changed to) Amazon DB hosting, you will need that hostname instead! 1 people think this answer is useful In my case, I was importing a new database, and I wasnt able to connect again after that. Finally I realized that was a space problem. So you can delete the last database and expand you hard drive or what I did, restored a snapshot of my virtual machine. Just in case someone thinks that is useful 0 people think this answer is useful ran into this issue while trying to connect mysql in SSH client, found adding the socket path to the command helpful when switching between sockets is necessary. > mysql -u user -p --socket=/path/to/mysql5143.sock 0 people think this answer is useful This is a problem if you are running out of disk space. Solution is to free some space from the HDD. If you are running MySQL at LINUX check the free space of HDD with the command disk free : df if you are getting something like that : Filesystem 1K-blocks Used Available Use% Mounted on /dev/sda2 5162828 4902260 0 100% / udev 156676 84 156592 1% /dev /dev/sda3 3107124 70844 2878444 3% /home Then this is the problem and now you have the solution! Since mysql.sock wants to be created at the mysql folder which is almost always under the root folder could not achieve it because lack of space. If you are periodicaly give the ls command under the mysql directory (at openSUSE 11.1 is at /var/lib/mysql) you will get something like : hostname:/var/lib/mysql # .protected IT files ibdata1 mysqld.log systemtemp .tmp NEWS greekDB mysql mysqld.pid test ARXEIO TEMP1 ib_logfile0 mysql.sock polis The mysql.sock file appearing and disappearing often (you must to try allot with the ls to hit a instance with the mysql.sock file on folder). This caused by not enough disk space. I hope that i will help some people!!!! Thanks! 0 people think this answer is useful I had to disable explicit_defaults_for_timestamp from my.cnf. 0 people think this answer is useful Try first 2, 3 solutions. Error is stil popup & If you can not find /var/lib/mysql/mysql.sock find /var/ -name mysql.sock Check the space available in /var/ df If the directory is full remove some unusefull files/directories rm /var/cache/* Probably your issue will sorted now. mysql --host=mysql-{LETTER} --user={LETTER}{GROUP ID}admin -p	2021-01-22 22:18:08	{"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": 0.17064633965492249, "perplexity": 8958.648816169603}, "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/1610703531429.49/warc/CC-MAIN-20210122210653-20210123000653-00197.warc.gz"}
https://gmatclub.com/forum/the-parallelogram-shown-has-four-sides-of-equal-length-what-is-the-207449.html	It is currently 17 Dec 2017, 23:20 ### 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 # Events & Promotions ###### Events & Promotions in June Open Detailed Calendar # The parallelogram shown has four sides of equal length. What is the Author Message TAGS: ### Hide Tags Math Expert Joined: 02 Sep 2009 Posts: 42670 Kudos [?]: 136007 [5], given: 12723 The parallelogram shown has four sides of equal length. What is the [#permalink] ### Show Tags 20 Oct 2015, 23:19 5 KUDOS Expert's post 40 This post was BOOKMARKED 00:00 Difficulty: 75% (hard) Question Stats: 63% (01:47) correct 37% (01:47) wrong based on 879 sessions ### HideShow timer Statistics The parallelogram shown has four sides of equal length. What is the ratio of the length of the shorter diagonal to the length of the longer diagonal? (A) 1/2 (B) $$\frac{1}{\sqrt{2}}$$ (C) $$\frac{1}{2\sqrt{2}}$$ (D) $$\frac{1}{\sqrt{3}}$$ (E) $$\frac{1}{2\sqrt{3}}$$ Kudos for a correct solution. [Reveal] Spoiler: Attachment: 2015-10-21_1114.png [ 3.11 KiB \| Viewed 18950 times ] [Reveal] Spoiler: OA _________________ Kudos [?]: 136007 [5], given: 12723 Intern Joined: 30 Jan 2015 Posts: 2 Kudos [?]: 13 [5], given: 13 GMAT 1: 650 Q47 V33 The parallelogram shown has four sides of equal length. What is the [#permalink] ### Show Tags 21 Oct 2015, 00:15 5 KUDOS 7 This post was BOOKMARKED IMO D. The parallelogram has four equal sides (x), if the angles is 60° for one, the opposite angle is 60° and the other two are 120° each (360-60-60/2). Therefore, the parallelogram diagonal bisects every angle to form four different 30-60-90 rectangle triangles (ratio: $$x : 2x : x\sqrt{3}$$). Longer diagonal = $$2x\sqrt{3}$$ Shorter diagonal = $$2x$$ Therefore the ratio is $$\frac{2x}{2x\sqrt{3}}$$ = $$\frac{1}{\sqrt{3}}$$ Kudos [?]: 13 [5], given: 13 Verbal Forum Moderator Status: Greatness begins beyond your comfort zone Joined: 08 Dec 2013 Posts: 1851 Kudos [?]: 1075 [1], given: 90 Location: India Concentration: General Management, Strategy GPA: 3.2 WE: Information Technology (Consulting) The parallelogram shown has four sides of equal length. What is the [#permalink] ### Show Tags 21 Oct 2015, 00:29 1 KUDOS Opposite angles of a parallegram are equal and sum of adjacent angles is 180 degree . Each of the angles adjacent to 60 is 120 . The shorter diagonal divies the parallelogram divides into 2 isosceles triangles . Since all the sides of the parallelogram are equal , we get an equilateral triangle. Therefore the shorter diagonal will be equal to length of sides of parallelogram = x The longer diagonal divides parallelogram into 2 isoceles trianges with one angle 120 and the other 2 angles equal to 30 . Since, the diagonals of a parallelogram are perpendicular bisectors ,the 2 diagonals divide the parallelogram into 4 - 30-60-90 triangles. The sides of a 30-60- 90 triangle are in ratio 1:(3^(1/2)):2 Sin 60=(y/2)/x Where y= length of the longer diagonal =>(3^(1/2))/2=(y/2)/x =>x/y=1/(3^(1/2)) _________________ When everything seems to be going against you, remember that the airplane takes off against the wind, not with it. - Henry Ford The Moment You Think About Giving Up, Think Of The Reason Why You Held On So Long +1 Kudos if you find this post helpful Kudos [?]: 1075 [1], given: 90 SC Moderator Joined: 13 Apr 2015 Posts: 1511 Kudos [?]: 1240 [0], given: 896 Location: India Concentration: Strategy, General Management WE: Information Technology (Consulting) Re: The parallelogram shown has four sides of equal length. What is the [#permalink] ### Show Tags 21 Oct 2015, 04:00 Let ABCD be our parallelogram with A = 60 degree and AB=BC=CD=AD=1 Opposite angles of a parallelogram are equal. So C = 60 degree. Sum of adjacent angles = 180 degree. So B = 120 degree and D = 120 degree. AC and BD be the diagonal. Applying cosine rule (a^2 = b^2 + c^2 - 2bc(cos(included angle)), AC^2 = 1 + 1 - 2cos(120) = 2 - 2(-0.5) = 3 --> AC = sqrt(3) BD^2 = 1 + 1 - 2cos(60) = 2 - 2(0.5) = 1 --> BD = 1 Length of shorter diagonal to length of longer diagonal = 1:sqrt(3). Ans: D Kudos [?]: 1240 [0], given: 896 Retired Moderator Joined: 29 Apr 2015 Posts: 888 Kudos [?]: 1930 [3], given: 302 Location: Switzerland Concentration: Economics, Finance Schools: LBS MIF '19 WE: Asset Management (Investment Banking) Re: The parallelogram shown has four sides of equal length. What is the [#permalink] ### Show Tags 21 Oct 2015, 08:17 3 KUDOS Bunuel wrote: The parallelogram shown has four sides of equal length. What is the ratio of the length of the shorter diagonal to the length of the longer diagonal? (A) 1/2 (B) $$\frac{1}{\sqrt{2}}$$ (C) $$\frac{1}{2\sqrt{2}}$$ (D) $$\frac{1}{\sqrt{3}}$$ (E) $$\frac{1}{2\sqrt{3}}$$ Kudos for a correct solution. [Reveal] Spoiler: Attachment: 2015-10-21_1114.png The figure must be a rhombus. Draw the diagonals on your scratch paper. You will now have two equiliteral triangles. The shorter diagonal is one side of such an equiliteral triangle. Let's say s=1. To get the longer side of the diagonal of the rhombus, you have 2height of the equiliteral triangle. If you split an equiliteral triangle into two parts, you will have two 30:60:90 triangles. The height of the equiliteral triangle if s = 1 is $$0.5\sqrt{3}$$ and the long diagonal will be $$20.5\sqrt{3}$$ So you have $$1:\sqrt{3}$$ _________________ Saving was yesterday, heat up the gmatclub.forum's sentiment by spending KUDOS! PS Please send me PM if I do not respond to your question within 24 hours. Kudos [?]: 1930 [3], given: 302 Retired Moderator Joined: 12 Aug 2015 Posts: 2226 Kudos [?]: 914 [0], given: 616 GRE 1: 323 Q169 V154 Re: The parallelogram shown has four sides of equal length. What is the [#permalink] ### Show Tags 03 Apr 2016, 20:28 Here is what i did Firstly the given \|\|gm has all sides equal => Rhombus Now shorter diagonal is equal to the side of the \|\|gm as the equilateral triangle is formed. Now to find the Longer diagonal i used the property that diagonals of rhombus bisect each other at a right angle.. so longer diagonal => √3 x side hence the ratio => 1/3 or 1:3 _________________ Give me a hell yeah ...!!!!! Kudos [?]: 914 [0], given: 616 SVP Joined: 06 Nov 2014 Posts: 1903 Kudos [?]: 552 [0], given: 23 Re: The parallelogram shown has four sides of equal length. What is the [#permalink] ### Show Tags 06 Jun 2016, 07:44 2 This post was BOOKMARKED Assume the length of each side = x The other angles would be 60, 120 and 120. Diagonal divides the parallelogram into half Hence the shorter diagonal will make two equilateral triangles Length of the shorter diagonal = length of side = x For the longer diagonal. The diagonal will divide the parallelogram in to isosceles triangles with angles 30, 120, 30 Dropping a perpendicular from top most point on to the diagonal, we have two triangles with angles 30, 60 and 90 with base as half the length of diagonal Hypotenuse = x, Cos 30 = base/x base = (√3/2)x Hence the length of the diagonal = √3x Ratio of the shorter to the longer diagonal = x : √3x = 1: √3 Correct Option: D Kudos [?]: 552 [0], given: 23 Manager Joined: 21 Sep 2015 Posts: 82 Kudos [?]: 127 [9], given: 323 Location: India GMAT 1: 730 Q48 V42 GMAT 2: 750 Q50 V41 Re: The parallelogram shown has four sides of equal length. What is the [#permalink] ### Show Tags 08 Jun 2016, 03:31 9 KUDOS 3 This post was BOOKMARKED The parallelogram shown has four sides of equal length. Inference : It is a rhombus. Properties of Rhombus : 1) Diagonals are not congruent 2) Diagonals act as angle bisectors 3) Diagonals intersect at right angles and bisect each other Hence we get a 30-60-90 triangle as shown below. Characteristic property of this triangle is that sides are in the ratio of 1:[ sqrt 3 ]:2 Shortest diagonal is the diagonal opposite the 30 degree angle and longest diagonal is the the one opposite 60 degree angle Therefore ratio of shortest to longest is 1: sqrt 3 Attachments Q139.png [ 6.59 KiB \| Viewed 13512 times ] _________________ Appreciate any KUDOS given ! Kudos [?]: 127 [9], given: 323 Intern Joined: 26 Nov 2015 Posts: 31 Kudos [?]: 19 [1], given: 5 Re: The parallelogram shown has four sides of equal length. What is the [#permalink] ### Show Tags 08 Jun 2016, 03:53 1 KUDOS Lets assume that side = 1 now the shorter diagonal will belong to squeezed quadrilateral with one side = 1 - 1/2 = 1/2( being a 30-60-90 triangle on the right side) & one side = (3)^0.5/2 Longer diagonal will belong to enlarged quadrilateral with one side = 1 + 1/2 = 3/2 (being a 30-60-90 triangle but side will add) & one side = (3)^0.5/2 to find the diagonal we will apply Pythagoras to both rectangles Squeezed Quadrilateral : ( 1/4 + 3/4) ^ 0.5 = 1 Enlarged Quadrilateral : ( 9/4 + 3/4 ) ^ 0.5 = 3^0.5 so the ratio= 1/ (3)^0.5 _________________ PS : I do mind kudos Kudos [?]: 19 [1], given: 5 Intern Joined: 13 Jun 2016 Posts: 19 Kudos [?]: 1 [0], given: 2 Re: The parallelogram shown has four sides of equal length. What is the [#permalink] ### Show Tags 05 Oct 2016, 00:05 A novice question: For parallelogram, i understand the rules of relationship for 30:60:90, opposite angels are congruent etc. But how can we ascertain that when we split the parallelogram in to two parts , the 60 and 120 degrees angles on the vertices split in to exactly HALF? (i.e. 30 , 60 degrees) Is that a math rule i've forgotten? Kudos [?]: 1 [0], given: 2 Math Expert Joined: 02 Sep 2009 Posts: 42670 Kudos [?]: 136007 [0], given: 12723 Re: The parallelogram shown has four sides of equal length. What is the [#permalink] ### Show Tags 05 Oct 2016, 01:09 xnthic wrote: A novice question: For parallelogram, i understand the rules of relationship for 30:60:90, opposite angels are congruent etc. But how can we ascertain that when we split the parallelogram in to two parts , the 60 and 120 degrees angles on the vertices split in to exactly HALF? (i.e. 30 , 60 degrees) Is that a math rule i've forgotten? Usually diagonals of a parallelogram do not bisect the angles but here we have special kind of parallelogram - a rhombus, where the diagonals bisect the angles. For more check here: math-polygons-87336.html Hope it helps. _________________ Kudos [?]: 136007 [0], given: 12723 Manager Joined: 03 Jan 2017 Posts: 196 Kudos [?]: 9 [0], given: 4 Re: The parallelogram shown has four sides of equal length. What is the [#permalink] ### Show Tags 25 Mar 2017, 09:17 the shorter diagonal is 1, because 60 can be calculated for the triangle->all sides are the same the longer one is square root of 3, because if we split triangles we will get to ration x:2x:x(3)^1/2, because this is triangle of 30:60:90 angles 2x is 1. hence our side is 21/23^1/2 Kudos [?]: 9 [0], given: 4 Senior Manager Status: Professional GMAT Tutor Affiliations: AB, cum laude, Harvard University (Class of '02) Joined: 10 Jul 2015 Posts: 448 Kudos [?]: 527 [0], given: 59 Location: United States (CA) Age: 38 GMAT 1: 770 Q47 V48 GMAT 2: 730 Q44 V47 GMAT 3: 750 Q50 V42 GRE 1: 337 Q168 V169 WE: Education (Education) Re: The parallelogram shown has four sides of equal length. What is the [#permalink] ### Show Tags 30 May 2017, 15:24 Top Contributor Attached is a visual that should help. Attachments Screen Shot 2017-05-30 at 4.07.28 PM.png [ 107.12 KiB \| Viewed 6857 times ] _________________ Harvard grad and 770 GMAT scorer, offering high-quality private GMAT tutoring, both in-person and online via Skype, since 2002. GMAT Action Plan - McElroy Tutoring Kudos [?]: 527 [0], given: 59 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7799 Kudos [?]: 18155 [4], given: 236 Location: Pune, India Re: The parallelogram shown has four sides of equal length. What is the [#permalink] ### Show Tags 30 May 2017, 21:44 4 KUDOS Expert's post Bunuel wrote: The parallelogram shown has four sides of equal length. What is the ratio of the length of the shorter diagonal to the length of the longer diagonal? (A) 1/2 (B) $$\frac{1}{\sqrt{2}}$$ (C) $$\frac{1}{2\sqrt{2}}$$ (D) $$\frac{1}{\sqrt{3}}$$ (E) $$\frac{1}{2\sqrt{3}}$$ Kudos for a correct solution. [Reveal] Spoiler: Attachment: 2015-10-21_1114.png In a parallelogram, the opposite angles are equal. So angle opposite to 60 degrees is also 60. Draw the shorter diagonal. Since the sides are equal, we see that we get two congruent equilateral triangles. The shorter diagonal is the side of each equilateral triangle and the longer diagonal is twice the altitude of each equilateral triangle. We know that if the side of an equilateral triangle is s, its altitude is $$\sqrt{3}s/2$$ and hence twice of its altitude is $$\sqrt{3}s$$. Required Ratio $$= s/\sqrt{3}s = 1/\sqrt{3}$$ _________________ Karishma Veritas Prep \| GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for \$199 Veritas Prep Reviews Kudos [?]: 18155 [4], given: 236 Intern Joined: 01 May 2017 Posts: 9 Kudos [?]: 1 [0], given: 8 Re: The parallelogram shown has four sides of equal length. What is the [#permalink] ### Show Tags 16 Nov 2017, 18:57 A low level approach of assuming that each side is of length 1 and that we have two equilateral triangles joint together: Attachments tri.PNG [ 27.74 KiB \| Viewed 1821 times ] Kudos [?]: 1 [0], given: 8 Target Test Prep Representative Status: Founder & CEO Affiliations: Target Test Prep Joined: 14 Oct 2015 Posts: 1945 Kudos [?]: 1025 [1], given: 3 Location: United States (CA) Re: The parallelogram shown has four sides of equal length. What is the [#permalink] ### Show Tags 20 Nov 2017, 11:38 1 KUDOS Expert's post Bunuel wrote: The parallelogram shown has four sides of equal length. What is the ratio of the length of the shorter diagonal to the length of the longer diagonal? (A) 1/2 (B) $$\frac{1}{\sqrt{2}}$$ (C) $$\frac{1}{2\sqrt{2}}$$ (D) $$\frac{1}{\sqrt{3}}$$ (E) $$\frac{1}{2\sqrt{3}}$$ Kudos for a correct solution. [Reveal] Spoiler: Attachment: 2015-10-21_1114.png We are given a parallelogram with equal sides, and we must determine the ratio of the length of the shorter diagonal to that of the longer diagonal. Since the sides are all equal, we know we have a rhombus, and the diagonals are perpendicular. Let’s sketch this diagram below. We should see that the diagonals bisect each angle of the rhombus, and thus we have created four 30-60-90 right triangles. Using our side ratio of a 30-60-90 right triangle, we have: x : x√3 : 2x Let’s use this side ratio to determine the lengths of each diagonal in terms of x. We can see that the length of the shorter diagonal is x + x = 2x, and the length of the longer diagonal is x√3 + x√3 = 2x√3. Thus, the ratio of the length of the shorter diagonal to the length of the longer diagonal is: (2x)/(2x√3) = 1/√3 _________________ Scott Woodbury-Stewart Founder and CEO GMAT Quant Self-Study Course 500+ lessons 3000+ practice problems 800+ HD solutions Kudos [?]: 1025 [1], given: 3 Re: The parallelogram shown has four sides of equal length. What is the [#permalink] 20 Nov 2017, 11:38 Display posts from previous: Sort by	2017-12-18 07:20:13	{"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": 0.6266382932662964, "perplexity": 4361.486785139579}, "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/1512948609934.85/warc/CC-MAIN-20171218063927-20171218085927-00599.warc.gz"}
https://paradigms.oregonstate.edu/problem/412/	## Icecream Mass • group Flux through a Cone group Small Group Activity 30 min. ##### Flux through a Cone Static Fields 2021 (4 years) Integration Sequence Students calculate the flux from the vector field $\vec{F} = C\, z\, \hat{z}$ through a right cone of height $H$ and radius $R$ . • assignment Cone Surface assignment Homework ##### Cone Surface Static Fields 2023 (6 years) • Find $dA$ on the surface of an (open) cone in both cylindrical and spherical coordinates. Hint: Be smart about how you coordinatize the cone. • Using integration, find the surface area of an (open) cone with height $H$ and radius $R$. Do this problem in both cylindrical and spherical coordinates. • keyboard Electric field for a waffle cone of charge keyboard Computational Activity 120 min. ##### Electric field for a waffle cone of charge Computational Physics Lab II 2022 Students integrate numerically to find the electric field due to a cone of surface charge, and then visualize the result. This integral can be done in either spherical or cylindrical coordinates, giving students a chance to reason about which coordinate system would be more convenient. • assignment Find Area/Volume from $d\vec{r}$ assignment Homework ##### Find Area/Volume from $d\vec{r}$ Static Fields 2023 (5 years) Start with $d\vec{r}$ in rectangular, cylindrical, and spherical coordinates. Use these expressions to write the scalar area elements $dA$ (for different coordinate equals constant surfaces) and the volume element $d\tau$. It might help you to think of the following surfaces: The various sides of a rectangular box, a finite cylinder with a top and a bottom, a half cylinder, and a hemisphere with both a curved and a flat side, and a cone. 1. Rectangular: \begin{align} dA&=\\ d\tau&= \end{align} 2. Cylindrical: \begin{align} dA&=\\ d\tau&= \end{align} 3. Spherical: \begin{align} dA&=\\ d\tau&= \end{align} • assignment Cube Charge assignment Homework ##### Cube Charge charge density Integration Sequence Static Fields 2023 (6 years) 1. Charge is distributed throughout the volume of a dielectric cube with charge density $\rho=\beta z^2$, where $z$ is the height from the bottom of the cube, and where each side of the cube has length $L$. What is the total charge inside the cube? Do this problem in two ways as both a single integral and as a triple integral. 2. On a different cube: Charge is distributed on the surface of a cube with charge density $\sigma=\alpha z$ where $z$ is the height from the bottom of the cube, and where each side of the cube has length $L$. What is the total charge on the cube? Don't forget about the top and bottom of the cube. • assignment Current in a Wire assignment Homework ##### Current in a Wire Static Fields 2023 (4 years) The current density in a cylindrical wire of radius $R$ is given by $\vec{J}(\vec{r})=\alpha s^3\cos^2\phi\,\hat{z}$. Find the total current in the wire. • assignment Current from a Spinning Cylinder assignment Homework ##### Current from a Spinning Cylinder A solid cylinder with radius $R$ and height $H$ has its base on the $x,y$-plane and is symmetric around the $z$-axis. There is a fixed volume charge density on the cylinder $\rho=\alpha z$. If the cylinder is spinning with period $T$: 1. Find the volume current density. 2. Find the total current. • assignment Einstein condensation temperature assignment Homework ##### Einstein condensation temperature Einstein condensation Density Thermal and Statistical Physics 2020 Einstein condensation temperature Starting from the density of free particle orbitals per unit energy range \begin{align} \mathcal{D}(\varepsilon) = \frac{V}{4\pi^2}\left(\frac{2M}{\hbar^2}\right)^{\frac32}\varepsilon^{\frac12} \end{align} show that the lowest temperature at which the total number of atoms in excited states is equal to the total number of atoms is \begin{align} T_E &= \frac1{k_B} \frac{\hbar^2}{2M} \left( \frac{N}{V} \frac{4\pi^2}{\int_0^\infty\frac{\sqrt{\xi}}{e^\xi-1}d\xi} \right)^{\frac23} T_E &= \end{align} The infinite sum may be numerically evaluated to be 2.612. Note that the number derived by integrating over the density of states, since the density of states includes all the states except the ground state. Note: This problem is solved in the text itself. I intend to discuss Bose-Einstein condensation in class, but will not derive this result. • assignment Energy of a relativistic Fermi gas assignment Homework ##### Energy of a relativistic Fermi gas Fermi gas Relativity Thermal and Statistical Physics 2020 For electrons with an energy $\varepsilon\gg mc^2$, where $m$ is the mass of the electron, the energy is given by $\varepsilon\approx pc$ where $p$ is the momentum. For electrons in a cube of volume $V=L^3$ the momentum takes the same values as for a non-relativistic particle in a box. 1. Show that in this extreme relativistic limit the Fermi energy of a gas of $N$ electrons is given by \begin{align} \varepsilon_F &= \hbar\pi c\left(\frac{3n}{\pi}\right)^{\frac13} \end{align} where $n\equiv \frac{N}{V}$ is the number density. 2. Show that the total energy of the ground state of the gas is \begin{align} U_0 &= \frac34 N\varepsilon_F \end{align} • assignment Potential energy of gas in gravitational field assignment Homework ##### Potential energy of gas in gravitational field Potential energy Heat capacity Thermal and Statistical Physics 2020 Consider a column of atoms each of mass $M$ at temperature $T$ in a uniform gravitational field $g$. Find the thermal average potential energy per atom. The thermal average kinetic energy is independent of height. Find the total heat capacity per atom. The total heat capacity is the sum of contributions from the kinetic energy and from the potential energy. Take the zero of the gravitational energy at the bottom $h=0$ of the column. Integrate from $h=0$ to $h=\infty$. You may assume the gas is ideal. • Static Fields 2023 (6 years) Use integration to find the total mass of the icecream in a packed cone (both the cone and the hemisphere of icecream on top).	2023-03-24 23:29:47	{"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": 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": 7, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9701729416847229, "perplexity": 638.9328229540002}, "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/1679296945289.9/warc/CC-MAIN-20230324211121-20230325001121-00634.warc.gz"}
http://www.jstor.org/stable/2043510	## Access You are not currently logged in. Access JSTOR through your library or other institution: ## If You Use a Screen Reader This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader. # Absolutely Convergent Fourier Series of Distributions Proceedings of the American Mathematical Society Vol. 83, No. 2 (Oct., 1981), pp. 276-278 DOI: 10.2307/2043510 Stable URL: http://www.jstor.org/stable/2043510 Page Count: 3 Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader. Preview not available ## Abstract Let $S$ be a distribution (in the sense of L. Schwartz) defined on the circle $T$, and suppose that $S$ is equal to a function in $L^\infty$ on an open interval of $T$. A necessary and sufficient condition is given in order that the Fourier series of $S$ converges absolutely. • 276 • 277 • 278	2017-02-20 14:19:44	{"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": 0.7111611366271973, "perplexity": 1339.629760369872}, "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-2017-09/segments/1487501170562.59/warc/CC-MAIN-20170219104610-00254-ip-10-171-10-108.ec2.internal.warc.gz"}
https://docs.lucedaphotonics.com/reference/device_sim/index.html	Device simulation reference¶ All functions and classes related to device simulation are namespaced under i3.device_sim. For a tutorial running device simulations with IPKISS, please check the running device simulations tutorial. device_sim.SimulationGeometry Defines the geometry of the electromagnetic simulation. device_sim.SMatrixOutput Output an SMatrix device_sim.LumericalFDTDSimulation Define a simulation for use with Lumerical’s FDTD Solutions. device_sim.CSTTDSimulation Define a simulation for use with CST Studio Suite ® (Time Domain solver). device_sim.Port A combined monitor and source, for use in a physical solver device_sim.Macro Defines a tool-specific macro which will be executed at a specified stage device_sim.MacroFile Defines a macro to be loaded from file device_sim.camfr_guided_modes Calculates the neff for guided modes for a material stack using the camfr mode solver. device_sim.camfr_mode_fields Calculates the fields for the modes for a material stack using the camfr mode solver. device_sim.camfr_compute_stack_neff Computes the effective index of mode in the material stack at the given wavelength. device_sim.camfr_stack_expr_for_structure For a given layout, call virtual fabrication and return a camfr.Expression that can be used to run a simulation. device_sim.get_material_stacks Convenience function that returns material stacks used to fabricate the structure.	2021-06-12 18:35:48	{"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": 0.5185471177101135, "perplexity": 6113.497103555603}, "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-2021-25/segments/1623487586239.2/warc/CC-MAIN-20210612162957-20210612192957-00154.warc.gz"}
https://www.encyclopediaofmath.org/index.php?title=Brauer_second_main_theorem&oldid=18936	Brauer second main theorem (diff) ← Older revision \| Latest revision (diff) \| Newer revision → (diff) For notation and definitions, see Brauer first main theorem. Let be an element of whose order is a power of . The -section of associated to is the set of all elements of whose -part is conjugate to . Brauer second main theorem relates the values of irreducible characters of on the -section associated to to values of characters in certain blocks of . Suppose that is an irreducible character of (cf. also Character of a group), afforded by the -free right -module , and belonging to the block (cf. also Defect group of a block). Let be a -element of , and let . For all -subgroups of , ; hence is defined for all blocks of . One can organize the block decomposition of as . Let be the projection of on , and let be the projection of on . The restriction of to can be decomposed as . If is the character of and is the character of , then of course for all . Brauer's second main theorem states that for all elements of order prime to , . Thus, the values of on the -section associated to are determined in the blocks of sent to by the Brauer correspondence (cf. also Brauer first main theorem). This theorem was first proved in [a1]. See also [a2], [a3], and [a4]. References [a1] R. Brauer, "Zur Darstellungstheorie der Gruppen endlicher Ordnung II" Math. Z. , 72 (1959) pp. 22–46 [a2] C. Curtis, I. Reiner, "Methods of representation theory" , II , Wiley (1987) [a3] W. Feit, "The representation theory of finite groups" , North-Holland (1982) [a4] H. Nagao, Y. Tsushima, "Representation of finite groups" , Acad. Press (1987) How to Cite This Entry: Brauer second main theorem. H. Ellers (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Brauer_second_main_theorem&oldid=18936 This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098	2019-10-22 05:46:57	{"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": 0.898766815662384, "perplexity": 2123.91190882337}, "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-2019-43/segments/1570987803441.95/warc/CC-MAIN-20191022053647-20191022081147-00274.warc.gz"}
http://tktermpaperzbvn.larryclark.us/e-mc-2.html	E mc 2 Generations have grown up knowing that the equation e=mc2 changed the shape of our world, but never understanding what it actually means, why it was so significant, and how it informs our daily lives today--governing, as it does, everything from the atomic bomb to a television's cathode ray tube. E = mc2: e = mc^2, equation in einstein's theory of special relativity that expresses the equivalence of mass and energy. 2008 release of the e=mc2 album the album name means (e) emancipation = (mc) mariah carey to the second power it is a play on einstein's famous mass energy equivalence formula. When deuterium and tritium nuclei collide and stick, the leftovers, (a helium nucleus and a neutron) are slightly lighter than the original nuclei. Check out the e = mc2 calculator to learn the most famous equation of all times. I have been having trouble distinguishing these two equations and figuring out which one is correct i have watched a video that says that $e^2=(mc^2)^2+(pc)^2$ is correct, but i do not know why. Mass can be converted into pure energy this is the second meaning of the equation, where e = mc 2 tells us exactly how much energy you get from converting mass. Mass to energy calculator, matter to energy calculator, albert einstein's theory of relativity, e equals mc squared, e=mc2, e=mc. Mc2 prides itself on having a diveresed client base-which in turn fuels the overall creative marketing process. Getting to grips with einstein's famous equation can be a bit of an assault course for the novice, but, says alok jha, the reader is in supremely capable hands with brian cox and jeff forshaw. No equation is more famous than e = mc 2, and few are simpler indeed, the immortal equation's fame rests largely on that utter simplicity: the energy e of a system is equal to its mass m multiplied by c2, the speed of light squared the equation's message is that the mass of a system measures. (physorg) university of arizona physicist andrei lebed has stirred the physics community with an intriguing idea yet to be tested experimentally: the world's most iconic equation, albert einstein's e=mc2, may be correct. Mc2: a full-service, integrated marketing & design agency with a studied fusion of smart media planning, innovative design & artful publicity. Albert einstein 'e=mc2' to see what your friends thought of this quote, please sign up. Looking for online definition of e=mc2 or what e=mc2 stands for e=mc2 is listed in the world's largest and most authoritative dictionary database of abbreviations and acronyms. Home about faq lessons downloads links further reading: the derivation of e=mc 2 perhaps the most famous equation of all time is e = mc 2 the equation is a direct result of the theory of special relativity, but what does it mean and how did einstein find it. Einstein took so many years to prove that energy is not mc^3 but mc^2, but isn't it obvious that e=mc^3 is dimensionally wrong. E=mc is the eleventh studio album by american singer and songwriter mariah carey it was released in the united states on april 15, 2008 by island records. Researchers say that soon it will be possible to smash photons together to create matter in the laboratory. E mc 2 Teenage spy mckeyla teams up with three other super-smart girls to become secret agents who use their science and tech skills to save the day secret agent mckeyla mcalister knew the mission was tough when she took it but her new friends are making it downright impossible project mc : part 2. The units of e = mc^2 explained in an easy-to-follow way - how energy, mass and the speed of light are related. E = mc2 energy equals mass times the speed of light squared e= energy m=mass c= the speed of light 2= squared (a number times itself) a famous. For a very long time telangana people were ashamed to speak telangana in front of others, and the practice continues even now though it is declining. 2018 mc 2, an mch company thank you for contacting mc we will be in touch with you very shortly. E = \gamma mc^2 [/tex] plug in for gamma zhermes, aug 7, 2010 aug 7, 2010 #3 altabeh kaleidoscope said. An easy-to-follow explanation of e = mc^2 - meaning, units and solving with many worked examples. E mc 2 Rated 5/5 based on 25 review	2018-08-20 18:28:00	{"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": 0.38248351216316223, "perplexity": 1904.7845416620555}, "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-2018-34/segments/1534221216724.69/warc/CC-MAIN-20180820180043-20180820200043-00016.warc.gz"}
https://www.jiskha.com/questions/1734830/how-do-we-evaluate-limit-of-lim-x-0-ln-x-1-2-x-1-i-tried-using-the	Math-Limits How do we evaluate limit of, lim x-> 0 [ln(x+1)/( (2^x) - 1)] I tried using the substitution x+1 = e^k , when x tends to 0 so does k, which gave out, lim k->0 [ k/((2^((e^k) - 1)) -1 ) ] which I simplified into( for the ease of use let e^k =a) lim k->0 [k/( ((2^a)/2) - 1)] lim k->0 [2k/((2^a) -2 )] which sums up to, (20)/((2^(e^0)) - 2) (20)/(2-2) which gives out an undefined value as 0/0 How do we solve this? 1. 👍 2. 👎 3. 👁 1. How about good old L'Hopital's rule lim x-> 0 [ln(x+1)/( (2^x) - 1)] = lim x-> 0 [1/(x+1) / (ln22^x) = (1/1)/(ln2 ( 1) = 1/ln2 = appr 1.44269.. check: let x = .0001 in the original = ln(1.0001) / (2^.0001 - 1) 1. 👍 2. 👎 Similar Questions 1. Math Evaluate the limit using the appropriate Limit Law(s). (If an answer does not exist, enter DNE.) lim t → −1 (t2 + 1)^4(t + 3)^5 2. Calculus Limits Question: If lim(f(x)/x)=-5 as x approaches 0, then lim(x^2(f(-1/x^2))) as x approaches infinity is equal to (a) 5 (b) -5 (c) -infinity (d) 1/5 (e) none of these The answer key says (a) 5. So this is what I know: Since 3. Calculus The area A of the region S that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles. A = lim n → ∞ [f(x1)Δx + f(x2)Δx + . . . + f(xn)Δx] Use this definition to 4. Calc 1 Evaluate the limit. lim x → 1 ln x/sin 5πx 1. Calculus Evaluate the limit. as x approaches infinity, lim sqrt(x^2+6x+3)-x 1. Evaluate: lim x->infinity(x^4-7x+9)/(4+5x+x^3) 0 1/4 1 4 The limit does not exist. 2. Evaluate: lim x->infinity (2^x+x^3)/(x^2+3^x) 0 1 3/2 2/3 The limit does not exist. 3. lim x->0 (x^3-7x+9)/(4^x+x^3) 0 1/4 1 **9 The 3. Calculus Show that limit as n approaches infinity of (1+x/n)^n=e^x for any x>0... Should i use the formula e= lim as x->0 (1+x)^(1/x) or e= lim as x->infinity (1+1/n)^n Am i able to substitute in x/n for x? and then say that e lim x ->0 4. Calculus Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.) lim t^2-64/2t^2+17t+8 t→−8 Please show steps! 1. Calculus Find the limit. lim 5-x/(x^2-25) x-->5 Here is the work I have so far: lim 5-x/(x^2-25) = lim 5-x/(x-5)(x+5) x-->5 x-->5 lim (1/x+5) = lim 1/10 x-->5 x-->5 I just wanted to double check with someone and see if the answer is 2. Calculus Find the indicated limits. If the limit does not exist, so state, or use the symbol + ∞ or - ∞. f(x) = { 2 - x if x ≤ 3 { -1 + 3x - x^2 if x > 3 a) lim 3+ f(x) x->3 b) lim 3- f(x) x->3 c) lim f(x) x->3 d) lim ∞ f(x) x->3 3. Calculus Evaluate the limit using L'Hospital's rule if necessary. lim as x goes to +infinity x^(6/x)	2021-01-27 19:34:56	{"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": 0.911760151386261, "perplexity": 4753.75253572728}, "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-2021-04/segments/1610704832583.88/warc/CC-MAIN-20210127183317-20210127213317-00730.warc.gz"}
https://www.deeplearningwizard.com/deep_learning/intro/	Skip to content # Deep Learning and Deep Reinforcement Learning Theory and Programming Tutorials¶ We'll be covering both CPU and GPU implementations of deep learning and deep reinforcement learning algorithms If you would like a more visual and guided experience, feel free to take our video course. Work in progress This open-source portion is still a work in progress, it may be sparse in explanation as traditionally all our explanation are done via video. Stay tuned while we gradually upload our tutorials and notes. Feel free to contact Ritchie Ng if you would like to contribute via our Facebook page. Also take note that these notes are best used as a referral. This is because we have yet to expand it comprehensively to be a stand-alone guide. Go head and take our video course that provides a much easier, proven-to-work, experience. All of our code allows you to run in a notebook for this deep learning section. Please use a jupyter notebook and run the examples from the start of the page to the end. Remember to Right mouse click > Open image in new tab if you would like to zoom into the diagrams if you find them too small. Clarifications These are some clarifications we would like to highlight. • When we state linear function, more specifically we meant affine function that comprises a linear function and a constant. We did this initially to make it easier as "linear function" was easier to digest. • For all diagrams that says valid padding, they refer to no padding such that your output size will be smaller than your input size. • For all diagrams that says same padding, they refer to zero padding (padding your input with zeroes) such that your output size will be equal to your input size. Errors to be corrected As we are rapidly prototyping there may be some errors. For these errors stated here, they will be corrected soon. Feel free to report bugs, corrections or improvements on our Github repository. If you did not go through all the materials, you would not be familiar with these, so go through them and come back to review these changes. • For all diagrams that says dot product, they refer to matrix product.	2020-01-22 05:38:14	{"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": 0.2620418965816498, "perplexity": 921.051093904208}, "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-2020-05/segments/1579250606696.26/warc/CC-MAIN-20200122042145-20200122071145-00065.warc.gz"}
http://mathhelpforum.com/calculus/171858-how-could-i-show.html	# Math Help - How could I show this? 1. ## How could I show this? Show by means of an example that lim x->a [f(x) + g(x)] may exist even though neither lim x->a f(x) nor lim x->a g(x) exists. How could I show this? 2. One option is to take a function $f$, for which the limit at $x=a$ does not exist, and then take $g(x) = -f(x)$.	2014-07-25 16:00:16	{"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": 3, "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": 0.8939182162284851, "perplexity": 376.61238186573155}, "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-2014-23/segments/1405997894319.36/warc/CC-MAIN-20140722025814-00106-ip-10-33-131-23.ec2.internal.warc.gz"}
https://kerodon.net/tag/01RF	# Kerodon $\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$ Definition 5.0.0.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories and let $f: X \rightarrow Y$ be a morphism in the category $\operatorname{\mathcal{E}}$. • We say that $f$ is $U$-cartesian if, for every object $W \in \operatorname{\mathcal{E}}$, the diagram of sets $\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(W,X) \ar [r]^-{f \circ } \ar [d]^-{U} & \operatorname{Hom}_{\operatorname{\mathcal{E}}}(W,Y) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(W), U(X) ) \ar [r]^-{U(f) \circ } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(W), U(Y) ) }$ is a pullback square. • We say that $f$ is $U$-cocartesian if, for every object $Z \in \operatorname{\mathcal{D}}$, the diagram of sets $\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(Y,Z) \ar [r]^-{\circ f} \ar [d]^-{U} & \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(Y), U(Z) ) \ar [r]^-{\circ U(f)} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Z) ) }$ is a pullback square.	2021-03-02 07:04:30	{"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": 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": 0.9970806837081909, "perplexity": 255.0293971105512}, "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/1614178363782.40/warc/CC-MAIN-20210302065019-20210302095019-00046.warc.gz"}
http://cds.cern.ch/collection/COMPASS%20Conference%20Proceedings?ln=fr	# COMPASS Conference Proceedings Derniers ajouts: 2017-02-25 07:55 The $t$-dependence of the pure DVCS cross section at COMPASS / Joerg, Philipp (Freiburg U.) /COMPASS Collaboration The key reactions to study the Generalised Parton Distributions are Deeply Virtual Compton Scattering (DVCS) and Deeply Virtual Meson Production (DVMP). At COMPASS, these processes are investigated using a high intensity muon beam with a momentum of 160\,GeV/c and a 2.5\,m-long liquid hydrogen target. [...] arXiv:1702.06315.- SISSA, 2016-10-10 - 7 p. - Published in : PoS: DIS2016 (2016) , pp. 235 Fulltext: PDF; Preprint: PDF; External links: PoS server; 00007 COMPASS results for the t-slope parameter (\textbf{left}) and the extracted transverse size of the partonic distribution of the nucleon (\textbf{right}), compared with previous HERA measurements.; 00003 Exclusivity variables: The whole Monte Carlo estimate is shown in red while in grey only the $\pi^0$ contamination is shown. The variables $\Delta \varphi$, $\Delta p_{\bot}$ and $M_X^2$ are defined in equation \ref{eq_phi}-\ref{eq_m} while $\Delta z_{A}$ encodes a reverse vertex pointing. The blue dotted lines indicate the applied cuts.; 00006 COMPASS results for the t-slope parameter (\textbf{left}) and the extracted transverse size of the partonic distribution of the nucleon (\textbf{right}), compared with previous HERA measurements.; 00001 Exclusivity variables: The whole Monte Carlo estimate is shown in red while in grey only the $\pi^0$ contamination is shown. The variables $\Delta \varphi$, $\Delta p_{\bot}$ and $M_X^2$ are defined in equation \ref{eq_phi}-\ref{eq_m} while $\Delta z_{A}$ encodes a reverse vertex pointing. The blue dotted lines indicate the applied cuts.; 00002 Exclusivity variables: The whole Monte Carlo estimate is shown in red while in grey only the $\pi^0$ contamination is shown. The variables $\Delta \varphi$, $\Delta p_{\bot}$ and $M_X^2$ are defined in equation \ref{eq_phi}-\ref{eq_m} while $\Delta z_{A}$ encodes a reverse vertex pointing. The blue dotted lines indicate the applied cuts.; 00000 Exclusivity variables: The whole Monte Carlo estimate is shown in red while in grey only the $\pi^0$ contamination is shown. The variables $\Delta \varphi$, $\Delta p_{\bot}$ and $M_X^2$ are defined in equation \ref{eq_phi}-\ref{eq_m} while $\Delta z_{A}$ encodes a reverse vertex pointing. The blue dotted lines indicate the applied cuts.; 00005 The $t$-differential DVCS cross section as a function of $t$; 00004 Acceptance correction factors as a function of $t$, $Q^2$ and $\nu$.; Fulltext In : 24th International Workshop on Deep Inelastic Scattering and related subjects, Hamburg, Germany, 11 - 15 Apr 2016, pp.235 2017-02-25 07:55 Measurement of the exclusive ${\pi}^0$ muoproduction cross section at COMPASS / Gorzellik, Matthias (Freiburg U.) At COMPASS DVCS and DVMP processes are studied in order to probe the partonic structure of the nucleon by constraining GPD models. [...] arXiv:1702.06293. - 4 p. Preprint 2017-02-10 07:37 COMPASS measurement of the $P_T$ weighted Sivers asymmetry / Bradamante, Franco (INFN, Trieste) The SIDIS transverse spin asymmetries weighted with powers of $P_T$, the hadron transverse momentum in the $\gamma N$ reference system,have been introduced already twenty years ago and are considered quite interesting. [...] arXiv:1702.00621. - 4 p. Preprint 2016-10-08 07:48 The spectrum of light isovector mesons with $C=+1$ from the COMPASS experiment / Paul, Stephan (Munich, Tech. U.) Based on the largest event sample of diffractively produced $\pi^-\pi^-\pi^+$, obtained by a pion beam of $190~\rm{GeV/c}$ momentum, the COMPASS collaboration has performed the most advanced partial wave analysis on multi-body final states, using the isobar model. The large number of waves included in the analysis reduces truncation effects. [...] arXiv:1610.01231.- 2016 - 9 p. - Published in : JPS Conf. Proc. 13 (2017) 010018 Fulltext: PDF; External link: Preprint In : Meson-Nucleon Physics and the Structure of the Nucleon, Kyoto, Japan, 25 - 30 Jul 2016, pp.010018 2016-09-29 01:12 Photons interacting with Hadrons at COMPASS / Friedrich, Jan Michael (Technische Universitaet Muenchen (DE)) /COMPASS NA58 Collaboration COMPASS is a multi-purpose experiment operated at CERN for investigations of the strong interaction from low to medium energy scales. At low energies, photon-pion interactions provide input for understanding the effective dynamics of strongly bound states of quarks and gluons, most prominently the recent measurement of the polarisability of the charged pion. [...] COMPASS-PROC-2016-001.- Geneva : CERN, 2015 - 8. Fulltext: PDF; 2016-09-28 07:58 Extraction of the $\pi^+\pi^-$ Subsystem in Diffractively Produced $\pi^-\pi^+\pi^-$ at COMPASS / Krinner, Fabian (Munich, Tech. U.) /for the COMPASS collaboration The COMPASS experiment at CERN has collected a large data sample of 50 million diffractively produced $\pi^-\pi^+\pi^-$ events using a $190\,$GeV$/c$ negatively charged hadron beam. The partial-wave analysis (PWA) of these high-precision data reveals previously unseen details. [...] arXiv:1609.08514.- 2017 - 5 p. - Published in : JPS Conf.Proc. 13 (2017) 020028 Preprint: PDF; External link: Preprint In : Meson-Nucleon Physics and the Structure of the Nucleon, Kyoto, Japan, 25 - 30 Jul 2016, pp.020028 2016-07-27 06:00 Tests of Chiral perturbation theory with COMPASS / Friedrich, Jan M (CERN) /COMPASS Collaboration The COMPASS experiment at CERN accesses pion-photon reactions via the Primakoff effect., where high-energetic pions react with the quasi-real photon field surrounding the target nuclei. When a single real photon is produced, pion Compton scattering is accessed and from the measured cross-section shape, the pion polarisability is determined. [...] 2014 - 4 p. - Published in : EPJ Web Conf. 73 (2014) 05011 Fulltext: PDF; In : 13th International Conference on Meson-Nucleon Physics and the Structure of the Nucleon, Rome, Italy, 30 Sep - 4 Oct 2013, pp.05011 2016-06-08 06:45 New results from COMPASS / Kabuß, Eva-Maria (Mainz U.) /COMPASS Collaboration An overview on recent COMPASS results is given, including the extraction of the longitudinal spin structure functions interpreted with a NLO QCD fit, new results on the gluon polarisation and a measurement of pion and kaon multiplicities with a LO extraction of quark-to-hadron fragmentation functions SISSA, 2015 - 6 p. - Published in : PoS: EPS-HEP2015 (2015) , pp. 502 Fulltext: PDF; External link: Published version from PoS In : European Physical Society Conference on High Energy Physics 2015, Vienna, Austria, 22 - 29 Jul 2015, pp.502 2016-06-08 06:45 Generalized Parton Distributions program at COMPASS / Fuchey, Eric (Saclay) /COMPASS Collaboration SISSA, 2015 - Published in : PoS: QCDEV2015 (2015) , pp. 048 Fulltext: PDF; External link: Published version from PoS In : QCD Evolution Workshop, Newport News, VA, USA, 26-30 May 2015, pp.048 2016-06-08 06:45 Precision Hadron Spectroscopy at COMPASS - The Scalar Isoscalar Meson Spectrum - / Austregesilo, Alexander (Munich, Tech. U.) /COMPASS Collaboration SISSA, 2015 - Published in : PoS: Bormio2015 (2015) , pp. 022 Fulltext: PDF; External link: Published version from PoS In : 53rd International Winter Meeting on Nuclear Physics, Bormio, Italy, 26 - 30 Jan 2015, pp.022	2018-03-17 22:03:21	{"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": 0.7915108799934387, "perplexity": 5669.75405916656}, "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-2018-13/segments/1521257645362.1/warc/CC-MAIN-20180317214032-20180317234032-00443.warc.gz"}
https://www.mathway.com/examples/algebra/analytic-geometry/finding-the-square-or-rectangle-area-given-three-points?id=694	# Algebra Examples Find the Square or Rectangle Area , , The distances between the given points are the length (l), width (w), and diagonal (d). The distance between the three corners is: and and and The diagonal is the longest distance and the width is the shortest distance, which means the and . The area of a rectangle is . Multiply by to get . We're sorry, we were unable to process your request at this time Step-by-step work + explanations • Step-by-step work • Detailed explanations • Access anywhere Access the steps on both the Mathway website and mobile apps $--.--/month$--.--/year (--%)	2018-02-22 13:03:54	{"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": 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": 0.5763549208641052, "perplexity": 3421.5274362331884}, "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-2018-09/segments/1518891814105.6/warc/CC-MAIN-20180222120939-20180222140939-00626.warc.gz"}
http://www.thestudentroom.co.uk/showthread.php?t=2043061&page=2&p=38378834	You are Here: Home # Trig question Tweet Maths and statistics discussion, revision, exam and homework help. Announcements Posted on TSR launches Learn Together! - Our new subscription to help improve your learning 16-05-2013 IMPORTANT: You must wait until midnight (morning exams)/4.30AM (afternoon exams) to discuss Edexcel exams and until 1pm/6pm the following day for STEP and IB exams. Please read before posting, including for rules for practical and oral exams. 28-04-2013 1. Re: Trig question Well, if we factorize the LHS, we can further restrict the domain to but I don't see how that helps... Do you have a context for this question, Plato's Trousers? Last edited by aznkid66; 27-06-2012 at 21:51. 2. Re: Trig question While I've got this equation written down (again) I'll post it. I'm not sure it's much help but it's somewhere safe to write it down. and the factorisation 3. Re: Trig question (Original post by BabyMaths) While I've got this equation written down (again) I'll post it. I'm not sure it's much help but it's somewhere safe to write it down. and the factorisation The factorisation is not right. Expanded you get - 4. Re: Trig question (Original post by SecondHand) The factorisation is not right. Expanded you get - The factorisation is right. I just wrote the wrong equation. 5. Re: Trig question Yeah, I got the same equation (all signs flipped). The determinant of the quadratic is negative, right? So the quadratic has no real solutions. Then we can...use the cubic formula and find the real solution(s) to c, and cancel out or double the solutions when we put c=cos(±x+2pik) into the domain [0,pi/2]. Sum-to-product, quintic factorization, and cubic formula...I'm still waiting for the context that might lead us to a more elegant solution. :P 6. Re: Trig question (Original post by aznkid66) Yeah, I got the same equation (all signs flipped). The determinant of the quadratic is negative, right? So the quadratic has no real solutions. Then we can...use the cubic formula and find the real solution(s) to c, and cancel out or double the solutions when we put c=cos(±x+2pik) into the domain [0,pi/2]. Sum-to-product, quintic factorization, and cubic formula...I'm still waiting for the context that might lead us to a more elegant solution. :P The determinant is positive, 4-(44-1)=8 7. Re: Trig question (Original post by SecondHand) The determinant is positive, 4-(44-1)=8 D'oh. I always mess up when I take the negative of a negative. So cosx=(1/4)±(1/sqrt(8)) ...that looks nice :\| 8. Re: Trig question (Original post by aznkid66) D'oh. I always mess up when I take the negative of a negative. So cosx=(1/4)±(1/sqrt(8)) ...that looks nice :\| You just need to show that only two solutions exist, not find the solutions. 9. Re: Trig question (Original post by SecondHand) You just need to show that only two solutions exist, not find the solutions. Well, cubic aside, even after finding there exists two unique solutions for c we don't know how many unique x values of each solution c=cosx are in the domain of 0≤x≤pi/2. Last edited by aznkid66; 28-06-2012 at 03:01. 10. Re: Trig question Could you not just say, within the interval, there is only one point of inflection (and then find that point which is blow -1/2), hence there can only be two points since you've already proved the graph is continuous. Last edited by djpailo; 28-06-2012 at 11:45. 11. Re: Trig question (Original post by djpailo) Could you not just say, within the interval, there is only one point of inflection (and then find that point which is blow -1/2), hence there can only be two points since you've already proved the graph is continuous. But how do you prove that there is only one through graphical analysis? 12. Re: Trig question (Original post by djpailo) Could you not just say, within the interval, there is only one point of inflection (and then find that point which is blow -1/2), hence there can only be two points since you've already proved the graph is continuous. Yes. This is the method they are looking for. 13. Re: Trig question (Original post by Plato's Trousers) Yes. This is the method they are looking for. So, without calculus? 14. Re: Trig question (Original post by ghostwalker) So, without calculus? yes, I think it's basically about finding values of the function either side of zero and then inferring that (because the function is continuous) that it must have zeroes between them	2013-05-20 09:30:17	{"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": 0.8706517815589905, "perplexity": 2032.1193850851573}, "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-2013-20/segments/1368698693943/warc/CC-MAIN-20130516100453-00018-ip-10-60-113-184.ec2.internal.warc.gz"}
https://www.lesswrong.com/users/avoropaev/replies	# All of AVoropaev's Comments + Replies The case for hypocrisy But that's a fix to a global problem that you won't fix anyway. What you can do is allocate some resources to fixing a lesser problem "this guy had nothing to eat today". It seems to me that your argument proves too much -- when faced with a problem that you can fix you can always say "it is a part of a bigger problem that I can't fix" and do nothing. The case for hypocrisy What do you mean by 'real fix' here? What if said that real-real fix requires changing human nature and materialization of food and other goods out of nowhere? That might be more effective fix, but it is unlikely to happen in near future and it is unclear how you can make it happen. Donating money now might be less effective, but it is somehow that you can actually do. 1Gerald Monroe4moA real fix is forcing everyone in a large area to contribute to fixing a problem. If enough people can't be compelled to contribute the problem can't be fixed. Doing something that costs you resources but doesn't fix the problem and negatively affects you vs others who aren't contributing but are competing with you isn't a viable option. In prisoners dilemma you may preach always cooperate but you have to defect if your counterparty won't play fair. Similarly warren Buffett can preach that billionaires should pay more taxes but not pay any extra voluntarily until all billionaires have to. In Defence of Spock Detailed categorizations of mental phenomena sounds useful. Is there a way for me to learn that without reading religious texts? 1frcassarino5moQualia Research institute is working on building a catalogue of qualia iirc. Julia Galef and Matt Yglesias on bioethics and "ethics expertise" How can you check proof of any interesting statement about real world using only math? The best you can do is check for mathematical mistakes. 3Gerald Monroe6mo"what do they claim to know and how do they know it" No amount of credentials or formal experience makes an expert not wrong if they do not have high quality evidence, that they have shown, to get their conclusions from. And an algorithm formally proven to be correct that they show they are using. Or in the challenge trials : ethicist claims to value human life. A challenge trial only risks the lives of a few people, where even if they die it would have saved hundreds of thousands. In this case the " basic math" is one of multiplication and quantities, showing the "experts" don't know anything. As you might notice, ethicists do not have high quality information as input to generate their conclusions from. Without that information you cannot expect more than expensive bullshitting. "Ethics" today is practiced by reading ancient texts and more modern arguments, many of which have cousins with religion. But ethics is not philosophy. It is actually a math problem. Ultimately, there are things you claim to value ("terminal values"). There are actions you can consider doing. Some actions have an expected value that with a greater score on the things you care about, and some actions have a lesser expected value. Any other action but taking the one with the highest expected value (factoring in variance), is UNETHICAL. Yes, professional ethicists today are probably mostly all liars and charlatans, no more qualified than a water douser. I think EY worked down to this conclusion in a sequence but this is the simple answer. One general rule of thumb if you didn't read the above: if an expert claims to know what they are doing, look at the evidence they are using. I don't know the anatomy of the human body enough to gainsay an orthopedic surgeon, but I'm going to trust the one that actually looks at a CT scan over one that palpates my broken limb and reads from some 50 year old book. Doesn't matter if the second one went to the most credible medical school and has 50 year Extracting Money from Causal Decision Theorists I assume you mean that I assume P(money in Bi \| buyer chooses Bi )=0.25? Yes, I assume this, although really I assume that the seller's prediction is accurate with probability 0.75 and that she fills the boxes according to the specified procedure. From this, it then follows that P(money in Bi \| buyer chooses Bi )=0.25. Yes, you are right. Sorry. Why would it be a logical contradiction? Do you think Newcomb's problem also requires a logical contradiction? Okay, it probably isn't a contradiction, because the situation "Buyer writes his decision and it is common... (read more) Extracting Money from Causal Decision Theorists I've skimmed over the beginning of your paper, and I think there might be several problems with it. 1. I don't see where it is explicitly stated, but I think information "seller's prediction is accurate with probability 0,75" is supposed to be common knowledge. Is it even possible for a non-trivial probabilistic prediction to be a common knowledge? Like, not as in some real-life situation, but as in this condition not being logical contradiction? I am not a specialist on this subject, but it looks like a logical contradiction. And you can prove absolutel 4Caspar428mo>I think information "seller's prediction is accurate with probability 0,75" is supposed to be common knowledge [https://en.wikipedia.org/wiki/Common_knowledge_(logic)]. Yes, correct! >Is it even possible for a non-trivial probabilistic prediction to be a common knowledge? Like, not as in some real-life situation, but as in this condition not being logical contradiction? I am not a specialist on this subject, but it looks like a logical contradiction. And you can prove absolutely anything if your premise contains contradiction. Why would it be a logical contradiction? Do you think Newcomb's problem also requires a logical contradiction? Note that in neither of these cases does the predictor tell the agent the result of a prediction about the agent. >What kinds of mistakes does seller make? For the purpose of the paper it doesn't really matter what beliefs anyone has about how the errors are distributed. But you could imagine that the buyer is some piece of computer code and that the seller has an identical copy of that code. To make a prediction, the seller runs the code. Then she flips a coin twice. If the coin does not come up Tails twice, she just uses that prediction and fills the boxes accordingly. If the coin does come up Tails twice, she uses a third coin flip to determine whether to (falsely) predict one of the two other options that the agent can choose from. And then you get the 0.75, 0.125, 0.125 distribution you describe. And you could assume that this is common knowledge. Of course, for the exact CDT expected utilities, it does matter how the errors are distributed. If the errors are primarily "None" predictions, then the boxes should be expected to contain more money and the CDT expected utilities of buying will be higher. But for the exploitation scheme, it's enough to show that the CDT expected utilities of buying are strictly positive. >When you write "$1−P (money in Bi \| buyer chooses Bi ) ·$3 = $1 − 0.25 ·$3 = \$0.25.", you assume that P(m What's the big deal about Bayes' Theorem? I've skimmed over A Technical Explanation of Technical Explanation (you can make links end do over stuff by selecting the text you want to edit (as if you want to copy it); if your browser is compatible, toolbar should appear). I think that's the first time in my life when I've found out that I need to know more math to understand non-mathematical text. The text is not about Bayes' Theorem, but it is about application of probability theory to reasoning, which is relevant to my question. As far as I understand, Yudkowski writes about the same algorithm that... (read more) What's the big deal about Bayes' Theorem? That's interesting. I've heard about probabilistic modal logics, but didn't know that not only logics are working towards statisticians, but also vice versa. Is there some book or videocourse accessible to mathematical undergraduates? What's the big deal about Bayes' Theorem? This formula is not Bayes' Theorem, but it is a similar simple formula from probability theory, so I'm still interested in how you can use it in daily life. Writing P(x\|D) implies that x and D are the same kind of object (data about some physical process?) and there are probably a lot of subtle problems in defining hypothesis as a "set of things that happen if it is true" (especially if you want to have hypotheses that involve probabilities). Use of this formula allows you to update probabilities you prescribe to hypotheses, but it is no... (read more) The above formula is usually called "odds form of Bayes formula". We get the standard form by letting in the odds form, and we get the odds form from the standard form by dividing it by itself for two hypotheses ( cancels out). The serious problem with the standard form of Bayes is the term, which is usually hard to estimate (as we don't get to choose what is). We can try to get rid of it by expanding but that's also no good, because now we need to know . One way to state the problem... (read more)	2021-09-23 06:46:56	{"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": 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": 0.6029451489448547, "perplexity": 852.7171282541667}, "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/1631780057417.10/warc/CC-MAIN-20210923044248-20210923074248-00204.warc.gz"}
https://cstheory.stackexchange.com/questions/37965/what-else-besides-the-usual-can-be-said-about-a-scott-information-system-if-th	# What else (besides the usual) can be said about a Scott Information System if the constructed domain is required to be Hausdorff? As per subject, if a Scott domain with $T_2$ topology is to be constructed from a Scott information system, then what besides the usual definitional requirements, e.g., https://en.wikipedia.org/wiki/Scott_information_system , does the information system have to satisfy? Information systems are more completely discussed in (Scott's original paper) https://www.researchgate.net/publication/220897586_Domains_for_Denotational_Semantics (note: the original paper is followed by a three-page bibliography missing from this pdf) containing a section "Topological Considerations" starting on page 586, where "Hausdorff" is briefly mentioned on page 588 with respect to (what Scott calls) "total elements", $Tot_A$, assuming no unique top. But the domain itself, what Scott calls $\mid A\mid$, is still $T_0$ in general. And I'm asking for additional definitional requirements so that $\mid A\mid$, when contsructed from the corresponding information system, is "automatically" $T_2$. • As Andrew points out, the only way to get $T_2$ for the entire domain is to make it trivial. But also note that apart from Scott topology on a domain there is also the Lawson topology on a domain, which is Hausdorff. – Andrej Bauer Apr 12 '17 at 18:50 • Thanks, @Kaveh for your comment about Stone duality, though you seem to have deleted it. Scott's "brief mention" that I cited above indeed also says "compact totally disconnected Hausdorff space", and I'd imagine you're likely right that this suggests some connection with the Stone representation theorem (though google's not immediately coughing it up). – John Forkosh Apr 13 '17 at 7:52 • Thanks @AndrejBauer (and also thanks for comment below). And I think you may be right, as follows. If for every data token $a\in A$ there's a complement $a^\perp\in A$, maybe analogous to $a\subseteq\mathbb{N}\;\mbox{r.e.}\Longrightarrow \mathbb{N}\backslash a\;\mbox{r.e.}$, then would that induce a Lawson topology on the corresponding constructed domain??? (Haven't tried to answer that myself yet; only just now saw your comment and had the preceding thought.) – John Forkosh Apr 13 '17 at 8:04 • Note that the maximal set of an (algebraic) domain is totally disconnected, whereas Stone duality gives you zero-dimensional spaces. These are two different concepts, and indeed we get different kinds of spaces. There is no direct relation to Stone duality, but of course one can cook up a slightly less direct one (as always). – Andrej Bauer Apr 13 '17 at 8:27 • Thanks again, @AndrejBauer . I'm now thinking your and A.P.'s remarks below are likely setting me on the right track -- think about the space of total elements rather than the domain. That gives more opportunity/flexibility to (your words) "cook up" interesting stuff. – John Forkosh Apr 14 '17 at 8:24 Since the least element $\bot$ of any Scott domain is a compactification point $-$ the only open set containing it is the whole space $-$ the Scott topology is never Hausdorff, unless it is trivial. An information system $(A,Con,\vdash)$ induces a trivial domain iff it is itself trivial: $Con = \{\emptyset\}$. Otherwise, there exists at least one point above $\bot$. The total elements of the Scott domain can indeed be given their own topology sometimes, which may be $T_2$ or even metrizable.	2021-04-13 20:15:35	{"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": 0.9195312857627869, "perplexity": 559.8724098772832}, "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-2021-17/segments/1618038074941.13/warc/CC-MAIN-20210413183055-20210413213055-00533.warc.gz"}
https://de.maplesoft.com/support/help/maple/view.aspx?path=OreTools/AdjointRing&L=G	OreTools construct the adjoint of a given Ore polynomial ring compute the adjoint Ore polynomial in a given Ore ring Parameters Poly - Ore polynomial; to define an Ore polynomial, use the OrePoly structure. A - Ore ring; to define an Ore ring, use the SetOreRing function. Description • The AdjointOrePoly(Poly, A) calling sequence computes the adjoint Ore polynomial of the polynomial Poly in A. • An Ore polynomial ring is defined vi SetOreRing. For a description of the adjoint of an Ore polynomial ring, see OreAlgebra. Examples > $\mathrm{with}\left(\mathrm{OreTools}\right):$ > $\mathrm{with}\left(\mathrm{OreTools}\left[\mathrm{Properties}\right]\right):$ Define the shift polynomial ring. > $A≔\mathrm{SetOreRing}\left(n,'\mathrm{shift}'\right)$ ${A}{≔}{\mathrm{UnivariateOreRing}}{}\left({n}{,}{\mathrm{shift}}\right)$ (1) Construct the adjoint Ore polynomial ring B of A. > $B≔\mathrm{AdjointRing}\left(A\right)$ ${B}{≔}{\mathrm{Adj}}{}\left({\mathrm{UnivariateOreRing}}{}\left({n}{,}{\mathrm{shift}}\right)\right)$ (2) Construct the adjoint Ore polynomial ring C of B. The ring C must be the same as A. > $C≔\mathrm{AdjointRing}\left(B\right)$ ${C}{≔}{\mathrm{UnivariateOreRing}}{}\left({n}{,}{\mathrm{shift}}\right)$ (3) > $\mathrm{GetSigma}\left(A\right)\left(s\left(n\right),n\right)=\mathrm{GetSigma}\left(C\right)\left(s\left(n\right),n\right)$ ${s}{}\left({n}{+}{1}\right){=}{s}{}\left({n}{+}{1}\right)$ (4) > $\mathrm{GetSigmaInverse}\left(A\right)\left(s\left(n\right),n\right)=\mathrm{GetSigmaInverse}\left(C\right)\left(s\left(n\right),n\right)$ ${s}{}\left({n}{-}{1}\right){=}{s}{}\left({n}{-}{1}\right)$ (5) > $\mathrm{Getdelta}\left(A\right)\left(s\left(n\right),n\right)=\mathrm{Getdelta}\left(C\right)\left(s\left(n\right),n\right)$ ${0}{=}{0}$ (6) Define two Ore polynomials P1 and P2 in A. > $\mathrm{P1}≔\mathrm{OrePoly}\left(n+1,n\right);$$\mathrm{P2}≔\mathrm{OrePoly}\left(1,n+1\right)$ ${\mathrm{P1}}{≔}{\mathrm{OrePoly}}{}\left({n}{+}{1}{,}{n}\right)$ ${\mathrm{P2}}{≔}{\mathrm{OrePoly}}{}\left({1}{,}{n}{+}{1}\right)$ (7) Compute the adjoint operators of P1 and P2 in A. > $\mathrm{adjP1}≔\mathrm{AdjointOrePoly}\left(\mathrm{P1},A\right)$ ${\mathrm{adjP1}}{≔}{\mathrm{OrePoly}}{}\left({n}{+}{1}{,}{n}{-}{1}\right)$ (8) > $\mathrm{adjP2}≔\mathrm{AdjointOrePoly}\left(\mathrm{P2},A\right)$ ${\mathrm{adjP2}}{≔}{\mathrm{OrePoly}}{}\left({1}{,}{n}\right)$ (9) > $\mathrm{Multiply}\left(\mathrm{adjP1},\mathrm{adjP2},B\right)$ ${\mathrm{OrePoly}}{}\left({n}{+}{1}{,}{{n}}^{{2}}{+}{2}{}{n}{-}{1}{,}{\left({n}{-}{1}\right)}^{{2}}\right)$ (10)	2022-06-27 15:28:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 24, "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": 0.9865561723709106, "perplexity": 2708.3982842238197}, "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/1656103334753.21/warc/CC-MAIN-20220627134424-20220627164424-00031.warc.gz"}
https://physics.aps.org/articles/v15/s106	Synopsis # Identifying a Galactic Particle Accelerator Physics 15, s106 An analysis of 12 years of gamma-ray observations has allowed researchers to pinpoint a Galactic source of high-energy cosmic rays. Cosmic rays constantly bombard Earth from all directions. The energy spectrum of these particles—which are mostly protons and other atomic nuclei—roughly follows a power law. Most of the particles having energies of a few GeV or less, but a spectral feature at around 1 PeV hints at a fraction of particles accelerated to much higher energies. These high-energy cosmic rays are thought to originate from within our Galaxy, but the identity of the accelerators, or “PeVatrons,” that produce them remains a mystery (see Viewpoint: Signs of PeVatrons in Gamma-Ray Haze). Now, after analyzing 12 years of data taken by the Fermi Large Area Telescope (Fermi-LAT), researchers point to a supernova remnant (SNR) as a potential source [1]. The precise origin of cosmic rays is difficult to pin down because the trajectories of these charged particles are perturbed by interstellar magnetic fields. Astronomers map cosmic-ray sources indirectly by observing the gamma radiation created when cosmic rays interact with interstellar material near to where astronomers think the particles were created. But the very-high-energy radiation that might signpost a PeVatron can also be generated by other processes, such as inverse Compton scattering of cosmic background radiation by relativistic electrons. Ke Fang, at the University of Wisconsin-Madison, and colleagues studied G106.3 + 2.7, an SNR about 2600 light years away, seeking a mechanism that could fit both the gamma radiation detected from it by Fermi-LAT and the x rays and radio waves measured by other observatories. They found that the data strongly support the SNR being a PeVatron, while being incompatible with an inverse-Compton-scattering origin for the radiation. Fang says that she hopes that G106.3 + 2.7 will be the first of many galactic PeVatrons to be discovered. –Marric Stephens Marric Stephens is a Corresponding Editor for Physics Magazine based in Bristol, UK. ## References 1. K. Fang et al., “Evidence for PeV proton acceleration from Fermi-LAT observations of SNR G106.3 + 2.7,” Phys. Rev. Lett. 129, 071101 (2022). ## Related Articles Astrophysics ### Mini Interferometers Offer Impressive Sensitivity A sensor containing thumbnail-sized interferometers might help astronomers detect gravitational waves emitted from certain black hole mergers. Read More » Astrophysics ### To Hear or Not to Hear Overtones in Black Hole Mergers A new analysis questions whether researchers found so-called overtones in the first detected gravitational-wave signal from a black hole merger. Read More » Astrophysics ### Experiment Sees Elusive Magnetic-Fluid Instability Magnetorotational instability—a process that might explain the dynamics of astrophysical accretion disks—has finally been observed in the laboratory. Read More »	2022-10-03 19:07:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.562930703163147, "perplexity": 2271.194071729702}, "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/1664030337428.0/warc/CC-MAIN-20221003164901-20221003194901-00767.warc.gz"}
https://www.physicsforums.com/threads/taylor-series-and-random-variables.770783/	# Homework Help: Taylor Series and Random Variables 1. Sep 14, 2014 ### GottaLoveMath 1. The problem statement, all variables and given/known data A standard procedure for finding an approximate mean and variance of a function of a variable is to use a Taylor Expansion for the function about the mean of the variable. Suppose the variable is y, and that its mean and standard deviation are "u" and "o". f(y) = f(u) + f'(u)(y-u) + f''(u)(((y-u)^2)/2!)) + f'''(u)((y-u)^3)/3!)) + ... Consider the case of f(.) as e^(.). By taking the expectation of both sides of this equation, explain why the bias correction factor given in Equation A is an overcorrection if the residual series has a negative skewness, where skewness p of a random variable y is defined by p = E((y-u)^3)/(o^3) Equation A = x^hat_t = e^(m_t + s_t)*e^((1/2)(o^2)) where x_t is observed series, m_t is the trend, s_t is seasonal effect 2. Relevant equations 3. The attempt at a solution Im not even really sure where to start. If someone could point me in the right direction, it would be greatly appreciated 2. Sep 14, 2014 $$E(f(y)) = f(\mu) + f'(\mu)E(y-\mu) + \frac{f'(\mu)}{2!} E((y-\mu)^2) + \frac{f'''(\mu)}{3!} E((y-\mu)^3)$$	2018-05-28 10:11:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7933908700942993, "perplexity": 629.0317827873677}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794872766.96/warc/CC-MAIN-20180528091637-20180528111637-00574.warc.gz"}
https://brilliant.org/discussions/thread/help-needed-10/	# Help needed!! I want help in a question that i came across $$\displaystyle\lim_{x \rightarrow 0}(\sum _{ r=0 }^{ n }{ (-1)^{ r }.^{ n }C_{ r } } (\sum _{ k=0 }^{ n-r }{ ^{ n-r }{ C }_{ k }{ 2 }^{ k }{ x }^{ k } } )(x^{ 2 }-x)^r) ^{\frac{1}{x}}$$ If the value of the limit = $e^{\lambda n}$. Find $\lambda$ My work: $\Rightarrow\displaystyle\lim_{x \rightarrow 0}(\sum _{ r=0 }^{ n }{ (-1)^{ r }.^{ n }C_{ r } } (1+2x)^{n-r}(x^{ 2 }-x)^r)^ {\frac{1}{x}}$ $\Rightarrow\displaystyle\lim_{x \rightarrow 0}(\sum _{ r=0 }^{ n }{^{ n }C_{ r } } (1+2x)^{n-r}(-1)^{r}(x^{ 2 }-x)^r)^{\frac{1}{x}}$ $\Rightarrow \displaystyle\lim_{x \rightarrow 0}(\sum _{ r=0 }^{ n }{^{ n }C_{ r } } (1+2x)^{n-r}(x-x^2)^r)^{\frac{1}{x}}$ $\Rightarrow\displaystyle\lim_{x \rightarrow 0}(1+3x-x^2)^{\frac{n}{x}}$ How to calculate this limit? Note by Gautam Sharma 6 years, 4 months ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events. MarkdownAppears as italics or _italics_ italics bold or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting. 2 \times 3 $2 \times 3$ 2^{34} $2^{34}$ a_{i-1} $a_{i-1}$ \frac{2}{3} $\frac{2}{3}$ \sqrt{2} $\sqrt{2}$ \sum_{i=1}^3 $\sum_{i=1}^3$ \sin \theta $\sin \theta$ \boxed{123} $\boxed{123}$ Sort by: Answer Should be $\lambda =3$ . It is ${ 1 }^{ \infty }$ form. - 6 years, 4 months ago change limit as $L={ e }^{ n\lim _{ x\rightarrow 0 }{ \cfrac { \ln { (1+3x-{ x }^{ 2 }) } }{ x } } }$ now you can use expansion as suggested by taylor series. Else this is 0/0 form so you can use L-hospital rule , if you had studied it yet. Else There is direct formula for ${ 1 }^{ \infty }$ form : $L=\lim _{ x\rightarrow a }{ { f\left( x \right) }^{ g\left( x \right) } } ={ e }^{ \lim _{ x\rightarrow a }{ { (f\left( x \right) -1)g\left( x \right) } } }\\ here:\quad \lim _{ x\rightarrow a }{ { f\left( x \right) } } =1\quad \& \quad \lim _{ x\rightarrow a }{ { g\left( x \right) } } =\infty \quad$ - 6 years, 4 months ago Oh thanks - 6 years, 4 months ago Yeah but explain. - 6 years, 4 months ago You can use the fact that $\displaystyle \lim_{x \rightarrow 0} f(x)^{g(x)} = \displaystyle \lim_{x \rightarrow 0} e^{g(x)(f(x) - 1)}$ - 6 years, 4 months ago Thanks - 6 years, 4 months ago	2021-08-04 01:13:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 20, "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": 0.9918429851531982, "perplexity": 3914.583370740582}, "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/1627046154486.47/warc/CC-MAIN-20210803222541-20210804012541-00325.warc.gz"}
http://mathhelpforum.com/trigonometry/212566-sine-graph-equation-print.html	# Sine Graph Equation • Feb 4th 2013, 02:09 PM shazi Sine Graph Equation I need help! Thanks in advanced! Attachment 26858 The curve above is the graph of a sinusoidal function. It goes through the point (0,1) and (4,1). Find a sinusoidal function that matches the given graph. • Feb 4th 2013, 02:38 PM Plato Re: Sine Graph Equation Quote: Originally Posted by shazi I need help! Thanks in advanced! Attachment 26858 The curve above is the graph of a sinusoidal function. It goes through the point (0,1) and (4,1). Find a sinusoidal function that matches the given graph. You are looking for three numbers $a\sin(bx)+c$ to do that. Hint: $\frac{-\pi}{2}$ is one of them. • Feb 4th 2013, 02:42 PM shazi Re: Sine Graph Equation I am pretty sure c is 1, but how do I find a or b? • Feb 4th 2013, 02:44 PM Plato Re: Sine Graph Equation Quote: Originally Posted by shazi I am pretty sure c is 1, but how do I find a or b? • Feb 4th 2013, 02:46 PM shazi Re: Sine Graph Equation I did, but I still don't know how • Feb 4th 2013, 03:06 PM Plato Re: Sine Graph Equation Quote: Originally Posted by shazi I did, but I still don't know how Take a graphing calculator and try some possibilities. You say $c=1$ and I told you that $-\frac{\pi}{2}$ is one other answer.	2017-07-25 23:23:06	{"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": 4, "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": 0.8928420543670654, "perplexity": 1902.9775611558105}, "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-2017-30/segments/1500549425407.14/warc/CC-MAIN-20170725222357-20170726002357-00238.warc.gz"}
https://stsievert.com/blog/2015/12/09/inverse-part-2/	This post is a part 2 of a 3 part series: Part I, Part II, Part III We often have fewer measurements than unknowns, which happens all the time in genomics and medical imaging. For example, we might be collecting 8,000 gene measurements in 300 patients and we’d like to determine which ones are most important in determining cancer. This means that we typically have an underdetermined system because we’re collecting more measurement than unknowns. This is an unfavorable situation – there are infinitely may solutions to this problem. However, in the case of breast cancer, biological intuition might tell us that most of the 8,000 genes aren’t important and have zero important in cancer expression. How do we enforce that most of the variables are 0? This post will try and give intuition for the problem formulation and dig into the algorithm to solve the posed problem. I’ll use a real-world cancer dataset1 to predict which genes are important for cancer expression. It should be noted that we’re more concerned with the type of solution we obtain rather than how well it performs. This post will build of Part I of this series. In this post, we will only change the type of solution we obtain by changing the regularization parameter. In Part I, we saw that we could get an acceptable solution by introducing a regularization parameter, $\norm{\xb}_2^2$. In this post, we’ll examine changing that to $\norm{\xb}_1$. Before getting started, we’ll have to use norms, as they provide a nice syntax for working with vectors and matrices. We define the $\ell_p$ norms as $$\norm{\xb}_p = \parens{\sum \abs{x_i}^p}^{1/p}$$ which means that $\ell_2$ norm of $\xb$ is $\norm{\xb}_2 := \sqrt{\sum_i x_i^2}$ (meaning $\norm{\xb}_2^2 = \sum_i x_i^2$) and $\ell_1$ norm of $\xb$ is $\norm{\xb}_1 := \sum_i \abs{x_i}$. This definition doesn’t define $\ell_0$ norms, but we define it to be the number of non-zero terms. ## LASSO problem formulation Through external information, we know that most of our solution is 0. Therefore, we want to limit the number of non-zeros in our solution. We can enforce adding a penalty for the number of non-zeros. That is, after observing $\yb$ and $\Xb$, we’re trying to find a solution $\widehat{\wb}$ that minimizes the squared error and has a small number of zeros. We can do this by adding a penalty on the number of non-zeros: \align{ \widehat{\wb} &= \arg \min_\wb \norm{\yb - \Xb\wb}_2^2 + \lambda\sum_i 1_{w_i \not= 0} } but this problem is exceptionally hard to solve because this is non-convex and NP-hard. The best algorithms to solve this by allowing $k$ variables to be non-zero and runs through all $2^n$ ways to do that. This takes exponential time… how can we find a more efficient method? The closest convex relaxation of the $\ell_0$ norm is the $\ell_1$ norm. In our new problem with the $\ell_1$ norm as defined above, we can make the $\ell_1$ norm small by making many of the terms zero. This means we’re trying to solve $$\widehat{\wb} = \arg \min_\wb \norm{\yb - \Xb\wb}_2^2 + \lambda\norm{\wb}_1$$ The type of regularization matters characterizes the signal we get as output. In this problem formulation, we are including the term $\norm{\wb}_1$ or the $\ell_1$ regularization parameter. This gives a much different result than the $\ell_2$ regularization parameter $\norm{\wb}_2^2$. To help see this, I have developed an interactive widget that highlights the difference between $\ell_1$ and $\ell_2$ regularization! The type of regularization parameter we use matters a ton – there’s a huge difference in the output when using $\norm{\wb}^2_2$ instead of $\norm{\wb}_1$. Why? We can think about the physical interpretations. If trying to optimize for the engine to buy and normalizing for appropriate units, we might use the • $\ell_2$ norm if we’re trying to use as little gas as possible. This corresponds to using as little energy as possible. This makes sense, because energy typically comes in squares (i.e., kinetic energy is $\frac{1}{2}mv^2$. See Table 3 at tauday.com for more examples). • $\ell_0$ or $\ell_1$ norm if we want to run the engine as little as possible. We don’t care about how much gas we use, just how long it’s running. Because the engine is off most of the time, this corresponds to a sparse solution. • $\ell_\infty$ norm, or the maximum element in a vector. This would correspond to being limited to how much power we can use. We can have the engine on as long as we want and use as much gas as we want. For example, the state could regulate that cars have to be less powerful than some limit. To help provide more intuition, I have provided a 2D example using an interactive widget. In this 2D example, we can think that $\Xb = [c_1, c_2]$ and $y$ as a scalar. We’re trying to find $\wb = [w_1, w_2]^T \in \R^2$, but we only have one measurement; there are infinitely many solutions. • All possible solutions to $y = c_1 w_1 + c_2 w_2$ are graphed by the purple line. We know $y$ and are trying to estimate $w_1$ and $w_2$ from our knowledge of $c_1$ and $c_2$. • The $\ell_1$ solution vector and the the $\ell_2$ solution vector are in blue and green. The norm balls for the solution vectors are also graphed. When we change $c_1$, we see that the solution tends to minimize the distance between the norm ball and the line of all possible solutions. We can see when we increase $\lambda$, our estimate gets smaller. This makes sense because we are placing more emphasis on this value, and it reaches it’s optimum at the origin. We can think of this optimization for $\ell_2$ and $\ell_1$ as minimizing the distance between the norm balls and the line of all possible solutions. We see that the $\ell_1$ norm tends to give solutions with more zeros in them such as $(1, 0)$ or $(0, 1)$. The $\ell_2$ solution gives more non-zero values off the axis which means, by definition, the $\ell_1$ solution is more sparse than the $\ell_2$ solution. Now that we know what tool to use we can start to tackle this cancer problem! ## Predicting breast cancer In a class this semester Rob Nowak and Laurent Lessard introduced a breast cancer dataset described in a New England Journal of Medicine article.2 This dataset tests cancerous and health patients for gene expression levels (295 patients in total, roughly 8000 genes). We’d like to design a predictor for breast cancer based off levels of gene expression. In this dataset, we observe if someone has cancer or not, indicated by $\yb$, with each element being $\pm 1$ indicating the presence of cancer. For these 295 patients, this dataset also provides tests the expression levels of 8,000 genes, as expressed by $\Xb$. The $i$th row in $\Xb$ corresponds to $\yb_i$ – it contains the gene expression levels for patient $i$. In this problem, we’d like to determine how important each gene is for cancer. We will assign a weight to each gene, and a weight of 0 means it’s not at all important. We will assume that the underlying model takes the sign of our prediction, or $\yb = \sign{\left(\Xb\wb\right)}$ where $\sign{\left(\cdot\right)}$ is applied element-wise. We will solve this problem formulation we saw above, the LASSO problem formulation: $$\widehat{\wb} = \arg \min_\wb \norm{\yb - \Xb\wb}_2^2 + \lambda \norm{\wb}_1$$ As we saw above, it will encourage that most $w_i$’s are 0, or have no importance in determining if someone has breast cancer or not. We saw above why this formulation made sense, but now let’s see how to solve it! The solution to this optimization problem has no closed form solution meaning we have to find our solution iteratively. We’ll choose to solve this with an alternating minimization method (a method for biconvex optimization). This method utilizes the fact that both the error term and the regularization parameter term are positive. Given two positive parameters, it’s natural to optimize one then the other to minimize their sum. If you were minimizing the product, it’d be natural to drive one as close to 0 as possible. A method of alternating optimization is the proximal gradient method with rigorous justification in academic papers.3456 Given suitable loss functions and regularization parameters, this method does two steps (typically in a for-loop): 1. Take a step towards the solution that minimizes the error as defined by the loss function. This takes a step in the negative gradient (scalar case: derivative) direction because the gradient/derivative points to the direction the function increases. 2. Enforces that the solution that minimizes the loss function should also minimize the regularization function. This takes the solutions found in (1) and enforces that they must be acceptable. These steps can be written as $$\bm{z} = \wb_k - \tau \nabla F\left(\wb_k\right))$$ $$\wb_{k+1} = \arg \min_\wb \norm{\wb - \bm{z}} + \tau \lambda \norm{\wb}_1$$ where $\nabla F(\wb_k)$ represents the gradient of $\norm{\yb - \Xb\wb}$ at the point $\wb_k$ (which represents the estimate at iteration $k$) and $\tau$ represents some step size. The equations correspond to steps (1) and (2) above. The derivations are in the appendix and results in $$\bm{z} = \wb_k - \tau \cdot 2 \Xb^T \left(\Xb\wb - \yb\right)$$ $$\wb_{k+1} = \textrm{sign}(\bm{z}) \cdot \left(\abs{\bm{z}} - \tau\lambda/2\right)_+$$ where $(\cdot)_+$ is the soft thresholding operator that keeps the positive elements of the input and sets the rest to $0$. All the operations ($\textrm{sign}$, $(\cdot)_+$, multiplication, etc) are done element-wise. We can implement these functions as follows: def proximity(z, threshold): """ The prox operator for L1 regularization/LASSO. Returns sign(z) * (abs(z) - threshold)_+ where (.)_+ is the soft thresholding operator """ x_hat = np.abs(z) - threshold/2 x_hat[x_hat < 0] = 0 return x_hat * np.sign(z) """ Computes the gradient of least squares loss at the point x """ return 2A.T @ (A@x - y) We can also implement the alternating minimization. As equations $(1)$ and $(2)$ mention, the output of the gradient step is fed into the proximity operator. # X is a fat matrix, y a label vector, y_i \in {-1, 1} X, y = breast_cancer_data() # our initial guess; most of the values stay zero w_k = np.zeros(X.shape[1]) tau = 1 / np.linalg.norm(X)2 # guarantees convergence for k in range(int(1e3)): z = w_k - tau gradient(X, w_k, y) w_k = proximity(z, tau*lambda) # binary classification tends to take the sign # (we're only interested in the proprieties of the weights here, not y_hat) y_hat = np.sign(A @ w_k) After wrapping this function in a class, this turns into the basics of sklearn.linear_model.Lasso. I have done this and the source is available on GitHub. When we find the optimal weights, we don’t want to use all the data. We can save a set of training data and never see it until we test on it. Using only 80% of the dataset to train our weights, we get the following set of weights! With this $\lambda$, this prediction gives accuracy of 81.36% when predicting on 20% dataset of the data reserved for testing. If we had a balanced dataset (we don’t) we’d get accuracy of 50% by flipping a coin. While we don’t have ideal accuracy, we do have a solution with many zeros which is what we set out to do. The content of this blog post was finding sparse solutions, but how can we improve these results? We are performing binary classification – we’re predicting either “yes” or “no”… but we don’t really use that information. We just naïvely used a least squares loss which penalizes points that are easy to classify and too far on the correct side of the decision boundary! The next post will focus on support vector machines, classifiers that don’t have punish accuracies that are too correct. They do this by using hinge loss and logistic loss. ## Appendix For ease, we will drop the bold face math, meaning $x := \xb, A := \Ab, y:=\yb$. Also note that all operators are evaluated element-wise (expect matrix multiplication). This applies to $(\cdot)_+$, and use element-wise multiplication and sign operators. This “proof” details the iterative soft thresholding algorithm. This method can be accelerated by the algorithms FISTA7 or FASTA8 by choosing a different step size at each iteration with $\tau_k$. For rigorous justification why the proximal gradient method is justified, see academic papers.3456 Given $\phi(x) = \norm{Ax - y}_2^2$, the gradient is $2A^T(Ax - y)$. Proof: We can choose to represent squared error as $(Ax - y)^T (Ax-y)$. Then using intuition from scalar calculus and some gradient identities, \begin{aligned} x + y &= 1\\ x &= y \end{aligned} \begin{aligned} \nabla \phi(x) &= \frac{d (Ax - y)}{dx} \cdot 2\cdot (Ax - y)\\ &= 2 A^T (Ax - y) \end{aligned} ### Proximity operator Given the proximity operator for $\ell_1$ regularization as $\phi(x) = \norm{y - x}_2^2 + \lambda\norm{x}_1$, the optimum solution is given by $\widehat{x} = \sign(y)\left(\abs{y} - \lambda/2\right)_+$ where $(\cdot)_+$ denotes the soft thresholding operator. Proof: The definition of the proximity operator results in a separable equation that allows us to write \begin{aligned} \phi(x) &= \norm{y - x}_2^2 + \lambda \norm{x_i}_1\\ &= \sum_i (y_i - x_i)^2 + \lambda \abs{x_i} \end{aligned} This equation can be minimized by minimizing each term separately. $$\pder{\phi(x)_i}{x_i} = -2(y_i - x_i) + \lambda\pder{\abs{x_i}}{x_i}$$ $$\pder{x}{y}$$ This last term on the end, $\pder{\abs{x_i}}{x_i}$ is tricky: at $x_i = 0$, this term is not differentiable. After using subgradients to skirt around that fact, we can say that $\pder{\abs{x_i}}{x_i} = \sign(x_i) = \pm 1$ which makes sense when we’re not at the origin. This function is convex which allows us to set the derivative to 0 and find the global minima. $$\pder{\phi(x)_i}{x_i} = 0 = -2(y_i - x_i) + \lambda\sign(x_i)$$ $y_i > 0$ implies that $x_i > 0$ which allows us to write $$\Align{ x_i &= y_i - \frac{\lambda}{2}\sign(y_i)\\ &= \sign(y_i) (\abs{y_i} - \lambda/2) }$$ But when $\abs{y_i} < \lambda/2$ we run into difficulty because $y_i > 0 \implies x_i > 0$. To get around this fact, we set all the values where $\abs{y_i} < 0$ to 0 and subtract $\lambda/2$ from the rest (this is detailed further in a StackExchange post). With this, we can now write $$x_i = \sign(y_i) \cdot (\abs{y_i} - \lambda/2)_+$$ where $(x)_+ := \max(x, 0)$. 1. This data set is detailed in the section titled Predicting Breast Cancer 2. One of the trickiest issues is finding real world data to apply your problems too. 3. Wright, S. J., Nowak, R. D., & Figueiredo, M. A. (2009). Sparse reconstruction by separable approximation. Signal Processing, IEEE Transactions on, 57(7), 2479-2493. 2 4. Hale, E. T., Yin, W., & Zhang, Y. (2007). A fixed-point continuation method for l1-regularized minimization with applications to compressed sensing. CAAM TR07-07, Rice University, 43, 44. 2 5. Daubechies, I., Defrise, M., & De Mol, C. (2003). An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. arXiv preprint math/0307152. 2 6. Figueiredo, M. A., Bioucas-Dias, J. M., & Nowak, R. D. (2007). Majorization–minimization algorithms for wavelet-based image restoration. Image Processing, IEEE Transactions on, 16(12), 2980-2991. 2 7. Beck, A., and Teboulle, M.. “A fast iterative shrinkage-thresholding algorithm for linear inverse problems.” SIAM journal on imaging sciences 2.1 (2009): 183-202. 8. Goldstein, T., Christoph S., and Richard B.A field guide to forward-backward splitting with a FASTA implementation.” arXiv preprint arXiv:1411.3406 (2014)., website	2021-07-27 22:35:09	{"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": 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": 0.9069812893867493, "perplexity": 493.94344980840333}, "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-2021-31/segments/1627046153491.18/warc/CC-MAIN-20210727202227-20210727232227-00053.warc.gz"}
http://math.stackexchange.com/questions/162271/generating-the-sorgenfrey-topology-by-mappings-into-0-1-and-on-continuous	# Generating the Sorgenfrey topology by mappings into $\{0,1\}$, and on continuous images of the Sorgenfrey line 1. Show that the topology of the Sorgenfrey line can be generated be a family of mappings into a two-point discrete space. 2. Verify that the Sorgenfrey line can be mapped onto $D(\aleph_0)$ but cannot be mapped onto $D(\mathfrak{c})$ (where $D(\kappa)$ is the discrete space of cardinality $\kappa$). - 1. For each real $a$ define a function $f_a : \mathbb{R} \to \{ 0,1 \}$ by $$f_a (x) = \begin{cases}0, &\text{if } x < a \\ 1, &\text{if }x \geq a.\end{cases}$$ Show that the topology on $\mathbb{R}$ generated by this family of functions is the Sorgenfrey line. 2. (a) Consider $\mathbb{Z}$ with the discrete topology. Define $f : \mathbb{R} \to \mathbb{Z}$ by $f (x) = \lfloor x \rfloor$ (where $\lfloor x \rfloor$, the floor of $x$, denotes the greatest integer not (strictly) greater than $x$). Show that this is continuous (and onto). (b) Note that the Sorgenfrey line is separable (consider $\mathbb{Q}$, the set of rationals), and recall that any continuous image of a separable space is separable. Is $D(\mathfrak{c})$ separable?	2015-09-02 10:54:13	{"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": 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": 0.9816530346870422, "perplexity": 131.98503711835835}, "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/1440645261055.52/warc/CC-MAIN-20150827031421-00039-ip-10-171-96-226.ec2.internal.warc.gz"}
https://physics.aps.org/synopsis-for/10.1103/PhysRevC.98.041302	# Synopsis: Changing the Shape of a Zirconium Nucleus As zirconium gains neutrons, its nucleus morphs in shape—changing from a soccer ball to an American football—and researchers have found the exact moment of the transition. Atomic nuclei aren’t always spherical. The shape of a nucleus can change when neutrons are added to an element to create a heavier isotope. The distortion also occurs when nucleons are shuffled into different nuclear orbits. Zirconium is a prime example of an element with a shape-shifting nucleus: The ground state of zirconium-96, which contains 40 protons and 56 neutrons, is spherical, while that of zirconium-100, with its extra four neutrons, is shaped more like an American football. Now, a team has found the isotope—zirconium-98—in which the shape of zirconium’s nucleus transitions. The result could enable precise predictions of the energies and lifetimes of exotic nuclei that cannot be produced on Earth. Waldemar Witt of the Technical University of Darmstadt, Germany, and colleagues fired a beam of zirconium-98 isotopes at a platinum foil to nudge their nuclei into higher energy states. They then measured the spectrum of gamma rays emitted as the isotopes lost energy. Using the data, the team deduced the transition probabilities for two different energy decay paths of the isotopes. These probabilities are known to depend on the number of nucleons that participate in the decay process, which allowed the team to identify each nucleus created and use additional analysis to infer the shapes of the nuclei. Their measurements show that the ground state of zirconium-98 is spherical, while the first excited state is deformed. The energy difference between the spherical and deformed nuclei is the smallest measured to date. Thus the team says that their measurements pinpoint the shape transition. This research is published in Physical Review C. –Christopher Crockett Christopher Crockett is a freelance writer based in Arlington, Virginia. More Features » ### Announcements More Announcements » Nuclear Physics Graphene ## Next Synopsis Atomic and Molecular Physics ## Related Articles Nuclear Physics ### Synopsis: Detecting Nuclear Decay with Recoil Small particles levitated in an optical trap can recoil from radioactive decays in a way that identifies their nuclear composition, a new theoretical study suggests. Read More » Nuclear Physics ### Synopsis: The Fastest Alpha Emitter The detection of unusually fast alpha emission from a heavy isotope could lead to new ways of testing the nuclear shell model. Read More »	2018-11-13 23:23:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28758224844932556, "perplexity": 1944.621279997937}, "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-2018-47/segments/1542039741510.42/warc/CC-MAIN-20181113215316-20181114001316-00075.warc.gz"}
https://physics.aps.org/articles/v6/16	# Viewpoint: Revisiting Thermodynamic Efficiency Physics 6, 16 Breaking time-reversal symmetry in a thermoelectric device affects its efficiency in unexpected ways. Thermodynamic engines convert heat into useful work. Testing the optimal efficiency of these machines has been at the forefront of scientific developments ever since 1824, when Sadi Carnot showed that in a simple engine undergoing a reversible thermodynamic cycle, the ratio between used and wasted heat must be less than the ratio of the absolute temperatures of the cold and the hot reservoir. This Carnot limit is simply a statement of the second law of thermodynamics. On a practical level, knowing the ideal efficiency of engines and other devices helps us compare the advantages of using and developing one technology versus another. This is particularly important today, as the world faces energy challenges that could be mitigated by using available resources more efficiently. At the same time, studying the efficiency of thermodynamic engines helps us understand fundamental ideas, such as the relationship between the work an engine performs and the information gained or lost in the process, or the implications of a system being microscopically versus macroscopically reversible in time. In this spirit, Kay Brandner at the University of Stuttgart, Germany, and colleagues [1] report in Physical Review Letters their calculated efficiency of a simple thermoelectric device that converts heat to electrical current (Fig. 1). They show that when the device operates in an external magnetic field—a condition that breaks time-reversal symmetry for the motion of electrons—the efficiency is significantly lower than previous studies predicted. The lower bound on efficiency occurs, they argue, because in addition to the requirement that entropy be greater than or equal to zero, charge must be conserved—a point that was missed in earlier work. Their findings improve our understanding of thermoelectric efficiency and may one day influence the design of thermoelectric devices for real-world applications. The Carnot engine undergoes two isothermal and two adiabatic changes, and runs sufficiently slowly that the thermodynamic state of the engine at each step along the way is well defined. Carnot engines have an efficiency $ηC$ that depends on the temperatures of the two thermal reservoirs providing heat into and out of the engine ( $ηC=1-Tcold/Thot$). As typically framed, the Carnot cycle is considered macroscopically reversible, since it is infinitely slow and produces no entropy. In contrast, Brandner et al. consider the implications of reversibility and irreversibility on a microscopic scale, taking into account the individual trajectories of all of the electrons in a simple thermoelectric device (Fig. 1). Unlike the Carnot engine, this device operates in an out-of-equilibrium steady state, transforming the current of heat to an electric current at a finite voltage. The efficiency of thermoelectric devices like the one Brander et al. consider depends on three material properties: the thermopower $S$, which is the voltage produced by a temperature difference, the electric conductivity $σ$, and the heat conductivity $κ$. How the combination of these material properties combine to determine the thermoelectric’s efficiency can be found from the so-called Onsager matrix $Lij$, which connects the heat current ( $Jq$) and electrical current ( $Je$), to a pair of thermodynamic gradients, namely of temperature $T$, ( $Fq=-∇T/T2$), and of electrochemical potential $μ$, ( $Fe=∇μ/T$): $Jq=LqqFq+LqeFe$ and $Je=LeqFq+LeeFe$. Lars Onsager [2], in work that led to his Nobel Prize in Chemistry, and later Hendrik Casimir [3], showed that the principle of microscopic time-reversibility means that $Lqe=Leq$, while breaking time-reversibility, say by applying a magnetic field $B$, yields the general Onsager-Casimir relation $Lqe(B)=Leq(-B)$. In the case that the system is microscopically reversible ( $B=0$), the efficiency of the thermoelectric steady-state engine depends on the so-called figure of merit [4], $ZT=(σS2/κ)T=Leq2/detL$. $ZT$ becomes larger the more singular the matrix $L$ becomes, and in the limit $ZT$ approaches infinity, one attains the Carnot efficiency. The only condition imposed by the second law of thermodynamics is that the entropy produced by the system be greater than or equal to zero, which simply implies the positivity of the matrix $L$, or, positivity of $ZT$. Optimizing $ZT$ is a significant challenge in material science [5]. So far, practical values of $ZT∼1$ are still too low for thermoelectric technology to compete with cycle-based engines or refrigerators. Notable exceptions are nanoscale devices [6] and heating or cooling devices where a specialized function, and not efficiency, is most important. In view of these practical limitations, Benenti et al. [7] broke new ground by investigating bounds on thermoelectric efficiency in models where microscopic time-reversibility was broken (nonzero $B$). They derived a general expression for the thermoelectric efficiency $η$ in terms of two dimensionless parameters, the generalized figure of merit, $y=LqeLeq/(detL)$, and the asymmetry parameter, $x=Leq/Lqe$. Considering only limitations imposed by the second law, they observed that a range of $x$ and $y$ values were possible. Significantly, they observed that it was possible to obtain Carnot’s efficiency for any value of asymmetry $\|x\|>1$. However, it remained unclear if their model contained all of the ingredients necessary to be considered realistic. In their new work, Brandner et al. add this missing ingredient by focusing on a simple steady-state model with three terminals—or contacts—to heat and charge reservoirs (Fig. 1). In their model, only two of the terminals are connected to a source and drain of heat and electric charge, whereas the third terminal is merely a probe that does not, on average, exchange particles (in this case, charge) and energy with the environment. The reason that adding this third terminal is necessary is that the Onsager matrix for the simplest, two-terminal devices with noninteracting particles is always strictly symmetric ( $x=1$), even in the presence of magnetic fields. The minimal model that can break this symmetry has to have three terminals. Brandner et al. essentially used a mathematical trick in which they re-parametrized the Onsager matrix in terms of a new matrix $K$, the positivity of which is now a consequence of the law of conservation of charge. The requirement that $K$ be positive puts a constraint on the matrix elements of the Onsager matrix that is much stronger than the requirement from the second law and allows for no obvious anomalies, and means the efficiency cannot be as high as Benenti et al. predicted. Strong evidence that Brandner et al.’s work will apply to real world models comes from Ref. [8], in which the authors numerically calculated the efficiency of a specific, model device and found the same value that Brandner et al. now find as an upper bound. Brandner et al.’s result is not completely bad news about the efficiency of thermoelectric devices. Systems with microscopic reversible dynamics can be shown to have a maximum theoretical efficiency (the Curzon-Ahlborn limit [9]) of $1/2ηC$ if they operate at maximal output power. (Strictly speaking, this limits only holds when the difference of temperatures between hot and cold reservoirs is small with respect to absolute temperatures [10].) Using similar assumptions, Brandner et al. show that switching on a magnetic field or other vector potential, increases the optimal efficiency of their device at maximum power output to $4/7ηC$. This fundamental result has potentially interesting practical implications for developing small-scale heat engines or refrigerators with little compromise between efficiency and power. Brandner et al.’s results immediately open up two other, challenging questions. For one, does a similar bound on efficiency exist in more general transport models with strong particle-particle interactions? Two, what should we expect in models with more than three terminals? ## References 1. K. Brandner, K. Saito, and U. Seifert, “Strong Bounds on Onsager Coefficients and Efficiency for Three-Terminal Thermoelectric Transport in a Magnetic Field,” Phys. Rev. Lett. 110, 070603 (2013) 2. L. Onsager, “Reciprocal Relations in Irreversible Processes. I.,” Phys. Rev. 37, 405 (1931); "Reciprocal Relations in Irreversible Processes. II., ” 38, 2265 (1931) 3. H. B. G. Casimir, ”On Onsager’s Principle of Microscopic Reversibility,” Rev. Mod. Phys. 17, 343 (1948) 4. G. Mahan, B. Sales, and J. Sharp, “Thermoelectric Materials: New Approaches to an Old Problem,” Phys. Today 50, No. 3, 42 (1997) 5. W. Kim, J. Zide, A. Grossard, D. Klenov, S. Stemmer, A. Shakouri, and A. Majumdar, “Thermal Conductivity Reduction and Thermoelectric Figure of Merit Increase by Embedding Nanoparticles in Crystalline Semiconductors,” Phys. Rev.  Lett. 96, 045901 (2006) 6. T. E. Humphrey and H. Linke, “Reversible Thermoelectric Nanomaterials,” Phys. Rev. Lett. 94, 096601 (2005) 7. G. Benenti, K. Saito, and G. Casati, “Thermodynamic Bounds on Efficiency for Systems with Broken Time-Reversal  Symmetry,” Phys. Rev. Lett. 106, 230602 (2011) 8. B. Balachandran, G. Benenti, and G. Casati, “Efficiency of Three-Terminal Thermoelectric Transport Under Broken Time-Reversal Symmetry,” arXiv:1301.1570 9. F. L. Curzon and B. Ahlborn, “Efficiency of a Carnot Engine at Maximum Power Output,” Am. J. Phys. 43, 22 (1975) 10. C. Van den Broeck, “Thermodynamic Efficiency at Maximum Power,” Phys. Rev. Lett. 95, 190602 (2005) Tomaž Prosen has been a full professor of theoretical physics at University of Ljubljana since 2008, where he leads a group on nonequilibrium quantum and statistical physics. He obtained his Ph.D. in 1995 at the same university. His main current research interest is in fundamental problems in dynamical systems, quantum many-body dynamics, nonequilibrium statistical mechanics, and transport theory. ## Related Articles Statistical Physics ### Synopsis: Fluctuation Theorems Tested with a Levitating Bead Motion measurements of a nanosized bead—held aloft in an optical trap—confirm thermodynamic theories that describe fluctuations of microscopic objects. Read More » Statistical Physics ### Synopsis: The Benefits of Starting Anew Starting over can increase the chances of reaching a desired outcome among many possibilities, as demonstrated by a new statistical analysis. Read More » Statistical Physics	2018-02-25 19:34:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 40, "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": 0.72252357006073, "perplexity": 981.6784243469854}, "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-2018-09/segments/1518891816912.94/warc/CC-MAIN-20180225190023-20180225210023-00750.warc.gz"}
http://gohugo.io/tutorials/installing-on-mac/	# Installing Hugo on a Mac This tutorial aims to be a complete guide to installing Hugo on your Mac computer. ## Assumptions 1. You know how to open a terminal window. 2. You’re running a modern 64-bit Mac. 3. You will use ~/Sites as the starting point for your site. ## Pick Your Method There are three ways to install Hugo on your Mac computer: the brew utility, from the distribution, or from source. There’s no “best” way to do this. You should use the method that works best for your use case. There are pros and cons for each. 1. Brew is the simplest and least work to maintain. The drawbacks aren’t severe. The default package will be for the most recent release, so it will not have bug-fixes until the next release (unless you install it with the --HEAD option). The release to brew may lag a few days behind because it has to be coordinated with another team. Still, I’d recommend brew if you want to work from a stable, widely used source. It works well and is really easy to update. 2. Downloading the tarball and installing from it is also easy. You have to have a few more command line skills. Updates are easy, too. You just repeat the process with the new binary. This gives you the flexibility to have multiple versions on your computer. If you don’t want to use brew, then the binary is a good choice. 3. Compiling from source is the most work. The advantage is that you don’t have to wait for a release to add features or bug fixes. The disadvantage is that you need to spend more time managing the setup. It’s not a lot, but it’s more than with the other two options. Since this is a “beginner” how-to, I’m going to cover the first two options in detail and go over the third more quickly. ## Brew ### Step 1: Install brew if you haven’t already Go to the brew website, http://brew.sh/, and follow the directions there. The most important step is: ruby -e "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/master/install)" When I did this, I had some problems with directory permissions. Searches on Google pointed me to pages that walked me through updating permissions on the /usr/local directory. Seemed scary, but it’s worked well since. ### Step 2: Run the brew command to install hugo First, update the formulae and Homebrew itself by running: $ brew update Then, install Hugo using Homebrew by running: $brew install hugo ==> Downloading https://homebrew.bintray.com/bottles/hugo-0.13_1.yosemite.bottle.tar.gz ######################################################################## 100.0% ==> Pouring hugo-0.13_1.yosemite.bottle.tar.gz 🍺 /usr/local/Cellar/hugo/0.13_1: 4 files, 14M (Note: Replace brew install hugo with brew install hugo --HEAD if you want the absolute latest version in development, but beware—there might be bugs!) Brew should have updated your path to include Hugo. Confirm by opening a new terminal window and running a few commands: $ # show the location of the hugo executable $which hugo /usr/local/bin/hugo$ # show the installed version $ls -l$( which hugo ) lrwxr-xr-x 1 mdhender admin 30 Mar 28 22:19 /usr/local/bin/hugo -> ../Cellar/hugo/0.13_1/bin/hugo $# verify that hugo runs correctly$ hugo version Hugo Static Site Generator v0.13 BuildDate: 2015-03-09T21:34:47-05:00 ### Step 3: You’re Done You’ve installed Hugo. Now you need to set up your site. Read the Quickstart guide, explore the rest of the documentation, and if you still have questions just ask! ## From Tarball ### Step 1: Decide on the location When installing from the tarball, you have to decide if you’re going to install the binary in /usr/local/bin or in your home directory. There are three camps on this: 1. Install it in /usr/local/bin so that all the users on your system have access to it. This is a good idea because it’s a fairly standard place for executables. The downside is that you may need elevated privileges to put software into that location. Also, if there are multiple users on your system, they will all run the same version. Sometimes this can be an issue if you want to try out a new release. 2. Install it in ~/bin so that only you can execute it. This is a good idea because it’s easy to do, easy to maintain, and doesn’t require elevated privileges. The downside is that only you can run Hugo. If there are other users on your site, they have to maintain their own copies. That can lead to people running different versions. of course, this does make it easier for you to experiment with different releases. 3. Install it in your sites directory. This is not a bad idea if you have only one site that you’re building. It keeps every thing in a single place. If you want to try out new releases, you can just make a copy of the entire site, update the Hugo executable, and have it. All three locations will work for you. I’m going to document the second option, mostly because I’m comfortable with it. 1. Open https://github.com/spf13/hugo/releases in your browser. 2. Find the current release by scrolling down and looking for the green tag that reads “Latest Release.” 3. Download the current tarball for the Mac. The name will be something like hugo_X.Y_osx-64bit.tgz, where X.YY is the release number. 4. By default, the tarball will be saved to your ~/Downloads directory. If you chose to use a different location, you’ll need to change that in the following steps. Verify that the tarball wasn’t corrupted during the download: $tar tvf ~/Downloads/hugo_X.Y_osx-64bit.tgz -rwxrwxrwx 0 0 0 0 Feb 22 04:02 hugo_X.Y_osx-64bit/hugo_X.Y_osx-64bit.tgz -rwxrwxrwx 0 0 0 0 Feb 22 03:24 hugo_X.Y_osx-64bit/README.md -rwxrwxrwx 0 0 0 0 Jan 30 18:48 hugo_X.Y_osx-64bit/LICENSE.md The .md files are documentation. The other file is the executable. ### Step 4: Install into your bin directory $ # create the directory if needed $mkdir -p ~/bin$ # make it the working directory $cd ~/bin$ # extract the tarball $tar -xvzf ~/Downloads/hugo_X.Y_osx-64bit.tgz Archive: hugo_X.Y_osx-64bit.tgz x ./ x ./hugo x ./LICENSE.md x ./README.md$ # verify that it runs $./hugo version Hugo Static Site Generator v0.13 BuildDate: 2015-02-22T04:02:30-06:00 You may need to add your bin directory to your PATH variable. The which command will check for us. If it can find hugo, it will print the full path to it. Otherwise, it will not print anything. $ # check if hugo is in the path $which hugo /Users/USERNAME/bin/hugo If hugo is not in your PATH, add it by updating your ~/.bash_profile file. First, start up an editor: $ nano ~/.bash_profile Add a line to update your PATH variable: export PATH=$PATH:$HOME/bin Then save the file by pressing Control-X, then Y to save the file and return to the prompt. Close the terminal and then open a new terminal to pick up the changes to your profile. Verify by running the which hugo command again. ### Step 5: You’re Done You’ve installed Hugo. Now you need to set up your site. Read the Quickstart guide, explore the rest of the documentation, and if you still have questions just ask! ## Building from Source If you want to compile Hugo yourself, you’ll need Go, which is also available from Homebrew: brew install go. ### Step 1: Get the Source If you want to compile a specific version, go to https://github.com/spf13/hugo/releases and download the source code for the version of your choice. If you want to compile Hugo with all the latest changes (which might include bugs), clone the Hugo repository: git clone https://github.com/spf13/hugo ### Step 2: Compiling Make the directory containing the source your working directory, then fetch Hugo’s dependencies: mkdir -p src/github.com/spf13 ln -sf $(pwd) src/github.com/spf13/hugo # set the build path for Go export GOPATH=$(pwd) go get This will fetch the absolute latest version of the dependencies, so if Hugo fails to build it may be because the author of a dependency introduced a breaking change. Then compile: go build -o hugo main.go Then place the hugo executable somewhere in your \$PATH. ### Step 3: You’re Done You probably know where to go from here.	2017-01-21 06:21:23	{"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": 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": 0.2044440060853958, "perplexity": 1755.3055221391362}, "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-2017-04/segments/1484560280929.91/warc/CC-MAIN-20170116095120-00275-ip-10-171-10-70.ec2.internal.warc.gz"}
https://plainmath.net/differential-equations/54869-solve-the-first-order-differential-equations-x-2-plus-frac-dy-dx-equal-xy	Annette Sabin 2022-01-22 Solve the first-order differential equations: $\left({x}^{2}+1\right)\frac{dy}{dx}=xy$ Ana Robertson Expert Solution: $\left({x}^{2}+1\right)\frac{dy}{dx}=xy$ This is a separable linear first order differential which is of the form $N\left(y\right)dy=M\left(x\right)dx$ Intagrating both sides $\int \frac{1}{y}dy=\int \frac{x}{{x}^{2}+1}dx$ Let ${x}^{2}+1=u$ So, Do you have a similar question?	2023-02-01 23:05:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 31, "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": 0.7786270380020142, "perplexity": 4548.355830557283}, "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-2023-06/segments/1674764499953.47/warc/CC-MAIN-20230201211725-20230202001725-00572.warc.gz"}
https://marketplace.sasview.org/models/sphere/?page=2	# Sphere Name Description Category Upload Date Author Score Verified Spherical Sld Definition Similarly to the onion, this model provides the form factor, $P(q)$, for a multi-shell sphere, where the interface between the each neighboring shells can be described by the error... Sphere 07 Sep 2017 sasview 0 Adsorbed Layer Definition This model describes the scattering from a layer of surfactant or polymer adsorbed on large, smooth, notionally spherical particles under the conditions that (i) the particles (cor... Sphere 07 Sep 2017 sasview 0 Vesicle Definition This model provides the form factor, P(q), for an unilamellar vesicle and is effectively identical to the hollow sphere reparameterized to be more intuitive for a vesicle and nor... Sphere 07 Sep 2017 sasview 0 Core Multi Shell Definition This model is a trivial extension of the CoreShell function to a larger number of shells. The scattering length density profile for the default sld values (w/ 4 shells). ... Sphere 07 Sep 2017 sasview 0 Core Shell Sphere .. _core_shell_sphere: This model provides the form factor, $P(q)$, for a spherical particle with a core-shell structure. The form factor is normalized by the particle volume. For information... Sphere 07 Sep 2017 sasview 0 Page 2 of 2	2022-11-28 01:05:15	{"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": 0.7520980834960938, "perplexity": 3808.5351023936764}, "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-2022-49/segments/1669446710462.59/warc/CC-MAIN-20221128002256-20221128032256-00164.warc.gz"}
https://www.geteasysolution.com/50t-4t%5E2-120=0	# 50t-4t^2-120=0 ## Simple and best practice solution for 50t-4t^2-120=0 equation. Check how easy it is, and learn it for the future. Our solution is simple, and easy to understand, so dont hesitate to use it as a solution of your homework. If it's not what You are looking for type in the equation solver your own equation and let us solve it. ## Solution for 50t-4t^2-120=0 equation: 50t-4t^2-120=0 a = -4; b = 50; c = -120;Δ = b2-4acΔ = 502-4·(-4)·(-120)Δ = 580The delta value is higher than zero, so the equation has two solutionsWe use following formulas to calculate our solutions:$t_{1}=\frac{-b-\sqrt{\Delta}}{2a}$$t_{2}=\frac{-b+\sqrt{\Delta}}{2a}$The end solution: $\sqrt{\Delta}=\sqrt{580}=\sqrt{4145}=\sqrt{4}\sqrt{145}=2\sqrt{145}$ $t_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(50)-2\sqrt{145}}{2-4}=\frac{-50-2\sqrt{145}}{-8}$ $t_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(50)+2\sqrt{145}}{2-4}=\frac{-50+2\sqrt{145}}{-8}$ `	2019-06-25 04:56:00	{"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": 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": 0.32953941822052, "perplexity": 1075.9355008569066}, "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-2019-26/segments/1560627999787.0/warc/CC-MAIN-20190625031825-20190625053825-00312.warc.gz"}
http://meetings.aps.org/Meeting/APR05/Event/28941	### Session E7: Electroweak Physics with W's and Z's 3:30 PM–5:06 PM, Saturday, April 16, 2005 Marriott Tampa Waterside Room: Room 1 Chair: John Laiho, Fermilab Abstract ID: BAPS.2005.APR.E7.6 ### Abstract: E7.00006 : A New Method to Measure the Ratio of W and Z Cross Sections 4:42 PM–4:54 PM Preview Abstract MathJax On \| Off Abstract #### Authors: Katherine Copic (University of Michigan) Victoria Martin (Northwestern University) Michael Schmitt (Northwestern University) We present a new method for the measurement of the ratio, $R$, of cross sections for~$W$ and~$Z$ production in $p\bar{p}$~collisions at the Fermilab Tevatron. The leptonic signals from~$W$ and~$Z$ decays are treated in a strictly parallel fashion, and the~$p_T$ spectrum of the lepton is used to infer their relative rates. A preliminary measurement has been obtained from data collected with the CDF~II detector corresponding to approximately $350$~pb$^{-1}$. Using this value for~$R$ we extract a value for the width of the $W$~boson. To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2005.APR.E7.6	2013-05-22 12:28:41	{"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": 0.6829401254653931, "perplexity": 5754.829960750146}, "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-2013-20/segments/1368701670866/warc/CC-MAIN-20130516105430-00074-ip-10-60-113-184.ec2.internal.warc.gz"}
https://www.nature.com/articles/s41598-019-54012-5?error=cookies_not_supported&code=ae76f26b-0ca6-45f9-aeec-e53c97f6eb11	Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. # Spatiotemporal characteristics of an attacker’s strategy to pass a defender effectively in a computer-based one-on-one task ## Abstract For modern humans, chase-and-escape behaviors are fundamental skills in many sports. A critical factor related to the success or failure of chase-and-escape is the visuomotor delay. Recent studies on sensorimotor decision making have shown that humans can incorporate their own visuomotor delay into their decisions. However, the relationship between the decision of an attacker and the visuomotor delay of a defender is still unknown. Here, we conducted a one-on-one chase-and-escape task for humans and investigated the characteristics of the direction changes of the attacker and the responses of the defender. Our results showed that the direction change of the attacker has two characteristics: uniformity of spatial distribution and bimodality of temporal distribution. In addition, we showed that the response of the defender did not depend on the position but it was delayed to the direction change of the attacker with a short interval. These results suggest that the characteristics of direction change of an attacker increased unpredictability, and it could be useful for preventing the predictive response of the defender and to receive the benefit of an extra response delay of tens of milliseconds, respectively. ## Introduction Our decisions are often made in competitive interactions with others. In such situations, agents must achieve their own purpose in the interaction despite the conflicting purposes of other agents. A typical example of a competitive interaction is chase-and-escape behavior. Chase-and-escape interactions are ubiquitous in nature (predator–prey), and similarly feature in many sports such as football, rugby, and basketball (defender–attacker)1,2,3. In these interactions, the decisions of the agents on when and where to move are essential for survival and success4,5. Thus, it is important to consider how the agents make such decisions in these competitive interactions. Decisions regarding direction changes are important in chase-and-escape interactions. Previous studies on many animals, ranging from dragonflies6,7,8, to fish9 and dogs10, to humans11,12,13, have shown that a strategy called constant bearing is commonly employed to intercept a moving target, such as prey, a frisbee, or a flying ball. When the speed of the pursuer is equal to or faster than that of the evader, this interception strategy is theoretically unbeatable without a sensorimotor delay14. This time delay, which is the latency from the sensory input to the motor output, is inevitable in animals and can be several hundred milliseconds in humans15. Consequently, from the perspective of the evader, lengthening (or at least not shortening) the visuomotor delay (response time) of the pursuer will lead to a successful escape16. One possible evader strategy is to increase the unpredictability of its actions. In several studies on interceptive behaviors in severe time constraints such as in cricket, tennis, and handball, it has been reported that a pursuer (receiver or goalkeeper) anticipates the movement of the opponent using prior knowledge (experience), which is a probability distribution accumulated by long periods of observation17,18,19. For example, in a handball penalty shot, when there is a probability bias in the shot direction of the opponent, the goalkeeper is more likely to respond in that direction than to respond as if the shot of the opponent is equally likely to occur in either direction19. These studies suggested that the pursuer should use the situational (event) probability information to anticipate the movement of the opponent in interceptive behaviors. Given that the defender predicts the movement of the attacker to compensate for its own visuomotor delay, it should be useful for the attacker to increase the unpredictability of its own movements against the defender. Another possible strategy of the evader is to increase efficiency. In the psychological refractory period paradigm, in which two stimuli are presented in close succession, it has been shown that the response time to the second stimulus increases if the time between stimuli is short (400 ms or less)20,21,22,23,24. From the stimulus-response viewpoint, if the two consecutive movements change the directions of the attacker, such as moving right to left to right, the response of the defender to the second direction change could be delayed. Given that the response of defender to the direction change of the attacker is delayed in such a situation, the attacker could iterate effective direction changes. To address these possibilities, we examined probability distributions regarding the direction change of the attacker and the response time of the defender. In this study, we conducted a computer-based one-on-one task, which eliminates kinematic information to focus on the situational probability information. Our results suggest that the direction changes of the attacker have effective spatiotemporal characteristics that combine both of the above two possibilities to pass the defender. ## Methods ### Participants Twelve participants (mean age ±SD = 24.9 ± 2.3 years) who exercised regularly participated in the experiment. All participants were right-handed, had normal or corrected-to-normal vision, and were neurologically healthy. They each received 1000 yen per hour as a reward. The study was conducted in accordance with the Declaration of Helsinki and approved by the Ethics Committee of the University of Tokyo of Arts and Sciences. Informed consent was received from each participant before the experiments. In both experiments, participants were recruited in pairs, and each member of each pair took on the roles of both the attacker (evader) and the defender (pursuer) in turn. No statistical methods were used to predetermine sample sizes. To estimate the achieved statistical power (1 $$-\beta$$) a post hoc analysis was carried out. Considering the spatial uniformity of the frequency of the direction changes of the attacker as one of the primary outcomes, the achieved power was 0.26 for two-way repeated measures ANOVA. The relatively low power achieved is discussed as a limitation of the present study. ### Experimental apparatus Participants were seated in chairs and manipulated the joystick of a controller (Xbox One) to control a disk (filled circle) on the screen. The central position of the disk was recorded on a computer (Sony SVF152C16N) running Psychtoolbox 3.0 software at a frequency of 60 frames/s and a resolution of 1366 × 768 pixels. The stimuli were presented in a lit room on a 15.5 in (34.3 × 19.3 cm) screen. The participants were seated at a viewing distance of 50 cm. A partition prevented direct viewing of the hands of the other player. ### Experimental design Participants controlled either a red disk representing an attacker or a blue disk representing a defender on the screen. The dimensions of the onscreen court were 33.1 cm × 16.6 cm (width × height). The diameter of each disk was 1.0 cm. The start location of the attacker (red disk) was the upper middle, and that of the defender (blue disk) was the center of the court. The objective of the attacker was to move past the defender and reach the end line, which was on the lower side of the court (a yellow line) behind the defender. The objective of the defender was to catch the attacker without the attacker reaching the end line. We regarded a “catch” as a situation in which the outer edges of the disks were in contact. If the attacker left the bounds of the court, the trial was deemed a defensive success. The velocity of each agent was determined by the degree of inclination of the joystick on the respective controllers. The maximum speeds of the attacker and the defender were set equally. The experimental task began with a start cue. No additional instructions, such as a time limit, were given to the participants. To provide feedback on the result of each trial to the participants, when the attacker reached the end line (a successful attack), a high-pitched beep was played. Conversely, when the defender caught the attacker, or the attacker left the bounds of the court (a successful defense), a low-pitched beep was played. The number of successful attacks was indicated at the end of each block; blocks consisted of 30 trials. ### Experimental conditions We used two experimental conditions in this study (slow and fast) to examine the speed dependency of agent decision making. In previous studies, it has been reported that the required movement speed (time constraint) may affect the interception strategies (reactive or predictive) of the agent18,25. In a one-on-one sports situation, as the movement speeds of the agent increase, it becomes more difficult for the defender to intercept the attacker26. Thus, to test the possibility that changes in the strategy of the defender (and changes in the strategy of the attacker corresponding to that of the defender) depended on the movement speeds of the agent, we set two speed conditions. The slow condition was set so that the defender could easily intercept the attacker, and the fast condition was set so that the possibility of interception (successful defense) or penetration (successful attack) of the two agents was balanced. The minimum speed of the agents (both attacker and defender) was 0 cm/s in both conditions, and the maximum speed of the agents was 3 cm/s in the slow condition and 4.5 cm/s in the fast condition. There were 40 warm-up trials and 240 experimental trials per pair of participants. The experimental trials were presented in 8 blocks of 30 trials each. For each condition, participants performed 4 blocks (60 trials in each role of attacker and defender; 120 trials total). The role of each participant was randomized between blocks, and the experimental condition was randomized between pairs. The experiment lasted approximately an hour. ### Data analysis All data analysis was performed in MATLAB (MathWorks). We recorded the onscreen X and Y positions of the attacker and defender during the trial. We analyzed only the positional data collected while the absolute angle between the defender and the attacker was in the range of 0–180° to exclude situations in which the defender had given up trying to catch the attacker. The change-direction time was defined as the time when the velocity in the X direction crossed zero, and the position at that time was defined as the change-direction position. The response time was defined as the time between the change-direction time of the attacker and that of the defender. We distinguished between positive and negative X velocities. We limited the range of response times from 0 ms to 500 ms and removed any response times longer than 500 ms from the analyses to exclude responses in which the defender had given up trying to catch the attacker. The proportion of responses passed this criterion was 0.99 ± 0.01. The frequency was defined as the average number of occurrences per second. In other words, the frequency was calculated by dividing the number of direction changes by the time over which they spent. The standard deviation (SD) of the frequency was calculated using the frequency value in each X column. In this case, we divided the court into 10 X columns and used the frequency values in the middle 6 columns to exclude the columns containing missing values. These missing values indicated that the attacker did not move to those X columns. In the analysis of the temporal aspect, the direction change was classified into two types depending on the time elapsed from the previous direction change: short-interval direction changes (<350 ms) and long-interval direction changes (>350 ms). The boundary between them was 350 ms in both conditions. To determine this boundary, we fitted a mixture model containing a multimodal Gaussian distribution. The number of Gaussian distributions was 4, which was determined using Akaike’s Information Criterion. We estimated the parameters of this model using maximum likelihood estimates. ### Statistical analysis To compare the successful attack rates between conditions, we used the paired t-test because the normality assumption was accepted by Lilliefors test. To compare the frequency of the direction changes made by the attacker, a two-way repeated measures ANOVA with the factors of speed condition (slow, fast) and X position was used. In this comparison, the X position was divided into 10 columns, and we used the middle 6 columns in the statistical test to remove the columns containing missing values caused by the attacker never changing direction in those columns. To compare the SD of the frequency of direction changes of attacker, a two-way repeated measures ANOVA with the factors of speed condition and X position and with the factors of speed condition and number of trials was used. In this case, the term “number of trials” indicated the number of cumulative trials, and the comparison was made among 0 to 10, 0 to 20, 0 to 30, 0 to 40, 0 to 50, and 0 to 60 trials. To compare the response time of the defender, two-way repeated measures ANOVAs with the factors of speed condition and number of trials was used. In this comparison, the X position was divided into 10 columns, and we use the middle 4 columns in the statistical test to remove columns containing missing values caused by the defender failing to respond to the direction change of the attacker in those columns. In this case, the term “number of trials” indicated 10 trials, and the comparison was made among 0 to 10, 11 to 20, 21 to 30, 31 to 40, 41 to 50, and 51 to 60 trials. To compare the response time of the defender, a two-way repeated measures ANOVA with the factors of speed condition and the time interval (time from the previous direction change of the attacker) was used. In this case, we removed one participant from the statistical test because the participant included missing values caused by the opponent (attacker) never changing direction during the time interval. In the ANOVAs, Greenhouse-Geisser correction was applied for the violations of sphericity assumption in the Mauchly test. Multiple comparisons with Bonferroni correction were applied in the post-hoc analysis. To compare the distribution of positions of change-direction with the short-interval and that with the long-interval, the Kolmogorov–Smirnov test was used. The significance level was set at p < 0.05 and adjusted with the Bonferroni correction in the multiple comparisons. The effect size was estimated using Cohen’s d for t-test and multiple comparison and eta-squared (η2) for ANOVA. All data are reported as mean ± SD across subjects. Statistical analyses were performed using the R version 3.5.1 and Gpower version 3.127. ## Results The successful attack rates were 14.2 ± 13.2% in the slow condition and 51.8 ± 11.8% in the fast condition (t11 = 7.37, p = 1.4 × 10−5, d = 3.02). We illustrated the overwriting of the trajectories of the attackers from all trials in each condition (Fig. 1c,k) and the 2D-histogram (i.e., heatmap) at each position (Fig. 1e,m). In this case, the court was divided into the 968 (44 × 22) cells of 30 square pixels each. Then, we plotted the change-direction positions of the attackers (Fig. 1d,l) and the 2D-histogram at each position (Fig. 1f,n). The frequency of direction changes at each position was almost uniform (Fig. 1g,o). We focused on the data in the X direction and quantified them. Figure 1h,p show the time that the attacker spent within each X column, and Fig. 1I,q show the numbers of direction changes. Figure 1j,r show the frequency of direction changes. A two-way (condition and X position) repeated measures ANOVA revealed a main effect of the condition (F1, 11 = 13.29, p = 0.0039, η2 = 0.068). Notably, however, the main effect of the X position and the interaction between these factors were not significant (F5, 55 = 1.905, p = 0.11, η2 = 0.028; F5, 55 = 0.791, p = 0.56, η2 = 0.0093). Figure 2a,c show typical examples of attacker data for a participant in each condition. Each figure shows the cumulative changes in the time spent in the column (left), in the number of direction changes (middle), and in the frequency (right) of the attacker in each X column through all trials. In the SD of the frequency between X columns, a two-way (condition and trials) repeated measures ANOVA revealed a main effect of trials (F1.97, 21.68 = 11.72, p = 4.0 × 10−4, η2 = 0.21). The conditions and the interaction between these factors were not significant (F1, 11 = 1.51, p = 0.25, η2 = 0.029, 1 $$-\beta$$ = 0.54; F2.31, 25.37 = 1.171, p = 0.34, η2 = 0.013). As the main effect of the trials was significant, we conducted post hoc analysis (with Bonferroni correction), and the results revealed that the SD of the frequency up to 10 trials was significantly larger than the others, except for up to 20 trials. (Fig. 2b,d; ps < 0.0013, ds > 0.74). This result indicates that the SD of the frequency decreased initially and did not change thereafter. Figure 3a,e show the mean response times of the defender to the direction changes of the attacker at each position. A two-way (condition and X position) repeated measures ANOVA revealed that the main effects and the interaction were not significant (Fig. 3b,f; F1, 11 = 2.37, p = 0.15, η2 = 0.047; F1.52, 16.69 = 0.69, p = 0.48, η2 = 0.013; F3, 33 = 1.68, p = 0.19, η2 = 0.014). Figure 3c,g show typical histogram examples of the response times of the defender in each of the 10 trials, which competed with the participant in Fig. 2a,c, respectively. A two-way (condition and trials) repeated measures ANOVA revealed that the main effects and the interaction were not significant (Fig. 3d,h; F1, 11 = 2.31, p = 0.16, η2 = 0.047; F5, 55 = 0.641, p = 0.67, η2 = 0.0094; F5, 55 = 1.141, p = 0.35, η2 = 0.013). We next examined the time interval of the direction changes of the attacker. Figure 4a,f show the relative frequency of change-direction times from the previous change-direction time. The distributions were bimodal in both conditions, and thus, we classified the directions changes into two types: short-interval and long-interval (red and blue parts in Fig. 4a,f, respectively). The boundary between them was 350 ms in both conditions (see Methods). For the response time of the defender to the direction change of the attacker, a two-way (condition and time interval) repeated measures ANOVA revealed a main effect of time interval (F4, 40 = 17.2, p = 2.8 × 10−8, η2 = 0.32). The condition and the interaction between these factors were not significant (F1, 10 = 0.80, p = 0.39, η2 = 0.0059; F1.62, 40 = 0.31, p = 0.69, η2 = 0.0053). As the main effect of trials was significant, we conducted post hoc analysis (with Bonferroni correction), and the results revealed that the response time of the defender to the short-interval direction changes of the attacker was significantly greater than to the others (Fig. 4c,h; ps < 1.7 × 10−6, ds > 1.4). The response times to the direction change with short-interval and with long-interval were 317 ± 45 and 270 ± 20 ms in the slow condition, and those were 304 ± 25 and 264 ± 17 ms in the fast condition, respectively. The short-interval change-direction positions were distributed more in the center of the field compared with those with long intervals (Fig. d,e,i,j; p = 2.1 × 10−33). ## Discussion Here, we explore how agents make their decisions in complex interactions such as those seen in a variety of sports situations. In this study, we focused on the probability distribution regarding direction changes in chase-and-escape interactions and elucidated the corresponding spatiotemporal characteristics. For the spatial aspect, the frequency of the direction changes of the attacker (evader) was almost uniformly distributed. On the other hand, for the temporal aspect, the relative frequency of direction changes of the attacker showed a bimodal distribution. The frequency of the direction changes of the attacker at each position on the court was approximately uniform. This spatial uniformity means that the probability of the evader changing or not changing its movement direction was almost constant regardless of its position on the court. This characteristic would be useful for the attacker to maximize the uncertainty in its direction change. The bias of the frequency of direction changes of the attacker rapidly approached a uniform distribution over time (see also Figs. S1, 2). This result indicates that the predictability for the defender using spatial probability information does not improve, even if information on the direction changes of the attacker had been accumulated. As a result, it would be difficult for the defender to anticipate in which direction the attacker would change its movement. Our results show that the proportion of direction change converging to a certain value is similar to matching pennies, in which players stochastically choose between two alternatives (heads or tails) with 50% in a mixed strategy Nash equilibrium in the framework of Game Theory28,29,30. These results suggest that the attacker randomizes its own actions so that they cannot be predicted by the defender. The response time of the defender to the direction change of the attacker did not differ according to the position on the court and did not shorten over time (see also Figs. S3, 4). These results suggest that it would be difficult for the defender to respond predictively to the direction change of the attacker using situational probability information and support the idea that action selections with equal probabilities increase unpredictability against the opponent28,29,30. The relative frequency of the time interval in the direction changes of the attacker showed a bimodal distribution. The peak of the distribution of short-interval direction changes was 200 ms, which, assuming that the human visuomotor delay is 200–300 ms, is not enough time for the attacker to gain feedback regarding the response of the defender to its own action31. Thus, this short-interval direction change (i.e., two consecutive directions) would be a feedforward control. In this case, the movement direction of the attacker in the X direction is the same as the original movement direction (i.e., right to left to right or vice versa), and thus, if the defender does not respond to the first direction change, the situation worsens for the attacker. Consequently, to execute two consecutive direction changes, the attacker relies on the defender responding to the first direction change. On the other hand, for long-interval direction changes, the attacker has enough time to gain feedback and can consider the observed information regarding the response of the defender. From a temporal standpoint, there are two types of attacker direction changes; namely, short-interval direction changes with feedforward control and long-interval direction changes with feedback control. In addition, the shapes of bimodal distribution were somewhat different between individuals but were approximately consistent within individuals, namely, between conditions (see Figs. S5, 6). This may reflect an individual playing preferences or habits. The response of the defender to the short-interval direction change of the attacker was delayed compared with the response to the long-interval changes. Generally, it has been shown that humans respond quickly to frequent stimuli32. Based on this finding, the response time of the defender to short-interval direction changes should shorten, but contrarily, the response time lengthened in this study (see also Figs. S7, 8). One possible cause for this extra response delay is an interference of sensorimotor processing. In the double-stimulation paradigm, a phenomenon called the psychological refractory period has been demonstrated, in which the second response is delayed for two consecutive stimuli20,21,22,23,24. It is thought that this phenomenon is caused by overlapping of the response to the first stimulus and the response to the second stimulus. Because inter-stimulus intervals were distributed around 200 ms in the short-interval direction changes, a similar delay may occur to the response of the defender. In addition, it should be noted that, because the long-tailed distribution of long-interval changes increases the mean direction change time interval, this bimodal distribution can enhance the psychological refractory period effect by decreasing the temporal predictability of the direction changes. Interestingly, the temporal characteristics in bimodal distribution, which is a high frequency of direction change with short-interval and a long-tail of that with long intervals were more prominent in the group with high attack success rate than that with low attack success rate (see Fig. S9). These results indicate that the high frequency of direction change with a short interval is one of the characteristics of the high group, but the successful attack rate could not be explained by the frequency. This might be because it is more important to how combine the direction changes with short and long intervals rather than the frequency itself. Our experiment was different from sport situations in several points. One was the viewpoint, that is, egocentric view in sport situations and bird’s eye view in our computer-based task. For example, in sport situations, when an attacker changes movement direction, the position of image would move from left to right (or vice versa) in the field of view of the defender. On the other hand, in our task, the position of the image in the field of view is little changed. That is, the body of defender acts as a reference frame in sport situations and does not act in our task. This difference in visual information could affect cognitive processing (e.g., S-R associations33,34,35,36,37) and could change the response time of defender. Another was the kinematic factors, that is, whole body movement in sport situations and finger movement in our task. In general, direction changes in locomotion are mechanically constrained38,39,40,41, and it cannot change movement direction suddenly in sport situations, unlike our task. This difference in mechanical constraints also could affect their interaction. Thus, further considerations by integrating sensory and motor factors would be necessary for a better understanding of effective attack behaviors in sport situations. A limitation of the present study is the relatively small sample size. Regarding the result that the frequency of direction changes of the attacker at each position on the court was approximately uniform, it may be explained by the lack of samples. That is, we cannot exclude that absence of the significant effect of the X position and the interaction in the frequency of direction changes is associated with the number of participants in the present study. As a result, the result of the present study should be considered carefully, as it represents a pilot study. In summary, our results showed that the decisions regarding direction changes of the evader have two characteristics: spatial uniformity and temporal bimodality. The former result is consistent with the findings of non-human studies that suggested the effectiveness of a strategy that increases unpredictability42,43, indicating that an element of unpredictability is key to successful escape across species. The latter result indicated that the attacker repeated effective behaviors (i.e., two consecutive direction changes), which lengthened the response time of the defender. Our results suggest that these characteristics could be useful for preventing the predictive response of the pursuer and to gain the benefit of an extra response delay of tens of milliseconds. ## Data availability The datasets generated and analyzed during the current study are available from the corresponding authors on reasonable request. The sample code for conducting the experimental task of the current study is available through https://doi.org/10.6084/m9.figshare.9784622. ## References 1. 1. Lee, D. N., Georgopoulos, A. P., Clark, M. J., Craig, C. M. & Port, N. 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Detecting deception in movement: The case of the side-step in rugby. PLoS One 7, 1–13 (2012). 40. 40. Dickinson, M. H. How Animals Move: An Integrative View. Science (80-.). 288, 100–106 (2000). 41. 41. Fujii, K., Yoshioka, S., Isaka, T. & Kouzaki, M. The preparatory state of ground reaction forces in defending against a dribbler in a basketball 1-on-1 dribble subphase. Sport. Biomech. 14, 28–44 (2015). 42. 42. Humphries, D. A. & Driver, P. M. Protean defence by prey animals. Oecologia 5, 285–302 (1970). 43. 43. Domenici, P., Blagburn, J. M. & Bacon, J. P. Animal escapology II: escape trajectory case studies. J. Exp. Biol. 214, 2474–2494 (2011). ## Acknowledgements This work was supported by a Grant-in- Aid for Japan Society for the Promotion of Science Fellows (No. 17J10922) to K.T. and a Grant-in-Aid for Scientific Research (No. 25242059) and JST-Mirai Program (No. JPMJMI18C7) to K.K. ## Author information Authors ### Contributions K.T., M.S. and K.K. designed the experiment. K.T. performed the experiment. K.T., M.S. and K.K. analyzed data and wrote the paper. ### Corresponding authors Correspondence to Kazushi Tsutsui or Kazutoshi Kudo. ## Ethics declarations ### Competing interests The authors declare no competing interests. Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. ## Rights and permissions Reprints and Permissions Tsutsui, K., Shinya, M. & Kudo, K. Spatiotemporal characteristics of an attacker’s strategy to pass a defender effectively in a computer-based one-on-one task. Sci Rep 9, 17260 (2019). https://doi.org/10.1038/s41598-019-54012-5 • Accepted: • Published: • ### Flexible prediction of opponent motion with internal representation in interception behavior • Kazushi Tsutsui • Keisuke Fujii • Kazuya Takeda Biological Cybernetics (2021)	2021-11-27 06:19:59	{"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": 0.5851937532424927, "perplexity": 1795.1424603978912}, "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-2021-49/segments/1637964358118.13/warc/CC-MAIN-20211127043716-20211127073716-00024.warc.gz"}
http://mathhelpforum.com/calculus/207516-find-compounding-interest-other-problems.html	Math Help - Find compounding interest & other problems 1. Find compounding interest & other problems Hey all I'm completly new to this site and I have to say I am truly thrilled to find such a community. Here are my questions: 1. A painting purchased in 1998 for $75,000 is esitmated to be worth v(t)= 75,000e^t/5 dollars after t years. At what rate will the painting be appreciating in 2003? In 2003, the painting will be appeciaiting at$____ per year. 2. Let P(t) be the population (in millions) of a certain city t years after 1990, and suppose thta P(t) satisfies the differential equation P'= .02(t), P(0)= 7. a. Find the formula for P(t) b. What was the initial popluation in 1990? c. What is the growth constant? d. What was the population in 2000? e. Use the differential equation to determine how fast the population is growing when it reaches 8 million people. f. How large is the population when it is growing at the rate of 190,000 people per year? 3. Suppose that an investment grows at a CONTINUOUS rate of 9% rate each year. In how many years will the value of the investment double? 2. Re: Find compounding interest & other problems 1.) You want to compute: $\frac{dv}{dt}$ Then use the value for $t$ in 2003. 2.) You are given the IVP: $\frac{dP}{dt}=0.02t$ where $P(0)=7$ a) solve the IVP. b) You are given this as the initial condition. c) The term growth constant is usually used with exponential growth, in my experience. This model is not exponential. d) Using the result from part a), find P(10). e) Using the result from part a), solve $P(t)=8$ for $t$, then use this in the ODE. f) Use $P'(t)=0.19$ to find $t$, then use this in the solution from part a). 3.) Solve $2=(1.09)^t$ for $t$.	2014-09-16 10:46:47	{"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": 10, "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": 0.5014342665672302, "perplexity": 870.8662481837277}, "config": {"markdown_headings": false, "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-2014-41/segments/1410657114204.83/warc/CC-MAIN-20140914011154-00207-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"}
https://pressbooks.bccampus.ca/collegephysics/chapter/linear-momentum-and-force/	Chapter 8 Linear Momentum and Collisions # 52 8.1 Linear Momentum and Force ### Summary • Define linear momentum. • Explain the relationship between momentum and force. • State Newton’s second law of motion in terms of momentum. • Calculate momentum given mass and velocity. # Linear Momentum The scientific definition of linear momentum is consistent with most people’s intuitive understanding of momentum: a large, fast-moving object has greater momentum than a smaller, slower object. Linear momentum is defined as the product of a system’s mass multiplied by its velocity. In symbols, linear momentum is expressed as $\boldsymbol{\textbf{p}=m\textbf{v}}.$ Momentum is directly proportional to the object’s mass and also its velocity. Thus the greater an object’s mass or the greater its velocity, the greater its momentum. Momentum$\textbf{p}$is a vector having the same direction as the velocity$\textbf{v}.$The SI unit for momentum is$\textbf{kg}\cdotp\textbf{m/s}.$ ### LINEAR MOMENTUM Linear momentum is defined as the product of a system’s mass multiplied by its velocity: $\boldsymbol{\textbf{p}=m\textbf{v}}.$ ### Example 1: Calculating Momentum: A Football Player and a Football (a) Calculate the momentum of a 110-kg football player running at 8.00 m/s. (b) Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m/s. Strategy No information is given regarding direction, and so we can calculate only the magnitude of the momentum,$\boldsymbol{p}.$(As usual, a symbol that is in italics is a magnitude, whereas one that is italicized, boldfaced, and has an arrow is a vector.) In both parts of this example, the magnitude of momentum can be calculated directly from the definition of momentum given in the equation, which becomes $\boldsymbol{p=mv}$ when only magnitudes are considered. Solution for (a) To determine the momentum of the player, substitute the known values for the player’s mass and speed into the equation. $\boldsymbol{p_{\textbf{player}}=(110\textbf{ kg})(8.00\textbf{ m/s})=880\textbf{ kg}\cdotp\textbf{m/s}}$ Solution for (b) To determine the momentum of the ball, substitute the known values for the ball’s mass and speed into the equation. $\boldsymbol{p_{\textbf{ball}}=(0.410\textbf{ kg})(25.0\textbf{ m/s})=10.3\textbf{ kg}\cdotp\textbf{m/s}}$ The ratio of the player’s momentum to that of the ball is $\boldsymbol{\frac{p_{\textbf{player}}}{p_{\textbf{ball}}}}$$\boldsymbol{=}$$\boldsymbol{\frac{880}{10.3}}$$\boldsymbol{=85.9}.$ Discussion Although the ball has greater velocity, the player has a much greater mass. Thus the momentum of the player is much greater than the momentum of the football, as you might guess. As a result, the player’s motion is only slightly affected if he catches the ball. We shall quantify what happens in such collisions in terms of momentum in later sections. # Momentum and Newton’s Second Law The importance of momentum, unlike the importance of energy, was recognized early in the development of classical physics. Momentum was deemed so important that it was called the “quantity of motion.” Newton actually stated his second law of motion in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes. Using symbols, this law is $\boldsymbol{\textbf{F}_{\textbf{net}}\:=}$$\boldsymbol{\frac{\Delta\textbf{p}}{\Delta{t}}},$ where$\textbf{F}_{\textbf{net}}$is the net external force,$\boldsymbol{\Delta\textbf{p}}$is the change in momentum, and$\boldsymbol{\Delta{t}}$is the change in time. ### NEWTON’S SECOND LAW OF MOTION IN TERMS OF MOMENTUM The net external force equals the change in momentum of a system divided by the time over which it changes. $\boldsymbol{\textbf{F}_{\textbf{net}}\:=}$$\boldsymbol{\frac{\Delta\textbf{p}}{\Delta{t}}}$ ### MAKING CONNECTIONS: FORCE AND MOMENTUM Force and momentum are intimately related. Force acting over time can change momentum, and Newton’s second law of motion, can be stated in its most broadly applicable form in terms of momentum. Momentum continues to be a key concept in the study of atomic and subatomic particles in quantum mechanics. This statement of Newton’s second law of motion includes the more familiar$\boldsymbol{\textbf{F}_{\textbf{net}}=m\textbf{a}}$as a special case. We can derive this form as follows. First, note that the change in momentum$\boldsymbol{\Delta\textbf{p}}$is given by $\boldsymbol{\Delta\textbf{p}=\Delta(m\textbf{v})}.$ If the mass of the system is constant, then $\boldsymbol{\Delta(m\textbf{v})=m\Delta\textbf{v}}.$ So that for constant mass, Newton’s second law of motion becomes $\boldsymbol{\textbf{F}_{\textbf{net}}\:=}$$\boldsymbol{\frac{\Delta\textbf{p}}{\Delta{t}}}$$\boldsymbol{=}$$\boldsymbol{\frac{m\Delta\textbf{v}}{\Delta{t}}}.$ Because$\boldsymbol{\frac{\Delta\textbf{v}}{\Delta{t}}=a},$we get the familiar equation $\boldsymbol{\textbf{F}_{\textbf{net}}=m\textbf{a}}$ when the mass of the system is constant. Newton’s second law of motion stated in terms of momentum is more generally applicable because it can be applied to systems where the mass is changing, such as rockets, as well as to systems of constant mass. We will consider systems with varying mass in some detail; however, the relationship between momentum and force remains useful when mass is constant, such as in the following example. ### Example 2: Calculating Force: Venus Williams’ Racquet During the 2007 French Open, Venus Williams hit the fastest recorded serve in a premier women’s match, reaching a speed of 58 m/s (209 km/h). What is the average force exerted on the 0.057-kg tennis ball by Venus Williams’ racquet, assuming that the ball’s speed just after impact is 58 m/s, that the initial horizontal component of the velocity before impact is negligible, and that the ball remained in contact with the racquet for 5.0 ms (milliseconds)? Strategy This problem involves only one dimension because the ball starts from having no horizontal velocity component before impact. Newton’s second law stated in terms of momentum is then written as $\boldsymbol{\textbf{F}_{\textbf{net}}\:=}$$\boldsymbol{\frac{\Delta\textbf{p}}{\Delta{t}}}.$ As noted above, when mass is constant, the change in momentum is given by $\boldsymbol{\Delta{p}=m\Delta{v}=m(v_{\textbf{f}}-v_{\textbf{i}})}.$ In this example, the velocity just after impact and the change in time are given; thus, once$\boldsymbol{\Delta{p}}$is calculated,$\boldsymbol{F_{\textbf{net}}=\frac{\Delta{p}}{\Delta{t}}}$can be used to find the force. Solution To determine the change in momentum, substitute the values for the initial and final velocities into the equation above. $\begin{array}{lcl} \boldsymbol{\Delta{p}} & \boldsymbol{=} & \boldsymbol{m(v_{\textbf{f}}-v_{\textbf{i}})} \\ {} & \boldsymbol{=} & \boldsymbol{(0.057\textbf{ kg})(58\textbf{ m/s}-0\textbf{ m/s})} \\ {} & \boldsymbol{=} & \boldsymbol{3.306\textbf{ kg}\cdotp\textbf{m/s}\approx3.3\textbf{ kg}\cdotp\textbf{m/s}} \end{array}$ Now the magnitude of the net external force can determined by using$\boldsymbol{F_{\textbf{net}}=\frac{\Delta{p}}{\Delta{t}}}:$ $\begin{array}{lcl} \boldsymbol{F_{\textbf{net}}} & \boldsymbol{=} & \boldsymbol{\frac{\Delta{p}}{\Delta{t}}=\frac{3.306\textbf{ kg}\cdotp\textbf{m/s}}{5.0\times10^{-3}\textbf{ s}}} \\ {} & \boldsymbol{=} & \boldsymbol{661\textbf{ N}\approx660\textbf{ N,}} \end{array}$ where we have retained only two significant figures in the final step. Discussion This quantity was the average force exerted by Venus Williams’ racquet on the tennis ball during its brief impact (note that the ball also experienced the 0.56-N force of gravity, but that force was not due to the racquet). This problem could also be solved by first finding the acceleration and then using$\boldsymbol{F_{\textbf{net}}=ma},$ but one additional step would be required compared with the strategy used in this example. # Section Summary • Linear momentum (momentum for brevity) is defined as the product of a system’s mass multiplied by its velocity. • In symbols, linear momentum$\textbf{p}$is defined to be $\boldsymbol{\textbf{p}=m\textbf{v}},$ where$\boldsymbol{m}$is the mass of the system and$\textbf{v}$is its velocity. • The SI unit for momentum is$\textbf{kg}\cdotp\textbf{m/s}.$ • Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes. • In symbols, Newton’s second law of motion is defined to be $\boldsymbol{\textbf{F}_{\textbf{net}}\:=}$$\boldsymbol{\frac{\Delta\textbf{p}}{\Delta{t}}},$ $\textbf{F}_{\textbf{net}}$is the net external force,$\boldsymbol{\Delta\textbf{p}}$is the change in momentum, and$\boldsymbol{\Delta{t}}$is the change time. ### Conceptual Questions 1: An object that has a small mass and an object that has a large mass have the same momentum. Which object has the largest kinetic energy? 2: An object that has a small mass and an object that has a large mass have the same kinetic energy. Which mass has the largest momentum? 3: Professional Application Football coaches advise players to block, hit, and tackle with their feet on the ground rather than by leaping through the air. Using the concepts of momentum, work, and energy, explain how a football player can be more effective with his feet on the ground. 4: How can a small force impart the same momentum to an object as a large force? ### Problems & Exercises 1: (a) Calculate the momentum of a 2000-kg elephant charging a hunter at a speed of$\boldsymbol{7.50\textbf{ m/s}}.$(b) Compare the elephant’s momentum with the momentum of a 0.0400-kg tranquilizer dart fired at a speed of$\boldsymbol{600\textbf{ m/s}}.$(c) What is the momentum of the 90.0-kg hunter running at$\boldsymbol{7.40\textbf{ m/s}}$after missing the elephant? 2: (a) What is the mass of a large ship that has a momentum of$\boldsymbol{1.60\times10^9\textbf{ kg}\cdotp\textbf{m/s}},$when the ship is moving at a speed of$\boldsymbol{48.0\textbf{ km/h}}?$(b) Compare the ship’s momentum to the momentum of a 1100-kg artillery shell fired at a speed of$\boldsymbol{1200\textbf{ m/s}}.$ 3: (a) At what speed would a$\boldsymbol{2.00\times10^4\textbf{-kg}}$airplane have to fly to have a momentum of$\boldsymbol{1.60\times10^9\textbf{ kg}\cdotp\textbf{m/s}}$(the same as the ship’s momentum in the problem above)? (b) What is the plane’s momentum when it is taking off at a speed of$\boldsymbol{60.0\textbf{ m/s}}?$(c) If the ship is an aircraft carrier that launches these airplanes with a catapult, discuss the implications of your answer to (b) as it relates to recoil effects of the catapult on the ship. 4: (a) What is the momentum of a garbage truck that is$\boldsymbol{1.20\times10^4\textbf{ kg}}$and is moving at$\boldsymbol{10.0\textbf{ m/s}}?$(b) At what speed would an 8.00-kg trash can have the same momentum as the truck? 5: A runaway train car that has a mass of 15,000 kg travels at a speed of$\boldsymbol{5.4\textbf{ m/s}}$down a track. Compute the time required for a force of 1500 N to bring the car to rest. 6: The mass of Earth is$\boldsymbol{5.972\times10^{24}\textbf{ kg}}$and its orbital radius is an average of$\boldsymbol{1.496\times10^{11}\textbf{ m}}.$Calculate its linear momentum. ## Glossary linear momentum the product of mass and velocity second law of motion physical law that states that the net external force equals the change in momentum of a system divided by the time over which it changes ### Solutions Problems & Exercises 1: (a)$\boldsymbol{1.50\times10^4\textbf{ kg}\cdotp\textbf{m/s}}$ (b) 625 to 1 (c)$\boldsymbol{6.66\times10^2\textbf{kg}\cdotp\textbf{m/s}}$ 3: (a)$\boldsymbol{8.00\times10^4\textbf{ m/s}}$ (b)$\boldsymbol{1.20\times10^6\textbf{ kg}\cdotp\textbf{m/s}}$ (c) Because the momentum of the airplane is 3 orders of magnitude smaller than of the ship, the ship will not recoil very much. The recoil would be$\boldsymbol{-0.0100\textbf{ m/s}},$which is probably not noticeable. 5: 54 s	2021-09-17 22:23:18	{"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": 0.8280340433120728, "perplexity": 960.0839172483328}, "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-2021-39/segments/1631780055808.78/warc/CC-MAIN-20210917212307-20210918002307-00254.warc.gz"}
http://mathhelpforum.com/math-puzzles/115142-novice-needs-help-print.html	# novice needs help • Nov 17th 2009, 05:05 AM mathguy9 novice needs help I am needing help with the following six puzzles. Any and all help would be greatly appreciated. 1. Here is a long division sum, showing all the work, and the result. Simple, right? The sum works out exactly, with no remainder. The slight complication is, that all the numbers have been replaced with letters on a random basis. However, one letter always represents the same number. Can you reconstruct the original sum? (The problem is ADGAAHD divided by AB) (CDEFG is the answer) ______CDEFG_________________ AJK ------------------------------------------- AKA AAG --------------------------------------------- FA JF ------------------------------------------------ AGH AEE -------------------------------------------------- FD 2. 6-2 18 8-7 84 12-4 ? 3. Can you figure out the reasoning of these numbers, and replace the question mark with the correct number? 5 3 8 7 12 15 49 56 3 9 4 12 18 27 36 ? 4.Take a look at these digital "watches". By cracking the logic that connects them, you should be able to figure out what time should be on watch number five. 15.14.01 12.18.00 08.26.58 03.42.55 ??.??.?? 5.Thirty-six Sixty-four Seventy-two Twenty-five Eighty-one What is the odd number out? 6. Can you figure out the logic behind these number "squares" and find the number that replaces the question mark? 9 6 5 10 4 6 ? 5 4 2 3 7 8 11 12 7 (Think of each set of four numbers as a square. 9 = top left, 4 = bottom left, 6 = top right, 2 = bottom right, and so on, it might help to write them down.) Thank you again for any and all help mathguy • Nov 17th 2009, 07:34 PM Soroban Hello, mathguy9! I hope I interpreted #1 correctly . . . Quote: 1. Here is a long division sum, showing all the work, and the result. The divison works out exactly with no remainder. All the digits have been replaced with letters on a random basis. However, one letter always represents the same number. Can you reconstruct the original division? . . $\begin{array}{cccccccccc} & & & & & C & D & E & F & G \\ & & & -- & -- & -- & -- & -- & -- & -- \\ A & B & \| & A & D & G & A & A & H & D \\ & & & A & J & K \\ & & & -- & -- & -- \\ & & & & A & K & A \\ & & & & A & A & G \\ & & & & -- & -- & -- \\ & & & & & & F & A \\ & & & & & & J & F \\ & & & & & & -- & -- \\ \end{array}$ . . . . . . . . . . . . . . . . . . . . . $\begin{array}{cccccccccc} & & & & & & A & G & H \\ & & & & & & A & E & E \\ & & & & & & -- & -- & -- \\ & & & & & & & & F & D \\ & & & & & & & & F & D \\ & & & & & & & & -- & -- \end{array}$ . . $\begin{array}{ccc} A\:=\:1 & & F \:=\:7 \\ B \:=\:9 && G \:=\:4 \\ C\:=\:8 && H\:=\:0 \\ D\:=\:6 && J\:=\:5 \\ E\:=\:3 && K\:=\: 2 \end{array}$ . . $\begin{array}{cccccccccc} & & & & & 8 & 6 & 3 & 7 & 4 \\ & & & -- & -- & -- & -- & -- & -- & -- \\ 1 & 9 & \| & 1 & 6 & 4 & 1 & 1 & 0 & 6 \\ & & & 1 & 5 & 2 \\ & & & -- & -- & -- \\ & & & & 1 & 2 & 1 \\ & & & & 1 & 1 & 4 \\ & & & & -- & -- & -- \\ & & & & & & 7 & 1 \\ & & & & & & 5 & 7 \\ & & & & & & -- & -- \\ \end{array}$ . . . . . . . . . . . . . . . . . . . . . $\begin{array}{cccccccccc} & & & & & & 1 & 4 & 0 \\ & & & & & & 1 & 3 & 3 \\ & & & & & & -- & -- & -- \\ & & & & & & & & 7 & 6 \\ & & & & & & & & 7 & 6 \\ & & & & & & & & -- & -- \end{array}$ • Nov 17th 2009, 08:17 PM Wilmer Quote: Originally Posted by mathguy9 5.Thirty-six Sixty-four Seventy-two Twenty-five Eighty-one What is the odd number out? Question really means: which number does not belong ? Easy...LOOK carefully! • Nov 18th 2009, 07:06 AM Soroban Hello, mathguy9! I think I have #4 . . . Quote: 4. Take a look at these digital "watches". By cracking the logic that connects them, you should be able to figure out what time should be on watch number five. . . $\begin{array}{ccccc}\text{H} && \text{M} && \text{S} \\ \hline 15 & : & 14 & : & 01 \\ 12 & : & 18 & : & 00 \\ 08 & : & 26 & : & 58 \\ 03 & : & 42 &:& 55 \\ ? &:& ? &:& ? \end{array}$ The consecutive differences of the Hours suggest this sequence: . $\begin{array}{c\|ccccccccc}\text{Hours} & 15 && 12 && 8 && 3 && {\color{blue}\text{-}3} \\ \hline \text{Difference} && \text{-}3 && \text{-}4 && \text{-}5 && {\color{blue}\text{-}6} \end{array}$ The consecutive differences of the Minutes suggest this sequence: . $\begin{array}{c\|ccccccccc}\text{Minutes} & 14 && 18 && 26 && 42 && {\color{blue}74} \\ \hline \text{Difference} && +4 && +8 && +16 && {\color{blue}+32} \end{array}$ The consecutive differences of the Seconds suggest this sequence: . $\begin{array}{c\|ccccccccc}\text{Seconds} & 01 && 00 && 58 && 55 && {\color{blue}51} \\ \hline \text{Difference} && \text{-}1 && \text{-}2 && \text{-}3 && {\color{blue}\text{-}4} \end{array}$ $\begin{array}{cccccccc}\text{The final time is:} & -3 & : & 74 &:& 51 \\ \text{which convets to:} & 09 &:& 74 &:& 51 \\ \text{and finally:} & 10 &:& 14 &:& 51 \end{array}$ • Nov 18th 2009, 08:19 AM bbeckett Quote: 3. Can you figure out the reasoning of these numbers, and replace the question mark with the correct number? 5 3 8 7 12 15 49 56 3 9 4 12 18 27 36 ? The answer is 48. The numbers are in four squares of four numbers, so the first square is: 5 3 let this be: a b 12 15 c d You will find that in each square, c = (a-1)b and d=ab. In the final square, 412=48 • Nov 18th 2009, 06:36 PM mathguy9 you guys are so great. Thank you so much for all your help. • Nov 30th 2009, 11:09 AM wonderboy1953 For the sixth question The answer is 9. Here's the pattern: 64 - 29 = 24 - 18 = 6 103 - 75 = 30 - 35 = -5 68 - 411 = 48 - 44 = 4 512 - 7*9 = 60 - 63 = -3 To caution, patterns are useful but sometimes more than one pattern can match up with the problem at hand so keep a sharp eye open for that. • Nov 30th 2009, 01:46 PM Soroban Hello, mathguy9! I have five answers to problem #5. Quote: 5. . $\begin{array}{c}\text{Thirty-six} \\ \text{Sixty-four} \\ \text{Seventy-two} \\ \text{Twenty-five} \\ \text{Eighty-one} \end{array}$ . . What is the odd number out? 36 - the only number with 9 divisors. 64 - the only number whose sum of digits is even. 72 - the only number which is not a square. 25 - the only number less than 33. 81 - the only number greater than 77.	2017-03-25 11:55:44	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 11, "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": 0.7339058518409729, "perplexity": 704.8162392803257}, "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-2017-13/segments/1490218188924.7/warc/CC-MAIN-20170322212948-00354-ip-10-233-31-227.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/2105218/find-cdf-of-random-variable-which-depends-on-other-variable	# Find CDF of random variable which depends on other variable Let $X$ be a random variable with uniform distribution on $[-1, 1]$. Find the CDF of random variable Y given by the following formula: $Y = \left\{\begin{matrix} -\frac{1}{2},& X < - \frac{1}{2}\\ X,& -\frac{1}{2} \leq X \leq \frac{1}{4}\\ \frac{1}{4}, & X > \frac{1}{4} \end{matrix}\right.$ So I've found PDF and CDF of $X$: $f_X(x) = \begin{cases} \frac{1}{2}, & x \in [-1, 1]\\ 0, & \text{otherwise} \end{cases}$ $F_X(a) = \int_{-\infty}^{a} f_X(x) dx = \left\{\begin{matrix} 0, & a \leq -1\\ \frac{a+1}{2}, & a \in (-1, 1) \\ 1, & a \geq 1 \end{matrix}\right.$ I tried to find Y's CDF by: $F_Y(a) = P(Y \leq a)$ $= P(-\frac{1}{2} \leq a, X < - \frac{1}{2}) + P(X \leq a, - \frac{1}{2} \leq X \leq \frac{1}{4}) + P(\frac{1}{4} \leq a, X > -\frac{1}{4})$ But what should I do next? I'm finding such CDF for the first time and my notes say I need to consider a few different cases, but I have no clue what they should look like and how to do it. Any tips would be helpful. Tip: You can find a CDF, but not a pdf, for all points on the support. Notice that $Y$ will have a support of the interval $[-\tfrac 12;\tfrac 14]$, but will have a probability mass at the two end points, with a probability density in the interval between. It is a mixed distribution. That is $~\mathsf P(Y{=}-\tfrac 12)~=~\mathsf P(-1{\leq}X{<}-\tfrac 12)~$ and $~\mathsf P(Y{=}\tfrac 14)~=~\mathsf P(\tfrac 14{<}X{\leq}1)~$. So then $$F_Y(a)~=~\mathsf P(Y\leq a) ~=~ \begin{cases} 0 &:& a<-\tfrac 12\\ \tfrac {2a+3}4 & : & -\tfrac 12\leq a < \tfrac 14 \\ 1 &:& \tfrac 14 \leq a \end{cases}$$ If you've peeked, notice the step discontinuities. Note that $P(Y\leq y) = \int_\mathbb{R} P(Y \leq Y \| X = x) f_X(x) dx$ by the law of total probability. You can split the integral up by $x< -1/2$, $-1/2 \leq x \leq 1/4$ and $x> 1/4$ and do it.	2019-06-18 05:22:21	{"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": 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": 0.7058497071266174, "perplexity": 92.96205612652767}, "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-2019-26/segments/1560627998607.18/warc/CC-MAIN-20190618043259-20190618065259-00141.warc.gz"}
http://gmatclub.com/forum/donald-plans-to-invest-x-dollars-in-a-savings-account-that-82702.html?fl=similar	Find all School-related info fast with the new School-Specific MBA Forum It is currently 15 Apr 2014, 20:58 ### 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 # Events & Promotions ###### Events & Promotions in June Open Detailed Calendar # Donald plans to invest x dollars in a savings account that Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: Manager Joined: 29 May 2008 Posts: 117 Followers: 1 Kudos [?]: 7 [1] , given: 0 Donald plans to invest x dollars in a savings account that [#permalink] 19 Aug 2009, 05:44 1 KUDOS 00:00 Difficulty: 35% (medium) Question Stats: 61% (02:16) correct 38% (01:31) wrong based on 83 sessions Donald plans to invest x dollars in a savings account that pays interest at an annual rate of 8% compounded quarterly. Approximately what amount is the minimum that Donald will need to invest to earn over $100 in interest within 6 months? A. 1500 B. 1750 C. 2000 D. 2500 E. 3000 [Reveal] Spoiler: OA Last edited by Bunuel on 07 Jul 2013, 00:20, edited 1 time in total. Renamed the topic, edited the question and added the OA. Manager Joined: 17 Dec 2007 Posts: 107 Followers: 0 Kudos [?]: 49 [0], given: 8 Re: Compounded Interest [#permalink] 19 Aug 2009, 08:18 TheRob wrote: Sorry i posted a rate problem with the wrong title Here you have the real problem of comound interest, I kind of have the idea but I don not know how to apply the formula Donald plans to invest x dollars in a savings account that pays interest at an annual rate of 8% compounded quarterly. Approximately what amount is the minimum that Donald will need to invest to earn over$100 in interest within 6 months? $1500$1750 $2000$2500 $3000 Compound interest formula A = P ( 1+r/n)power nt given, n= 4 (quaterly);r =.08 the approach is substitution, our interest requirement is 100$ after 6 months, 2 compounding period. interest per compounding period is 2% lets take 1500, after 3 months interest accumulated is 30$, total amount is 1530 after 6 months, interest is 30.6$ and total is 1560.6$, so not 1500 1500 & 1750 have a difference of 250$ only , but the expected interest different is around 40$hence you can straightaway rule out 1750 2000 is again can be ruled out as approx 4% interest yeilds only 80$ 2500$is a good bet, first 3 months it earns 50$ as interest, next 3 months it will earn 51$as interest. hence answer is D Director Joined: 01 Apr 2008 Posts: 911 Schools: IIM Lucknow (IPMX) - Class of 2014 Followers: 10 Kudos [?]: 161 [1] , given: 18 Re: Compounded Interest [#permalink] 19 Aug 2009, 09:34 1 This post received KUDOS E. For CI: Final amount = Principal(1+ r/n)^nt Let principal = x, then final amount = x+100 x+100 = x[1+ 0.02]^2 x+100 = 1.04x 0.04x=100 x=2500 so if 2500 is invested CI will be exactly 100. To earn more interest more principal should be invested. So E. Manager Joined: 29 May 2008 Posts: 117 Followers: 1 Kudos [?]: 7 [0], given: 0 Re: Compounded Interest [#permalink] 19 Aug 2009, 10:57 Thank you veyr much here is the book explanation The formula for calculating compound interest is A = P(1 + r/n)nt where the variables represent the following: A = amount of money accumulated after t years (principal + interest) P = principal investment r = interest rate (annual) n = number of times per year interest is compounded t = number of years In this case, x represents the unknown principal, r = 8%, n = 4 since the compounding is done quarterly, and t = .5 since the time frame in question is half a year (6 months). You can solve this problem without using compound interest. 8% interest over half a year, however that interest is compounded, is approximately 4% interest. So, to compute the principal, it's actually a very simple calculation: 100 = .04x 2500 = x The correct answer is D. Director Joined: 01 Apr 2008 Posts: 911 Schools: IIM Lucknow (IPMX) - Class of 2014 Followers: 10 Kudos [?]: 161 [0], given: 18 Re: Compounded Interest [#permalink] 19 Aug 2009, 21:20 Unfortunately I dont agree with the OE if I am understanding the question correctly. If we invest 2500, then CI will ONLY be 100. Question asks "to earn over$100" how much amount is to be invested. Considering rate of interest, number of years and everything else to be the same, the only way to earn CI>100 is to increase the Principal Amount because CI is directly proportional to Principal Amount. On test day I would have chosen E for sure. What is the source though? Manager Joined: 09 Aug 2009 Posts: 57 Followers: 1 Kudos [?]: 4 [0], given: 1 Re: Compounded Interest [#permalink] 19 Aug 2009, 22:14 Economist wrote: Unfortunately I dont agree with the OE if I am understanding the question correctly. If we invest 2500, then CI will ONLY be 100. Question asks "to earn over $100" how much amount is to be invested. Considering rate of interest, number of years and everything else to be the same, the only way to earn CI>100 is to increase the Principal Amount because CI is directly proportional to Principal Amount. On test day I would have chosen E for sure. What is the source though? For P=2500, first 3 months you earn 50$ as interest, next 3 months it will earn 51$as interest. So total 101$. I guess D should be fine Director Joined: 01 Apr 2008 Posts: 911 Schools: IIM Lucknow (IPMX) - Class of 2014 Followers: 10 Kudos [?]: 161 [0], given: 18 Re: Compounded Interest [#permalink] 19 Aug 2009, 22:47 Economist wrote: E. For CI: Final amount = Principal(1+ r/n)^nt Let principal = x, then final amount = x+100 x+100 = x[1+ 0.02]^2 x+100 = 1.04x 0.04x=100 x=2500 so if 2500 is invested CI will be exactly 100. To earn more interest more principal should be invested. So E. OK. I got the problem, the reason is that I rounded off 1.02^2 in the above calculation. Precisely it is 1.0404 and then we get x= 2475. So anything >2475 will give CI >100 The .xx04 made the difference Hence, either I should be very precise and not round off for CI problems or solve via back tracking. Manager Joined: 18 Feb 2010 Posts: 174 Schools: ISB Followers: 3 Kudos [?]: 96 [2] , given: 0 Re: Compounded Interest [#permalink] 06 Mar 2010, 23:27 2 KUDOS TheRob wrote: Sorry i posted a rate problem with the wrong title Here you have the real problem of comound interest, I kind of have the idea but I don not know how to apply the formula Donald plans to invest x dollars in a savings account that pays interest at an annual rate of 8% compounded quarterly. Approximately what amount is the minimum that Donald will need to invest to earn over $100 in interest within 6 months?$1500 $1750$2000 $2500$3000 Lets solve it under 30 seconds... 8% compounded quarterly = 2% per quarter = 4% for half year if 4% is 100 then 100% would be 2500.....Answer D. What say.... Hope this works !!! _________________ CONSIDER AWARDING KUDOS IF MY POST HELPS !!! Manager Joined: 04 Dec 2011 Posts: 82 Schools: Smith '16 (I) Followers: 0 Kudos [?]: 9 [0], given: 13 Re: Compounded Interest [#permalink] 06 Jul 2013, 23:21 what wrong am I doing, can someone point out? {x(1+.02)^2 } -x > 100 taking x common on LHS (x)(1.02)^2 -1>100 (x)(1.02)^2 >101 (x) 1.04 >101 x= 101\1.04 I think i am doing some silly calculation mistake here, can someone point me out pl. _________________ Life is very similar to a boxing ring. Defeat is not final when you fall down… It is final when you refuse to get up and fight back! 1 Kudos = 1 thanks Nikhil Intern Joined: 01 Jul 2013 Posts: 2 Location: United States Concentration: General Management, Strategy Schools: Insead '14, ISB '15 GPA: 4 WE: Business Development (Computer Software) Followers: 0 Kudos [?]: 6 [0], given: 0 Re: Compounded Interest [#permalink] 07 Jul 2013, 00:26 nikhil007 wrote: what wrong am I doing, can someone point out? {x(1+.02)^2 } -x > 100 taking x common on LHS (x)(1.02)^2 -1>100 (x)(1.02)^2 >101 (x) 1.04 >101 x= 101\1.04 I think i am doing some silly calculation mistake here, can someone point me out pl. Hi Nikhil, why don't use the simple formula: CI = P(R/N)^NT P= x CI = 100 N= 4 T = 1/2 = 0.5 => P = 100/ (.08/4)^(40.5) => P = 100/ (.02)^2 => p = 100/ (4 10^-4) => P = 10^6/4 => P = 25,000 Last edited by arunraj on 07 Jul 2013, 03:19, edited 1 time in total. Math Expert Joined: 02 Sep 2009 Posts: 17278 Followers: 2862 Kudos [?]: 18313 [3] , given: 2345 Re: Compounded Interest [#permalink] 07 Jul 2013, 00:37 3 KUDOS Expert's post nikhil007 wrote: what wrong am I doing, can someone point out? {x(1+.02)^2 } -x > 100 taking x common on LHS (x)(1.02)^2 -1>100 (x)(1.02)^2 >101 (x) 1.04 >101 x= 101\1.04 I think i am doing some silly calculation mistake here, can someone point me out pl. Algebra: x(1+0.02)^2 -x > 100 x(1.02^2-1)>100 x0.0404>100 x>\frac{100}{0.0404}\approx{2475.25} Donald plans to invest x dollars in a savings account that pays interest at an annual rate of 8% compounded quarterly. Approximately what amount is the minimum that Donald will need to invest to earn over \$100 in interest within 6 months? A. 1500 B. 1750 C. 2000 D. 2500 E. 3000 Annual rate of 8% compounded quarterly is approximately 4% in 6 months (a bit more). x0.04=100 --> x=2500. Similar questions to practice: john-deposited-10-000-to-open-a-new-savings-account-that-135825.html on-the-first-of-the-year-james-invested-x-dollars-at-128825.html marcus-deposited-8-000-to-open-a-new-savings-account-that-128395.html jolene-entered-an-18-month-investment-contract-that-127308.html alex-deposited-x-dollars-into-a-new-account-126459.html michelle-deposited-a-certain-sum-of-money-in-a-savings-138273.html leona-bought-a-1-year-10-000-certificate-of-deposit-that-143742.html Theory: math-number-theory-percents-91708.html Hope it helps. _________________ Manager Joined: 04 Dec 2011 Posts: 82 Schools: Smith '16 (I) Followers: 0 Kudos [?]: 9 [0], given: 13 Re: Donald plans to invest x dollars in a savings account that [#permalink] 07 Jul 2013, 02:33 Damn..when will I stop making these silly mistakes.. _________________ Life is very similar to a boxing ring. Defeat is not final when you fall down… It is final when you refuse to get up and fight back! 1 Kudos = 1 thanks Nikhil Senior Manager Joined: 03 Dec 2012 Posts: 368 Followers: 0 Kudos [?]: 13 [0], given: 291 Re: Compounded Interest [#permalink] 28 Nov 2013, 01:35 arunraj wrote: nikhil007 wrote: what wrong am I doing, can someone point out? {x(1+.02)^2 } -x > 100 taking x common on LHS (x)(1.02)^2 -1>100 (x)(1.02)^2 >101 (x) 1.04 >101 x= 101\1.04 I think i am doing some silly calculation mistake here, can someone point me out pl. Hi Nikhil, why don't use the simple formula: CI = P(R/N)^NT P= x CI = 100 N= 4 T = 1/2 = 0.5 => P = 100/ (.08/4)^(40.5) => P = 100/ (.02)^2 => p = 100/ (4 10^-4) => P = 10^6/4 => P = 25,000 I think you have quoted the incorrect formula for CI. Re: Compounded Interest [#permalink] 28 Nov 2013, 01:35 Similar topics Replies Last post Similar Topics: Person A invests x dollars and B invests y dollars. A earns 4 06 Jan 2005, 09:45 Larry saves x dollars per month. Will Larry s total savings 9 05 Oct 2005, 02:11 If Ann saves x dollars each week and Beth saves y dollars 3 23 Jun 2006, 22:55 Larry saves x dollars per month. Will Larry s total savings 1 01 Nov 2007, 16:19 Grace makes an initial deposit of x dollars into a savings 3 25 May 2011, 10:55 Display posts from previous: Sort by # Donald plans to invest x dollars in a savings account that Question banks Downloads My Bookmarks Reviews Important topics Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.	2014-04-16 04:58:12	{"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": 0.3217886686325073, "perplexity": 10161.786538958662}, "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-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00026-ip-10-147-4-33.ec2.internal.warc.gz"}
https://www.bartleby.com/solution-answer/chapter-13-problem-1cqq-principles-of-microeconomics-mindtap-course-list-8th-edition/9781305971493/xavier-opens-up-a-lemonade-stand-for-two-hours-he-spends-10-for-ingredients-and-sells-60-worth-of/1eeb9120-98d8-11e8-ada4-0ee91056875a	# Xavier opens up a lemonade stand for two hours. He spends $10 for ingredients and sells$60 worth of lemonade. In the same two hours, he could have mowed his neighbor's lawn for $40. Xavier has an accounting profit of ________ and an economic profit of _______. a.$50, $10 b.$90, $50 c.$10, $50 d.$50, $90 BuyFindarrow_forward ### Principles of Microeconomics (Mind... 8th Edition N. Gregory Mankiw Publisher: Cengage Learning ISBN: 9781305971493 #### Solutions Chapter Section BuyFindarrow_forward ### Principles of Microeconomics (Mind... 8th Edition N. Gregory Mankiw Publisher: Cengage Learning ISBN: 9781305971493 Chapter 13, Problem 1CQQ Textbook Problem 192 views ## Xavier opens up a lemonade stand for two hours. He spends$10 for ingredients and sells $60 worth of lemonade. In the same two hours, he could have mowed his neighbor's lawn for$40. Xavier has an accounting profit of ________ and an economic profit of _______.a. $50,$10b. $90,$50c. $10,$50d. $50,$90 To determine Accounting profit and economic profit. Option ‘a’ is correct. ### Explanation of Solution Option (a): When Xavier spend $10 for ingredients and sell$60 worth of lemonade, then the accounting profit will be (Total revenueTotal explicit cost) (total revenue – total explicit cost) $50($60$10). Economic profit is (Total revenue(Implicit cost +Explicit cost))$10 ($60($50+$10)) . Therefore, the value of accounting profit is$50 and the economic profit is $10. Option (b): Xavier accounting profit is not$90, because an accounting profit is the total revenue minus the total explicit cost. Economic profit is not $50, because economic profit is total revenue minus implicit and explicit cost. Thus, option ‘b’ is incorrect. Option (c): When Xavier spends$10 for ingredients and sells $60 worth of lemonade, Xavier accounting profit will not be$10, because an accounting profit is the total revenue minus the total explicit cost. Economic profit is not $50, because economic profit is the total revenue minus implicit and explicit cost. Thus, option ‘c’ is incorrect. Option (d): When Xavier spend$10 for ingredients and sell $60 worth of lemonade, then the accounting profit will be (total revenue – total explicit cost)$50($60$10) . But the economic profit is not \$90, because the economic profit is (Total revenue(Implicit cost +Explicit cost)) . Thus, option ‘d’ is incorrect. Economics Concept Introduction Concept introduction: Accounting profit: Accounting profit refers to the total revenue minus total explicit cost Economic profit: Economic profit refers to the total revenue minus implicit and explicit cost. #### The Solution to Your Study Problems Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees! Get Started Find more solutions based on key concepts	2020-02-16 20:13:43	{"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": 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": 0.802950382232666, "perplexity": 8243.34285031452}, "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-2020-10/segments/1581875141396.22/warc/CC-MAIN-20200216182139-20200216212139-00478.warc.gz"}
https://schools.aglasem.com/ncert/ncert-solutions-class-7-english-chapter-4/	NCERT Solutions Class 7 English (Honeycomb) Chapter 4 The Ashes that Made Trees Bloom – Here are all the NCERT solutions for Class 7 English Chapter 4. This solution contains questions, answers, images, explanations of the complete chapter 4 titled The Ashes that Made Trees Bloom of English taught in class 7. If you are a student of class 7 who is using NCERT Textbook to study English, then you must come across chapter 4 The Ashes that Made Trees Bloom. After you have studied lesson, you must be looking for answers of its questions. Here you can get complete NCERT Solutions for Class 7 English Chapter 4 The Ashes that Made Trees Bloom in one place. ## NCERT Solutions Class 7 English 1 Chapter 4 The Ashes That Made Trees Bloom Here on AglaSem Schools, you can access to NCERT Book Solutions in free pdf for English 1 for Class 7 so that you can refer them as and when required. The NCERT Solutions to the questions after every unit of NCERT textbooks aimed at helping students solving difficult questions. For a better understanding of this chapter, you should also see summary of Chapter 4 The Ashes That Made Trees Bloom , English 1, Class 7. Class 7 Subject English 1 Book Honeycomb Chapter Number 4 Chapter Name The Ashes That Made Trees Bloom ### NCERT Solutions Class 7 English 1 chapter 4 The Ashes That Made Trees Bloom Class 7, English 1 chapter 4, The Ashes That Made Trees Bloom solutions are given below in PDF format. You can view them online or download PDF file for future use. ### The Ashes That Made Trees Bloom Q.1: Why did the neighbours kill the dog? Ans : The neighbours dragged the dog around their garden to find a treasure for themselves. When the dog stopped near a pine tree and started scratching the ground, they dug happily hoping to find a treasure. When they saw that there was nothing in there except a dead kitten, they became furious at the dog. They kicked it and beat it to death. They killed it because it did not help them find a Treasure. Q.2: Mark the right item. (i) The old farmer and his wife loved the dog (a) because it helped them in their day-to-day work. (b) as if it was their own baby. (c) as they were kind to all living beings. (ii) When the old couple became rich, they (a) gave the dog better food. (b) invited their greedy neighbours to a feast. (c) lived comfortably and were generous towards their poor neighbours. (iii) The greedy couple borrowed the mill and the mortar to make (a) rice pastry and bean sauce. (b) magic ash to win rewards. (c) a pile of gold. Ans : (i) The old farmer and his wife loved the dog as if it was their own baby. (ii) When the old couple became rich, they lived comfortably and were generous towards their poor neighbours. (iii) The greedy couple borrowed the mill and the mortar to make a pile of gold. Q.1: The old farmer is a kind person. What evidence of his kindness do you find in the first two paragraphs. Ans : The old farmer was a kind person. He loved his dog as if it was his own baby. He fed it with fish with his own chopsticks and all the boiled rice it wanted. He was patient and kind to everything that had life, and often dug up a sod on purpose to give food to the birds. Q.2: What did the dog do to lead the farmer to the hidden gold? Ans : The dog came running to the farmer, putting its paws against his legs and motioning with its head to some spot behind. He thought it was only playing and did not mind it. However, the dog kept on whining and running to and fro for some minutes. Then the farmer followed it a few yards to a place where it began scratching the ground. The farmer thought there was a bone or a bit of fish buried there and therefore, struck his hoe in the earth. However, what he found was a pile of gold. Q.3: (i) How did the spirit of the dog help the farmer first? (ii) How did it help him next? Ans : (i) The spirit of the dog asked the farmer to cut down the pine tree over its grave, and make from it a mortar for rice pastry and a mill for bean sauce. The farmer did so. Some time close to the New Year, the farmer wanted to make some rice pastry. When the rice was boiled, his wife put it into the mortar and he pounded the mass into dough. When the pastry was ready for baking, the whole mass turned into a heap of gold coins. Similarly, when beans were ground in the hand-mill, gold started dropping from it like rain and in a few minutes, the tub under the mill was filled with gold. (ii) Informing the farmer about how his wicked neighbours had burned the hand mill, the spirit of the dog asked him to take the ashes of the mill and sprinkle them on withered trees to make them bloom. The old man did so and found to his delight that the words of the spirit were indeed true. The bare cherry tree in his garden sprouted blossoms when a pinch of the ashes were sprinkled on it. Later, he was rewarded by the daimio for making an old withered cherry tree blossom once again. Q.4: Why did the daimio reward the farmer but punish his neighbour for the same act? Ans : The daimio rewarded the farmer for making an old withered cherry tree blossom once again. Like the farmer, his greedy neighbour also sprinkled ashes over a withered cherry tree. However, the result this time was different. The tree did not blossom, while the wind blew the dust into the noses and eyes of the daimio and his wife. This was the reason why the greedy neighbour was punished. Q.1: Read the following conversation. RAVI : What are you doing? MRIDU : I’m reading a book. RAVI : Who wrote it? MRIDU : Ruskin Bond. RAVI : Where did you find it? MRIDU : In the library. Notice that ‘what’, ‘who’, ‘where’, are question words. Questions that require information begin with question words. Some other question words are ‘when’, ‘why’, ‘where’, ‘which’ and ‘how’. Remember that Read the following paragraph and frame questions on the italicised phrases. Anil is in school. I am in school too. Anil is sitting in the left row. He is reading a book. Anil’s friend is sitting in the second row. He is sharpening his pencil. The teacher is writing on the blackboard. Children are writing in their copybooks. Some children are looking out of the window. (i) ____________________________________________________________ (ii) ____________________________________________________________ (iii)____________________________________________________________ (iv) ___________________________________________________________ (v) ____________________________________________________________ (vi) ____________________________________________________________ (vii) ____________________________________________________________ Ans : (i) Where is Anil? (ii) Where is Anil sitting? (iii) What is Anil doing? (iv) Where is Anil's friend sitting? (v) Who is writing on the blackboard? (vi) What are some children doing? Q.2: Write appropriate question words in the blank spaces in the following dialogue. NEHA : __________ did you get this book? SHEELA : Yesterday morning. NEHA : _________ is your sister crying? SHEELA : Because she has lost her doll. NEHA : __________ room is this, yours or hers? SHEELA : It’s ours. NEHA : __________ do you go to school? SHEELA : We walk to school. It is nearby. Ans : NEHA: When did you get this book? SHEELA: Yesterday morning. NEHA: Why is your sister crying? SHEELA: Because she has lost her doll. NEHA: Whose room is this, yours or hers? SHEELA: It's ours. NEHA: How do you go to school? SHEELA: We walk to school. It is nearby. Q.3: Fill in the blanks with the words given in the box. (i) My friend lost his chemistry book. Now he doesn’t know _______________ to do and _______________ to look for it. (ii) There are so many toys in the shops. Neena can’t decide _______________ one to buy. (iii) You don’t know the way to my school. Ask the policeman _______________ to get there. (iv) You should decide soon _______________ to start building your house. (v) Do you know ________________ to ride a bicycle? I don’t remember _______________ and _______________ I learnt it. (vi) “You should know _______________ to talk and ______________ to keep your mouth shut,” the teacher advised Anil. Ans : (i) My friend lost his chemistry book. Now he doesn’t know what to do and where to look for it. (ii) There are so many toys in the shops. Neena can’t decide Which one to buy. (iii) You don’t know the way to my school. Ask the policeman how to get there. (iv) You should decide soon where to start building your house. (v) Do you know how to ride a bicycle? I don’t remember When and where I learnt it. (vi) “You should know When to talk and when to keep your mouth shut,” the teacher advised Anil. Q.4: Add im- or in- to each of the following words and use them in place of the italicised words in the sentences given below. (i) The project appears very difficult at first sight but it can be completed if we work very hard. (ii) He lacks competence. That’s why he can’t keep any job for more than a year. (iii) “Don’t lose patience. Your letter will come one day,” the postman told me. (iv) That’s not a proper remark to make under the circumstances. (v) He appears to be without sensitivity. In fact, he is very emotional. Ans : (i) The project appears impossible at first sight but it can be completed if we work very hard. (ii) He is incompetent. That’s why he can’t keep any job for more than a year. (iii) “Don’t be impatient. Your letter will come one day,” the postman told me. (iv) That’s an improper remark to make under the circumstances. (v) He appears to be insensitive. In fact, he is very emotional. Q.5: Read the following sentences. It was a cold morning and stars still glowed in the sky. An old man was walking along the road. The words in italics are articles. ‘A’ and ‘an’ are indefinite articles and ‘the’ is the definite article. ‘A’ is used before a singular countable noun. ‘An’ is used before a word that begins with a vowel. Use a, an or the in the blanks. There was once _______________ play which became very successful _______________ . famous actor was acting in it. In _______________ play his role was that of aristocrat who had been imprisoned in _______________ castle for twenty years. In _______________ last act of _______________ play someone would come on _______________ stage with _______________ letter which he would hand over to prisoner. Even though aristocrat was not expected to read _______________ letter at each performance, he always insisted that _______________ letter be written out from beginning to end. Ans : There was once a play which became very successful. A famous actor was acting in it. In the play his role was that of an aristocrat who had been imprisoned in a castle for twenty years. In the last act of the play someone would come on the stage with a letter which he would hand over to the prisoner. Even though the aristocrat was not expected to read the letter at each performance, he always insisted that the letter be written out from beginning to end. Q.6: Encircle the correct article. Nina was looking for ( a / the) job. After many interviews she got (a / the ) job she was looking for. A : Would you like (a/an/the) apple or (a/an/the) banana? B : I’d like (a/an/the) apple, please. A : Take (a/an/the) red one in (a/an/the) fruit bowl. You may take (a/an/the) orange also, if you like. B : Which one? A : (A/An/The) one beside (a/an/the) banana. Ans : A : Would you like an apple or a banana? B : I’d like an apple, please. A : Take the red one in the fruit bowl. You may take an orange also, if you like. B : Which one? A : The one beside the banana. Q.1: Do you remember an anecdote or a story about a greedy or jealous person and the unhappy result of his/her action? Narrate the story to others in your class. Here is one for you to read. Seeing an old man planting a fig tree, the king asked why he was doing this. The man replied that he might live to eat the fruit, and, even if he did not, his son would enjoy the figs. “Well,” said the king, “if you do live to eat the fruit of this tree, please let me know.” The man promised to do so, and sure enough, before too long, the tree grew and bore fruit. Packing some fine figs in a basket, the old man set out for the palace to meet the king. The king accepted the gift and gave orders that the old man’s basket be filled with gold. Now, next door to the old man, there lived a greedy old man jealous of his neighbour’s good fortune. He also packed some figs in a basket and took them to the palace in the hope of getting gold. The king, on learning the man’s motive, ordered him to stand in the compound and had him pelted with figs. The old man returned home and told his wife the sad story. She consoled him by saying, “You should be thankful that our neighbour did not grow coconuts.” Ans : There was a farmer. He had a hen. He loved his hen very much. It was a gifted hen because it laid one golden egg everyday. The farmer used to go to the market and sell that egg. It was the source of his livelihood and happy life. His earning from the golden egg began to grow. Now he was living a good life. But as he grew rich, his greed also grew. One day he thought that their may be some treasure of gold in the hen's stomach. If he got all the gold, he would be richer. So he decided to get it one day. He cut out the hen's stomach. But he found nothing inside the stomach except some blood and flesh. The hen was dead. Then the farmer realized what he had done. But now he was helpless. He himself killed the source of his all happiness due to his greed. Q.2: Put each of the following in the correct order. Then use them appropriately to fill the blanks in the paragraph that follows. Use correct punctuation marks. English and Hindi/both/in/he writes and only/a few short stories/many books in English/ in Hindi is/my Hindi/than my English/much better Ravi Kant is a writer, and _________________________________. Of course, he is much happier writing in English than in Hindi. He has written ________________________________. I find his books a little hard to understand ________________________________. Ans : 1. he writes both in English and Hindi 2 many books in English and only few short stories in Hindi 3. My Hindi is much better than my English Q.3: Are you fond of reading stories? Did you read one last month? If not, read one or two and then write a paragraph about the story. Use the following hints. title of the story name of author how many characters • which one you liked some details of the story main point(s) as you understand it Tell your friends why they should also read it. Ans : Last week I read a story from Panchtantra. It was a simple story of only two characters. In the story there was a hermit. He was very kind and pious. He devoted all his time in his prayer. One day he was having his bath in the river. Then he began to offer his prayer to the sun god. While he was praying he saw an ant drowning in the water. The hermit thought to save it. The ant bit in the palm of the hermit. It was a painful bite. But the hermit again tried to save it. Again and again it tried to save it. But the ant bit him repeatedly. A man was watching all this from a distance. Finally he saw that the hermit brought the ant out of water. The man was surprised at the patience of the hermit. He asked the hermit why he did so. The hermit replied that it is the nature of the ant to bite any one. But a hermit's nature is to do welfare to all. So he saved the ant. One can't change one's nature. It is really a nice story. All should read it. Q.1: Discuss these questions in small groups before you answer them. (i) When is a grown-up likely to say this? Don’t talk with your mouth full. (ii) When are you likely to be told this? Say thank you. (iii) When do you think an adult would say this? No one thinks you are funny. Ans : 1. The grown-ups are likely to tell the children not to talk while their mouth is full of food. 2 The children are likely to be reminded to Say thank you when they receive a gift or a favour from someone. 3. Adults are likely to tell children, 'no one thinks you are funny' when the children are too shy to speak or perform before the others. Q.2: The last two lines of the poem are not prohibitions or instructions. What is the adult now asking the child to do? Do you think the poet is suggesting that this is unreasonable? Why? Ans : The adult is now asking the child to think independently. The poet finds this entirely unreasonable because the young child has not been trained to use his mind. He has only been trained to follow the instructions given by the adults. Q.3: Why do you think grown-ups say the kind of things mentioned in the poem? Is it important that they teach children good manners, and how to behave in public? Ans : The adults constantly give instructions to their children for various reasons. They try to train them to behave in a decent, well-mannered and Sophisticated way. This, however, robs away their childlike innocence. Q.4: If you had to make some rules for grown-ups to follow, what would you say? Make at least five such rules. Arrange the lines as in a poem. Ans : Don't dictate and impose your rules. Your ways and thinking is different from ours and so are the times. Don't talk over the phone while eating. Please spend some time with us. Did you find NCERT Solutions Class 7 English 1 chapter 4 The Ashes That Made Trees Bloom helpful? If yes, please comment below. Also please like, and share it with your friends! ### NCERT Solutions Class 7 English 1 chapter 4 The Ashes That Made Trees Bloom- Video You can also watch the video solutions of NCERT Class7 English 1 chapter 4 The Ashes That Made Trees Bloom here. Video – will be available soon. If you liked the video, please subscribe to our YouTube channel so that you can get more such interesting and useful study resources. ### Download NCERT Solutions Class 7 English 1 chapter 4 The Ashes That Made Trees Bloom In PDF Format You can also download here the NCERT Solutions Class 7 English 1 chapter 4 The Ashes That Made Trees Bloom in PDF format.	2019-07-20 01:30:54	{"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": 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": 0.2720126211643219, "perplexity": 3369.4901752593705}, "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/1563195526401.41/warc/CC-MAIN-20190720004131-20190720030131-00025.warc.gz"}
https://www.wptricks.com/question/php-hide-categories-that-are-not-used-in-the-post-type/	## php – Hide categories that are not used in the post type Question I have a situation where I need to display categories on the Archive Page of CPTs, but if the category is not used anywhere in the CPT, it needs to be excluded from the list. For example, I have a CPT for “Books” and “Movies”. I created a post inside the Books CPT and attached it to the “Horror” category. I also created another post in the “Books” CPT and removed all the categories from it. Now, when I add the below code to the archive page, it will show the “Horror” category in the “Books” Archive page as well as on the “Movies” Archive page. <?PHP $categories = get_categories( array( 'orderby' => 'name', 'order' => 'ASC' ) ); foreach ($categories as $category) { echo '<li class="filter-tab-list archive__filterizr-btn" data-filter="' .$category->term_id . '"> ' . \$category->name . ' </li>'; } ?> Is there any way I can exclude the empty category from the “Movies” archive page?	2023-01-31 09:58:28	{"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": 0.7079323530197144, "perplexity": 1495.0900747265914}, "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/1674764499857.57/warc/CC-MAIN-20230131091122-20230131121122-00509.warc.gz"}
https://developer.mozilla.org/en-US/docs/Web/CSS/transform-function/scaleY	scaleY() The `scaleY()` CSS function modifies the ordinate of each element point by a constant factor except if this scale factor is `1`, in which case the function is the identity transform. The scaling is not isotropic and the angles of the element are not conserved. `scaleY(sy)` is a shorthand for `scale(1, sy)` or for `scale3d(1, sy, 1)`. `scaleY(-1)` defines an axial symmetry with a horizontal axis passing by the origin (as specified by the `transform-origin` property). Syntax ```scaleY(s) ``` Values s Is a `<number>` representing the scaling factor to apply on the ordinate of each point of the element. Cartesian coordinates on ℝ2 Homogeneous coordinates on ℝℙ2 Cartesian coordinates on ℝ3 Homogeneous coordinates on ℝℙ3 $\left(\begin{array}{c}10\\ 0& s\end{array}\right)$ $\left(\begin{array}{cc}10& 0\\ 0s& 0\\ 0& 0& 1\end{array}\right)$ $\left(\begin{array}{cc}10& 0\\ 0s& 0\\ 0& 0& 1\end{array}\right)$ $\left(\begin{array}{ccc}10& 0& 0\\ 0s& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$ `[1 0 0 s 0 0]` Examples HTML ```<p>foo</p> <p class="transformed">bar</p>``` CSS ```p { width: 50px; height: 50px; background-color: teal; } .transformed { transform: scaleY(2); background-color: blue; } ``` Document Tags and Contributors Tags: Contributors to this page: Sebastianz, moali1., SphinxKnight Last updated by: Sebastianz,	2016-10-26 12:02:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 4, "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": 0.7088828086853027, "perplexity": 1647.7998629246579}, "config": {"markdown_headings": false, "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-2016-44/segments/1476988720941.32/warc/CC-MAIN-20161020183840-00435-ip-10-171-6-4.ec2.internal.warc.gz"}
http://mathhelpforum.com/calculus/114601-sea-turtles.html	# Math Help - Sea turtles!!!! 1. ## Sea turtles!!!! Suppose a species of sea turtles is confined to an island and has a current population of 70 turtles. If the population, N, is modeled by N=70(5+2t)/(5+.04t) , t≥0 where t is the time in years, how many turtles are predicated to be on the island in 38 years? ok well i thought this was pretty simple, but it seemed that i was wrong. i plugged 38 into the formula like so N=70(5+2(38)/(5+.04(38) and got 1520.4 which did not match the answer in the back of the book. what did i do wrong??? 2. $N=\frac{70(5+2t)}{5+0.04t}$ $t \geq 0$ We want $N$ when $t = 38$ : $N=\frac{70(5+2 \times 38)}{5+0.04 \times 38}$ $N=\frac{70 \times 81}{6,52}$ $N=\frac{5670}{6,52}$ $N \approx 869,6$ (2 dp) Is that right ? What was the answer in the book anyway, knowing it should help ... 3. Originally Posted by Bacterius $N=\frac{70(5+2t)}{5+0.04t}$ $t \geq 0$ We want $N$ when $t = 38$ : $N=\frac{70(5+2 \times 38)}{5+0.04 \times 38}$ $N=\frac{70 \times 81}{6,52}$ $N=\frac{5670}{6,52}$ $N \approx 869,6$ (2 dp) Is that right ? What was the answer in the book anyway, knowing it should help ... yes that is the answer in the back. but i wonder what i did wrong?? i guess i was the problem. thank you. 4. Originally Posted by VNVeteran yes that is the answer in the back. but i wonder what i did wrong?? i guess i was the problem. thank you. I believe you did a calculation error, maybe because of misplaced brackets. It happens sometimes.	2016-02-08 11:01:06	{"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": 16, "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": 0.6386441588401794, "perplexity": 781.5425829840143}, "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-2016-07/segments/1454701152987.97/warc/CC-MAIN-20160205193912-00134-ip-10-236-182-209.ec2.internal.warc.gz"}
http://physics.stackexchange.com/questions/51375/why-is-the-spring-constant-w-p0/51377	# Why is the spring constant $W_p''(0)$? According to my physics book, the spring constant can be calculated from knowing the potential energy, with the formula $k = W_p''(0)$. I don't really understand why, and the book doesn't explain it any further. How do I know the two are equal? - The key concept is small oscilation. It is very easy to see this via Lagrangian Formalism, so we write the Lagrangian for our problem $$L(x,v)=\frac{1}{2}mv^2-W(x)$$ We expand the potential in series about the equilibrium point $x_0$ so that $x-x_0$ is a small oscillation in some way, say $\displaystyle\frac{x-x0}{x}<<1$ $$W(x)=W(x_0)+W'(x_0)(x-x_0)+\frac{1}{2}W''(x-x_0)^2 +O((x-x_0)^3)$$ Why truncate the series at $(x-x_0)^3$. Because $(x-x_0)^2$ gives the first non trivial contribution: $W(x_0)$ is a constant, and it does not contribute to the equations of motion, because the motion is governed by the derivatives of the Lagrangian. On the other hand $W'(x_0)=0$ by hypothesis so we can write $$W(x)\approx \frac{1}{2}W''(x_0)(x-x_0)^2$$ and hence $$L(x,v)=\frac{1}{2}mv^2-\frac{1}{2}W''(x_0)(x-x_0)^2$$ this is the Lagrangian for a spring with constant $W''(x_0)$. Remember the notation meaning, $W''(x_0)$ it is a number, the second derivative evaluated at $x_0$, it is not a function. - The oscillator potential is the one quadratic in the degree of freedom, let's call it $x$. If you expand a potential energy term $W$ via Taylor series, you get $$W(x)=W(0)+W'(0)\cdot x+\tfrac{1}{2}W''(0)\cdot x^2+\ \dots$$ and here you can identify $W''(0)$ as the spring constant $k$. Remark: The first term is just a constant energy value and the second one must be taken zero, assuming $x=0$ is a stable position for your system. So effectvely $\bar W(x)=\tfrac{k}{2}\cdot x^2+O(x^3)$. -	2015-05-06 08:32:19	{"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": 0.9649541974067688, "perplexity": 139.59777093229826}, "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-2015-18/segments/1430458431586.52/warc/CC-MAIN-20150501053351-00048-ip-10-235-10-82.ec2.internal.warc.gz"}
https://solvedlib.com/n/inb-prator-nota-043nst02e02ganina-wauadaskeu-konsissicnnaasencarcertdin,15459702	# Inb* Prator Nota 043nst02E02Ganina WauaDaskeu? KonsissIcnnaasencarcertdin ccede temTorercereTour lontarteanoniduaiWgevd?CreaieConeideKlhcllFher Wnevdiquarothen lireups with * 4s [omard ] WevdNon suppote the rorter ##### Cu-Au alloy of"X 0.85 SH And H in an environment of gases ambient temperature. Oo Lzs when the alloy Cu and HS 2Cu + H,S = Cu,s + H pure reaction Sno Pxz creates in equilibrium 2,98x10-5 Pxzs when detected as 2Cu+HS-Cu S+H standard free energy change of the reaction depending on the temperature 4G_ = 52593 + 26.02TlogT + 70,8ST a) Calculate the activity of copper in the alloy: b) in the same alloy 527 Do Au activity 0.12 As measured as, calculate the mixture free energy of the alloy Cu-Au alloy of "X 0.85 SH And H in an environment of gases ambient temperature. Oo Lzs when the alloy Cu and HS 2Cu + H,S = Cu,s + H pure reaction Sno Pxz creates in equilibrium 2,98x10-5 Pxzs when detected as 2Cu+HS-Cu S+H standard free energy change of the reaction depending on the temperatur... ##### PART II: Given only these promises, supply the conclusion that will make the argument deductively valid.... PART II: Given only these promises, supply the conclusion that will make the argument deductively valid. No plants are mammals A pasque flower is a plant. Therefore 2. All men are mortal. Socrates is a man. Therefore, If animals are sentient, then animals deserve moral consideration. If animals dese... ##### 6. Ordered: Morphine gr 1/6 IM Q4h prn severe pain. Available: Morphine gr 74 per 0.5mL... 6. Ordered: Morphine gr 1/6 IM Q4h prn severe pain. Available: Morphine gr 74 per 0.5mL How much will you give the patient? - 0.3 ml 0.33 undes I ne... ##### Q2 (6 points)Find the volume of the solid that is enclosed by the cone 2 = Vz? Fry2 and the sphere 22 + y? + 22 = 4by using Spherical coordinates. Q2 (6 points) Find the volume of the solid that is enclosed by the cone 2 = Vz? Fry2 and the sphere 22 + y? + 22 = 4by using Spherical coordinates.... ##### The managerial decision-making process has which of the following as its third step?A. Review, analyze and evaluate the results of the decision.B. Decide, based upon the analysis, the best course of action.C. Identify alternative courses of action to achieve a goal or solve a problem.D. Perform a comprehensive differential (differential) analysis of potential solutions. The managerial decision-making process has which of the following as its third step? A. Review, analyze and evaluate the results of the decision. B. Decide, based upon the analysis, the best course of action. C. Identify alternative courses of action to achieve a goal or solve a problem. D. Perf... ##### Question 15 2 pts Which of the following is true about Enuresis? It is usually outgrown... Question 15 2 pts Which of the following is true about Enuresis? It is usually outgrown Its a sign that a child is angry A sign of poor parenting more common among girls Question 16 2 pts Piaget used the three mountain test to study in children. Egocentrism Morality Theory of Mind Confusion in menta... ##### 0 FeCalculate the thcorctical yicld of iron(III) oxide (FezOz)thcorctical yieldThe rcaction produccs 88 g ot Fcz 0 What is the percemt yield of the rcaction?percent yield 0 Fe Calculate the thcorctical yicld of iron(III) oxide (FezOz) thcorctical yield The rcaction produccs 88 g ot Fcz 0 What is the percemt yield of the rcaction? percent yield... ##### Prison Education In a federal prison,inmates can select to complete high school, take college courses,or do neither. The following survey results were obtained usingages of the inmates.AgeHigh School CoursesCollege CoursesNeitherUnder 305410845630 and Over2937367Choose a prisoner at random. Find the following probabilities.Round the answers to at least three decimal places.Part 1 of 3The probability that the prisonerdoes not take classes is Prison Education In a federal prison, inmates can select to complete high school, take college courses, or do neither. The following survey results were obtained using ages of the inmates. Age High School Courses College Courses Neither Under 30 54 108 456 30 and Over 29 37 367 Choose a prisoner a... ##### Evaluate 3 following inegrals3u12 24~/c dxdzdy Evaluate 3 following inegrals 3u12 24 ~/c dxdzdy... ##### Q.3- Define "Graetz number for ciru lar tube with constant wall temperature, At what "okie' of... Q.3- Define "Graetz number for ciru lar tube with constant wall temperature, At what "okie' of Groetz number we can assume the flow site the tube be fully-developed. What is the Nusself number (a constant) in this case when the flow is fully-devolused: (tve the Fepure in the test book.) ... ##### Draw the structure of the cyclic compound that is produced when acetone treated with 1,3-propanedithiol in the presence of an acid catalyst:HS SH 1,3-propanedithiolAcetoneEdit~s H;C FCH; Draw the structure of the cyclic compound that is produced when acetone treated with 1,3-propanedithiol in the presence of an acid catalyst: HS SH 1,3-propanedithiol Acetone Edit ~s H;C FCH;... ##### Dixon Development began operations in December 2021. When lots for industrial development are sold, Dixon recognizes... Dixon Development began operations in December 2021. When lots for industrial development are sold, Dixon recognizes income for financial reporting purposes in the year of the sale. For some lots, Dixon recognizes income for tax purposes when collected. Income recognized for financial reporting purp... ##### Consider the following initial value problem, in which an input of large amplitude and short duration... Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function: x' + x = 9+5(t – 5), x(0) = 0. In the following parts, use h(t – c) for the Heaviside function he(t) if necessary. a. Find the Laplace transfo... ##### (bl;) &kij 114 JlgulThe interval of convergence of the power senes solution of t"y" + ty' +y = Ocentered at xo = Lis (0.1) (23) (0.2) (-1,1)ABD (bl;) &kij 1 14 Jlgul The interval of convergence of the power senes solution of t"y" + ty' +y = Ocentered at xo = Lis (0.1) (23) (0.2) (-1,1) A B D...	2022-05-22 20:24:59	{"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": 0.6240829229354858, "perplexity": 8344.9694421606}, "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-2022-21/segments/1652662546071.13/warc/CC-MAIN-20220522190453-20220522220453-00179.warc.gz"}
https://proofwiki.org/wiki/Parity_Function_is_Homomorphism	Parity Function is Homomorphism Theorem Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi \in S_n$. Let $\map \sgn \pi$ be the sign of $\pi$. Let the parity function of $\pi$ be defined as: Parity of $\pi = \begin{cases} \mathrm {Even} & : \map \sgn \pi = 1 \\ \mathrm {Odd} & : \map \sgn \pi = -1 \end{cases}$ The mapping $\sgn: S_n \to C_2$, where $C_2$ is the cyclic group of order 2, is a homomorphism. Proof We need to show that: $\forall \pi, \rho \in S_n: \map \sgn \pi \, \map \sgn \rho = \map \sgn {\pi \rho}$ Let $\Delta_n$ be an arbitrary product of differences. $\ds \map \sgn {\pi \rho} \Delta_n$ $=$ $\ds \pi \rho \cdot \Delta_n$ Definition of Sign of Permutation $\ds$ $=$ $\ds \pi \cdot \paren {\rho \cdot \Delta_n}$ Permutation on Polynomial is Group Action $\ds$ $=$ $\ds \pi \cdot \paren {\map \sgn \rho \cdot \Delta_n}$ Definition of Sign of Permutation $\ds$ $=$ $\ds \pi \, \map \sgn \rho \cdot \Delta_n$ Permutation on Polynomial is Group Action $\ds$ $=$ $\ds \map \sgn \pi \, \map \sgn \rho \Delta_n$ Definition of Sign of Permutation As $\struct {\set {1, -1}, \times}$ is the parity group, the result follows immediately. $\blacksquare$	2021-12-08 09:55:04	{"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": 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": 0.9923983216285706, "perplexity": 275.5540280064359}, "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-2021-49/segments/1637964363465.47/warc/CC-MAIN-20211208083545-20211208113545-00410.warc.gz"}
https://demo.formulasearchengine.com/wiki/Conditional_sentence	# Conditional sentence {{#invoke:Hatnote\|hatnote}} Conditional sentences are sentences expressing factual implications, or hypothetical situations and their consequences. They are so called because the validity of the main clause of the sentence is conditional on the existence of certain circumstances, which may be expressed in a dependent clause or may be understood from the context. A full conditional sentence (one which expresses the condition as well as its consequences) therefore contains two clauses: the dependent clause expressing the condition, called the protasis; and the main clause expressing the consequence, called the apodosis.[1] An example of such a sentence (in English) is the following: If it rains, the picnic will be cancelled. Here the condition is expressed by the clause "If it rains", this being the protasis, while the consequence is expressed by "the picnic will be cancelled", this being the apodosis. (The protasis may either precede or follow the apodosis; it is equally possible to say "The picnic will be cancelled if it rains".) In terms of logic, the protasis corresponds to the antecedent, and the apodosis to the consequent. Languages use a variety of grammatical forms and constructions in conditional sentences. The forms of verbs used in the protasis and apodosis are often subject to particular rules as regards their tense and mood. Many languages have a specialized type of verb form called the conditional mood – broadly equivalent in meaning to the English "would (do something)" – for use in some types of conditional sentence. ## Types of conditional sentence There are various ways of classifying conditional sentences. One distinction is between those that state an implication between facts, and those that set up and refer to a hypothetical situation. There is also the distinction between conditionals that are considered factual or predictive, and those that are considered counterfactual or speculative (referring to a situation that did not or does not really exist). ### Implicative and predictive A conditional sentence expressing an implication (also called a factual conditional sentence) essentially states that if one fact holds, then so does another. (If the sentence is not a declarative sentence, then the consequence may be expressed as an order or a question rather than a statement.) The facts are usually stated in whatever grammatical tense is appropriate to them; there are not normally special tense or mood patterns for this type of conditional sentence. Such sentences may be used to express a certainty, a universal statement, a law of science, etc. (in these cases if may often be replaced by when): If you heat water to 100 degrees, it boils. If the sea is stormy, the waves are high. They can also be used for logical deductions about particular circumstances (which can be in various mixtures of past, present and future): If it's raining here now, then it was raining on the West Coast this morning. If it's raining now, then your laundry is getting wet. If it's raining now, there will be mushrooms to be picked next week. If he locked the door, then Kitty is trapped inside. A predictive conditional sentence concerns a situation dependent on a hypothetical (but entirely possible) future event. The consequence is normally also a statement about the future, although it may also be a consequent statement about present or past time (or a question or order). If I become President, I'll lower taxes. If it rains this afternoon, everybody will stay home. If it rains this afternoon, then yesterday's weather forecast was wrong. If it rains this afternoon, your garden party is doomed. What will you do if he invites you? If you see them, shoot! ### Counterfactual {{#invoke:main\|main}} In a counterfactual or speculative[2] conditional sentence, a situation is described as dependent on a condition that is known to be false, or presented as unlikely. The time frame of the hypothetical situation may be past, present or future, and the time frame of the condition does not always correspond to that of the consequence. For example: If I were king, I could have you thrown in the dungeon. If I won the lottery, I would buy a car. If he said that to me, I would run away. If you had called me, I would have come. If you had done your job properly, we wouldn't be in this mess now. The difference in meaning between a "counterfactual" conditional with a future time frame, and a "predictive" conditional as described in the previous section, may be slight. For example, there is no great practical difference in meaning between "If it rained tomorrow, I would cancel the match" and "If it rains tomorrow, I will cancel the match". It is in the counterfactual type of conditional sentence that the grammatical form called the conditional mood (meaning something like the English "would ...") is most often found. For the uses of particular verb forms and grammatical structures in the various types and parts of conditional sentences in certain languages, see the following sections. ## Grammar of conditional sentences Languages have different rules concerning the grammatical structure of conditional sentences. These may concern the syntactic structure of the condition clause (protasis) and consequence (apodosis), as well as the forms of verbs used in them (particularly their tense and mood). Rules for English and certain other languages are described below; more information can be found in the articles on the grammars of individual languages. (Some languages are also described in the article on the conditional mood.) ### English {{#invoke:main\|main}} In English conditional sentences, the condition clause (protasis) is most commonly introduced by the conjunction if, or sometimes other conjunctions or expressions such as unless, provided (that), providing (that) and as long as. Certain condition clauses can also be formulated using inversion without any conjunction (should you fail...; were he to die...; had they helped us...). In English language teaching, conditional sentences are often classified under the headings zero conditional, first conditional (or conditional I), second conditional (or conditional II), third conditional (or conditional III) and mixed conditional, according to the grammatical pattern followed.[3] A range of variations on these structures are possible. #### Zero conditional "Zero conditional" refers to conditional sentences that express a simple implication (see above section), particularly when both clauses are in the present tense: If you don't eat for a long time, you become hungry. This form of the conditional expresses the idea that a universally known fact is being described: If you touch a flame, you burn yourself. The act of burning oneself only happens on the condition of the first clause being completed. However such sentences can be formulated with a variety of tenses (and moods), as appropriate to the situation. #### First conditional "First conditional" refers to predictive conditional sentences (see above section); here, normally, the condition is expressed using the present tense and the consequence using the future: If you make a mistake, someone will let you know. #### Second conditional "Second conditional" refers to the pattern where the condition clause is in the past tense, and the consequence in conditional mood (using would or, in the first person and rarely, should). This is used for hypothetical, counterfactual situations in a present or future time frame (where the condition expressed is known to be false or is presented as unlikely). If I liked parties, I would attend more of them. If it were to rain tomorrow, I would dance in the street. The past tense used in the condition clause is historically the past subjunctive; however in modern English this is identical to the past indicative except in certain dialects in the case of the verb be (first and third person singular), where the indicative is was and the subjunctive were. In this case either form may be used (was is more colloquial, and were more formal, although the phrase if I were you is common in colloquial language too): If I (he, she, it) was/were rich, there would be plenty of money available for this project. #### Third conditional "Third conditional" is the pattern where the condition clause is in the past perfect, and the consequence is expressed using the conditional perfect. This is used to refer to hypothetical, counterfactual (or believed likely to be counterfactual) situations in the past If you had called me, I would have come. #### Mixed conditionals "Mixed conditional" usually refers to a mixture of the second and third conditionals (the counterfactual patterns). Here either the condition or the consequence, but not both, has a past time reference: If you had done your job properly, we wouldn't be in this mess now. If we were soldiers, we wouldn't have done it like that. ### Latin Conditional sentences in Latin are traditionally classified into three categories, based on grammatical structure. • simple conditions (factual or logical implications) • present tense [if present indicative then indicative] • past tense [if perfect/imperfect indicative then indicative] • future conditions • "future more vivid" [if future indicative then future indicative] • "future less vivid" [if present subjunctive then present subjunctive] • contrafactual conditions • "present contrary-to-fact" [if imperfect subjunctive then imperfect subjunctive] • "past contrary-to-fact" [if pluperfect subjunctive then pluperfect subjunctive] ### French In French, the conjunction corresponding to "if" is si. The use of tenses is quite similar to English: • In implicative conditional sentences, the present tense (or other appropriate tense, mood, etc.) is used in both clauses. • In predictive conditional sentences, the future tense or imperative generally appears in the main clause, but the condition clause is formed with the present tense (as in English). This contrasts with dependent clauses introduced by certain other conjunctions, such as quand ("when"), where French uses the future (while English has the present). • In counterfactual conditional sentences, the imperfect is used to express the condition (where English similarly uses the past tense). The main clause contains the conditional mood (e.g. j'arriverais, "I would arrive"). • In counterfactual conditional sentences with a past time frame, the condition is expressed using the pluperfect e.g. (s'il avait attendu, "if he had waited"), and the consequence with the conditional perfect (e.g. je l'aurais vu, "I would have seen him"). Again these verb forms parallel those used in English. As in English, certain mixtures and variations of these patterns are possible. See also French verbs. ### Italian Italian uses the following patterns (the equivalent of "if" is se): • Present tense (or other as appropriate) in both parts of an implicative conditional. • Future tense in both parts of a predictive conditional sentence (the future is not replaced with the present in condition clauses as in English or French). • In a counterfactual conditional, the imperfect subjunctive is used for the condition, and the conditional mood for the main clause. A more informal equivalent is to use the imperfect indicative in both parts. • In a counterfactual conditional with past time frame, the pluperfect subjunctive is used for the condition, and the past conditional (conditional perfect) for the main clause. ### Slavic languages In Slavic languages, such as Russian, clauses in conditional sentences generally appear in their natural tense (future tense for future reference, etc.) However, for counterfactuals, a conditional/subjunctive marker such as the Russian бы by generally appears in both condition and consequent clauses, and this normally accompanies the past tense form of the verb. See Russian grammar, Bulgarian grammar, etc. for more detail. ## Logic While the material conditional operator used in logic (i.e.${\displaystyle \scriptstyle p\Rightarrow q}$) is sometimes read aloud in the form of a conditional sentence (i.e. "if p, then q"), the intuitive interpretation of conditional statements in natural language does not always correspond to the definition of this mathematical relation. Modelling the meaning of real conditional statements requires the definition of an indicative conditional, and contrary-to-fact statements require a counterfactual conditional operator, formalized in modal logic.	2020-05-29 12:49:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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": 0.5147405862808228, "perplexity": 2435.6063115567795}, "config": {"markdown_headings": true, "markdown_code": false, "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/1590347404857.23/warc/CC-MAIN-20200529121120-20200529151120-00244.warc.gz"}
http://devdriven.com/category/programming-languages/ruby-programming-languages/	# Ruby #each .vs. #inject in the Presence of Side-Effects Note the difference between: numbers.inject(0) {\| sum, x \| sum + x } .vs. sum = 0 numbers.each {\| x \| sum += x } sum … the former is obviously preferred. However: If the #inject result value is mutated and the result value is used elsewhere (or implicitly returned), use a local variable with #each, instead of #inject. # Discrete Event Probablities My colleague, David, had an interesting problem: “I want to simulate exceptional events (e.g. I/O errors) at predictable rates. For example, if an error should occur 33% of the time, it should be predictable like: ERROR, ok, ok, ERROR, ok, ok, ... Or at 50% error rate: ERROR, ok, ERROR, ok, ... devdriven: “Pseudo-random function p(t) in range [0,1) at time t and P is the probability of the event (e.g. I/O error) then: event?(t) = p(t) < P. Pseudo-random numbers are predictable if you know the seed." David: “Not really what I had in mind. There’s got to be a better way…” # Recovering From the Unrecoverable In some programming languages, asynchronous events can occur at any time. For instance, in Ruby, there are subclasses of Exception that can be raised at any time — there are few lines of code safe from interruption. Some of exceptions, due to their cause, are not recoverable at all. # Ruby: REE RUBY_HEAP allocation parameter and page alignment improvements. Use mmap() to allocate heaps by default. Use mmap() MAP_ANON, instead of /dev/zero. Align mmap() size to pagesize. Align heap allocation size to pagesize. Expand heap slotlimit to fit in aligned allocation. New \$RUBY_HEAP_* options: RUBY_HEAP_INIT_SLOTS=N initial number of slots per heap. RUBY_HEAP_MIN_SLOTS=N value is independent of RUBY_HEAP_INIT_SLOTS. RUBY_HEAP_MAX_SLOTS=N max number slots for a heap. RUBY_HEAP_PAGESIZE=bytes defaults to PAGESIZE or 4096. RUBY_HEAP_SLOTS_INCREMENT=N allow 0. Refactor set_gc_parameters(). https://github.com/kstephens/ruby-enterprise-1.8.7-2011.03/tree/master-gc_tuning_parameter_improvements https://github.com/kstephens/ruby-enterprise-1.8.7-2011.03/commit/692fd3c6a1d11bc6b4caeb5e167c35de7bd22d11 # Ruby: Thread stack leak patch accepted into REE. This patch reduces the stack buffer memory footprint of dead Threads as early as possible, rather than waiting until the Thread can be GCed. This is applicable only to the zero-copy context switch patch. http://blog.phusion.nl/2011/02/12/ruby-enterprise-edition-1-8-7-2011-01-released/ # ChicagoRuby Ruby Code Tweaks slides, code and video The slides from my ChicagoRuby 2010/05/04 presentation : http://kurtstephens.com/pub/ruby/ruby_code_tweaks/ All the raw data used to generate the graph should be referenced in the slides. The code used to generate the slides is here: http://github.com/kstephens/ruby_code_tweaks I’m looking to increase the set of code “Problems” to cover other tiny code idioms and platform issues, for example: regular expressions, numerics, etc. If you have ideas, take a look at the code and contact me. Justin Love gave a fantastic presentation on lambda and closure. Thanks to everyone who came — hope it was helpful. Video from the talk: # Ruby 1.8: Improved Rational performance by 15% This should also speed up DateTime. This will not help 1.9 performance. The attached file is based on MRI 1.8.6 rational.rb. > ruby rational_performance.rb user system total real test_it 32.930000 3.030000 35.960000 ( 35.971832) test_it 33.840000 2.910000 36.750000 ( 36.758585) test_it ks_rational 29.110000 2.460000 31.570000 ( 31.572762) Overview: • case x; when Foo; ...; end is faster than if Foo.kind_of?(x). • Avoid recuring on ephemeral objects. • Avoid local variables, in-line expressions. • Avoid return in tail position. • String interpolation: "#{x}/#{y}" is faster than x.to_s + "/" + y.to_s. • Implement #-, #zero?, #nonzero? natively. • #abs returns self if > 0. MRI 1.8.7 patch to follow shortly.	2017-02-26 23:34:25	{"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": 0.2680477499961853, "perplexity": 9493.215768128137}, "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-2017-09/segments/1487501172156.69/warc/CC-MAIN-20170219104612-00486-ip-10-171-10-108.ec2.internal.warc.gz"}
http://www.lmfdb.org/GaloisGroup/38T5	# Properties Label 38T5 Order $$114$$ n $$38$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $D_{19}:C_3$ ## Group action invariants Degree $n$ : $38$ Transitive number $t$ : $5$ Group : $D_{19}:C_3$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,34,29)(2,33,30)(3,9,13)(4,10,14)(5,23,35)(6,24,36)(7,38,19)(8,37,20)(11,27,25)(12,28,26)(15,18,32)(16,17,31), (1,3,27,12,10,24)(2,4,28,11,9,23)(5,13,34,8,38,17)(6,14,33,7,37,18)(15,20,29,36,32,21)(16,19,30,35,31,22)(25,26) $\|\Aut(F/K)\|$: $2$ ## Low degree resolvents \|G/N\|Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $C_6$ Resolvents shown for degrees $\leq 47$ ## Subfields Degree 2: $C_2$ Degree 19: $C_{19}:C_{6}$ ## Low degree siblings 19T4 Siblings are shown with degree $\leq 47$ A number field with this Galois group has no arithmetically equivalent fields. ## Conjugacy classes Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1$ $19$ $3$ $( 3,16,24)( 4,15,23)( 5,29, 7)( 6,30, 8)( 9,20,13)(10,19,14)(11,34,35) (12,33,36)(17,37,26)(18,38,25)(21,28,31)(22,27,32)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1$ $19$ $3$ $( 3,24,16)( 4,23,15)( 5, 7,29)( 6, 8,30)( 9,13,20)(10,14,19)(11,35,34) (12,36,33)(17,26,37)(18,25,38)(21,31,28)(22,32,27)$ $6, 6, 6, 6, 6, 6, 2$ $19$ $6$ $( 1, 2)( 3,18,16,38,24,25)( 4,17,15,37,23,26)( 5,33,29,36, 7,12) ( 6,34,30,35, 8,11)( 9,27,20,32,13,22)(10,28,19,31,14,21)$ $6, 6, 6, 6, 6, 6, 2$ $19$ $6$ $( 1, 2)( 3,25,24,38,16,18)( 4,26,23,37,15,17)( 5,12, 7,36,29,33) ( 6,11, 8,35,30,34)( 9,22,13,32,20,27)(10,21,14,31,19,28)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $19$ $2$ $( 1, 2)( 3,38)( 4,37)( 5,36)( 6,35)( 7,33)( 8,34)( 9,32)(10,31)(11,30)(12,29) (13,27)(14,28)(15,26)(16,25)(17,23)(18,24)(19,21)(20,22)$ $19, 19$ $6$ $19$ $( 1, 4, 5, 7,10,11,14,15,18,19,22,23,25,27,29,32,34,35,38)( 2, 3, 6, 8, 9,12, 13,16,17,20,21,24,26,28,30,31,33,36,37)$ $19, 19$ $6$ $19$ $( 1, 5,10,14,18,22,25,29,34,38, 4, 7,11,15,19,23,27,32,35)( 2, 6, 9,13,17,21, 26,30,33,37, 3, 8,12,16,20,24,28,31,36)$ $19, 19$ $6$ $19$ $( 1,10,18,25,34, 4,11,19,27,35, 5,14,22,29,38, 7,15,23,32)( 2, 9,17,26,33, 3, 12,20,28,36, 6,13,21,30,37, 8,16,24,31)$ ## Group invariants Order: $114=2 \cdot 3 \cdot 19$ Cyclic: No Abelian: No Solvable: Yes GAP id: [114, 1] Character table: 2 1 1 1 1 1 1 . . . 3 1 1 1 1 1 1 . . . 19 1 . . . . . 1 1 1 1a 3a 3b 6a 6b 2a 19a 19b 19c 2P 1a 3b 3a 3a 3b 1a 19b 19c 19a 3P 1a 1a 1a 2a 2a 2a 19b 19c 19a 5P 1a 3b 3a 6b 6a 2a 19b 19c 19a 7P 1a 3a 3b 6a 6b 2a 19a 19b 19c 11P 1a 3b 3a 6b 6a 2a 19a 19b 19c 13P 1a 3a 3b 6a 6b 2a 19c 19a 19b 17P 1a 3b 3a 6b 6a 2a 19b 19c 19a 19P 1a 3a 3b 6a 6b 2a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 -1 -1 -1 1 1 1 X.3 1 A /A -/A -A -1 1 1 1 X.4 1 /A A -A -/A -1 1 1 1 X.5 1 A /A /A A 1 1 1 1 X.6 1 /A A A /A 1 1 1 1 X.7 6 . . . . . B C D X.8 6 . . . . . C D B X.9 6 . . . . . D B C A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(19)^2+E(19)^3+E(19)^5+E(19)^14+E(19)^16+E(19)^17 C = E(19)^4+E(19)^6+E(19)^9+E(19)^10+E(19)^13+E(19)^15 D = E(19)+E(19)^7+E(19)^8+E(19)^11+E(19)^12+E(19)^18	2020-09-28 10:06:52	{"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": 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": 0.5236068367958069, "perplexity": 1237.5127916985136}, "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/1600401598891.71/warc/CC-MAIN-20200928073028-20200928103028-00699.warc.gz"}
https://www.physicsforums.com/threads/taylor-expansion-of-a-scalar-potential-field.852046/	# Taylor expansion of a scalar potential field 1. Jan 13, 2016 ### spaghetti3451 Consider the potential $U(\phi) = \frac{\lambda}{8}(\phi^{2}-a^{2})^{2}-\frac{\epsilon}{2a}(\phi - a)$, where $\phi$ is a scalar field and the mass dimensions of the couplings are: $[\lambda]=0$, $[a]=1$, and $[\epsilon]=4$. Expanding the field $\phi$ about the point $\phi=\phi_{-}$ ($\phi = \phi_{-}+ \varphi$) and keeping terms up to dimension four, we find $U(\varphi)=\frac{m^{2}}{2}\varphi^{2}-\eta\varphi^{3}+\frac{\lambda}{8}\varphi^{4}$, where $m^{2}=\frac{\lambda}{2}(3\phi_{-}^{2}-a^{2})$ and $\eta = \frac{\lambda}{2}\lvert\phi_{-}\lvert$. How do you derive this potential $U(\varphi)$ in explicit steps? Can you provide just the first two lines? I'll work out the rest for myself. 2. Jan 13, 2016 ### vanhees71 What's $\phi_-$? 3. Jan 13, 2016 ### spaghetti3451 $\phi_{-}$ is some fixed value of the potential $\phi$. 4. Jan 13, 2016 ### spaghetti3451 Alright, I have made some progress with the problem. $U(\varphi) = U(\phi)+U'(\phi)(\varphi-\phi)+\frac{U''(\phi)}{2}(\varphi-\phi)^{2}+\frac{U''(\phi)}{6}(\varphi-\phi)^{3} \cdots$ $=\frac{\lambda}{8}(\phi^{2}-a^{2})^{2}-\frac{\epsilon}{2a}(\phi - a)+\frac{1}{2}\bigg(\frac{\lambda}{4}(\phi^{2}-a^{2})(2\phi)-\frac{\epsilon}{2a}\bigg)(-\phi_{-})+\frac{1}{6}\bigg(\frac{3\lambda}{2}\phi^{2}-\frac{\lambda}{2}a^{2}\bigg)(-\phi_{-})^{2}$. Now, I'm not really sure how many terms to keep based on the comment 'keeping terms up to dimension four'. Can you help with that? 5. Jan 14, 2016 ### vanhees71 Well, the expression is of order 4 in the fields. So you just keep all terms, and I still don't get what your manipulations mean. The first line makes sense, because it's the Taylor expansion around $\phi$, but the 2nd line looks totally unrelated to the first. Usually you shift the field such that the linear term vanishes. Most convenient is to expand around a (local) minimum of the potential. 6. Jan 14, 2016 ### spaghetti3451 The paper from which I took this problem mentions 'keeping terms up to dimension four' - why do they keep terms up to dimension four? I think the answer to this question lies in your third comment. $U(\varphi) = U(\phi)+U'(\phi)(\varphi-\phi)+\frac{U''(\phi)}{2}(\varphi-\phi)^{2}+\frac{U''(\phi)}{6}(\varphi-\phi)^{3}+\frac{U'''(\phi)}{24}(\varphi-\phi)^{4} + \cdots$ $=\frac{\lambda}{8}(\phi^{2}-a^{2})^{2}-\frac{\epsilon}{2a}(\phi - a)+\bigg(\frac{\lambda}{4}(\phi^{2}-a^{2})(2\phi)-\frac{\epsilon}{2a}\bigg)(-\phi_{-})+\frac{1}{2}\bigg(\frac{3\lambda}{2}\phi^{2}-\frac{\lambda}{2}a^{2}\bigg)(-\phi_{-})^{2}+\frac{1}{6}\bigg(3\lambda\phi\bigg)(-\phi_{-})^{3}+\frac{1}{24}\bigg(3\lambda\bigg)(-\phi_{-})^{4}$. But, I think this is unnecessary in lieu of your next comment. $U(\varphi)=U(\phi_{-})+U'(\phi_{-})(\varphi-\phi_{-})+\frac{U''(\phi_{-})}{2}(\varphi-\phi_{-})^{2}+\frac{U''(\phi_{-})}{6}(\varphi-\phi_{-})^{3}+\frac{U'''(\phi_{-})}{24}(\varphi-\phi_{-})^{4} + \cdots$ Now, I did not mention it before, but $\phi=\phi_{-}$ is a local minimum, so that $U'(\phi_{-})=0$. Also, keeping terms to dimension four means that we truncate after the second power in $(\varphi-\phi_{-})$. Therefore, $U(\varphi)=U(\phi_{-})+\frac{U''(\phi_{-})}{2}(\varphi-\phi_{-})^{2}$. Am I correct so far? Also, why do we shift the field such that the linear term vanishes? Last edited: Jan 14, 2016 7. Jan 14, 2016 ### vanhees71 because it simplifies the further calculations a lot. In your case, if $U''(\phi_-) > 0$ for small deviations from $\phi_-$ you can treat the problem as a harmonic-oscillator problem, i.e., a linear differential equation.	2017-10-24 01:10:36	{"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": 0.865145742893219, "perplexity": 519.676086202687}, "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-2017-43/segments/1508187827662.87/warc/CC-MAIN-20171023235958-20171024015958-00516.warc.gz"}
https://brilliant.org/problems/basic-trigo-iv/	# Basic trigo - IV Geometry Level 1 If ABCD is a cyclic quadrilateral, find the value of $$\cos { A } +\cos { B } +\cos { C } +\cos { D }$$ ×	2017-05-27 19:36:28	{"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": 0.2612483501434326, "perplexity": 6364.671770165255}, "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/1495463609054.55/warc/CC-MAIN-20170527191102-20170527211102-00480.warc.gz"}
https://ch.mathworks.com/help/physmod/sdl/ref/variabletranslationaldamper.html	# Variable Translational Damper Translational viscous damper with variable damping coefficient ## Library Couplings & Drives/Springs & Dampers ## Description The block represents a translational viscous damper with variable damping coefficient. A physical signal input port provides the magnitude of the translational damping coefficient. The magnitude of the damping force is equal to the product of the physical signal input and the relative linear velocity between the two translational conserving ports. A minimum damping coefficient prevents non-physical negative damping values. The translational damping force satisfies the following expression: `$F=\left\{\begin{array}{cc}B.v,& B\ge {B}_{\mathrm{min}}\\ {B}_{\mathrm{min}}\cdot v,& B<{B}_{\mathrm{min}}\end{array},$` The parameters are: • F — Force transmitted through the translational damper between the two translational conserving ports • B — Viscous damping coefficient • Bmin — Minimum allowed damping coefficient • v — Relative linear velocity measured between the two translational conserving ports • vR — Linear velocity of port R • vC — Linear velocity of port C The block applies equal and opposite damping forces on the two translational conserving ports. The sign of the damping force acting on port R is equal to the sign of the relative linear velocity. A positive linear velocity corresponds to a positive damping force acting on port R, and a negative damping force of equal magnitude acting on port C. ### Assumptions and Limitations • The block represents strictly viscous damping. • The value of the damping coefficient must be greater than or equal to zero. ## Ports `B` Physical signal input port representing the variable translational damping coefficient. `C` Translational conserving port. `R` Translational conserving port. ## Parameters Minimum damping coefficient Minimum value allowed for the variable damping coefficient. The physical signal input saturates below the specified value. The parameter must be greater than or equal to zero. The default value is `0` `N/(m/s)`. ## Version History Introduced in R2013a	2022-08-14 10:07:15	{"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": 0.8637223839759827, "perplexity": 2550.935041339575}, "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-2022-33/segments/1659882572021.17/warc/CC-MAIN-20220814083156-20220814113156-00622.warc.gz"}
https://www.physicsforums.com/threads/using-boiling-temperature-as-a-proxy-for-vapor-pressure.470186/	# Using boiling temperature as a proxy for vapor pressure Gold Member So, we know that isopropanol is more volatile than water. We know that since isopropanol has a higher vapor pressure (40 mmHg at 23.8 degrees Celsius, whereas water is 17.5 mmHg at 20 degrees Celsius). We also know that isopropanol has a lower boiling point than water. Now, are there molecules that have higher vapor pressures than water and higher boiling points than water? The chart at http://upload.wikimedia.org/wikipedia/commons/9/96/Vapor_Pressure_Chart.png shows very few intersections. But I wonder - are there cases where the vapor pressure curves for two molecules do intersect? Also, how do we objectively compare the vapor pressures between two molecules? Do people try to measure the vapor pressure for a specific number of moles of a molecule? (and set the number of moles of N2 equal to the number of moles of H2O, if they want to compare the vapor pressures between the two molecules?) Mapes Homework Helper Gold Member Now, are there molecules that have higher vapor pressures than water and higher boiling points than water? The boiling point is the temperature at which the vapor pressure equals 1 atm, so not simultaneously, no. But I wonder - are there cases where the vapor pressure curves for two molecules do intersect? The vapor pressure $p$ should scale as $$p\propto \exp\left[-L\left(\frac{1}{T}-{\frac{1}{T_\mathrm{B}}\right)\right]$$ where $L$ is the latent heat and $T_\mathrm{B}$ is the boiling temperature. So it would seem that this could occur with two materials if one had the higher latent heat and the other had the higher boiling temperature.	2021-05-06 01:05:54	{"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": 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": 0.6721278429031372, "perplexity": 813.3142571999966}, "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/1620243988724.75/warc/CC-MAIN-20210505234449-20210506024449-00582.warc.gz"}
https://ncatlab.org/homotopytypetheory/show/diff/equivalence	# Homotopy Type Theory equivalence (changes) Showing changes from revision #0 to #1: Added \| Removed \| Changed ## Idea A function $f : A \to B$ is an equivalence if it has inverses whose composition with $f$ is homotopic to the corresponding identity map. ## Definition Let $A,B$ be types, and $f : A \to B$ a function. We define the property of $f$ being an equivalence as follows: $isequiv(f) \equiv \left( \sum_{g : B \to A} f \circ g \sim id_B \right) \times \left( \sum_{h : B \to A} h \circ f \sim id_A \right)$ We define the type of equivalences from $A$ to $B$ as $(A \simeq B) \equiv \sum_{f : A \to B} isequiv(f)$ or, phrased differently, the type of witnesses to $A$ and $B$ being equivalent types.	2022-01-29 04:06:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 11, "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": 0.7331146597862244, "perplexity": 597.2112854852838}, "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/1642320299927.25/warc/CC-MAIN-20220129032406-20220129062406-00357.warc.gz"}
https://www.zigya.com/study/book?class=11&board=bsem&subject=Physics&book=Physics+Part+I&chapter=Motion+in+A+Plane&q_type=&q_topic=Projectile+Motion&q_category=&question_id=PHEN11039748	ï»¿ A particle is projected with velocityÂ Â at such an angle that it just clears twoÂ walls of equal height h which are at a distance 2h from each other. Show that time of passing between the two walls isÂ from Physics Motion in A Plane Class 11 Manipur Board ### Book Store Currently only available for. CBSE Gujarat Board Haryana Board ### Previous Year Papers Download the PDF Question Papers Free for off line practice and view the Solutions online. Currently only available for. Class 10 Class 12 A particle is projected with velocityÂ Â at such an angle that it just clears twoÂ walls of equal height h which are at a distance 2h from each other. Show that time of passing between the two walls isÂ Given,Â Velocity with which the particle is projected = 2Â Height of the wall = h Distance between the walls = 2h Let the particle be projected at angle Î¸. Time taken by particle to cross the two walls each of height 'h' is equal to difference of instants when it would be at a height h above ground. The kinematics of motion of particle in vertical direction is given by,Â That is,Â Â is the time of passing between the two walls.Â 249 Views Give three examples of vector quantities. Force, impulse and momentum. 865 Views What is a vector quantity? A physical quantity that requires direction along with magnitude, for its complete specification is called a vector quantity. 835 Views Give three examples of scalar quantities. Mass, temperature and energy 769 Views What are the basic characteristics that a quantity must possess so that it may be a vector quantity? A quantity must possess the direction and must follow the vector axioms. Any quantity that follows the vector axioms are classified as vectors.Â 814 Views What is a scalar quantity? A physical quantity that requires only magnitude for its complete specification is called a scalar quantity. 1212 Views Do a good deed today Refer a friend to Zigya	2018-06-18 05:43:48	{"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": 0.6061925292015076, "perplexity": 2012.5331873836844}, "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/1529267860089.11/warc/CC-MAIN-20180618051104-20180618070843-00003.warc.gz"}
https://www.betterment.com/resources/retirement/planning-ahead/you-arent-saving-enough-for-retirement/	Only 52 percent of Americans expressed confidence that they will be comfortable in retirement, according to a March 2012 survey conducted by the Employee Benefit Research Institute.¹ Chances are people know they’re not putting enough aside or they simply have no idea how much they will need to continue the life they lead and the one they want for the future. ### How much do you really need to retire? According to the report, 56% of Americans haven’t even performed a retirement needs calculation. Most people haven’t even taken the time to figure out how much they will need to retire. If you haven’t actually sat down and done the math (preferably with the help of a retirement calculator), chances are that you aren’t saving enough for your retirement. ### It always comes back to savings Despite our constant speculation over market movements—in the end —one of the biggest predictors of whether or not you will have enough for retirement is how much you save. It’s that simple! Even if you never get around to doing your own calculations on how much you should save for retirement, at least do this: always save 20% of your income and invest it in a broadly diversified mix of stocks and bonds appropriate for your time horizon. Recognize the reality: You probably aren’t saving enough for retirement and look for ways to boost your retirement account contributions. You’ll be in a much better place when you retire. ### Try our Retirement Savings Calculator You can see what you are on track for with your current savings and adjust advanced settings like your retirement zip code and the age when you elect to receive social security. ¹http://www.ebri.org/pdf/surveys/rcs/2012/fs-01-rcs-12-fs1-conf.pdf More from Betterment:	2017-09-25 11:44:18	{"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": 0.1815258264541626, "perplexity": 1229.9197724178728}, "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-2017-39/segments/1505818691476.47/warc/CC-MAIN-20170925111643-20170925131643-00638.warc.gz"}
https://www.enotes.com/homework-help/how-write-cos-2x-terms-cos-x-240109	# how to write cos 2x in terms of cos x? The most straightforward way to obtain the expression for cos(2x) is by using the "cosine of the sum" formula: cos(x + y) = cosxcosy - sinxsiny. To get cos(2x), write 2x = x + x. Then, cos(2x) = cos(x + x ) =... Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime. The most straightforward way to obtain the expression for cos(2x) is by using the "cosine of the sum" formula: cos(x + y) = cosxcosy - sinxsiny. To get cos(2x), write 2x = x + x. Then, cos(2x) = cos(x + x) = cosxcosx - sinxsinx = cos² (x) - sin² (x) Then, from Pythagorean Identity, we can get express sine in terms of cosine: sin² (x) + cos² (x) = 1 sin² (x) = 1 - cos² (x) Plugging this into the formula for cos(2x), we get cos(2x) = cos² (x) - (1 - cos²(x)) = 2cos²(x) - 1. Alternatively, we could get the expression for cos(2x) in terms of sin(x): cos(2x) = (1 - sin²(x)) - sin²(x) = 1 - 2sin²(x) So, there are three formulas for the cosine of the double angle: cos(2x) = cos²(x) - sin²(x) = 1 - 2sin²(x) = 2cos²(x) - 1. The last one writes the cosine of 2x in terms of cosine of x. Approved by eNotes Editorial Team We know that the expression for cos ( x + y) is: cos (x + y) = cos x * cos y - sin x* sin y Now substituting x for both x and y we get cos ( x + x) = cos x * cos x - sin x * sin x => cos 2x = (cos x)² - (sin x)² Now we use the relation (cos x)² + (sin x)² = 1 which gives (sin x)² = 1 - (cos x)² So we can eliminate (sin x)² and get => cos 2x = (cos x)² - 1 + (cos x)² => cos 2x = 2( cos x)² - 1 Therefore in terms of cos x , cos 2x = 2( cos x)² - 1 Approved by eNotes Editorial Team	2021-11-30 22:20:06	{"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": 0.861464262008667, "perplexity": 4167.554893756461}, "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-2021-49/segments/1637964359073.63/warc/CC-MAIN-20211130201935-20211130231935-00185.warc.gz"}
https://stats.stackexchange.com/questions/99555/optimal-scaling-catreg-categorical-regression-for-imputed-data	# Optimal scaling / CATREG (categorical regression) for imputed data I have a data set with 5 different kinds of nutrient statuses and I want to see whether they are associated with categorical / ordinal grades at school. I have multiple covariates which I will included in the analyses. Due to missing values I have used the multiple imputation strategy (5 times). So now I have this databases and I want to do a optimal scaling regression (CATREG). However when I do this SPSS says: The following variables have values less than or equal to zero, which are considered as missing in this procedure: Smoking, etc. Why is this and what can I do about it? • Optimal scaling procedures require categorical data coded as positive integers. There is Discretize button in the procedure, to help you recode continuous variables automatically in a useful way, or you can recode manually. – ttnphns May 21 '14 at 15:34 • Positive or non-negative? – conjugateprior Jul 29 '15 at 16:14 1) Why is this? This is simply a peculiarity of this analysis. No substantive reason for it. 2) What can you do about it? First, what are the normal ranges of the (non missing) values on these variables? Do these include values of zero and below? If yes, then recode them. If no, then inspect the imputed data files and see how many values it concerns. If you imputed lots of missing values, chances are that by mere chance a handful of them end up at/below zero. In this case you could consider recoding the non imputed data files to values well above zero and then rerun the multiple imputation procedure. Imputed values at/below zero should now be much less likely. • Some variables are coded as either being there are not there (0 and 1) so if I would recode those to say 10 for no and 20 for yes that would solve the problem? I am also not quite sure what to do with this dicretize button, could anyone explain? – user45954 May 21 '14 at 19:50 • Well, given that CATREG considers values at/below zero as missing, you can't use codes 0-1 for dichotomies, right? Note that this has nothing to do with the imputation process, the problem is already there in your raw, unedited data. Forget about the discretize button. Even if you want to discretize anything, you don't want SPSS to decide for you where to draw the category boundaries. – RubenGeert May 22 '14 at 4:49 • Ok great :) I will tranform these variables. However if I do not use the discretize button, SPSS will use the default setting... is that ok then? – Inge May 22 '14 at 6:35 • And another question is it better to use ordinal or spline ordinal? – Inge May 22 '14 at 7:04 • 1) What do you want to discretize and why? In this discussion you haven't explicitly mentioned any continuous variables so far. 2) I'm not sure but I believe spline ordinal is a more restricted model than just ordinal. I don't know about the exact details. P.s. if you find answers/comments helpful you can upvote them. – RubenGeert May 22 '14 at 8:01	2019-06-25 22:22:11	{"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": 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": 0.5724495649337769, "perplexity": 1108.2829690758733}, "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-2019-26/segments/1560627999948.3/warc/CC-MAIN-20190625213113-20190625235113-00078.warc.gz"}
https://labs.tib.eu/arxiv/?author=A.%20Galindo	• Data Acquisition Architecture and Online Processing System for the HAWC gamma-ray observatory(1709.03751) Dec. 5, 2017 astro-ph.IM The High Altitude Water Cherenkov observatory (HAWC) is an air shower array devised for TeV gamma-ray astronomy. HAWC is located at an altitude of 4100 m a.s.l. in Sierra Negra, Mexico. HAWC consists of 300 Water Cherenkov Detectors, each instrumented with 4 photomultiplier tubes (PMTs). HAWC re-uses the Front-End Boards from the Milagro experiment to receive the PMT signals. These boards are used in combination with Time to Digital Converters (TDCs) to record the time and the amount of light in each PMT hit (light flash). A set of VME TDC modules (128 channels each) is operated in a continuous (dead time free) mode. The TDCs are read out via the VME bus by Single-Board Computers (SBCs), which in turn are connected to a gigabit Ethernet network. The complete system produces ~ 500 MB/s of raw data. A high-throughput data processing system has been designed and built to enable real-time data analysis. The system relies on off-the-shelf hardware components, an open-source software technology for data transfers (ZeroMQ) and a custom software framework for data analysis (AERIE). Multiple trigger and reconstruction algorithms can be combined and run on blocks of data in a parallel fashion, producing a set of output data streams which can be analyzed in real time with minimal latency (< 5 s). This paper provides an overview of the hardware set-up and an in-depth description of the software design, covering both the TDC data acquisition system and the real-time data processing system. The performance of these systems is also discussed. • Search for gamma-rays from the unusually bright GRB 130427A with the HAWC Gamma-ray Observatory(1410.1536) April 28, 2017 astro-ph.HE The first limits on the prompt emission from the long gamma-ray burst (GRB) 130427A in the $>100\nobreakspace\rm{GeV}$ energy band are reported. GRB 130427A was the most powerful burst ever detected with a redshift $z\lesssim0.5$ and featured the longest lasting emission above $100\nobreakspace\rm{MeV}$. The energy spectrum extends at least up to $95\nobreakspace\rm{GeV}$, clearly in the range observable by the High Altitude Water Cherenkov (HAWC) Gamma-ray Observatory, a new extensive air shower detector currently under construction in central Mexico. The burst occurred under unfavourable observation conditions, low in the sky and when HAWC was running 10% of the final detector. Based on the observed light curve at MeV-GeV energies, eight different time periods have been searched for prompt and delayed emission from this GRB. In all cases, no statistically significant excess of counts has been found and upper limits have been placed. It is shown that a similar GRB close to zenith would be easily detected by the full HAWC detector, which will be completed soon. The detection rate of the full HAWC detector may be as high as one to two GRBs per year. A detection could provide important information regarding the high energy processes at work and the observation of a possible cut-off beyond the $\mathit{Fermi}$-LAT energy range could be the signature of gamma-ray absorption, either in the GRB or along the line of sight due to the extragalactic background light. • The Latin American Giant Observatory: Contributions to the 34th International Cosmic Ray Conference (ICRC 2015)(1605.02151) May 7, 2016 astro-ph.IM, astro-ph.HE The Latin American Giant Observatory (LAGO) is an extended cosmic ray observatory composed by a network of water-Cherenkov detectors spanning over different sites located at significantly different altitudes (from sea level up to more than $5000$\,m a.s.l.) and latitudes across Latin America, covering a huge range of geomagnetic rigidity cut-offs and atmospheric absorption/reaction levels. This detection network is designed to measure the temporal evolution of the radiation flux at ground level with extreme detail. The LAGO project is mainly oriented to perform basic research in three branches: high energy phenomena, space weather and atmospheric radiation at ground level. LAGO is built and operated by the LAGO Collaboration, a non-centralized collaborative union of more than 30 institutions from ten countries. These are the contributions of the LAGO Collaboration to the 34th International Cosmic Ray Conference, 30 July - 6 August 2015, The Hague, The Netherlands • The Sensitivity of HAWC to High-Mass Dark Matter Annihilations(1405.1730) Dec. 9, 2014 hep-ph, astro-ph.CO, astro-ph.HE The High Altitude Water Cherenkov (HAWC) observatory is a wide field-of-view detector sensitive to gamma rays of 100 GeV to a few hundred TeV. Located in central Mexico at 19 degrees North latitude and 4100 m above sea level, HAWC will observe gamma rays and cosmic rays with an array of water Cherenkov detectors. The full HAWC array is scheduled to be operational in Spring 2015. In this paper, we study the HAWC sensitivity to the gamma-ray signatures of high-mass (multi- TeV) dark matter annihilation. The HAWC observatory will be sensitive to diverse searches for dark matter annihilation, including annihilation from extended dark matter sources, the diffuse gamma-ray emission from dark matter annihilation, and gamma-ray emission from non-luminous dark matter subhalos. Here we consider the HAWC sensitivity to a subset of these sources, including dwarf galaxies, the M31 galaxy, the Virgo cluster, and the Galactic center. We simulate the HAWC response to gamma rays from these sources in several well-motivated dark matter annihilation channels. If no gamma-ray excess is observed, we show the limits HAWC can place on the dark matter cross-section from these sources. In particular, in the case of dark matter annihilation into gauge bosons, HAWC will be able to detect a narrow range of dark matter masses to cross-sections below thermal. HAWC should also be sensitive to non-thermal cross-sections for masses up to nearly 1000 TeV. The constraints placed by HAWC on the dark matter cross-section from known sources should be competitive with current limits in the mass range where HAWC has similar sensitivity. HAWC can additionally explore higher dark matter masses than are currently constrained. • Observation of Small-scale Anisotropy in the Arrival Direction Distribution of TeV Cosmic Rays with HAWC(1408.4805) Oct. 10, 2014 astro-ph.HE The High-Altitude Water Cherenkov (HAWC) Observatory is sensitive to gamma rays and charged cosmic rays at TeV energies. The detector is still under construction, but data acquisition with the partially deployed detector started in 2013. An analysis of the cosmic-ray arrival direction distribution based on $4.9\times 10^{10}$ events recorded between June 2013 and February 2014 shows anisotropy at the $10^{-4}$ level on angular scales of about $10^\circ$. The HAWC cosmic-ray sky map exhibits three regions of significantly enhanced cosmic-ray flux; two of these regions were first reported by the Milagro experiment. A third region coincides with an excess recently reported by the ARGO-YBJ experiment. An angular power spectrum analysis of the sky shows that all terms up to $\ell=15$ contribute significantly to the excesses. • Milagro Limits and HAWC Sensitivity for the Rate-Density of Evaporating Primordial Black Holes(1407.1686) Oct. 6, 2014 astro-ph.HE Primordial Black Holes (PBHs) are gravitationally collapsed objects that may have been created by density fluctuations in the early universe and could have arbitrarily small masses down to the Planck scale. Hawking showed that due to quantum effects, a black hole has a temperature inversely proportional to its mass and will emit all species of fundamental particles thermally. PBHs with initial masses of ~5.0 x 10^14 g should be expiring in the present epoch with bursts of high-energy particles, including gamma radiation in the GeV - TeV energy range. The Milagro high energy observatory, which operated from 2000 to 2008, is sensitive to the high end of the PBH evaporation gamma-ray spectrum. Due to its large field-of-view, more than 90% duty cycle and sensitivity up to 100 TeV gamma rays, the Milagro observatory is well suited to perform a search for PBH bursts. Based on a search on the Milagro data, we report new PBH burst rate density upper limits over a range of PBH observation times. In addition, we report the sensitivity of the Milagro successor, the High Altitude Water Cherenkov (HAWC) observatory, to PBH evaporation events. • VAMOS was a prototype detector built in 2011 at an altitude of 4100m a.s.l. in the state of Puebla, Mexico. The aim of VAMOS was to finalize the design, construction techniques and data acquisition system of the HAWC observatory. HAWC is an air-shower array currently under construction at the same site of VAMOS with the purpose to study the TeV sky. The VAMOS setup included six water Cherenkov detectors and two different data acquisition systems. It was in operation between October 2011 and May 2012 with an average live time of 30%. Besides the scientific verification purposes, the eight months of data were used to obtain the results presented in this paper: the detector response to the Forbush decrease of March 2012, and the analysis of possible emission, at energies above 30 GeV, for long gamma-ray bursts GRB111016B and GRB120328B. • The HAWC Gamma-Ray Observatory: Sensitivity to Steady and Transient Sources of Gamma Rays(1310.0071) Sept. 30, 2013 astro-ph.HE The High-Altitude Water Cherenkov (HAWC) Gamma-Ray Observatory is designed to record air showers produced by cosmic rays and gamma rays between 100 GeV and 100 TeV. Because of its large field of view and high livetime, HAWC is well-suited to measure gamma rays from extended sources, diffuse emission, and transient sources. We describe the sensitivity of HAWC to emission from the extended Cygnus region as well as other types of galactic diffuse emission; searches for flares from gamma-ray bursts and active galactic nuclei; and the first measurement of the Crab Nebula with HAWC-30. • We describe measurements of GeV and TeV cosmic rays with the High-Altitude Water Cherenkov Gamma-Ray Observatory, or HAWC. The measurements include the observation of the shadow of the moon; the observation of small-scale and large-scale angular clustering of the TeV cosmic rays; the prospects for measurement of transient solar events with HAWC; and the observation of Forbush decreases with the HAWC engineering array and HAWC-30. • The HAWC Gamma-Ray Observatory: Dark Matter, Cosmology, and Fundamental Physics(1310.0073) Sept. 30, 2013 astro-ph.HE The High-Altitude Water Cherenkov Gamma Ray Observatory (HAWC) is designed to perform a synoptic survey of the TeV sky. The high energy coverage of the experiment will enable studies of fundamental physics beyond the Standard Model, and the large field of view of the detector will enable detailed studies of cosmologically significant backgrounds and magnetic fields. We describe the sensitivity of the full HAWC array to these phenomena in five contributions shown at the 33rd International Cosmic Ray Conference in Rio de Janeiro, Brazil (July 2013). • The HAWC Gamma-Ray Observatory: Design, Calibration, and Operation(1310.0074) Sept. 30, 2013 astro-ph.IM, astro-ph.HE The High-Altitude Water Cherenkov Gamma Ray Observatory (HAWC) is under construction 4100 meters above sea level at Sierra Negra, Mexico. We describe the design and cabling of the detector, the characterization of the photomultipliers, and the timing calibration system. We also outline a next-generation detector based on the water Cherenkov technique. • Sensitivity of the High Altitude Water Cherenkov Detector to Sources of Multi-TeV Gamma Rays(1306.5800) June 24, 2013 astro-ph.HE The High Altitude Water Cherenkov (HAWC) observatory is an array of large water Cherenkov detectors sensitive to gamma rays and hadronic cosmic rays in the energy band between 100 GeV and 100 TeV. The observatory will be used to measure high-energy protons and cosmic rays via detection of the energetic secondary particles reaching the ground when one of these particles interacts in the atmosphere above the detector. HAWC is under construction at a site 4100 meters above sea level on the northern slope of the volcano Sierra Negra, which is located in central Mexico at 19 degrees N latitude. It is scheduled for completion in 2014. In this paper we estimate the sensitivity of the HAWC instrument to point-like and extended sources of gamma rays. The source fluxes are modeled using both unbroken power laws and power laws with exponential cutoffs. HAWC, in one year, is sensitive to point sources with integral power-law spectra as low as 5x10^-13 cm^-2 sec^-1 above 2 TeV (approximately 50 mCrab) over 5 sr of the sky. This is a conservative estimate based on simple event parameters and is expected to improve as the data analysis techniques are refined. We discuss known TeV sources and the scientific contributions that HAWC can make to our understanding of particle acceleration in these sources. • Singular ways to search for the Higgs boson(1202.2552) Feb. 12, 2012 hep-ph The discovery or exclusion of the fundamental standard scalar is a hot topic, given the data of LEP, the Tevatron and the LHC, as well as the advanced status of the pertinent theoretical calculations. With the current statistics at the hadron colliders, the workhorse decay channel, at all relevant H masses, is H to WW, followed by W to light leptons. Using phase-space singularity techniques, we construct and study a plethora of "singularity variables" meant to facilitate the difficult tasks of separating signal and backgrounds and of measuring the mass of a putative signal. The simplest singularity variables are not invariant under boosts along the collider's axes and the simulation of their distributions requires a good understanding of parton distribution functions, perhaps not a serious shortcoming during the boson hunting season. The derivation of longitudinally boost-invariant variables, which are functions of the four charged-lepton observables that share this invariance, is quite elaborate. But their use is simple and they are, in a kinematical sense, optimal. • We present the sensitivity of HAWC to Gamma Ray Bursts (GRBs). HAWC is a very high-energy gamma-ray observatory currently under construction in Mexico at an altitude of 4100 m. It will observe atmospheric air showers via the water Cherenkov method. HAWC will consist of 300 large water tanks instrumented with 4 photomultipliers each. HAWC has two data acquisition (DAQ) systems. The main DAQ system reads out coincident signals in the tanks and reconstructs the direction and energy of individual atmospheric showers. The scaler DAQ counts the hits in each photomultiplier tube (PMT) in the detector and searches for a statistical excess over the noise of all PMTs. We show that HAWC has a realistic opportunity to observe the high-energy power law components of GRBs that extend at least up to 30 GeV, as it has been observed by Fermi LAT. The two DAQ systems have an energy threshold that is low enough to observe events similar to GRB 090510 and GRB 090902b with the characteristics observed by Fermi LAT. HAWC will provide information about the high-energy spectra of GRBs which in turn could help to understanding about e-pair attenuation in GRB jets, extragalactic background light absorption, as well as establishing the highest energy to which GRBs accelerate particles. • Measuring the W-Boson mass at a hadron collider: a study of phase-space singularity methods(1106.0396) July 21, 2011 hep-th, hep-ph The traditional method to measure the W-Boson mass at a hadron collider (more precisely, its ratio to the Z-mass) utilizes the distributions of three variables in events where the W decays into an electron or a muon: the charged-lepton transverse momentum, the missing transverse energy and the transverse mass of the lepton pair. We study the putative advantages of the additional measurement of a fourth variable: an improved phase-space singularity mass. This variable is statistically optimal, and simultaneously exploits the longitudinal- and transverse-momentum distributions of the charged lepton. Though the process we discuss is one of the simplest realistic ones involving just one unobservable particle, it is fairly non-trivial and constitutes a good "training" example for the scrutiny of phenomena involving invisible objects. Our graphical analysis of the phase space is akin to that of a Dalitz plot, extended to such processes. • Information and Computation: Classical and Quantum Aspects(quant-ph/0112105) Dec. 18, 2001 quant-ph, hep-th, cond-mat Quantum theory has found a new field of applications in the realm of information and computation during the recent years. This paper reviews how quantum physics allows information coding in classically unexpected and subtle nonlocal ways, as well as information processing with an efficiency largely surpassing that of the present and foreseeable classical computers. Some outstanding aspects of classical and quantum information theory will be addressed here. Quantum teleportation, dense coding, and quantum cryptography are discussed as a few samples of the impact of quanta in the transmission of information. Quantum logic gates and quantum algorithms are also discussed as instances of the improvement in information processing by a quantum computer. We provide finally some examples of current experimental realizations for quantum computers and future prospects.	2020-12-06 00:37:54	{"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": 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": 0.6275539398193359, "perplexity": 1924.8642033568542}, "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/1606141753148.92/warc/CC-MAIN-20201206002041-20201206032041-00336.warc.gz"}
https://questioncove.com/updates/4ea8bebde4b02eee39ccee84	Mathematics 83 Online OpenStudy (anonymous): can anyone help me on a quotient problem for Calculus on derivatives? OpenStudy (anonymous): whats the prob? OpenStudy (anonymous): y=(cosx)/(1-sinx) can you show the steps because I know I messed up somewhere but I just can extract exactly where. OpenStudy (amistre64): what have you tried? OpenStudy (anonymous): u know the quotient rule? OpenStudy (amistre64): $[\frac{t}{b}]'=\frac{bt'-b't}{b^2}$ OpenStudy (anonymous): Yeah I do know the quotient rule. OpenStudy (anonymous): yes amistre, u definetly know this:) OpenStudy (amistre64): ;) OpenStudy (anonymous): ok Dandy, apply it for ur prb OpenStudy (anonymous): Hmmm remember when you solve trig problems in calculus you have to simplify them using trig identity like Pythagorean theorem and double angel formula etc before you can eve get the correct answer OpenStudy (anonymous): Well right now I'm at (-sinx+sin^2x-cosx+cos^2x)/(-sinx)^2 OpenStudy (anonymous): and I tried simplifying witht he trig identities but I'm not sure if I am applying them accordingly OpenStudy (anonymous): y'=(1-sinx)(-sinx)-(cosx)(-cosx)/(1-sinx)^2 OpenStudy (anonymous): ohhhhhhhhhhhhhhhhh I see what I did wrong OpenStudy (anonymous): =(-sinx+sin^2x+cos^2)/(1-sinx)^2=-sinx+1/(1-sinx)^2 OpenStudy (anonymous): I did the derivative wrong for v, it should of been -cosx because 1 is a constant. OpenStudy (anonymous): yes :) OpenStudy (anonymous): okay I think i did another computation error. now I'm at the final stages and I have (-sinx+1)/(1-sinx)^2 Can't find your answer? Make a FREE account and ask your own questions, OR help others and earn volunteer hours! Join our real-time social learning platform and learn together with your friends! Can't find your answer? Make a FREE account and ask your own questions, OR help others and earn volunteer hours! Join our real-time social learning platform and learn together with your friends!	2022-11-27 02:43:19	{"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": 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": 0.6327354311943054, "perplexity": 4953.703149457645}, "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-2022-49/segments/1669446710155.67/warc/CC-MAIN-20221127005113-20221127035113-00223.warc.gz"}
https://www.springerprofessional.de/advances-in-digital-forensics-xiv/16083500	scroll identifier for mobile main-content ## Über dieses Buch Edited by: Gilbert Peterson and Sujeet Shenoi Digital forensics deals with the acquisition, preservation, examination, analysis and presentation of electronic evidence. Computer networks, cloud computing, smartphones, embedded devices and the Internet of Things have expanded the role of digital forensics beyond traditional computer crime investigations. Practically every crime now involves some aspect of digital evidence; digital forensics provides the techniques and tools to articulate this evidence in legal proceedings. Digital forensics also has myriad intelligence applications; furthermore, it has a vital role in information assurance - investigations of security breaches yield valuable information that can be used to design more secure and resilient systems. Advances in Digital Forensics XIV describes original research results and innovative applications in the discipline of digital forensics. In addition, it highlights some of the major technical and legal issues related to digital evidence and electronic crime investigations. The areas of coverage include: Themes and Issues; Forensic Techniques; Network Forensics; Cloud Forensics; and Mobile and Embedded Device Forensics. This book is the fourteenth volume in the annual series produced by the International Federation for Information Processing (IFIP) Working Group 11.9 on Digital Forensics, an international community of scientists, engineers and practitioners dedicated to advancing the state of the art of research and practice in digital forensics. The book contains a selection of nineteen edited papers from the Fourteenth Annual IFIP WG 11.9 International Conference on Digital Forensics, held in New Delhi, India in the winter of 2018. Advances in Digital Forensics XIV is an important resource for researchers, faculty members and graduate students, as well as for practitioners and individuals engaged in research and development efforts for the law enforcement and intelligence communities. Gilbert Peterson, Chair, IFIP WG 11.9 on Digital Forensics, is a Professor of Computer Engineering at the Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, USA. Sujeet Shenoi is the F.P. Walter Professor of Computer Science and a Professor of Chemical Engineering at the University of Tulsa, Tulsa, Oklahoma, USA. ## Inhaltsverzeichnis ### Measuring Evidential Weight in Digital Forensic Investigations Abstract This chapter describes a method for obtaining a quantitative measure of the relative weight of each individual item of evidence in a digital forensic investigation using a Bayesian network. The resulting evidential weights can then be used to determine a near-optimal, cost-effective triage scheme for the investigation in question. Richard Overill, Kam-Pui Chow ### Challenges, Opportunities and a Framework for Web Environment Forensics Abstract The web has evolved into a robust and ubiquitous platform, changing almost every aspect of people’s lives. The unique characteristics of the web pose new challenges to digital forensic investigators. For example, it is much more difficult to gain access to data that is stored online than it is to access data on the hard drive of a laptop. Despite the fact that data from the web is more challenging for forensic investigators to acquire and analyze, web environments continue to store more data than ever on behalf of users. This chapter discusses five critical challenges related to forensic investigations of web environments and explains their significance from a research perspective. It presents a framework for web environment forensics comprising four components: (i) evidence discovery and acquisition; (ii) analysis space reduction; (iii) timeline reconstruction; and (iv) structured formats. The framework components are non-sequential in nature, enabling forensic investigators to readily incorporate the framework in existing workflows. Each component is discussed in terms of how an investigator might use the component, the challenges that remain for the component, approaches related to the component and opportunities for researchers to enhance the component. Mike Mabey, Adam Doupé, Ziming Zhao, Gail-Joon Ahn ### Internet of Things Forensics – Challenges and a Case Study Abstract During this era of the Internet of Things, millions of devices such as automobiles, smoke detectors, watches, glasses and webcams are being connected to the Internet. The number of devices with the ability of monitor and collect data is continuously increasing. The Internet of Things enhances human comfort and convenience, but it raises serious questions related to security and privacy. It also creates significant challenges for digital investigators when they encounter Internet of Things devices in criminal scenes. In fact, current research focuses on security and privacy in Internet of Things environments as opposed to forensic acquisition and analysis techniques for Internet of Things devices. This chapter focuses on the major challenges with regard to Internet of Things forensics. A forensic approach for Internet of Things devices is presented using a smartwatch as a case study. Forensic artifacts retrieved from the smartwatch are analyzed and the evidence found is discussed with respect to the challenges facing Internet of Things forensics. ### Recovery of Forensic Artifacts from Deleted Jump Lists Abstract Jump lists, which were introduced in the Windows 7 desktop operating system, have attracted the interest of researchers and practitioners in the digital forensics community. The structure and forensic implications of jump lists have been explored widely. However, little attention has focused on anti-forensic activities such as jump list evidence modification and deletion. This chapter proposes a new methodology for identifying deleted entries in the Windows 10 AutoDest type of jump list files and recovering the deleted entries. The proposed methodology is best suited to scenarios where users intentionally delete jump list entries to hide evidence related to their activities. The chapter also examines how jump lists are impacted when software applications are installed and when the associated files are accessed by external storage devices. In particular, artifacts related to file access, such as the lists of most recently used and most frequently used files, file modification, access and creation timestamps, names of applications used to access files, file paths, volume names and serial numbers from where the files were accessed, can be recovered even after entries are removed from the jump lists and the software applications are uninstalled. The results demonstrate that the analysis of jump lists is immensely helpful in constructing the timelines of user activities on Windows 10 systems. Bhupendra Singh, Upasna Singh, Pankaj Sharma, Rajender Nath ### Obtaining Precision-Recall Trade-Offs in Fuzzy Searches of Large Email Corpora Abstract Fuzzy search is often used in digital forensic investigations to find words that are stringologically similar to a chosen keyword. However, a common complaint is the high rate of false positives in big data environments. This chapter describes the design and implementation of cedas, a novel constrained edit distance approximate string matching algorithm that provides complete control over the types and numbers of elementary edit operations considered in approximate matches. The unique flexibility of cedas facilitates fine-tuned control of precision-recall trade-offs. Specifically, searches can be constrained to the union of matches resulting from any exact edit combination of insertion, deletion and substitution operations performed on the search term. The flexibility is leveraged in experiments involving fuzzy searches of an inverted index of the Enron corpus, a large English email dataset, which reveal the specific edit operation constraints that should be applied to achieve valuable precision-recall trade-offs. The constraints that produce relatively high combinations of precision and recall are identified, along with the combinations of edit operations that cause precision to drop sharply and the combination of edit operation constraints that maximize recall without sacrificing precision substantially. These edit operation constraints are potentially valuable during the middle stages of a digital forensic investigation because precision has greater value in the early stages of an investigation while recall becomes more valuable in the later stages. Kyle Porter, Slobodan Petrovic ### Anti-Forensic Capacity and Detection Rating of Hidden Data in the Ext4 Filesystem Abstract The rise of cyber crime and the growing number of anti-forensic tools demand more research on combating anti-forensics. A prominent anti-forensic paradigm is the hiding of data at different abstraction layers, including the filesystem layer. This chapter evaluates various techniques for hiding data in the ext4 filesystem, which is commonly used by Android devices. The evaluation uses the capacity and detection rating metrics. Capacity reflects the quantity of data that can be concealed using a hiding technique. Detection rating is the difficulty of finding the concealed artifacts; specifically, the amount of effort required to discover the artifacts. Well-known data hiding techniques as well as new techniques proposed in this chapter are evaluated. Thomas Göbel, Harald Baier ### Detecting Data Leakage from Hard Copy Documents Abstract Document fraud has evolved to become a significant threat to individuals and organizations. Data leakage from hard copy documents is a common type of fraud. This chapter proposes a methodology for analyzing printed and photocopied versions of confidential documents to identify the source of a leak. The methodology incorporates a novel font pixel manipulation algorithm that embeds data in the pixels of certain characters of confidential documents in a manner that is imperceptible to the human eye. The embedded data is extracted from a leaked printed or photocopied document to identify the specific document that served as the source. The embedded data is robust in that it can withstand errors introduced by printing, scanning and photocopying documents. Experimental results demonstrate the efficiency, robustness and security of the methodology. Jijnasa Nayak, Shweta Singh, Saheb Chhabra, Gaurav Gupta, Monika Gupta, Garima Gupta ### Information-Entropy-Based DNS Tunnel Prediction Abstract DNS tunneling techniques are often used for malicious purposes. Network security mechanisms have struggled to detect DNS tunneling. Network forensic analysis has been proposed as a solution, but it is slow, invasive and tedious as network forensic analysis tools struggle to deal with undocumented and new network tunneling techniques. This chapter presents a method for supporting forensic analysis by automating the inference of tunneled protocols. The internal packet structure of DNS tunneling techniques is analyzed and the information entropy of various network protocols and their DNS tunneled equivalents are characterized. This provides the basis for a protocol prediction method that uses entropy distribution averaging. Experiments demonstrate that the method has a prediction accuracy of 75%. The method also preserves privacy because it only computes the information entropy and does not parse the actual tunneled content. Irvin Homem, Panagiotis Papapetrou, Spyridon Dosis ### Collecting Network Evidence Using Constrained Approximate Search Algorithms Abstract Intrusion detection systems are defensive tools that identify malicious activities in networks and hosts. In network forensics, investigators often study logs that store alerts generated by intrusion detection systems. This research focuses on Snort, a widely-used, open-source, misuse-based intrusion detection system that detects network intrusions based on a pre-defined set of attack signatures. When a security breach occurs, a forensic investigator typically starts by examining network log files. However, Snort cannot detect unknown attacks (i.e., zero-day attacks) even when they are similar to known attacks; as a result, an investigator may lose evidence in a criminal case. This chapter demonstrates the ease with which it is possible to defeat the detection of malicious activity by Snort and the possibility of using constrained approximate search algorithms instead of the default Snort search algorithm to collect evidence. Experimental results of the performance of constrained approximate search algorithms demonstrate that they are capable of detecting previously unknown attack attempts that are similar to known attacks. While the algorithms generate additional false positives, the number of false positives can be reduced by the careful choice of constraint values in the algorithms. Ambika Shrestha Chitrakar, Slobodan Petrovic ### Traffic Classification and Application Identification in Network Forensics Abstract Network traffic classification is an absolute necessity for network monitoring, security analyses and digital forensics. Without accurate traffic classification, the computational demands imposed by analyzing all the IP traffic flows are enormous. Classification can also reduce the number of flows that need to be examined and prioritized for analysis in forensic investigations. This chapter presents an automated feature elimination method based on a feature correlation matrix. Additionally, it proposes an enhanced statistical protocol identification method, which is compared against Bayesian network and random forests classification methods that offer high accuracy and acceptable performance. Each classification method is used with a subset of features that best suit the method. The methods are evaluated based on their ability to identify the application layer protocols and the applications themselves. Experiments demonstrate that the random forests classifier yields the most promising results whereas the proposed enhanced statistical protocol identification method provides an interesting trade-off between higher performance and slightly lower accuracy. Jan Pluskal, Ondrej Lichtner, Ondrej Rysavy ### Enabling Non-Expert Analysis OF Large Volumes OF Intercepted Network Traffic Abstract Telecommunications wiretaps are commonly used by law enforcement in criminal investigations. While phone-based wiretapping has seen considerable success, the same cannot be said for Internet taps. Large portions of intercepted Internet traffic are often encrypted, making it difficult to obtain useful information. The advent of the Internet of Things further complicates network wiretapping. In fact, the current level of complexity of intercepted network traffic is almost at the point where data cannot be analyzed without the active involvement of experts. Additionally, investigations typically focus on analyzing traffic in chronological order and predominately examine the data content of the intercepted traffic. This approach is overly arduous when the amount of data to be analyzed is very large. This chapter describes a novel approach for analyzing large amounts of intercepted network traffic based on traffic metadata. The approach significantly reduces the analysis time and provides useful insights and information to non-technical investigators. The approach is evaluated using a large sample of network traffic data. Erwin van de Wiel, Mark Scanlon, Nhien-An Le-Khac ### Hashing Incomplete and Unordered Network Streams Abstract Deep packet inspection typically uses MD5 whitelists/blacklists or regular expressions to identify viruses, malware and certain internal files in network traffic. Fuzzy hashing, also referred to as context-triggered piecewise hashing, can be used to compare two files and determine their level of similarity. This chapter presents the stream fuzzy hash algorithm that can hash files on the fly regardless of whether the input is unordered, incomplete or has an initially-undetermined length. The algorithm, which can generate a signature of appropriate length using a one-way process, reduces the computational complexity from $$O\left( n \log n\right)$$ to O(n). In a typical deep packet inspection scenario, the algorithm hashes files at the rate of 68 MB/s per CPU core and consumes no more than 5 KB of memory per file. The effectiveness of the stream fuzzy hash algorithm is evaluated using a publicly-available dataset. The results demonstrate that, unlike other fuzzy hash algorithms, the precision and recall of the stream fuzzy hash algorithm are not compromised when processing unordered and incomplete inputs. Chao Zheng, Xiang Li, Qingyun Liu, Yong Sun, Binxing Fang ### A Network Forensic Scheme Using Correntropy-Variation for Attack Detection Abstract Network forensic techniques help track cyber attacks by monitoring and analyzing network traffic. However, due to the large volumes of data in modern networks and sophisticated attacks that mimic normal behavior and/or erase traces to avoid detection, network attack investigations demand intelligent and efficient network forensic techniques. This chapter proposes a network forensic scheme for monitoring and investigating network-based attacks. The scheme captures and stores network traffic data, selects important network traffic features using the chi-square statistic and detects anomalous events using a novel correntropy-variation technique. An evaluation of the network forensic scheme employing the UNSW-NB15 dataset demonstrates its utility and high performance compared with three state-of-the-art approaches. Nour Moustafa, Jill Slay ### A Taxonomy of Cloud Endpoint Forensic Tools Abstract Cloud computing services can be accessed via browsers or client applications on networked devices such as desktop computers, laptops, tablets and smartphones, which are generally referred to as endpoint devices. Data relevant to forensic investigations may be stored on endpoint devices and/or at cloud service providers. When cloud services are accessed from an endpoint device, several files and folders are created on the device; the data can be accessed by a digital forensic investigator using various tools. An investigator may also use an application programming interface made available by a cloud service provider to obtain forensic information from the cloud related to objects, events and file metadata associated with a cloud user. This chapter presents a taxonomy of the forensic tools used to extract data from endpoint devices and from cloud service providers. The tool taxonomy provides investigators with an easily searchable catalog of tools that can meet their technical requirements during cloud forensic investigations. Anand Kumar Mishra, Emmanuel Pilli, Mahesh Govil ### A Layered Graphical Model for Cloud Forensic Mission Attack Impact Analysis Abstract Cyber attacks on the systems that support an enterprise’s mission can significantly impact its objectives. This chapter describes a layered graphical model designed to support forensic investigations by quantifying the mission impacts of cyber attacks. The model has three layers: (i) an upper layer that models operational tasks and their interdependencies that fulfill mission objectives; (ii) a middle layer that reconstructs attack scenarios based on the interrelationships of the available evidence; and (iii) a lower level that uses system calls executed in upper layer tasks in order to reconstruct missing attack steps when evidence is missing. The graphs constructed from the three layers are employed to compute the impacts of attacks on enterprise missions. The National Vulnerability Database – Common Vulnerability Scoring System scores and forensic investigator estimates are used to compute the mission impacts. A case study is presented to demonstrate the utility of the graphical model. Changwei Liu, Anoop Singhal, Duminda Wijesekera ### Forensic Analysis of Android Steganography Apps Abstract The processing power of smartphones supports steganographic algorithms that were considered to be too computationally intensive for handheld devices. Several steganography apps are now available on mobile phones to support covert communications using digital photographs. This chapter focuses on two key questions: How effectively can a steganography app be reverse engineered? How can this knowledge help improve the detection of steganographic images and other related files? Two Android steganography apps, PixelKnot and Da Vinci Secret Image, are analyzed. Experiments demonstrate that they are constructed in very different ways and provide different levels of security for hiding messages. The results of detecting steganography files, including images generated by the apps, using three software packages are presented. The results point to an urgent need for further research on reverse engineering steganography apps and detecting images produced by these apps. Wenhao Chen, Yangxiao Wang, Yong Guan, Jennifer Newman, Li Lin, Stephanie Reinders ### Automated Vulnerability Detection in Embedded Devices Abstract Embedded devices are widely used today and are rapidly being incorporated in the Internet of Things that will permeate every aspect of society. However, embedded devices have vulnerabilities such as buffer overflows, command injections and backdoors that are often undocumented. Malicious entities who discover these vulnerabilities could exploit them to gain control of embedded devices and conduct a variety of criminal activities. Due to the large number of embedded devices, non-standard codebases and complex control flows, it is extremely difficult to discover vulnerabilities using manual techniques. Current automated vulnerability detection tools typically use static analysis, but the detection accuracy is not high. Some tools employ code execution; however, this approach is inefficient, detects limited types of vulnerabilities and is restricted to specific architectures. Other tools use symbolic execution, but the level of automation is not high and the types of vulnerabilities they uncover are limited. This chapter evaluates several advanced vulnerability detection techniques used by current tools, especially those involving automated program analysis. These techniques are leveraged in a new automated vulnerability detection methodology for embedded devices. Danjun Liu, Yong Tang, Baosheng Wang, Wei Xie, Bo Yu ### A Forensic Logging System for Siemens Programmable Logic Controllers Abstract Critical infrastructure assets are monitored and managed by industrial control systems. In recent years, these systems have evolved to adopt common networking standards that expose them to cyber attacks. Since programmable logic controllers are core components of industrial control systems, forensic examinations of these devices are vital during responses to security incidents. However, programmable logic controller forensics is a challenging task because of the lack of effective logging systems. This chapter describes the design and implementation of a novel programmable logic controller logging system. Several tools are available for generating programmable logic controller audit logs; these tools monitor and record the values of programmable logic controller memory variables for diagnostic purposes. However, the logged information is inadequate for forensic investigations. To address this limitation, the logging system extracts data from Siemens S7 communications protocol traffic for forensic purposes. The extracted data is saved in an audit log file in an easy-to-read format that enables a forensic investigator to efficiently examine the activity of a programmable logic controller. Ken Yau, Kam-Pui Chow, Siu-Ming Yiu ### Enhancing the Security and Forensic Capabilities of Programmable Logic Controllers Abstract Industrial control systems are used to monitor and operate critical infrastructures. For decades, the security of industrial control systems was preserved by their use of proprietary hardware and software, and their physical separation from other networks. However, to reduce costs and enhance interconnectivity, modern industrial control systems increasingly use commodity hardware and software, and are connected to vendor and corporate networks, and even the Internet. These trends expose industrial control systems to risks that they were not designed to handle. This chapter describes a novel approach for enhancing industrial control system security and forensics by adding monitoring and logging mechanisms to programmable logic controllers, key components of industrial control systems. A proof-of-concept implementation is presented using a popular Siemens programmable logic controller. Experiments were conducted to compare the accuracy and performance impact of the proposed method versus the conventional programmable logic controller polling method. The experimental results demonstrate that the new method yields increased anomaly detection coverage and accuracy with only a small performance impact. Additionally, the new method increases the speed of anomaly detection and reduces network overhead, enabling forensic investigations of programmable logic controllers to be conducted more efficiently and effectively. Chun-Fai Chan, Kam-Pui Chow, Siu-Ming Yiu, Ken Yau Weitere Informationen	2019-07-18 19:22:08	{"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": 0.38580358028411865, "perplexity": 2988.3126338634256}, "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/1563195525793.19/warc/CC-MAIN-20190718190635-20190718212635-00247.warc.gz"}
https://www.physicsforums.com/threads/three-equations-two-unknowns.426629/	# Three equations, two unknowns 1. Sep 5, 2010 ### nobahar Hello! How do I determine if there is a solution to the following simultaneous equation?: 2a+3b=7 5a+7b=19 9a+5b=32 I have given a specific example but I was interested in the general case. I am confused because I have seen examples with two equations and two unkowns where it has been argued that, if, for example, I have: 2a+4b=x1 5a+6b=x2 Then: $$a=\frac{1}{2}(x_{2}-\frac{3}{2}x_{1})$$ $$b=\frac{1}{4}({\frac{5}{2}x_{1}-x_{2})$$ and I can plug in any x1 and any x2 and get an answer, as there is no division by zero, etc, and so there is no reason why I can't get an output for a and c. I can represent my original three equations in a similar format; presenting a as a combination of the outputs and b as a combination of the outputs. But when I put them in to the original equation it doesn't work! Has this something to do with an assumption being made? That with two equations and two unknowns, I can always find a solution for any x1 and x2 (provided that they're not parallel), but for three equations and two unknowns, there is not always going to be a solution? If this is the case, how do I ascertain that there is not a solution (without graphing the functions)? As always, any help appreciated. 2. Sep 5, 2010 ### debsuvra The simultaneous equations you gave here have conflicting values of a and b. Solving the first two equations, we can have a=8 and b=-3. On the other hand putting a=8 on 3rd equation doesn't satisfy b=-3. Since the equations are of first order, there will be single discreet solution for a and b. The third equation is conflicting with this. 3. Sep 5, 2010 ### Staff: Mentor If there are three equations in two unknowns there are basically two possibilities - either they are inconsistent, or one is redundant. Solve any two equations and put results into third - if it is satsified, equation was redundant, if not - it was inconsistent. 4. Sep 5, 2010 ### HallsofIvy Staff Emeritus Graphically, a single equation in two unknowns can be represented as a line in the plane. Two such equations, of course, represent two lines and, except in the unusual case that the lines are parallel, have a single point of intersection, giving a single solution to the two equations. Three equations in two unknowns represent three lines in the plane and now the "standard" situation is that two of the lines intersect in a single point while the third line intersects those two lines at point different from the original point- there is no point in common to the three lines and so no x, y that satisfy all three equations. 5. Sep 6, 2010 ### nobahar Thanks Debsuvra, Halls and Borek. Using these two pieces of information, it is possible that I can compute the unknowns using equations 1) and 2); 1) and 3); and 2) and 3) and arrive at appropriate values that satisfy two of the three equations, depending on which two of the three equations I use, but will not necessarily work for all three. Is this the only method of determining whether there is an answer. Solve for two and see if it works for the third? If I get a gobbeldygook answer such as 8=12, then there is no solution for all three? I found some information on Reduced Row Echelon Form, is this valid here? Many thanks for the responses. 6. Sep 6, 2010 ### HallsofIvy Staff Emeritus Yes, you can use "Row Echelon form". If one of the equations is a linear combination of the other two, your last row will be all "0"s and the solution is given by the first two rows. If not you will get something like "0 0 1" in the last row which corresponds to "0x+ 0y= 1" and is impossible. 7. Sep 6, 2010 ### nobahar Thanks again. This gets a little more confusing when introducing a fourth variable. I shall not attempt an explanation as they end up being long and often do not elicit a response! My question is, if I have three variables in three equations, therefore three planes, and which do not all intersect at a single point or line, if I then introduce a fouth variable, does it make it possible to solve a simultaneous equation where there is a solution to the three equations? It's impossible to imagine and I do not know where to begin to prove it algebraicly. Any responses would be appreciated. Many thanks.	2017-12-15 18:19:43	{"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": 0.7175276279449463, "perplexity": 290.346972214793}, "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-2017-51/segments/1512948577018.52/warc/CC-MAIN-20171215172833-20171215194833-00171.warc.gz"}
https://proofwiki.org/wiki/Leigh.Samphier/Sandbox/Matroid_Satisfies_Base_Axiom/Sufficient_Condition/Lemma/Lemma_3	Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 3 Theorem Let $S$ be a finite set. Let $B_1, B_2 \subseteq S$. Let $z \in B_1 \setminus B_2$. Let $y \in B_2 \setminus B_1$. Let $B_3 = \paren{B_2 \setminus \set y} \cup \set z$ Then: $\card{B_1 \cap B_3} = \card{B_1 \cap B_2} + 1$ Proof We have: $\ds B_3 \cap B_1$ $=$ $\ds \paren{\paren{B_2 \setminus \set y} \cup \set z} \cap B_1$ $\ds$ $=$ $\ds \paren{\paren{B_2 \setminus \set y} \cap B_1 } \cup \paren{ \set z \cap B_1}$ Intersection Distributes over Union $\ds$ $=$ $\ds \paren{\paren{B_2 \setminus \set y} \cap B_1 } \cup \set z$ As $z \in B_1$ $\ds$ $=$ $\ds \paren{\paren{B_2 \cap B_1} \setminus \paren{\set y \cap B_1} } \cup \set z$ Set Intersection Distributes over Set Difference $\ds$ $=$ $\ds \paren{\paren{B_2 \cap B_1} \setminus \O } \cup \set z$ As $y \notin B_1$ $\ds$ $=$ $\ds \paren{B_2 \cap B_1} \cup \set z$ Set Difference with Empty Set is Self $\ds \leadsto \ \$ $\ds \card{B_3 \cap B_1}$ $=$ $\ds \card{\paren{B_2 \cap B_1} \cup \set z}$ $\ds$ $=$ $\ds \card{\paren{B_2 \cap B_1} } + \card {\set z}$ As $z \notin B_2$ $\ds$ $=$ $\ds \card{\paren{B_2 \cap B_1} } + 1$ Cardinality of Singleton $\blacksquare$	2021-01-24 08:00:48	{"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": 0.9683465957641602, "perplexity": 218.23943883186118}, "config": {"markdown_headings": false, "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-2021-04/segments/1610703547475.44/warc/CC-MAIN-20210124075754-20210124105754-00215.warc.gz"}
https://www.journaltocs.ac.uk/index.php?action=browse&subAction=subjects&publisherID=87&journalID=2782&pageb=1&userQueryID=&sort=&local_page=1&sorType=DESC&sorCol=2	Subjects -> STATISTICS (Total: 130 journals) Showing 1 - 151 of 151 Journals sorted alphabetically Advances in Complex Systems (Followers: 11) Advances in Data Analysis and Classification (Followers: 62) Annals of Applied Statistics (Followers: 39) Applied Categorical Structures (Followers: 4) Argumentation et analyse du discours (Followers: 11) Asian Journal of Mathematics & Statistics (Followers: 8) AStA Advances in Statistical Analysis (Followers: 4) Australian & New Zealand Journal of Statistics (Followers: 13) Bernoulli (Followers: 9) Biometrical Journal (Followers: 11) Biometrics (Followers: 52) British Journal of Mathematical and Statistical Psychology (Followers: 18) Building Simulation (Followers: 2) Bulletin of Statistics (Followers: 4) CHANCE (Followers: 5) Communications in Statistics - Simulation and Computation (Followers: 9) Communications in Statistics - 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Computational Statistics (Followers: 1) Similar Journals Computational StatisticsJournal Prestige (SJR): 0.803 Citation Impact (citeScore): 1Number of Followers: 14 Hybrid journal (It can contain Open Access articles) ISSN (Print) 1613-9658 - ISSN (Online) 0943-4062 Published by Springer-Verlag [2656 journals] • Additive models with autoregressive symmetric errors based on penalized regression splines • Abstract: In this paper additive models with p-order autoregressive conditional symmetric errors based on penalized regression splines are proposed for modeling trend and seasonality in time series. The aim with this kind of approach is try to model the autocorrelation and seasonality properly to assess the existence of a significant trend. A backfitting iterative process jointly with a quasi-Newton algorithm are developed for estimating the additive components, the dispersion parameter and the autocorrelation coefficients. The effective degrees of freedom concerning the fitting are derived from an appropriate smoother. Inferential results and selection model procedures are proposed as well as some diagnostic methods, such as residual analysis based on the conditional quantile residual and sensitivity studies based on the local influence approach. Simulations studies are performed to assess the large sample behavior of the maximum penalized likelihood estimators. Finally, the methodology is applied for modeling the daily average temperature of San Francisco city from January 1995 to April 2020. PubDate: 2021-05-03 • A multigrid preconditioner for tensor product spline smoothing • Abstract: Penalized spline smoothing is a well-established, nonparametric regression method that is efficient for one and two covariates. Its extension to more than two covariates is straightforward but suffers from exponentially increasing memory demands and computational complexity, which brings the method to its numerical limit. Penalized spline smoothing with multiple covariates requires solving a large-scale, regularized least-squares problem where the occurring matrices do not fit into storage of common computer systems. To overcome this restriction, we introduce a matrix-free implementation of the conjugate gradient method. We further present a matrix-free implementation of a simple diagonal as well as more advanced geometric multigrid preconditioner to significantly speed up convergence of the conjugate gradient method. All algorithms require a negligible amount of memory and therefore allow for penalized spline smoothing with multiple covariates. Moreover, for arbitrary but fixed covariate dimension, we show grid independent convergence of the multigrid preconditioner which is fundamental to achieve algorithmic scalability. PubDate: 2021-04-30 • Direct statistical inference for finite Markov jump processes via the matrix exponential • Abstract: Given noisy, partial observations of a time-homogeneous, finite-statespace Markov chain, conceptually simple, direct statistical inference is available, in theory, via its rate matrix, or infinitesimal generator, $${\mathsf {Q}}$$ , since $$\exp ({\mathsf {Q}}t)$$ is the transition matrix over time t. However, perhaps because of inadequate tools for matrix exponentiation in programming languages commonly used amongst statisticians or a belief that the necessary calculations are prohibitively expensive, statistical inference for continuous-time Markov chains with a large but finite state space is typically conducted via particle MCMC or other relatively complex inference schemes. When, as in many applications $${\mathsf {Q}}$$ arises from a reaction network, it is usually sparse. We describe variations on known algorithms which allow fast, robust and accurate evaluation of the product of a non-negative vector with the exponential of a large, sparse rate matrix. Our implementation uses relatively recently developed, efficient, linear algebra tools that take advantage of such sparsity. We demonstrate the straightforward statistical application of the key algorithm on a model for the mixing of two alleles in a population and on the Susceptible-Infectious-Removed epidemic model. PubDate: 2021-04-19 • Automatic differentiation and maximal correlation of order statistics from discrete parents • Abstract: The maximal correlation is an attractive measure of dependence between the components of a random vector, however it presents the difficulty that its calculation is not easy. Here, we consider the case of bivariate vectors which components are order statistics from discrete distributions supported on $$N\ge 2$$ points. Except for the case $$N=2$$ , the maximal correlation does not have a closed form, so we propose the use of a gradient based optimization method. The gradient vector of the objective function, the correlation coefficient of pairs of order statistics, can be extraordinarily complicated and for that reason an automatic differentiation algorithm is proposed. PubDate: 2021-04-12 • Bootstrap joint prediction regions for sequences of missing values in spatio-temporal datasets • Abstract: Missing data reconstruction is a critical step in the analysis and mining of spatio-temporal data. However, few studies comprehensively consider missing data patterns, sample selection and spatio-temporal relationships. To take into account the uncertainty in the point forecast, some prediction intervals may be of interest. In particular, for (possibly long) missing sequences of consecutive time points, joint prediction regions are desirable. In this paper we propose a bootstrap resampling scheme to construct joint prediction regions that approximately contain missing paths of a time components in a spatio-temporal framework, with global probability $$1-\alpha$$ . In many applications, considering the coverage of the whole missing sample-path might appear too restrictive. To perceive more informative inference, we also derive smaller joint prediction regions that only contain all elements of missing paths up to a small number k of them with probability $$1-\alpha$$ . A simulation experiment is performed to validate the empirical performance of the proposed joint bootstrap prediction and to compare it with some alternative procedures based on a simple nominal coverage correction, loosely inspired by the Bonferroni approach, which are expected to work well standard scenarios. PubDate: 2021-04-05 • Bayesian sparse convex clustering via global-local shrinkage priors • Abstract: Sparse convex clustering is to group observations and conduct variable selection simultaneously in the framework of convex clustering. Although a weighted $$L_1$$ norm is usually employed for the regularization term in sparse convex clustering, its use increases the dependence on the data and reduces the estimation accuracy if the sample size is not sufficient. To tackle these problems, this paper proposes a Bayesian sparse convex clustering method based on the ideas of Bayesian lasso and global-local shrinkage priors. We introduce Gibbs sampling algorithms for our method using scale mixtures of normal distributions. The effectiveness of the proposed methods is shown in simulation studies and a real data analysis. PubDate: 2021-04-05 • A Bayesian algorithm based on auxiliary variables for estimating GRM with non-ignorable missing data • Abstract: In this paper, a highly effective Bayesian sampling algorithm based on auxiliary variables is used to estimate the graded response model with non-ignorable missing response data. Compared with the traditional marginal likelihood method and other Bayesian algorithms, the advantages of the new algorithm are discussed in detail. Based on the Markov Chain Monte Carlo samples from the posterior distributions, the deviance information criterion and the logarithm of the pseudomarignal likelihood are employed to compare the different missing mechanism models. Two simulation studies are conducted and a detailed analysis of the sexual compulsivity scale data is carried out to further illustrate the proposed methodology. PubDate: 2021-04-02 • GSDAR: a fast Newton algorithm for $$\ell _0$$ ℓ 0 regularized generalized linear models with statistical guarantee • Abstract: We propose a fast Newton algorithm for $$\ell _0$$ regularized high-dimensional generalized linear models based on support detection and root finding. We refer to the proposed method as GSDAR. GSDAR is developed based on the KKT conditions for $$\ell _0$$ -penalized maximum likelihood estimators and generates a sequence of solutions of the KKT system iteratively. We show that GSDAR can be equivalently formulated as a generalized Newton algorithm. Under a restricted invertibility condition on the likelihood function and a sparsity condition on the regression coefficient, we establish an explicit upper bound on the estimation errors of the solution sequence generated by GSDAR in supremum norm and show that it achieves the optimal order in finite iterations with high probability. Moreover, we show that the oracle estimator can be recovered with high probability if the target signal is above the detectable level. These results directly concern the solution sequence generated from the GSDAR algorithm, instead of a theoretically defined global solution. We conduct simulations and real data analysis to illustrate the effectiveness of the proposed method. PubDate: 2021-03-29 • A new two-parameter discrete poisson-generalized Lindley distribution with properties and applications to healthcare data sets • Abstract: Mixed-Poisson distributions have been used in many fields for modeling the over-dispersed count data sets. To open a new opportunity in modeling the over-dispersed count data sets, we introduce a new mixed-Poisson distribution using the generalized Lindley distribution as a mixing distribution. The moment and probability generating functions, factorial moments as well as skewness, and kurtosis measures are derived. Using the mean-parametrized version of the proposed distribution, we introduce a new count regression model which is an appropriate model for over-dispersed counts. The healthcare data sets are analyzed employing a new count regression model. We conclude that the new regression model works well in the case of over-dispersion. PubDate: 2021-03-28 • Mixture cure rate models with neural network estimated nonparametric components • Abstract: Survival data including potentially cured subjects are common in clinical studies and mixture cure rate models are often used for analysis. The non-cured probabilities are often predicted by non-parametric, high-dimensional, or even unstructured (e.g. image) predictors, which is a challenging task for traditional nonparametric methods such as spline and local kernel. We propose to use the neural network to model the nonparametric or unstructured predictors’ effect in cure rate models and retain the proportional hazards structure due to its explanatory ability. We estimate the parameters by Expectation–Maximization algorithm. Estimators are showed to be consistent. Simulation studies show good performance in both prediction and estimation. Finally, we analyze Open Access Series of Imaging Studies data to illustrate the practical use of our methods. PubDate: 2021-03-27 • Estimation of the association parameters in hierarchically clustered survival data by nested Archimedean copula functions • Abstract: Statisticians are frequently confronted with highly complex data such as clustered data, missing data or censored data. In this manuscript, we consider hierarchically clustered survival data. This type of data arises when a sample consists of clusters, and each cluster has several, correlated sub-clusters containing various, dependent survival times. Two approaches are commonly used to analysis such data and estimate the association between the survival times within a cluster and/or sub-cluster. The first approach is by using random effects in a frailty model while a second approach is by using copula models. Hereby we assume that the joint survival function is described by a copula function evaluated in the marginal survival functions of the different individuals within a cluster. In this manuscript, we introduce a copula model based on a nested Archimedean copula function for hierarchical survival data, where both the clusters and sub-clusters are allowed to be moderate to large and varying in size. We investigate one-stage, two-stage and three-stage parametric estimation procedures for the association parameters in this model. In a simulation study we check the finite sample properties of these estimators. Furthermore we illustrate the methods on a real life data-set on Chronic Granulomatous Disease. PubDate: 2021-03-26 • Weighted approximate Bayesian computation via Sanov’s theorem • Abstract: We consider the problem of sample degeneracy in Approximate Bayesian Computation. It arises when proposed values of the parameters, once given as input to the generative model, rarely lead to simulations resembling the observed data and are hence discarded. Such “poor” parameter proposals do not contribute at all to the representation of the parameter’s posterior distribution. This leads to a very large number of required simulations and/or a waste of computational resources, as well as to distortions in the computed posterior distribution. To mitigate this problem, we propose an algorithm, referred to as the Large Deviations Weighted Approximate Bayesian Computation algorithm, where, via Sanov’s Theorem, strictly positive weights are computed for all proposed parameters, thus avoiding the rejection step altogether. In order to derive a computable asymptotic approximation from Sanov’s result, we adopt the information theoretic “method of types” formulation of the method of Large Deviations, thus restricting our attention to models for i.i.d. discrete random variables. Finally, we experimentally evaluate our method through a proof-of-concept implementation. PubDate: 2021-03-22 • Algorithm for error-free determination of the variance of all contiguous subsequences and fixed-length contiguous subsequences for a sequence of industrial measurement data • Abstract: The article presents an algorithm for fast and error-free determination of statistics such as the arithmetic mean and variance of all contiguous subsequences and fixed-length contiguous subsequences for a sequence of industrial measurement data. Additionally, it shows that both floating-point and integer representation can be used to perform this kind of statistical calculations. The author proves a theorem on the number of bits of precision that an arithmetic type must have to guarantee error-free determination of the arithmetic mean and variance. The article also presents the extension of Welford’s formula for determining variance for the sliding window method—determining the variance of fixed-length contiguous subsequences. The section dedicated to implementation tests shows the running times of individual algorithms depending on the arithmetic type used. The research shows that the use of integers in calculations makes the determination of the aforementioned statistics much faster. PubDate: 2021-03-19 • A cluster-based taxonomy of bus crashes in the United States • Abstract: Accident taxonomy or classification can be used to direct the attention of policymakers to specific concerns in traffic safety, and can subsequently bring about effective regulatory change. Despite the widespread usage of accident taxonomy for general motor vehicle crashes, its use for analyzing bus crashes is limited. We apply a two-stage clustering-based approach based on self-organizing maps followed by neural gas clustering to construct a data-driven taxonomy of bus crashes. Using the 2005–2015 data from general estimates system, we identify four clusters and expose the qualitative traits that characterize four distinct types of bus crash. Our analysis suggests that cluster characteristics are largely stable over time. Consequently, we make targeted policy recommendations for each of the four subtypes of bus crash. PubDate: 2021-03-16 • Reduced-bias estimation of spatial autoregressive models with incompletely geocoded data • Abstract: The application of spatial Cliff–Ord models requires information about spatial coordinates of statistical units to be reliable, which is usually the case in the context of areal data. With micro-geographic point-level data, however, such information is inevitably affected by locational errors, that can be generated intentionally by the data producer for privacy protection or can be due to inaccuracy of the geocoding procedures. This unfortunate circumstance can potentially limit the use of the spatial autoregressive modelling framework for the analysis of micro data, as the presence of locational errors may have a non-negligible impact on the estimates of model parameters. This contribution aims at developing a strategy to reduce the bias and produce more reliable inference for spatial models with location errors. The proposed estimation strategy models both the spatial stochastic process and the coarsening mechanism by means of a marked point process. The model is fitted through the maximisation of a doubly-marginalised likelihood function of the marked point process, which cleans out the effects of coarsening. The validity of the proposed approach is assessed by means of a Monte Carlo simulation study under different real-case scenarios, whereas it is applied to real data on house prices. PubDate: 2021-03-13 • Jump Markov chains and rejection-free Metropolis algorithms • Abstract: We consider versions of the Metropolis algorithm which avoid the inefficiency of rejections. We first illustrate that a natural Uniform Selection algorithm might not converge to the correct distribution. We then analyse the use of Markov jump chains which avoid successive repetitions of the same state. After exploring the properties of jump chains, we show how they can exploit parallelism in computer hardware to produce more efficient samples. We apply our results to the Metropolis algorithm, to Parallel Tempering, to a Bayesian model, to a two-dimensional ferromagnetic 4 $$\times$$ 4 Ising model, and to a pseudo-marginal MCMC algorithm. PubDate: 2021-03-13 • Statistical inference for mixture GARCH models with financial application • Abstract: In this paper we consider mixture generalized autoregressive conditional heteroskedastic models, and propose a new iteration algorithm of type EM for the estimation of model parameters. The maximum likelihood estimates are shown to be consistent, and their asymptotic properties are investigated. More precisely, we derive simple expressions in closed form for the asymptotic covariance matrix and the expected Fisher information matrix of the ML estimator. Finally, we study the model selection and propose testing procedures. A simulation study and an application to financial real-series illustrate the results. PubDate: 2021-03-12 • Moving dynamic principal component analysis for non-stationary multivariate time series • Abstract: This paper proposes an extension of principal component analysis to non-stationary multivariate time series data. A criterion for determining the number of final retained components is proposed. An advance correlation matrix is developed to evaluate dynamic relationships among the chosen components. The theoretical properties of the proposed method are given. Many simulation experiments show our approach performs well on both stationary and non-stationary data. Real data examples are also presented as illustrations. We develop four packages using the statistical software R that contain the needed functions to obtain and assess the results of the proposed method. PubDate: 2021-03-07 • Spatial CART classification trees • Abstract: We propose to extend CART for bivariate marked point processes to provide a segmentation of the space into homogeneous areas for interaction between marks. While usual CART tree considers marginal distribution of the response variable at each node, the proposed algorithm, SpatCART, takes into account the spatial location of the observations in the splitting criterion. We introduce a dissimilarity index based on Ripley’s intertype K-function quantifying the interaction between two populations. This index used for the growing step of the CART strategy, leads to a heterogeneity function consistent with the original CART algorithm. Therefore the new variant is a way to explore spatial data as a bivariate marked point process using binary classification trees. The proposed procedure is implemented in an R package, and illustrated on simulated examples. SpatCART is finally applied to a tropical forest example. PubDate: 2021-03-04 • Correction to: The 8-parameter Fisher–Bingham distribution on the sphere • Abstract: In the original publication of the article, the corrections in Eq. (13) were missed, in which 2v − 1 was changed to 2v in the exponent. PubDate: 2021-03-01 DOI: 10.1007/s00180-020-01035-6 JournalTOCs School of Mathematical and Computer Sciences Heriot-Watt University Edinburgh, EH14 4AS, UK Email: journaltocs@hw.ac.uk Tel: +00 44 (0)131 4513762	2021-05-07 21:26:34	{"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": 0, "mathjax_display_tex": 2, "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": 0.7558085322380066, "perplexity": 2938.6172906015854}, "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/1620243988828.76/warc/CC-MAIN-20210507211141-20210508001141-00550.warc.gz"}
http://scipy.github.io/devdocs/generated/scipy.special.lqn.html	scipy.special.lqn¶ scipy.special.lqn(n, z)[source] Legendre function of the second kind. Compute sequence of Legendre functions of the second kind, Qn(z) and derivatives for all degrees from 0 to n (inclusive). References [1] Zhang, Shanjie and Jin, Jianming. “Computation of Special Functions”, John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html Previous topic scipy.special.lpn Next topic scipy.special.lpmn	2018-04-19 13:55:15	{"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": 0.9219360947608948, "perplexity": 5767.97586809868}, "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-2018-17/segments/1524125936969.10/warc/CC-MAIN-20180419130550-20180419150550-00555.warc.gz"}
https://www.albert.io/learn/ap-physics-1-and-2/question/fluid-flow-increasing-diameter	Limited access An ideal liquid is moving at a rate of $1.8\text{ m/s}$ through a horizontal $1.0\text{ inch}$ diameter pipe. What happens to the pressure of the liquid when the diameter is increased to $1.5\text{ inches}$? A The pressure increases. B The pressure decreases. C The pressure remains the same. D Not enough information is given to determine. Select an assignment template	2017-04-25 22:16:47	{"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": 0.23657484352588654, "perplexity": 2259.1617250582035}, "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-2017-17/segments/1492917120881.99/warc/CC-MAIN-20170423031200-00225-ip-10-145-167-34.ec2.internal.warc.gz"}
http://malomore.com/2022/08/15/rates-elasticities-with-each-other-a-good-linear/	11°C 11°C ## Malo More ### Rates Elasticities With each other a good Linear Consult Curve Rates Elasticities With each other a good Linear Consult Curve That it measure of suppleness, which is based on fee alter according to the typical worthy of of each and every variable anywhere between a couple things, is called arch elasticity . The fresh new arch suppleness means provides the virtue so it output the newest same flexibility whether we move from part A to area B or away from part B to indicate A. This is the approach we shall used to calculate elasticity. ## By using the average numbers and you will average rates in order to determine fee change, this new arc elasticity strategy prevents the need to specify brand new guidelines of one’s transform and you can, and so, gives us a similar address if or not we go from An inside B otherwise out of B to A To the arch suppleness method, we determine the price suppleness off request by using the average really worth regarding rates, $$\pub$$ , and the average value of quantity demanded, $$\bar$$ . We shall utilize the Greek-letter ? to help you imply “change in,” so that the change in quantity anywhere between a couple of affairs try ?Q and you will the change in cost is actually ?P. Today we are able to establish the fresh formula into the rates elasticity regarding demand because With the arch elasticity algorithm, the latest flexibility is the identical whether or not i move from point Good to suggest B or out-of part B to indicate A. Whenever we initiate on point B and you will go on to area A, i’ve: The brand new arch flexibility method provides a price out-of suppleness. It gives the worth of flexibility in the midpoint more good listing of transform, including the movement ranging from activities A good and B. Getting a precise formula away from elasticity, we would need consider the effect out-of a dependent variable in order to an extremely quick change in a separate adjustable. The fact that arch elasticities was approximate suggests an important standard rule into the figuring arc elasticities: we need to imagine simply brief changes in separate parameters. We simply cannot use the thought of arch elasticity so you’re able to high transform. Some other argument getting offered only brief alterations in measuring speed elasticities off consult will become clear within the next point. We are going to take a look at what are the results to help you speed elasticities once we disperse from 1 indicate several other collectively an effective linear consult contour. See that on the arc elasticity algorithm, the method having measuring a portion alter differs from the standard strategy that you iliar. One to strategy steps the latest fee improvement in an adjustable relative to its new really worth. Such as for instance, utilizing the standard strategy, whenever we move from section A to area B, we possibly may compute the latest fee improvement in amounts since 20,,one hundred thousand = 50%. The brand new payment change in speed might possibly be ?$0.10/$0.80 = ?several.5%. The cost suppleness regarding request carry out upcoming become 50%/(?several.5%) = ?cuatro.00. Heading from section B to point Good, not, would yield a different sort of elasticity. The fresh new fee improvement in wide variety could well be ?20,,100, or ?%. New fee change in rate could well be $0.10/$0.70 = %. The cost flexibility from consult carry out for this reason end up being ?%/% = ?dos.33.	2023-02-01 12:54:51	{"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": 0.39767464995384216, "perplexity": 2074.4575264333503}, "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/1674764499934.48/warc/CC-MAIN-20230201112816-20230201142816-00158.warc.gz"}
https://www.nature.com/articles/s41598-017-15313-9	Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. # Gpufit: An open-source toolkit for GPU-accelerated curve fitting ## Abstract We present a general purpose, open-source software library for estimation of non-linear parameters by the Levenberg-Marquardt algorithm. The software, Gpufit, runs on a Graphics Processing Unit (GPU) and executes computations in parallel, resulting in a significant gain in performance. We measured a speed increase of up to 42 times when comparing Gpufit with an identical CPU-based algorithm, with no loss of precision or accuracy. Gpufit is designed such that it is easily incorporated into existing applications or adapted for new ones. Multiple software interfaces, including to C, Python, and Matlab, ensure that Gpufit is accessible from most programming environments. The full source code is published as an open source software repository, making its function transparent to the user and facilitating future improvements and extensions. As a demonstration, we used Gpufit to accelerate an existing scientific image analysis package, yielding significantly improved processing times for super-resolution fluorescence microscopy datasets. ## Introduction Optimization algorithms are widely used in science and engineering. In particular, when comparing data with a model function, numerical optimization methods may be applied to establish the suitability of the model and to determine the parameters which best describe the observations. One such method, which is generally applicable to models which depend non-linearly on a set of parameters, is the Levenberg-Marquardt algorithm (LMA)1, and this has become a standard approach for non-linear least squares curve fitting2,3. Although the LMA is, in itself, an efficient algorithm, applications requiring many iterations of this procedure may encounter limitations due to the sheer number of calculations involved. The time required for the convergence of a fit, or a set of fits, can determine an application’s feasibility, e.g. in the context of real-time data processing and feedback systems. Alternatively, in the case of very large datasets, the time required to solve a particular optimization problem may prove impractical. In recent years, advanced graphics processing units (GPUs) and the development of general purpose GPU programming have enabled fast and parallelized computing by shifting calculations from the CPU to the GPU4. The large number of independent computing units available on modern GPUs enables the rapid execution of many instructions in parallel, with an overall computation power far exceeding that of a CPU. Languages such as CUDA C and OpenCL allow GPU-based programs to be developed in a manner similar to conventional software, but with an inherently parallelized structure. These developments have led to the creation of new GPU-accelerated tools, such as the MAGMA linear algebra library5, for example. Here, we present Gpufit: a GPU-accelerated implementation of the Levenberg-Marquardt algorithm. Gpufit was developed to meet the need for a high performance, general-purpose nonlinear curve fitting library which is publicly available and open source. As expected, this software exhibits significantly faster execution than the equivalent CPU-based code, with no loss of precision or accuracy. In this report we discuss the design of Gpufit, characterize its performance in comparison to other CPU-based and GPU-based algorithms, and demonstrate its use in a scientific data analysis application. ## Results The Gpufit library was designed to meet several criteria: (i) the software should make efficient use of the GPU resources in order to maximize execution speed, (ii) the interface should not require detailed knowledge of the GPU hardware, (iii) the source code should be modular and extendable, and (iv) the software should be accessible from multiple programming environments. Gpufit implements the LMA entirely on the GPU, minimizing data transfers between CPU and GPU memory. Copying memory between the CPU and the GPU is a slow operation, and it is most efficient to handle large blocks of data at once. Therefore, Gpufit was designed to process multiple fits simultaneously, each with the same model function and data size, but allowing for unique starting parameters for each fit. Large tasks (e.g. large numbers of fits) are divided into chunks, in order to balance processing and data transfer times. At the start of a calculation, a chunk of input data is copied to the GPU global memory, and upon completion the results are transferred back to CPU memory. GPU architecture is based on a set of parallel multiprocessors, which divide computations between blocks of processing threads, as illustrated in Supplementary Fig. S1. The efficiency of a GPU-based program depends on how these computing resources are used. While determining how best to parallelize the LMA, we found that different parts of the algorithm were most efficiently implemented with different parallelization strategies. For example, point-wise operations such as computation of the model function and its partial derivatives were more efficiently parallelized along the data coordinate index, meaning that each thread computes one model and derivative value at a particular coordinate. Other steps, such as the calculation of the Hessian matrix, were more efficiently parallelized along the index of the matrix element, i.e. with each thread assigned to calculate one element of the matrix. To accommodate the necessity for parallelizing different parts of the LMA in different ways, we structured Gpufit as a set of independent CUDA kernels, each responsible for a section of the algorithm. In this way, the blocks and threads of the GPU multiprocessors could be optimally allocated at each step. The details of the various parallelization schemes are documented in the Gpufit source code (see Code Availability). Another design consideration was how to distribute fits across the thread blocks of the GPU. Initially, we considered calculating one fit per thread block, and one data point per thread. However, this approach yielded poor results for small fit data sizes, due to the limited numbers of threads executed on each multiprocessor. Hence, the software was modified to allow the calculation of multiple fits in each thread block or, if necessary, to spread the calculation of a single fit over multiple thread blocks. Thus, the number of threads per block was maximized, yielding improved processing speeds without imposing restrictions on the size of each fit. Finally, we note that GPU shared memory (see Fig. S1) was used for the calculation of intermediate values requiring many reads and writes (e.g. the summation of chi-square), in order to take advantage of the faster performance of shared memory blocks. The computing resources of a GPU may vary significantly, depending on the details of the hardware. One of our aims was to avoid the necessity for any hardware-specific configuration parameters in the Gpufit interface. Therefore, Gpufit was designed to read the properties of the GPU at run-time, and automatically set parameters such as the number of blocks and threads available, and the number of fits to calculate simultaneously. This makes the use of a GPU transparent to the programmer, and allows the interface to be no different from that for a conventional curve fitting function. Moreover, it ensures that applications using Gpufit will perform optimally regardless of the machine on which they are running. The Gpufit source code is modular, such that fit functions and goodness-of-fit estimators are separate from the core sections of the code, and new functions or estimators may be added simply (see Supplementary Information). In its initial release, Gpufit includes two different fit estimators: the standard weighted least-squares estimator (LSE), and a maximum likelihood estimator (MLE) which provides better fit results when the input data is characterized by Poisson statistics6. The modular concept is illustrated schematically in Supplementary Fig. S2. This modularity allows Gpufit to be quickly adapted to new applications, or modified to accommodate future developments. Finally, Gpufit is written in C, CUDA C, and C++, and compiles to a Dynamic Link Library (DLL) providing a C interface, making it straightforward to call Gpufit from most programming environments. Furthermore, Gpufit bindings for Matlab and Python (e.g. the pyGpufit module) were implemented, forming part of the Gpufit distribution. ### Performance characterization We tested Gpufit by using the software to process randomized simulated datasets. The precision and accuracy of the fit results, as well as the number of fit iterations and the execution time, were measured. The input data consisted of 2D Gaussian functions defined by five parameters (see Supporting Information). Random noise (Gaussian or Poisson) was added to each data point. Test data was generated in Matlab and passed to the fit routines via their Matlab interfaces. ### Algorithm validation An initial step in characterizing Gpufit was to verify the correct operation of the algorithm. For this purpose, we tested Gpufit against the well-established MINPACK library7, evaluating the precision and accuracy of the fit results, and the convergence of the fit. Gaussian noise was added to the input data such that the signal to noise ratio (SNR) could be defined. Upon testing, the two packages yielded almost identical fit precision over a wide range of SNR values, and converged in a similar number of steps (Supplementary Fig. S3). At very high SNR, differences appear due to the limited numerical precision of floating point operations in CUDA (single precision) vs. MINPACK (double precision). Fit accuracy was checked by comparing the distributions of the fit parameters (Supplementary Fig. S4), and no systematic deviation between the output of Gpufit and MINPACK was detected. Together, these measurements demonstrate that Gpufit yields precise and accurate fit results, and converges similarly when compared to existing optimization software. ### Execution speed and fit precision The parallel architecture of the GPU enables significant speed improvements when a computation is amenable to being divided among multiple processors. To fairly evaluate the speed improvement obtained by shifting processing tasks from the CPU to the GPU, equivalent implementations of the LMA, running on both architectures, were required. We created a library called Cpufit to serve as the reference CPU-based fitting algorithm for testing purposes (see Methods). Cpufit implements precisely the same algorithm, section by section, as Gpufit. We verified that Cpufit and Gpufit returned numerically identical fit results given identical input data. A comparison of execution speeds, plotted against the number of fits per function call (N), is shown in Fig. 1. As expected, the fitting speed on the CPU remains constant as N changes, due to the sequential execution of each fit calculation. On the other hand, for Gpufit the speed increases with increasing N. The more fits that are calculated in one execution, the greater the benefit of parallelization. The crossover point at which use of the GPU becomes advantageous is, in this case, approximately $$N=130$$ fits. For very large N the speed of GPU processing saturates, indicating that GPU resources are fully utilized. At $$N={10}^{8}$$ fits per function call, we measured a processing speed of more than 4.5 million fits per second, approximately 42 times faster than the same algorithm executed on the CPU. Measurements of the execution time for each sub-section of the Cpufit and Gpufit code reveal the bottlenecks of the CPU-based algorithm, and how the computational workload is re-distributed on the GPU. For a set of $$5\times {10}^{6}$$ fits we measured the time duration of each step of the fitting process, and these results are shown in Fig. 2. For reference, program flowcharts for Cpufit and Gpufit, color coded according to processing time, are shown in Supplementary Fig. S5 and S6. In our tests, the most time-consuming task for Cpufit was the calculation of the model function and its derivatives, requiring more than one third of the total execution time. Gpufit completed the same calculation more than 100 times faster, due to the parallelization of the work. Similarly, all other steps in the fit process ran 10–100 times faster on the GPU. Gpufit includes additional operations, such as data transfers between CPU and GPU memory, however these did not impact performance when sufficient numbers of fits were processed. Given the parallel computing capability of the GPU, it was not surprising that Gpufit outperformed an equivalent algorithm running on the CPU. In order to verify that our code is efficiently implemented, we therefore tested Gpufit against another GPU-based fitting library: GPU-LMFit8. These tests were limited to smaller datasets because GPU-LMFit is available only as a closed-source, 32-bit binary package, restricting the size of the memory it can address. Figure 3a shows the speed of the Gpufit and GPU-LMFit libraries measured as a function of the number of fits per function call (N), with the speed of the MINPACK library shown for reference. Both packages exhibited similar scaling in speed as the number of fits and the data size varied, however, Gpufit showed faster performance for all conditions tested. As the data size per fit was increased (Fig. 3b), the speeds became more comparable, indicating that Gpufit makes more efficient use of GPU resources for smaller fits. The increased speed comes with no loss of precision, as it was also shown that the fit results returned by Gpufit and GPU-LMFit have virtually identical numerical precision (see Supplementary Fig. S7). When data is subject to counting statistics (i.e. when the noise has a Poisson distribution), curve fitting using an alternative goodness-of-fit measure based on maximum likelihood estimation (MLE) has been reported to yield more accurate fit results, when compared with least-squares fitting9,10. We tested the performance of this estimator using input data with simulated Poisson noise. The precision of the fit results, using either the unweighted LSE, weighted LSE, or MLE estimators, is shown in Supplementary Fig. S8. We found that the MLE estimator yielded better results than the LSE, particularly at small data values, although for larger values the two methods were approximately equivalent. ### Application to super-resolution microscopy Gpufit is well suited for rapid processing of large datasets, and its interface allows it to serve as a direct replacement for existing CPU-based fit routines. To demonstrate these capabilities in a real application, we integrated Gpufit into the image analysis pipeline of a super-resolution fluorescence microscopy experiment. Stochastic optical reconstruction microscopy (STORM) is a method for fluorescence imaging of biological samples, which obtains an image resolution significantly higher than the classical “diffraction limit” of far-field optical microscopy11. This method relies heavily on image processing: the generation of a single STORM image requires thousands to millions of individual fitting operations, and this task may require minutes to hours of computation before the final image is obtained. We integrated Gpufit into a recently published software package, Picasso12, which may be used to process raw STORM data into a super-resolved image of the sample. The Picasso software is written in Python, and is optimized to carry out the fit using a multi-threaded process running on all cores of the CPU. Moreover, Picasso uses “just in time” (JIT) compilation to optimize its execution speed. We modified this section of the Picasso source code, replacing the multi-core CPU-based fitting with a call to Gpufit, as illustrated schematically in Fig. 4a. Comparing the speed of Picasso before and after the modification demonstrates the benefit of Gpufit. When analyzing a raw STORM dataset (80000 images requiring 3.6 million fit operations), a fitting task which required 99.4 s with standard Picasso was completed in only 2.2 s after Gpufit was included, a 45-fold speed increase, with identical fit precision. The output STORM image is shown in Fig. 4b, and a comparison of the curve fitting time with and without Gpufit is shown in Fig. 4c. ## Discussion General purpose GPU computing is a relatively new technology, which is making an impact in many fields of science and engineering. The software introduced here, Gpufit, represents the first general purpose, open source implementation of a non-linear curve fitting algorithm for the GPU. It is intended as a compact, high performance optimization tool, easily modified and adapted for new tasks, which can be rapidly implemented within existing data analysis applications. In terms of performance, Gpufit exhibits similar precision and accuracy to other fitting libraries, but with significantly faster execution. In our measurements, curve fitting with Gpufit was approximately 42 times faster than the same algorithm running on the CPU. Gpufit also outperformed another GPU-based implementation of the LMA, GPU-LMFit, for all conditions tested. The absolute values of the timing results depend on the details of the fit and the computer hardware. Higher performance would be expected with a more powerful GPU (e.g. an Nvidia Tesla), or with multiple GPUs running in parallel. In addition to its speed, the principal advantages of Gpufit are its general purpose design, which may incorporate any model function or modified estimator, and the availability of the source code, which allows it to be compiled and run on multiple computing architectures. Since Gpufit automatically distributes the curve fitting tasks over the blocks and threads of the GPU multiprocessors, the user is not required to know the specifications of the hardware, thereby allowing Gpufit to be used as a “drop-in” replacement for existing fit functions. To demonstrate this, we modified Picasso to make use of Gpufit rather than its own multi-core CPU fitting code. Despite the fact that curve fitting in Picasso was already parallelized (by virtue of its use of multi-threaded processing) we found that Gpufit outperformed the built-in Picasso curve fitting by a factor of 45 times (Fig. 4). The ease with which Gpufit was integrated into Picasso, and the gain in performance, show that our original design goals were met. As of its initial release, the Gpufit package has several limitations. First, the fit model functions are built into the code at compilation time, and the addition or modification of a model function requires re-compilation of the source code. We also note that Gpufit requires the explicit calculation of the partial derivatives of the model, and expressions for these functions must be present in the code embodying the fit model function. However, as an open-source software project, we expect that Gpufit will continue to develop and improve, potentially removing these limitations in future versions. For example, runtime compilation of model functions written in CUDA would lift the requirement for re-building the source, and methods to approximate the derivatives numerically could also be introduced. Finally, there is the potential for porting Gpufit to other general-purpose GPU computing languages, such as OpenCL, thereby allowing the software to function on other GPU hardware platforms. Using an inexpensive graphics card and a standard PC, we achieved speeds higher than 4.5 million fits per second for data that is typical of STORM experiments (Fig. 1). Considering recent developments in localization based super-resolution methods, in particular experiments in which > 400 million individual fluorophore switching events were recorded13, data processing speed has become highly relevant. A tool such as Gpufit could make the difference between a researcher waiting minutes (with Gpufit) or hours (without) before the image can be examined. This rapid feedback has a greater importance than simply reducing waiting time – it enables the quick screening of samples and test conditions, ultimately leading to higher quality image data. Furthermore, we expect that Gpufit will facilitate the adoption of new, computationally demanding data analysis approaches, such as cubic spline fitting14, in order to further improve STORM image resolution. ## Conclusions We have developed a GPU-accelerated implementation of the Levenberg Marquardt algorithm, with significantly faster performance as compared with traditional CPU-based software. Gpufit is designed to be general purpose, and as such we expect it to be useful for diverse applications in science and engineering which may depend critically on rapid data processing, e.g. high-speed feedback systems and the recently described MINFLUX method for particle tracking and nanoscale imaging15. The Gpufit library is accessible from most programming environments, and the automatic configuration of GPU-specific parameters makes it simple to work with in practice. The modularity of Gpufit facilitates the addition of custom fit models or estimators, for example the mixed Gaussian/Poisson likelihood model required for accurately analyzing sCMOS camera data16. Finally, Gpufit makes efficient use of GPU computing resources by exploiting simple parallelization schemes as well as distributing fit tasks evenly over the multiprocessors of the GPU to achieve high performance. Using Gpufit with our own hardware, we obtained a 42-fold improvement in execution time for batch-processing of curve fitting tasks, with no loss of precision or accuracy. Gpufit is an open source software project, and the source code is available via the Github public software repository. Open source software development is advantageous from several standpoints: it enhances code integrity through review by users, and offers the possibility for users to fix bugs and add features17. In addition, open source code makes the workings of an algorithm transparent to the user, which may be a crucial factor when a “black box” software tool is not sufficient, such as in scientific data analysis applications. Finally, we note that Marquardt’s original paper introducing his algorithm, published in 1963, concludes with the following sentence: “A FORTRAN program … embodying the algorithm described in this paper is available as IBM Share Program No. 1428”1. The SHARE software library referred to by Marquardt was, in fact, an early form of open source software18, and we find it appropriate that Gpufit should be published in a similar manner. ## Methods ### Levenberg-Marquardt algorithm The LMA provides a general numerical procedure for fitting a non-linear model function to a set of data points. It may be considered as a combination of the method of steepest descent and Newton’s method, having a high probability of convergence even when the initial parameter estimates are poor, and fast convergence near the minimum. The standard algorithm, as described by Marquardt1, minimizes iteratively the general least squares equation: $${\chi }^{2}\,(\mathop{{a}}\limits^{{\rightharpoonup }})\,=\,\sum _{n=0}^{N-1}{({f}_{n}(\mathop{a}\limits^{\rightharpoonup })-{z}_{n})}^{2}$$ (1) where $${f}_{n}$$ are the model function values, $$\mathop{a}\limits^{\rightharpoonup }$$ is the vector of model parameters, and $${z}_{n}$$ are the set of N data points. To find the minimum of $${\chi }^{2}\,(\mathop{a}\limits^{\rightharpoonup })$$ (chi-square), the algorithm performs an iterative search of the parameter space for a coordinate where the gradient of the function equals zero, i.e. $$\nabla {\chi }^{2}\,(\mathop{a}\limits^{\rightharpoonup })=0$$. The gradient is approximated by a Taylor expansion: $$0\,=\,\nabla {\chi }^{2}\,({\mathop{a}\limits^{\rightharpoonup }}_{i}+{\mathop{\delta }\limits^{\rightharpoonup }}_{i})\,\approx \,\nabla {\chi }^{2}\,({\mathop{a}\limits^{\rightharpoonup }}_{i})\,+\,{H}_{{\chi }^{2}}\,({\mathop{a}\limits^{\rightharpoonup }}_{i})\,{\mathop{\delta }\limits^{\rightharpoonup }}_{i}$$ (2) where $${H}_{{\chi }^{2}}\,({\mathop{a}\limits^{\rightharpoonup }}_{i})$$ is the Hessian matrix of $${\chi }^{2}({\mathop{a}\limits^{\rightharpoonup }}_{i})$$, and $${\mathop{\delta }\limits^{\rightharpoonup }}_{i}$$ is a small correction to $${\mathop{a}\limits^{\rightharpoonup }}_{i}$$ (the index i corresponds to the iteration number). The expression for the Hessian includes the first and the second partial derivatives of $${\chi }^{2}\,(\mathop{a}\limits^{\rightharpoonup })$$, however terms containing the second derivatives are assumed negligible and ignored. Solving (2) for $${\mathop{\delta }\limits^{\rightharpoonup }}_{i}$$ yields the Newton step for the minimization of $${\chi }^{2}\,(\mathop{a}\limits^{\rightharpoonup })$$. Up to this point, the LMA is equivalent to Newton’s Method. A special characteristic of the LMA is the damping factor λ which controls the step size of each iteration by modifying the diagonal elements of the Hessian: $${H^{\prime} }_{{\chi }^{2},kl}\,(\mathop{a}\limits^{\rightharpoonup })\,=\,\{\begin{array}{lc}{H}_{{\chi }^{2},kl}\,(\mathop{a}\limits^{\rightharpoonup }), & {\rm{for}}\,k\ne l\\ {H}_{{\chi }^{2},kl}\,(\mathop{a}\limits^{\rightharpoonup })\cdot (1+\lambda ), & {\rm{for}}\,k=l\end{array}$$ (3) where k and l are the matrix indices. The positive factor λ is initialized with a small value, and thus initially the algorithm behaves like Newton’s method. As the algorithm iterates, if the value of chi-square in the latest iteration is smaller than in the previous step, λ is decreased by a constant factor v. Otherwise, λ is increased by the same factor. Increasing λ causes the LMA to tend towards the behavior of the method of steepest descent. In this manner, the LMA adjusts between the two methods, as the minimum is approached. By transposing equation (2) and applying the damping factor (3), a system of linear equations is obtained which may be solved for $${\mathop{\delta }\limits^{\rightharpoonup }}_{i}$$, e.g. by the Gauss-Jordan method3: $${\mathop{\delta }\limits^{\rightharpoonup }}_{i}\,=\,-H{{^{\prime} }_{{\chi }^{2}}}^{-1}\,({\mathop{a}\limits^{\rightharpoonup }}_{i})\cdot \nabla {\chi }^{2}\,({\mathop{a}\limits^{\rightharpoonup }}_{i}).$$ (4) If the iteration is successful (chi-square decreased), the difference $${\mathop{\delta }\limits^{\rightharpoonup }}_{i}$$ is added to the previous parameter values: $${\mathop{a}\limits^{\rightharpoonup }}_{i+1}\,=\,{\mathop{a}\limits^{\rightharpoonup }}_{i}\,+\,{\mathop{\delta }\limits^{\rightharpoonup }}_{i}.$$ (5) The damping factor λ is updated after each iteration, and convergence is tested. Any convenient convergence criterion may be used, but in general the overall convergence of the LMA depends on the relative size of the parameter adjustment in each iteration. As originally set out by Marquardt, with a reasonable choice of r and $$\varepsilon$$ (e.g. $$r={10}^{-3}$$ and $$\varepsilon ={10}^{-5}$$), the algorithm has converged when $$\frac{\|{\delta }_{i,j}\|}{r+\|{a}_{i,j}\|}\, < \,\varepsilon$$ (6) is satisfied for all parameters, where r is a small positive constant (to avoid division by zero), i is the iteration number, and j is the parameter index. ### Estimators of best fit Least squares estimation (LSE) is a common method for finding the parameters which yield the minimal deviation between observed data and a model function. The standard LMA minimizes the general LSE formula given by equation (1). However, it is also possible to include weighting factors in the calculation of chi-square, for example: $${\chi }_{{\rm{LSE}}\,}^{2}(\mathop{a}\limits^{\rightharpoonup })\,=\,{\sum _{n=0}^{N-1}(\frac{{f}_{n}(\mathop{a}\limits^{\rightharpoonup })-{z}_{n}}{{\sigma }_{n}})}^{2}$$ (7) where $${\sigma }_{n}$$ represents the uncertainty (standard deviation) of the data. This allows the precision of each data point to be taken into account. In cases where the uncertainties of the data points are Poisson distributed, a maximum likelihood estimator (MLE) yields more precise parameter estimates than the LSE9,10. In this situation it is beneficial to use an alternative estimator with the LMA. A procedure has been described6 in which the LSE formula (1) is replaced by the MLE equation for Poisson deviates, as follows: $${\chi }_{{\rm{M}}{\rm{L}}{\rm{E}}\,}^{2}(\mathop{a}\limits^{\rightharpoonup })\,=\,2[\sum _{n=0}^{N-1}({f}_{n}(\mathop{a}\limits^{\rightharpoonup })-{z}_{n})\,-\,\sum _{n=0,\,{z}_{n}\ne 0}^{N-1}{z}_{n}\,{\rm{l}}{\rm{n}}(\frac{{f}_{n}(\mathop{a}\limits^{\rightharpoonup })}{{z}_{n}})].$$ (8) Using this estimator within the context of the LMA is relatively simple to implement, requiring only the calculation of the gradient and Hessian matrix of $${\chi }_{{\rm{MLE}}\,}^{2}(\mathop{a}\limits^{\rightharpoonup })$$, and the calculation of $${\chi }_{{\rm{MLE}}\,}^{2}(\mathop{a}\limits^{\rightharpoonup })$$ itself. As before, in the calculation of the Hessian matrix, terms containing second partial derivatives are ignored. ### Gauss-Jordan elimination Gauss-Jordan elimination was used to solve the system of equations (4) in the LMA, and a parallelized Gauss-Jordan algorithm was developed for this purpose. The algorithm was parallelized according to the elements of the augmented matrix, with each thread responsible for one element of the matrix as the left side is transformed to reduced row echelon form. Partial pivoting (row swapping) was used to ensure precise and numerically stable calculations3. The sorting step in the pivot operation was accomplished on the GPU by means of a parallel bitonic merge sort19. The details of this algorithm are fully documented in the Gpufit source code. ### Software for comparison tests To evaluate its performance, Gpufit was compared against an equivalent CPU-based algorithm (Cpufit) and two other curve fitting libraries: MINPACK7 and GPU-LMFit8. Cpufit is a standard implementation of the LMA based on published examples1,3, which we wrote in C++ for execution on the CPU. C++ Minpack is an open-source C/C++ implementation of MINPACK which runs on the CPU (we used the function lmder() from this library)7,20. Both C++ Minpack and Cpufit were run in single CPU threads. GPU-LMFit is a closed-source implementation of the LMA (32-bit binary files are publicly available) which runs on the GPU and provides the option of using LSE or MLE as the estimator8. ### Computer hardware All tests were executed on a PC running the Windows 7 64-bit operating system and CUDA toolkit version 8.0. The PC hardware included an Intel Core i7 5820 K CPU, running at 3.3 GHz, and 64 GB of RAM. The graphics card was an NVIDIA GeForce GTX 1080 GPU with 8 GB of GDDR5X memory. All source codes, including the Cpufit, Gpufit, and C/C++ Minpack libraries, were compiled using Microsoft Visual Studio 2013, with compiler optimizations enabled (release mode). Additional details of the test conditions and instructions for compiling the Cpufit and Gpufit software libraries are provided in the Supplementary Information and the Gpufit documentation. ### Source code availability The source code for Gpufit, including external bindings, is available for download from a public software repository located at http://www.github.com/gpufit/Gpufit. The source code for the CPU-based reference algorithm, Cpufit, is also included in this repository. The Gpufit User’s Manual, describing how to use the software and containing instructions for building and modifying the source code, may be found online at http://gpufit.readthedocs.io. ### Data availability The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request. ## References 1. 1. Marquardt, D. W. An Algorithm for Least-Squares Estimation of Nonlinear Parameters. J. Soc. Ind. Appl. Math 11, 431–441 (1963). 2. 2. Moré, J. J. in Numerica l An al ysis Vol. 630 Lecture Notes in Mathematics (ed G. A. Watson) 105–116 (Springer Berlin, 1978). 3. 3. Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. Numerical Recipes in FORTRAN; The Art of Scientific Computing. (Cambridge University Press, 1993). 4. 4. Du, P. et al. From CUDA to OpenCL: Towards a performance-portable solution for multi-platform GPU programming. Parallel Comput. 38, 391–407 (2012). 5. 5. Dongarra, J. et al. in Numerical Computations with GPUs (ed V. Kindratenko) 3–28 (Springer International Publishing, 2014). 6. 6. Laurence, T. A. & Chromy, B. A. Efficient maximum likelihood estimator fitting of histograms. Nat. Methods 7, 338–339 (2010). 7. 7. Moré, J. J., Garbow, B. S. & Hillstrom, K. E. User Guide for MINPACK-1. (Argonne National Laboratory, 1980). 8. 8. Zhu, X. & Zhang, D. Efficient Parallel Levenberg-Marquardt Model Fitting towards Real-Time Automated Parametric Imaging Microscopy. PLoS One 8, e76665 (2013). 9. 9. Abraham, A. V., Ram, S., Chao, J., Ward, E. S. & Ober, R. J. Quantitative study of single molecule location estimation techniques. Opt. Express 17, 23352–23373 (2009). 10. 10. Smith, C. S., Joseph, N., Rieger, B. & Lidke, K. A. Fast, single-molecule localization that achieves theoretically minimum uncertainty. Nat. Methods 7, 373–375 (2010). 11. 11. Bates, M., Huang, B. & Zhuang, X. Super-resolution microscopy by nanoscale localization of photo-switchable fluorescent probes. Curr. Opin. Chem. Biol. 12, 505–514 (2008). 12. 12. Schnitzbauer, J., Strauss, M. T., Schlichthaerle, T., Schueder, F. & Jungmann, R. Super-resolution microscopy with DNA-PAINT. Nat. Protocols 12, 1198–1228 (2017). 13. 13. Legant, W. R. et al. High-density three-dimensional localization microscopy across large volumes. Nat. Methods 13, 359–365 (2016). 14. 14. Babcock, H. P. & Zhuang, X. Analyzing Single Molecule Localization Microscopy Data Using Cubic Splines. Sci. Rep. 7, 552 (2017). 15. 15. Balzarotti, F. et al. Nanometer resolution imaging and tracking of fluorescent molecules with minimal photon fluxes. Science (2016). 16. 16. Huang, F. et al. Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms. Nat. Methods 10, 653–658 (2013). 17. 17. Morin, A., Urban, J. & Sliz, P. A Quick Guide to Software Licensing for the Scientist-Programmer. PLoS Comp. Biol. 8, e1002598 (2012). 18. 18. Jones, C. The Technical and Social History of Software Engineering. (Pearson Education, 2013). 19. 19. Buck, I. & Purcell, T. in GPU Gems Vol. 1 (ed R. Fernando) 621–636 (Addison Wesley, 2004). 20. 20. Devernay, F. C/C++ Minpack, http://devernay.free.fr/hacks/cminpack/ (2007). ## Acknowledgements We would like to thank Dr. C. Wurm and Dr. E. Rothermel for the preparation of the sample used for STORM imaging, and also Dr. V. Cordes for providing the primary antibody against GP210. We thank N. Warmbold for contributions to an early version of the Gauss-Jordan algorithm used in the Gpufit source code. We thank Dr. M. Roose for excellent IT support. We thank Prof. Dr. S. W. Hell for generous support in the form of funding and equipment. M.B. gratefully acknowledges funding from the European Molecular Biology Organization (ALTF 800–2010) and the Max Planck Society. ## Author information Authors ### Contributions M.B., B.T., and B.S. conceived the project. A.P., B.T., J.K., and M.B. wrote the Cpufit and Gpufit source code. A.P. and M.B. carried out the quantitative evaluation of Gpufit. M.B. performed the STORM experiment. A.P. and J.K. wrote the external bindings for Gpufit. J.K. created the usage examples. J.K., A.P., M.B., and B.T. wrote the documentation. M.B., B.T., and B.S. supervised the research. M.B. wrote the manuscript. ### Corresponding author Correspondence to Mark Bates. ## Ethics declarations ### Competing Interests The authors declare that they have no competing interests. Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. ## Rights and permissions Reprints and Permissions Przybylski, A., Thiel, B., Keller-Findeisen, J. et al. Gpufit: An open-source toolkit for GPU-accelerated curve fitting. Sci Rep 7, 15722 (2017). https://doi.org/10.1038/s41598-017-15313-9 • Accepted: • Published: • ### Nanometric axial localization of single fluorescent molecules with modulated excitation • Pierre Jouchet • , Clément Cabriel • , Nicolas Bourg • , Marion Bardou • , Christian Poüs • , Emmanuel Fort • & Sandrine Lévêque-Fort Nature Photonics (2021) • ### Molecular-scale axial localization by repetitive optical selective exposure • Lusheng Gu • , Yuanyuan Li • , Shuwen Zhang • , Maoge Zhou • , Yanhong Xue • , Weixing Li • , Tao Xu • & Wei Ji Nature Methods (2021) • Vytautas Zickus • , Ming-Lo Wu • , Kazuhiro Morimoto • , Valentin Kapitany • , Areeba Fatima • , Alex Turpin • , Robert Insall • , Jamie Whitelaw • , Laura Machesky • , Claudio Bruschini • , Daniele Faccio • & Edoardo Charbon Scientific Reports (2020) • ### Evaluation of the ITER Real-Time Framework for Data Acquisition and Processing from Pulsed Gigasample Digitizers • , B. Jablonski • , P. Perek • & D. Makowski Journal of Fusion Energy (2020) • ### T2 and T2∗ mapping in ex situ porcine myocardium: myocardial intravariability, temporal stability and the effects of complete coronary occlusion • Bridgette Webb • , Martin Manninger • , Marlene Leoni • , Thomas Widek • , Martin Dobrovnik • , Daniel Scherr • , Rudolf Stollberger • & Thorsten Schwark International Journal of Legal Medicine (2020)	2021-09-24 01:19:20	{"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": 0, "mathjax_display_tex": 2, "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": 0.46171462535858154, "perplexity": 1893.6169413963623}, "config": {"markdown_headings": true, "markdown_code": false, "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/1631780057479.26/warc/CC-MAIN-20210923225758-20210924015758-00465.warc.gz"}
https://www.npsm-kps.org/main.html	pISSN 0374-4914 eISSN 2289-0041 • ### Research Paper 2020-12-31 #### Pixel Design Study of a CMOS Monolithic Active Pixel Sensor for the Charge Collection time Sanghyeon LEE, In-Kwon YOO* Abstract : A new silicon chip (INVESTIGATOR) manufactured for R\&D purposes has 134 mini-matrices with various pixel designs. Each matrix consists of 8 $\times$ 8 pixels, which put out 64 analogue signals at 65 MHz. The silicon pixel design is based on the newest technology of the complementary metal-oxide-semiconductor (CMOS) monolithic active pixel sensor (MAPS), which integrates the silicon sensor and the read-out circuitry in a pixel. The MAPS has advantages of low power consumption, high granularity of a pixel, and fast read-out. In this paper, the charge collection time for different pixel designs and reverse bias voltages is studied by using the INVESTIGATOR. The charge collection time is estimated by fitting the waveform for changing pixel pitch, reverse bias voltage, diameter of collection n-well diode, and spacing. Based on these results, we discuss the dependence of the relative depletion volume and the charge collection time on the pixel geometry. • ### Research Paper 2020-12-31 #### $\alpha$ + $^{116}$Sn and $^{6}$Li + $^{116}$Sn Elastic Scatterings at $E_{lab}=$ 240 MeV : Coulomb-modified Glauber Model Approach Yong Joo KIM* Abstract : We analyzed experimental data on elastic $\alpha$ + $^{116}$Sn and $^{6}$Li + $^{116}$Sn scatterings at $E_{lab}=$ 240 MeV within the framework of the Coulomb-modified Glauber model. The ingredients of the model used in this work were the nucleon-nucleon ($NN)$ amplitude and the densities of the colliding nuclei. The calculations included the effective $NN$ amplitude considering a $q^{4}$ component and the surface-matched Gaussian density of the target nucleus. The calculated results reproduced satisfactorily the structures of differential cross sections and agreed well with the experimental data. \ The oscillatory structures observed in the angular distributions were explained using the strong interference between the near-side and the far-side scattering amplitudes. We found that the introduction of both an effective $NN$ amplitude and a surface-matched Gaussian density plays an important role in providing a better description of the elastic data. • ### Research Paper 2020-11-30 #### Effects of Partial Substitution of Nitrate on the Superconducting Properties of TlSr$_{4}$Cu$_{2}$O$_{z}$(SO$_{4}$) Ho Keun LEE* Abstract : The effects of partial substitution of nitrate groups at the SO$_{4}$ sites on the superconducting properties of the TlSr$_{4}$Cu$_{2}$O$_{z}$(SO$_{4}$)$_{1-x}$(NO$_{3}$)$_{x}$ system have been investigated. X-ray diffraction data showed that the solubility limit of (NO$_{3}$) was about $x$ = 0.35. The change in the solubility limit of oxyanion for sulfate was discussed in connection with the characteristics of the oxyanion species. Within the solubility limit, the introduction of the nitrate groups did not change significantly the transition temperature of the pristine TlSr$_{4}$Cu$_{2}$O$_{z}$(SO$_{4}$) compound. However, we found that compared to the pristine compound, the introduction of the nitrate groups degraded the stability properties of the sample after annealing at 400$^{\circ}$C in Ar atmosphere. • ### Research Paper 2020-11-30 #### Fabrication of Epitaxial Cu$_{2}$O (111) Films from Cu(111) thin Films by Rapid Thermal Oxidation Miyeon CHEON, Yousil LEE, Sujae KIM et al. Abstract : Due to the energy gaps of copper from 2.1 to 2.7 eV, its high light absorption, its nontoxicity and its abundance on the earth, Cu$_{2}$O is an attractive material for use in various areas such as photovoltaic power generation, photocatalytic reactions, water photolysis, nonlinear optics, and gas sensing. Many researches efforts are being conducted to obtain high-quality Cu$_{2}$O thin films. In this study, high-quality, epitaxial Cu$_{2}$O (111) thin films were obtained via a relatively simple method, rapid thermal processes at high temperature of RF sputtered Cu (111) thin film on a sapphire substrate. XRD, SEM and UV-Vis spectroscopy measurements confirmed the high crystallinity of the Cu$_{2}$O (111) thin film oxidized for 30 minutes at a temperature of 800 $^{\circ}$C under an atmosphere of argon with 3 ppm of oxygen. Also, because of the high crystal-quality of the Cu$_{2}$O (111) thin films, blue and indigo energy gaps at room temperature were obtained from the absorption coefficient $\alpha$. The obtained energy band gaps are consistent with the theoretical values obtained from Cu$_{2}$O bulk structures. • ### Research Paper 2020-10-30 #### Far-infrared spectroscopic study on MAPbI$_3$ and MAPbBr$_3$ Jaeseung LIM, Sangheon PARK, Yu-Seong SEO et al. Abstract : MAPbX$_3$, is an organic-inorganic perovskite material system which can be applied in various areas such as magneto-optical data storage, solar cells, lasers, LEDs, etc. MAPbI$_3$ and MAPbBr$_3$ are known to undergo a cubic-to-tetragonal transition at temperatures of about 327 K and 220 K and a tetragonal-to-orthorhombic transition at about 150 K and 145 K, respectively. The transmittance spectra of pallet samples are measured in the far-infrared (FIR) region at various temperatures from room temperature to 80 K by using a Fourier transform infrared (FTIR) spectroscopy-type Bruker Vertex 80v spectrometer. The absorption coefficients are obtained and fitted by using the Drude-Lorentz model to obtain other optical constants including the electric permittivity, optical conductivity, and extinction coefficient. Then, the optical conductivity is fitted to obtain the position and damping coefficient of the longitudinal optical (LO) and the transverse optical (TO) phonons for the sample materials, which are used to calculate the electron-phonon coupling constants, polaron mass, and polaron radii. ## Current Issue December, 2020 \| Volume 70, No. 12 • ### Research Paper 2020-12-31 #### First-principles Study of Intrinsic Stability of Perovskite Halide MAPbI$_{3}$ Dokyun KIM, Chul Hong PARK* Abstract : We examined the stability of the MAPbI$_{3}$ perovskite through a first-principles total energy calculation and estimated the thermodynamic energy through phonon calculations. The total energies calculated for the ground state showed that the energy for the dissociation of MAPbI$_{3}$ into MAI and PbI$_{2}$ is small relative to the thermal energy at room temperature, indicating that MAPbI$_{3}$ is not stable for dissociation. We estimated the dependence of the thermodynamics free energies of MAPbI$_{3}$ and MAI and PbI$_{2}$ on the temperature. We found that the entropy effect can contribute to the stabilization of MAPbI$_{3}$, however the stability is not so robust. Because the PbI$_{2}$ has a two-dimensional layered structure, the entropy effect is small compared to that of the three-dimensional structure. • ### Research Paper 2020-12-31 #### Research on Microstrip Transmission Line Design for Microwave Memristive Devices Hanju LEE* Abstract : We conducted research on the microwave memristive response of a microstrip transmission line (MTL) coupled with an yttrium iron garnet (YIG) thin film. We investigated the memristive responses of YIG-MTL devices with MTL signal lines of different widths. Measurement results showed that a decrease in the width of the MTL signal line resulted in an improvement of the microwave memristive response of the YIG-MTL device. Based on transmission line theory, we showed that the enhancement of the memristive property of the MTL was due to an increase in the inductance that was caused by a decrease in the width of the signal line. The increase in the inductance of the MTL caused an increase in the strength of the microwave magnetic near field, and as a result, the magnetic coupling between the YIG and MTL was improved. The present result provides a useful way to improve the memristive response of a wave memristive device and should be helpful in the design of wave memristive microwave device. • ### Research Paper 2020-12-31 #### Up- and down-conversion Luminescence Characteristics of (Gd$_{0.85-x}$Yb$_{0.15}$)AlO$_{3}$:Er$_{x}^{3+}$ Phosphors Kang Hyun KIM, Soung Soo YI* Abstract : Er$^{3+}$and Yb$^{3+}$ co-doped Gd$_{(0.85-x)}$AlO$_{3}$:Yb$_{0.15}^{3+}$, Er$_{x}^{3+}$ ($x$ = 0.06, 0.09, 0.12 and 0.15) polycrystalline powders have been prepared by using a solid-state reaction method. The crystallinity of the powders shows a polycrystalline orthorhombic system. For the down-conversion photoluminescence, through the direct excitation of Er$^{3+}$ ions using a wide emission spectrum in the range of 420 ~ 500 nm from the host crystal and 520 ~ 570 nm from the Er$^{3+}$ ions has been observed. The up-conversion photoluminescence properties of Gd$_{(0.85-x)}$AlO$_{3}$:Yb$_{0.15}^{3+}$, Er$_{x}^{3+}$ phosphors were investigated in detail. The green and the red up-conversion emissions from the phosphors were observed under an excitation at 980 nm by using a semiconductor laser. The powders exhibited strong green and weak red up-conversion emission peaks at 545 and 657 nm, respectively. The luminescence intensity showed the different Er$^{3+}$ concentration dependencies for the up- and down-conversion luminescence behaviors due to the difference in their luminescence mechanisms. The maximum intensities for the up- and down-conversion luminescence occurred at Er$^{3+}$ ions 0.12 and 0.09 mol, respectively. • ### Research Paper 2020-12-31 #### Microwave Synthesis and Luminescent Properties of red Emitting Ca$_{3}$ZrSi$_{2}$O$_{9}$:Eu$^{3+}$ Phosphor for Latent Fingerprint Detection Woo Tae HONG, Hyun Kyoung YANG, Byung Kee MOON* Abstract : Ca$_{3}$ZrSi$_{2}$O$_{9}$:Eu$^{3+}$ (CZS:Eu$^{3+}$) phosphors were synthesized by using a microwave irradiation method. The crystal structure, surface morphology and luminescence properties of the phosphors were analyzed for different Eu$^{3+}$ concentrations. The CZS:Eu$^{3+}$ phosphor exhibited their strongest emission at 610 nm under 392-nm excitation. Because the CZS:Eu$^{3+}$ phosphor contains 7 mol% of Eu$^{3+}$, the luminescent intensity at the dominant emission wavelength was maximized. Considering these analysis we applied, the CZS:Eu$^{3+}$ phosphors to detect latent fingerprints on various substrate materials. Under UV excitation, the detected fingerprints showed a red emission with high resolution. The results of this study show that the CZS:Eu$^{3+}$ phosphor can be used as luminescent sensor to detect latent fingerprints. • ### Research Paper 2020-12-31 #### Synthesis and Characterization of Mn-doped CdSe Quantum dots Eun-Bee JUNG, Il-Gon KIM, Dong-Sun YOO Abstract : We fabricated Mn-doped CdSe QDs by mixing Mn-acetate (Mn(CH$_{3}$CO$_{2}$)4H$_{2}$O) with 0.5 mmol of cadmium oxide (CdO) in the stage of Cd$^{2+}$ precursor synthesis and investigated the effects of Mn doping. When the amount of Mn-acetate mixed was below 0.1 mmol, the properties of QDs doped like PL spectra and the XRD patterns, were not different from those of undoped QDs, and this resulted from the difference in reaction rates for Cd-Mn and Mn-oleate formation. On the other hand, when the amount of Mn-acetate mixed reached 0.5 mmol, the XRD patterns and the PL spectra showed that the doped QDs had a core-shell structure a CdSe core with a big band gap energy and a MnSe shell with small band gap energy, and in this core-shell structure, the surface passivation ligands changed from Cd-oleate of CdSe to Mn-oleate of MnSe. • ### Research Paper 2020-12-31 #### Pixel Design Study of a CMOS Monolithic Active Pixel Sensor for the Charge Collection time Sanghyeon LEE, In-Kwon YOO Abstract : A new silicon chip (INVESTIGATOR) manufactured for R\&D purposes has 134 mini-matrices with various pixel designs. Each matrix consists of 8 $\times$ 8 pixels, which put out 64 analogue signals at 65 MHz. The silicon pixel design is based on the newest technology of the complementary metal-oxide-semiconductor (CMOS) monolithic active pixel sensor (MAPS), which integrates the silicon sensor and the read-out circuitry in a pixel. The MAPS has advantages of low power consumption, high granularity of a pixel, and fast read-out. In this paper, the charge collection time for different pixel designs and reverse bias voltages is studied by using the INVESTIGATOR. The charge collection time is estimated by fitting the waveform for changing pixel pitch, reverse bias voltage, diameter of collection n-well diode, and spacing. Based on these results, we discuss the dependence of the relative depletion volume and the charge collection time on the pixel geometry. • ### Research Paper 2020-12-31 #### Growth Probability of an Additional Offspring with a Beneficial Reversal Allele in the Four-State Haploid Coupled Discrete-Time Mutation-Selection Model for a Finite Population Wonpyong GILL* Abstract : Growth probabilities of an additional offspring with a beneficial reversal allele were calculated by computer simulation for various population sizes, sequence lengths, selective advantages, and measuring parameters for a finite population in the four-state haploid coupled discrete-time mutation-selection (HCDMS) model. The mutation rates between all sequence elements were set to be equal. This study suggested that the boundary between the deterministic and the stochastic regions in the four-state HCDMS model could be determined by using the same criterion as that in the two-state HCDMS model. For various population sizes, sequence lengths, measuring parameters, and selective advantages, the growth probabilities in the stochastic region could be described using the theoretical formula for the growth probability in the Wright-Fisher two-allele model. • ### Research Paper 2020-12-31 #### Mesoscale Properties of Mutualistic Networks in Ecosystems Sang Hoon LEE* Abstract : Uncovering the structural properties of ecological networks is a crucial starting point when studying the system's stability in response to various types of perturbations. We analyze pollination and seed disposal networks, which are representative examples of mutualistic networks in ecosystems, in various scales. In particular, we examine mesoscale properties such as the nested structure, the core-periphery structure, and the community structure by statistically investigating their interrelationships with real network data. As a result of community detection on different scales, we find an absence of a meaningful hierarchy between networks, and a negative correlation between the modularity and the two other structures (nestedness and core-periphery-ness), which themselves are highly positively correlated. In addition, no characteristic scale for the communities is perceivable from the community-inconsistency analysis. Therefore, community structures, which are the most widely studied mesoscale structures of networks, are not, in fact, adequate to characterize mutualistic networks of this scale in ecosystems. • ### Research Paper 2020-12-31 #### $\alpha$ + $^{116}$Sn and $^{6}$Li + $^{116}$Sn Elastic Scatterings at $E_{lab}=$ 240 MeV : Coulomb-modified Glauber Model Approach Yong Joo KIM* Abstract : We analyzed experimental data on elastic $\alpha$ + $^{116}$Sn and $^{6}$Li + $^{116}$Sn scatterings at $E_{lab}=$ 240 MeV within the framework of the Coulomb-modified Glauber model. The ingredients of the model used in this work were the nucleon-nucleon ($NN)$ amplitude and the densities of the colliding nuclei. The calculations included the effective $NN$ amplitude considering a $q^{4}$ component and the surface-matched Gaussian density of the target nucleus. The calculated results reproduced satisfactorily the structures of differential cross sections and agreed well with the experimental data. \ The oscillatory structures observed in the angular distributions were explained using the strong interference between the near-side and the far-side scattering amplitudes. We found that the introduction of both an effective $NN$ amplitude and a surface-matched Gaussian density plays an important role in providing a better description of the elastic data. • ### Research Paper 2020-12-31 #### Content Analysis of Interference and Diffraction Presented in High School Physics Textbooks Bongwoo LEE* Abstract : The purpose of this study is to analyze the explanation of interference and diffraction presented in high school physics textbooks. The definitions and the principles of interference and diffraction, as well as examples of interference and diffraction, in 13 physics textbooks according to the 2015 revised curriculum were analyzed. The definitions of interference were similarly described for each textbooks, but inaccurate expressions were found in the section describing coherence and constructive/destructive interference as a phase. Regarding diffraction, the degree of explanation of Huygen’s principle differed among textbooks, and some textbooks used the concept of 'interference' in the process of explaining the principle of diffraction. Examples of interference and diffraction have been described in various examples, but the same phenomena have been explained by using different principles (interference, diffraction) depending on the textbook. Based on the results, we discuss the unnecessariness of the dichotomy of interference and diffraction, as well as the teaching method for interference and diffraction. • ### Research Paper 2020-12-31 #### Comparison of Characteristics of Research Paper Introduction by Research Types in Physics Education Kwanghee JO* Abstract : This study attempted to examine the similarities and differences in the research paper introduction description of Korean physics education according to the research type. To this end, we reviewed physics education papers recently published in the journal “New Physics: Sae Mull,” and selected research papers corresponding to three representative types. The number of sentences constituting the introduction part was the lowest in experiment development studies and the highest in conception survey studies among the three types. According to the genre analysis results, the arrangement of the steps or the ratio of moves in the introduction description for each research type was different. In the experiment development study, the step of presentation of current situation occupied the greatest ration in the introduction, and relatively more sentences were used to derive the research necessity. Textbook analysis studies tended to focus on the obligatory steps of introduction. Overall, there was relatively little variation was seen among the papers belonging to this type. An average of more than 20 sentences was allocated to the presentation of previous research and the specification the research topic in the conception survey study. Through this study, some of the characteristics of a research paper introduction were found to differ depending on the research types. • ### 2020-12-31 #### Landau States in Time-Dependent Magnetic Fields Won KIM, Sang Pyo KIM* Abstract : We present the oscillator representation of the Pauli Hamiltonian for a scalar charge in a magnetic field and find a basis that diagonalizes the Hamiltonian in the special case of a constant or slowly varying magnetic field. We show that the diagonalization of the new basis is a canonical transformation in phase space, which leads to the Pauli Hamiltonian and counts the degeneracy of the Landau levels. Finally, we obtain the Liouville-von Neumann equation for quantum invariants as the annihilation and the creation operators for a scalar charge in time-dependent magnetic fields. • ### Research Paper 2020-12-31 #### Twist-angle Dependence of the Ground Exciton Energy in Twisted Bilayer MoS$_2$ Junkyoung KIM, Haeun CHA, Inwoo PARK et al. Abstract : The exciton is an electron hole bound state in semiconductors and plays an important role in opto-electronic devices such as light-emitting diode. Therefore, a desire exists to control the exciton formation energy, which is closely related to characteristics of optoelectronic applications. Here, we report formation energy of ground exciton states in a MoS$_2$ twisted bilayer measured by using photoluminescence spectroscopy. By varying the twist angle, we found that the exciton formation energy was tunable within the energy range roughly between 1.87 eV at about 0$^{\circ}$ and 60$^{\circ}$ twist angles and 1.90 eV at about a 30$^{\circ}$ twist angle. The exciton formation energy is directly related to the band gap energy, and the band gap of a bilayer MoS$_2$ becomes smaller than that of a MoS$_2$ monolayer due to interlayer coupling. Our results can be explained by the fact that the interlayer distance is smaller and the interlayer coupling is larger at 0$^{\circ}$ and 60$^{\circ}$ twist angles than at 30$^{\circ}$ twist angle. The twist angle dependence of the exciton formation energy can also be qualitatively explained by the twist-angle-dependent direct band-gap energy from first-principles calculations results. • ### Research Paper 2020-12-31 #### Effect of Pulsed Magnetic Field Stimulation of Bio-Conductance Response on The Human Meridian Circulatory System Hyunsook LEE* Abstract : This study aims to investigate the effect of pulsed magnetic field (PMF) stimulus on the circulation in the pericardium meridian system (AMS) by means of EAV (electroacupuncture according to Voll) and the galvanic skin response (GSR). The PMF stimulus was applied to PC8 at least three times a week for one month in order to see the continuous effect of PMF stimulation. The Max value measured at every biological active point except PC9 was found to be close to the normal range of 45 ~ 58 one month later. Therefore, PMF stimulation seems to be very effective in improving the function of the tenon-tissue connected to the pericadium meridian in Oriental medicine. The magnetic flux change due to the PMF is thought to lead to a strong stimulus at the acupoints, which have a lower impedance than the surrounding tissues and affects the current in the cell tissue. For further use of our results in meridian diagnosis, additional experiments on the subject for diverse diseases are required, as is a study of the changes in PMF intensity, transition time and stimulation duration. • ### Research Paper 2020-12-31 #### Electromagnetic Lensing by Born-Infeld type Electric Charge Jin Young KIM* Abstract : Light traveling in an electromagnetic field is not affected by the presence of the electromagnetic field in Maxwell electrodynamics, which is linear. However, in quantum electrodynamics, nonlinear terms are created in the effective action by quantum fluctuation and the propagation of light is affected by the external electromagnetic field. In Born-Infeld electrodynamics introduced to solve the divergence of the electromagnetic field at the origin, the permittivity and the permeability of vacuum in the presence of an external electromagnetic field are not one by non-linearity. We study the bending of a light ray passing a strong external electric field in generalized Born-Infeld electrodynamics, which includes the quantum effect. When a light ray moves around electrically charged objects like a charged black hole or an atomic nucleus, the path of light can be bent by the gradient of the effective index of refraction. We use the trajectory equation based on geometric optics for weak lensing, in which the impact parameter is large compared to the characteristic length of the Born-Infeld parameter, to compute the bending angle. ## Current Issue • ### A Study on the Measurement and Analysis of the Impedance of Various Materials Jong-Ho PARK* New Phys.: Sae Mulli 2019; 69(1): 1-15 https://doi.org/10.3938/NPSM.69.1 • ### Study of the Shapes of CdSe/CdS Nanocrystals by Using Raman Spectroscopy and the Stokes Shift Seong-Wook PARK, Hoon-Hyun CHO, Dong-Sun YOO, Il-Gon KIM* New Phys.: Sae Mulli 2019; 69(1): 16-24 https://doi.org/10.3938/NPSM.69.16 • ### Dependences of the Near-Field Characteristics of the Nano-Gap Structure on the Difference between Pentagonal and Circular Nano-Wires: A Numerical Study Vasanthan DEVARAJ, Jong-Min LEE, Chun Tae KIM, Won-Geun KIM, Jin-Woo OH* New Phys.: Sae Mulli 2019; 69(1): 25-30 https://doi.org/10.3938/NPSM.69.25	2021-01-26 20:52:21	{"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": 0.5014485716819763, "perplexity": 2350.567864111824}, "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/1610704803737.78/warc/CC-MAIN-20210126202017-20210126232017-00171.warc.gz"}
https://www.impan.pl/en/publishing-house/banach-center-publications/all/96/0/86037/mathbf-q-adapted-quantum-stochastic-integrals-and-differentials-in-fock-scale	Publishing house / Banach Center Publications / All volumes $\mathbf{Q}$-adapted quantum stochastic integrals and differentials in Fock scale Volume 96 / 2011 Banach Center Publications 96 (2011), 51-66 MSC: Primary 60H99; Secondary 60G99. DOI: 10.4064/bc96-0-3 Abstract In this paper we first introduce the Fock–Guichardet formalism for the quantum stochastic (QS) integration, then the four fundamental processes of the dynamics are introduced in the canonical basis as the operator-valued measures, on a space-time $\sigma$-field $\mathfrak{F}_\mathbb{X}$, of the QS integration. Then rigorous analysis of the QS integrals is carried out, and continuity of the QS derivative $\mathbf{D}$ is proved. Finally, $\mathrm{Q}$-adapted dynamics is discussed, including Bosonic ($\mathrm{Q}=\mathrm{I}$), Fermionic ($\mathrm{Q}=-\mathrm{I}$), and monotone ($\mathrm{Q}=\mathrm{O}$) quantum dynamics. These may be of particular interest to quantum field theory, quantum open systems, and quantum theory of stochastic processes. Authors • Viacheslav BelavkinSchool of Mathematical Sciences, University Park Nottingham, NG7 2RD, UK e-mail • Matthew BrownSchool of Mathematical Sciences, University Park Nottingham, NG7 2RD, UK e-mail Search for IMPAN publications Query phrase too short. Type at least 4 characters.	2020-07-09 11:25:26	{"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": 0.6483901739120483, "perplexity": 2677.8250034749976}, "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-2020-29/segments/1593655899931.31/warc/CC-MAIN-20200709100539-20200709130539-00236.warc.gz"}
https://en.m.wikipedia.org/wiki/Burnside_theorem	# Burnside theorem William Burnside. In mathematics, Burnside theorem in group theory states that if G is a finite group of order ${\displaystyle p^{a}q^{b}\ }$ where p and q are prime numbers, and a and b are non-negative integers, then G is solvable. Hence each non-Abelian finite simple group has order divisible by at least three distinct primes. Generalization : The finite group ${\displaystyle G}$ is solvable if and only if for every divisor ${\displaystyle n}$ of ${\displaystyle \|G\|}$ such that ${\displaystyle \left(n,{\frac {\|G\|}{n}}\right)=1}$, ${\displaystyle G}$ has a subgroup of order ${\displaystyle n}$. ## History The theorem was proved by William Burnside (1904) using the representation theory of finite groups. Several special cases of it had previously been proved by Burnside, Jordan, and Frobenius. John Thompson pointed out that a proof avoiding the use of representation theory could be extracted from his work on the N-group theorem, and this was done explicitly by Goldschmidt (1970) for groups of odd order, and by Bender (1972) for groups of even order. Matsuyama (1973) simplified the proofs. ## Proof This proof is by contradiction. Let paqb be the smallest product of two prime powers, such that there is a non-solvable group G whose order is equal to this number. If G had a nontrivial proper normal subgroup H, then (because of the minimality of G), H and G/H would be solvable, so G as well, which would contradict our assumption. So G is simple. If a were zero, G would be a finite q-group, hence nilpotent, and therefore solvable. Similarly, G cannot be abelian, otherwise it would be nilpotent. As G is simple, its center must therefore be trivial. • There is an element g of G which has qd conjugates, for some d > 0. By the first statement of Sylow's theorem, G has a subgroup S of order pa. Because S is a nontrivial p-group, its center Z(S) is nontrivial. Fix a nontrivial element ${\displaystyle g\in Z(S)}$ . The number of conjugates of g is equal to the index of its stabilizer subgroup Gg, which divides the index qb of S (because S is a subgroup of Gg). Hence this number is of the form qd. Moreover, the integer d is strictly positive, since g is nontrivial and therefore not central in G. Let (χi)1≤i≤h be the family of irreducible characters of G over ℂ (here χ1 denotes the trivial character). Because g is not in the same conjugacy class as 1, the orthogonality relation for the columns of the group's character table gives: ${\displaystyle 0=\sum _{i=1}^{h}\chi _{i}(1)\chi _{i}(g)=1+\sum _{i=2}^{h}\chi _{i}(1)\chi _{i}(g).}$ Now the χi(g) are algebraic integers, because they are sums of roots of unity. If all the nontrivial irreducible characters which don't vanish at g take a value divisible by q at 1, we deduce that ${\displaystyle -{\frac {1}{q}}=\sum _{i\geq 2,~\chi _{i}(g)\neq 0}{\frac {\chi _{i}(1)}{q}}\chi _{i}(g)}$ is an algebraic integer (since it is a sum of integer multiples of algebraic integers), which is absurd. This proves the statement. • The complex number qdχ(g)/n is an algebraic integer. The set of integer-valued class functions on G, Z(ℤ[G]), is a commutative ring, finitely generated over ℤ. All of its elements are thus integral over ℤ, in particular the mapping u which takes the value 1 on the conjugacy class of g and 0 elsewhere. The mapping ${\displaystyle A:Z(\mathbb {Z} [G])\rightarrow {\mbox{End}}(\mathbb {C} ^{n})}$ which sends a class function f to ${\displaystyle \sum _{s\in G}f(s)\rho (s)}$ is a ring homomorphism. Because ρ(s)−1A(u)ρ(s)=A(u) for all s, Schur's lemma implies that A(u) is a homothety λIn. Its trace nλ is equal to ${\displaystyle \sum _{s\in G}f(s)\chi (s)=q^{d}\chi (g).}$ Because the homothety λIn is the homomorphic image of an integral element, this proves that the complex number λ = qdχ(g)/n is an algebraic integer. • The complex number χ(g)/n is an algebraic integer. Since q is relatively prime to n, by Bézout's identity there are two integers x and y such that: ${\displaystyle xq^{d}+yn=1\quad {\text{therefore}}\quad {\frac {\chi (g)}{n}}=x{\frac {q^{d}\chi (g)}{n}}+y\chi (g).}$ Because a linear combination with integer coefficients of algebraic integers is again an algebraic integer, this proves the statement. • The image of g, under the representation ρ, is a homothety. Let ζ be the complex number χ(g)/n. It is an algebraic integer, so its norm N(ζ) (i.e. the product of its conjugates, that is the roots of its minimal polynomial over ℚ) is a nonzero integer. Now ζ is the average of roots of unity (the eigenvalues of ρ(g)), hence so are its conjugates, so they all have an absolute value less than or equal to 1. Because the absolute value of their product N(ζ) is greater than or equal to 1, their absolute value must all be 1, in particular ζ, which means that the eigenvalues of ρ(g) are all equal, so ρ(g) is a homothety. • Conclusion Let N be the kernel of ρ. The homothety ρ(g) is central in Im(ρ) (which is canonically isomorphic to G/N), whereas g is not central in G. Consequently, the normal subgroup N of the simple group G is nontrivial, hence it is equal to G, which contradicts the fact that ρ is a nontrivial representation.	2019-06-24 22:24:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 14, "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": 0.9120131731033325, "perplexity": 397.14396228425335}, "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-2019-26/segments/1560627999740.32/warc/CC-MAIN-20190624211359-20190624233359-00346.warc.gz"}
https://zbmath.org/?q=an%3A1251.35078	# zbMATH — the first resource for mathematics Problem of second grade fluids in convex polyhedrons. (English) Zbl 1251.35078 The stationary problem of a grade-two fluid is studied in convex 3D polyhedron $$\Omega$$. The velocity $$v$$ and the pressure $$p$$ satisfy to the equations \begin{aligned} & -\nu\Delta v+\text{curl}(v-\alpha\Delta v)\times v+\nabla p=f,\quad \\ &\text{div}\,v=0\;\text{in}\;\Omega, v=g\quad \text{on}\;\partial\Omega,\end{aligned} where $$\nu>0$$ is the kinematic viscosity coefficient, $$\alpha\neq 0$$ is the normal stress module, $$g\cdot n=0$$. The problem is reformulated in an equivalent form using a transport equation. The solvability of the problem is proved for small data $$(f,g)$$. The Galerkin method is the base of the proof. Uniqueness is established for inner angles of a polyhedron smaller than $$\frac{3\pi}{4}$$. ##### MSC: 35Q35 PDEs in connection with fluid mechanics 76A05 Non-Newtonian fluids 35G30 Boundary value problems for nonlinear higher-order PDEs Full Text:	2022-01-22 14:17:22	{"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": 0.914767324924469, "perplexity": 1489.6925372272185}, "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/1642320303864.86/warc/CC-MAIN-20220122134127-20220122164127-00098.warc.gz"}
https://gitbook.rootwhois.cn/Alogrithm/Leetcode/187.%20Repeated%20DNA%20Sequences.html	# 187. Repeated DNA Sequences ## 1. Question The DNA sequence is composed of a series of nucleotides abbreviated as 'A', 'C', 'G', and 'T'. For example, "ACGAATTCCG" is a DNA sequence. When studying DNA, it is useful to identify repeated sequences within the DNA. Given a string s that represents a DNA sequence, return all the 10-letter-long sequences (substrings) that occur more than once in a DNA molecule. You may return the answer in any order. ## 2. Examples Example 1: Input: s = "AAAAACCCCCAAAAACCCCCCAAAAAGGGTTT" Output: ["AAAAACCCCC","CCCCCAAAAA"] Example 2: Input: s = "AAAAAAAAAAAAA" Output: ["AAAAAAAAAA"] ## 3. Constraints • 1 <= s.length <= 105 • s[i] is either 'A','C', 'G', or 'T'. ## 5. Solutions class Solution { public List<String> findRepeatedDnaSequences(String s) { List<String> ans = new ArrayList<>(); int n = s.length(); Map<String, Integer> map = new HashMap<>(); for (int i = 0; i + 10 <= n; i++) { String cur = s.substring(i, i + 10); int cnt = map.getOrDefault(cur, 0);	2022-12-01 23:27:26	{"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": 0.30330634117126465, "perplexity": 12430.17349245786}, "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/1669446710870.69/warc/CC-MAIN-20221201221914-20221202011914-00440.warc.gz"}
http://physics.stackexchange.com/questions/21873/moment-of-inertia-of-a-sector-about-its-center-point?answertab=active	# Moment of inertia of a sector about its center point? What is the moment of inertia of a pizza slice that has a radius r, an angle (radians) of theta, and a height of h about the center point perpendicular to the cheese plane? - It could possibly be like a cylinder? if $$I = (mr^2) / 2$$ for a cylander than maybe for a sector: $$I = (\theta m r^2) / 2$$ – Sam Mar 4 '12 at 21:42 Do you mean the entire moment of inertia tensor, or only about the axis through the "center point" (I assume this means center of mass) and perpendicular to the plane of the cheese/crust? – Mark Eichenlaub Mar 4 '12 at 21:42 I think, OP meant The moment of inertia about the axis through the "center point" (center of complete circle) and perpendicular to the plane. Using the forlmulae $I=\int r^2 dm$, you are receive the same formulae, as for disk $I=\frac{mr^2}{2}$, where $m$ - is the mass of pizza slice/ – Sergio Mar 4 '12 at 21:52 I assume you're talking about spinning the pizza around the axis a pizza-dough spinner would spin it. The moment of inertia of a solid disk is $\frac{mr^2}{2}$. If it spins, each slice contributes angular momentum proportional to $\theta$. The slice's mass is also proportional to $\theta$ so the moment of inertia of a slice of pizza about its tip is also $\frac{mr^2}{2}$ where $m$ is now the mass of the slice. We can use the parallel axis theorem to find the moment of inertia through the slice's center of mass. The center of mass is displaced a distance $\frac{2}{3}r\mathrm{sinc}\frac{\theta}{2}$ from the tip, so the moment of inertia through this axis is $$I = \left(\frac{1}{2} + (\frac{2}{3}\mathrm{sinc}\frac{\theta}{2})^2\right)mr^2$$ The location of the center of mass of the slice of pizza comes from a blog post I wrote a while ago that uses it to prove the identity -	2015-04-26 11:35:09	{"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": 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": 0.7616563439369202, "perplexity": 339.0853212992843}, "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-2015-18/segments/1429246654292.99/warc/CC-MAIN-20150417045734-00127-ip-10-235-10-82.ec2.internal.warc.gz"}
https://pipiwiki.com/wiki/Aristotle	# Aristotle Aristotle Roman copy in marble of a Greek bronze bust of Aristotle by Lysippos, c. 330 BC, with modern alabaster mantle Born384 BC[A] Died322 BC (aged 61–62) Spouse(s)Pythias EraAncient Greek philosophy RegionWestern philosophy School Notable studentsAlexander the Great, Theophrastus Main interests Notable ideas Aristotle (/ˈærɪstɒtəl/;[3] Greek: Ἀριστοτέλης Aristotélēs, pronounced [aristotélɛːs]; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Lyceum, the Peripatetic school of philosophy, and the Aristotelian tradition. His writings cover many subjects including physics, biology, zoology, metaphysics, logic, ethics, aesthetics, poetry, theatre, music, rhetoric, psychology, linguistics, economics, politics, and government. Aristotle provided a complex synthesis of the various philosophies existing prior to him. It was above all from his teachings that the West inherited its intellectual lexicon, as well as problems and methods of inquiry. As a result, his philosophy has exerted a unique influence on almost every form of knowledge in the West and it continues to be a subject of contemporary philosophical discussion. Little is known about his life. Aristotle was born in the city of Stagira in Northern Greece. His father, Nicomachus, died when Aristotle was a child, and he was brought up by a guardian. At seventeen or eighteen years of age he joined Plato's Academy in Athens and remained there until the age of thirty-seven (c. 347 BC).[4] Shortly after Plato died, Aristotle left Athens and, at the request of Philip II of Macedon, tutored Alexander the Great beginning in 343 BC.[5] He established a library in the Lyceum which helped him to produce many of his hundreds of books on papyrus scrolls. Though Aristotle wrote many elegant treatises and dialogues for publication, only around a third of his original output has survived, none of it intended for publication.[6] Aristotle's views on physical science profoundly shaped medieval scholarship. Their influence extended from Late Antiquity and the Early Middle Ages into the Renaissance, and were not replaced systematically until the Enlightenment and theories such as classical mechanics were developed. Some of Aristotle's zoological observations found in his biology, such as on the hectocotyl (reproductive) arm of the octopus, were disbelieved until the 19th century. His works contain the earliest known formal study of logic, studied by medieval scholars such as Peter Abelard and John Buridan. Aristotle's influence on logic also continued well into the 19th century. He influenced Judeo-Islamic philosophies (800–1400) during the Middle Ages, as well as Christian theology, especially the Neoplatonism of the Early Church and the scholastic tradition of the Catholic Church. Aristotle was revered among medieval Muslim scholars as "The First Teacher" and among medieval Christians like Thomas Aquinas as simply "The Philosopher". His ethics, though always influential, gained renewed interest with the modern advent of virtue ethics, such as in the thinking of Alasdair MacIntyre and Philippa Foot. ## Life School of Aristotle in Mieza, Macedonia, Greece In general, the details of Aristotle's life are not well-established. The biographies written in ancient times are often speculative and historians only agree on a few salient points.[B] Aristotle, whose name means "the best purpose" in Ancient Greek,[7] was born in 384 BC in Stagira, Chalcidice, about 55 km (34 miles) east of modern-day Thessaloniki.[8][9] His father Nicomachus was the personal physician to King Amyntas of Macedon. Both of Aristotle's parents died when he was about thirteen, and Proxenus of Atarneus became his guardian.[10] Although little information about Aristotle's childhood has survived, he probably spent some time within the Macedonian palace, making his first connections with the Macedonian monarchy.[11] At the age of seventeen or eighteen, Aristotle moved to Athens to continue his education at Plato's Academy.[12] He probably experienced the Eleusinian Mysteries as he wrote when describing the sights one viewed at the Eleusinian Mysteries, "to experience is to learn" [παθείν μαθεĩν].[13] Aristotle remained in Athens for nearly twenty years before leaving in 348/47 BC. The traditional story about his departure records that he was disappointed with the Academy's direction after control passed to Plato's nephew Speusippus, although it is possible that he feared the anti-Macedonian sentiments in Athens at that time and left before Plato died.[14] Aristotle then accompanied Xenocrates to the court of his friend Hermias of Atarneus in Asia Minor. After the death of Hermias, Aristotle travelled with his pupil Theophrastus to the island of Lesbos, where together they researched the botany and zoology of the island and its sheltered lagoon. While in Lesbos, Aristotle married Pythias, either Hermias's adoptive daughter or niece. She bore him a daughter, whom they also named Pythias. In 343 BC, Aristotle was invited by Philip II of Macedon to become the tutor to his son Alexander.[15][5] Portrait bust of Aristotle; an Imperial Roman (1st or 2nd century AD) copy of a lost bronze sculpture made by Lysippos Aristotle was appointed as the head of the royal academy of Macedon. During Aristotle's time in the Macedonian court, he gave lessons not only to Alexander, but also to two other future kings: Ptolemy and Cassander.[16] Aristotle encouraged Alexander toward eastern conquest, and Aristotle's own attitude towards Persia was unabashedly ethnocentric. In one famous example, he counsels Alexander to be "a leader to the Greeks and a despot to the barbarians, to look after the former as after friends and relatives, and to deal with the latter as with beasts or plants".[16] By 335 BC, Aristotle had returned to Athens, establishing his own school there known as the Lyceum. Aristotle conducted courses at the school for the next twelve years. While in Athens, his wife Pythias died and Aristotle became involved with Herpyllis of Stagira, who bore him a son whom he named after his father, Nicomachus. According to the Suda, he also had an erômenos, Palaephatus of Abydus.[17] This period in Athens, between 335 and 323 BC, is when Aristotle is believed to have composed many of his works.[5] He wrote many dialogues, of which only fragments have survived. Those works that have survived are in treatise form and were not, for the most part, intended for widespread publication; they are generally thought to be lecture aids for his students. His most important treatises include Physics, Metaphysics, Nicomachean Ethics, Politics, On the Soul and Poetics. Aristotle studied and made significant contributions to "logic, metaphysics, mathematics, physics, biology, botany, ethics, politics, agriculture, medicine, dance and theatre."[4] Near the end of his life, Alexander and Aristotle became estranged over Alexander's relationship with Persia and Persians. A widespread tradition in antiquity suspected Aristotle of playing a role in Alexander's death, but the only evidence of this is an unlikely claim made some six years after the death.[18] Following Alexander's death, anti-Macedonian sentiment in Athens was rekindled. In 322 BC, Demophilus and Eurymedon the Hierophant reportedly denounced Aristotle for impiety,[19] prompting him to flee to his mother's family estate in Chalcis, on Euboea, at which occasion he was said to have stated: "I will not allow the Athenians to sin twice against philosophy"[20][21][22] – a reference to Athens's trial and execution of Socrates. He died on Euboea of natural causes later that same year, having named his student Antipater as his chief executor and leaving a will in which he asked to be buried next to his wife.[23] ## Speculative philosophy ### Logic With the Prior Analytics, Aristotle is credited with the earliest study of formal logic,[24] and his conception of it was the dominant form of Western logic until 19th-century advances in mathematical logic.[25] Kant stated in the Critique of Pure Reason that with Aristotle logic reached its completion.[26] #### Organon One of Aristotle's types of syllogism[C] In words In terms[D] In equations[E] All men are mortal. All Greeks are men. All Greeks are mortal. M a P S a M S a P What we today call Aristotelian logic with its types of syllogism (methods of logical argument),[27] Aristotle himself would have labelled "analytics". The term "logic" he reserved to mean dialectics. Most of Aristotle's work is probably not in its original form, because it was most likely edited by students and later lecturers. The logical works of Aristotle were compiled into a set of six books called the Organon around 40 BC by Andronicus of Rhodes or others among his followers.[29] The books are: 1. Categories 2. On Interpretation 3. Prior Analytics 4. Posterior Analytics 5. Topics 6. On Sophistical Refutations Plato (left) and Aristotle in Raphael's 1509 fresco, The School of Athens. Aristotle holds his Nicomachean Ethics and gestures to the earth, representing his view in immanent realism, whilst Plato gestures to the heavens, indicating his Theory of Forms, and holds his Timaeus.[30][31] The order of the books (or the teachings from which they are composed) is not certain, but this list was derived from analysis of Aristotle's writings. It goes from the basics, the analysis of simple terms in the Categories, the analysis of propositions and their elementary relations in On Interpretation, to the study of more complex forms, namely, syllogisms (in the Analytics)[32][33] and dialectics (in the Topics and Sophistical Refutations). The first three treatises form the core of the logical theory stricto sensu: the grammar of the language of logic and the correct rules of reasoning. The Rhetoric is not conventionally included, but it states that it relies on the Topics.[34] ### Metaphysics The word "metaphysics" appears to have been coined by the first century AD editor who assembled various small selections of Aristotle's works to the treatise we know by the name Metaphysics.[35] Aristotle called it "first philosophy", and distinguished it from mathematics and natural science (physics) as the contemplative (theoretikē) philosophy which is "theological" and studies the divine. He wrote in his Metaphysics (1026a16): if there were no other independent things besides the composite natural ones, the study of nature would be the primary kind of knowledge; but if there is some motionless independent thing, the knowledge of this precedes it and is first philosophy, and it is universal in just this way, because it is first. And it belongs to this sort of philosophy to study being as being, both what it is and what belongs to it just by virtue of being.[36] #### Substance Aristotle examines the concepts of substance (ousia) and essence (to ti ên einai, "the what it was to be") in his Metaphysics (Book VII), and he concludes that a particular substance is a combination of both matter and form, a philosophical theory called hylomorphism. In Book VIII, he distinguishes the matter of the substance as the substratum, or the stuff of which it is composed. For example, the matter of a house is the bricks, stones, timbers etc., or whatever constitutes the potential house, while the form of the substance is the actual house, namely 'covering for bodies and chattels' or any other differentia that let us define something as a house. The formula that gives the components is the account of the matter, and the formula that gives the differentia is the account of the form.[37][38] ##### Immanent realism Plato's forms exist as universals, like the ideal form of an apple. For Aristotle, both matter and form belong to the individual thing (hylomorphism). Like his teacher Plato, Aristotle's philosophy aims at the universal. Aristotle's ontology places the universal (katholou) in particulars (kath' hekaston), things in the world, whereas for Plato the universal is a separately existing form which actual things imitate. For Aristotle, "form" is still what phenomena are based on, but is "instantiated" in a particular substance.[38] Plato argued that all things have a universal form, which could be either a property or a relation to other things. When we look at an apple, for example, we see an apple, and we can also analyse a form of an apple. In this distinction, there is a particular apple and a universal form of an apple. Moreover, we can place an apple next to a book, so that we can speak of both the book and apple as being next to each other. Plato argued that there are some universal forms that are not a part of particular things. For example, it is possible that there is no particular good in existence, but "good" is still a proper universal form. Aristotle disagreed with Plato on this point, arguing that all universals are instantiated at some period of time, and that there are no universals that are unattached to existing things. In addition, Aristotle disagreed with Plato about the location of universals. Where Plato spoke of the world of forms, a place where all universal forms subsist, Aristotle maintained that universals exist within each thing on which each universal is predicated. So, according to Aristotle, the form of apple exists within each apple, rather than in the world of the forms.[38][39] ##### Potentiality and actuality With regard to the change (kinesis) and its causes now, as he defines in his Physics and On Generation and Corruption 319b–320a, he distinguishes the coming to be from: 1. growth and diminution, which is change in quantity; 2. locomotion, which is change in space; and 3. alteration, which is change in quality. Aristotle argued that a capability like playing the flute could be acquired – the potential made actual – by learning. The coming to be is a change where nothing persists of which the resultant is a property. In that particular change he introduces the concept of potentiality (dynamis) and actuality (entelecheia) in association with the matter and the form. Referring to potentiality, this is what a thing is capable of doing, or being acted upon, if the conditions are right and it is not prevented by something else. For example, the seed of a plant in the soil is potentially (dynamei) plant, and if it is not prevented by something, it will become a plant. Potentially beings can either 'act' (poiein) or 'be acted upon' (paschein), which can be either innate or learned. For example, the eyes possess the potentiality of sight (innate – being acted upon), while the capability of playing the flute can be possessed by learning (exercise – acting). Actuality is the fulfilment of the end of the potentiality. Because the end (telos) is the principle of every change, and for the sake of the end exists potentiality, therefore actuality is the end. Referring then to our previous example, we could say that an actuality is when a plant does one of the activities that plants do.[38] For that for the sake of which (to hou heneka) a thing is, is its principle, and the becoming is for the sake of the end; and the actuality is the end, and it is for the sake of this that the potentiality is acquired. For animals do not see in order that they may have sight, but they have sight that they may see.[40] In summary, the matter used to make a house has potentiality to be a house and both the activity of building and the form of the final house are actualities, which is also a final cause or end. Then Aristotle proceeds and concludes that the actuality is prior to potentiality in formula, in time and in substantiality. With this definition of the particular substance (i.e., matter and form), Aristotle tries to solve the problem of the unity of the beings, for example, "what is it that makes a man one"? Since, according to Plato there are two Ideas: animal and biped, how then is man a unity? However, according to Aristotle, the potential being (matter) and the actual one (form) are one and the same.[38][41] ### Epistemology Aristotle's immanent realism means his epistemology is based on the study of things that exist or happen in the world, and rises to knowledge of the universal, whereas for Plato epistemology begins with knowledge of universal Forms (or ideas) and descends to knowledge of particular imitations of these.[34] Aristotle uses induction from examples alongside deduction, whereas Plato relies on deduction from a priori principles.[34] ## Natural philosophy Aristotle's "natural philosophy" spans a wide range of natural phenomena including those now covered by physics, biology and other natural sciences.[42] In Aristotle's terminology, "natural philosophy" is a branch of philosophy examining the phenomena of the natural world, and includes fields that would be regarded today as physics, biology and other natural sciences. Aristotle's work encompassed virtually all facets of intellectual inquiry. Aristotle makes philosophy in the broad sense coextensive with reasoning, which he also would describe as "science". Note, however, that his use of the term science carries a different meaning than that covered by the term "scientific method". For Aristotle, "all science (dianoia) is either practical, poetical or theoretical" (Metaphysics 1025b25). His practical science includes ethics and politics; his poetical science means the study of fine arts including poetry; his theoretical science covers physics, mathematics and metaphysics.[42] ### Physics The four classical elements (fire, air, water, earth) of Empedocles and Aristotle illustrated with a burning log. The log releases all four elements as it is destroyed. #### Five elements In his On Generation and Corruption, Aristotle related each of the four elements proposed earlier by Empedocles, Earth, Water, Air, and Fire, to two of the four sensible qualities, hot, cold, wet, and dry. In the Empedoclean scheme, all matter was made of the four elements, in differing proportions. Aristotle's scheme added the heavenly Aether, the divine substance of the heavenly spheres, stars and planets.[43] Aristotle's elements[43] Element Hot/Cold Wet/Dry Motion Modern state of matter Earth Cold Dry Down Solid Water Cold Wet Down Liquid Air Hot Wet Up Gas Fire Hot Dry Up Plasma Aether (divine substance) Circular (in heavens) #### Motion Aristotle describes two kinds of motion: "violent" or "unnatural motion", such as that of a thrown stone, in the Physics (254b10), and "natural motion", such as of a falling object, in On the Heavens (300a20). In violent motion, as soon as the agent stops causing it, the motion stops also; in other words, the natural state of an object is to be at rest,[44][F] since Aristotle does not address friction.[45] With this understanding, it can be observed that, as Aristotle stated, heavy objects (on the ground, say) require more force to make them move; and objects pushed with greater force move faster.[46][G] This would imply the equation[46] ${\displaystyle F=mv}$, incorrect in modern physics.[46] Natural motion depends on the element concerned: the aether naturally moves in a circle around the heavens,[H] while the 4 Empedoclean elements move vertically up (like fire, as is observed) or down (like earth) towards their natural resting places.[47][45][I] Aristotle's laws of motion. In Physics he states that objects fall at a speed proportional to their weight and inversely proportional to the density of the fluid they are immersed in.[45] This is a correct approximation for objects in Earth's gravitational field moving in air or water.[47] In the Physics (215a25), Aristotle effectively states a quantitative law, that the speed, v, of a falling body is proportional (say, with constant c) to its weight, W, and inversely proportional to the density,[J] ρ, of the fluid in which it is falling:[47][45] ${\displaystyle v=c{\frac {W}{\rho }}}$ Aristotle implies that in a vacuum the speed of fall would become infinite, and concludes from this apparent absurdity that a vacuum is not possible.[47][45] Opinions have varied on whether Aristotle intended to state quantitative laws. Henri Carteron held the "extreme view"[45] that Aristotle's concept of force was basically qualitative,[48] but other authors reject this.[45] Archimedes corrected Aristotle's theory that bodies move towards their natural resting places; metal boats can float if they displace enough water; floating depends in Archimedes' scheme on the mass and volume of the object, not as Aristotle thought its elementary composition.[47] Aristotle's writings on motion remained influential until the Early Modern period. John Philoponus (in the Middle Ages) and Galileo are said to have shown by experiment that Aristotle's claim that a heavier object falls faster than a lighter object is incorrect.[42] A contrary opinion is given by Carlo Rovelli, who argues that Aristotle's physics of motion is correct within its domain of validity, that of objects in the Earth's gravitational field immersed in a fluid such as air. In this system, heavy bodies in steady fall indeed travel faster than light ones (whether friction is ignored, or not[47]), and they do fall more slowly in a denser medium.[46][K] Newton's "forced" motion corresponds to Aristotle's "violent" motion with its external agent, but Aristotle's assumption that the agent's effect stops immediately it stops acting (e.g., the ball leaves the thrower's hand) has awkward consequences: he has to suppose that surrounding fluid helps to push the ball along to make it continue to rise even though the hand is no longer acting on it, resulting in the Medieval theory of impetus.[47] #### Four causes Aristotle argued by analogy with woodwork that a thing takes its form from four causes: in the case of a table, the wood used (material cause), its design (formal cause), the tools and techniques used (efficient cause), and its decorative or practical purpose (final cause).[49] Aristotle suggested that the reason for anything coming about can be attributed to four different types of simultaneously active factors. His term aitia is traditionally translated as "cause", but it does not always refer to temporal sequence; it might be better translated as "explanation", but the traditional rendering will be employed here.[50][51] • Material cause describes the material out of which something is composed. Thus the material cause of a table is wood. It is not about action. It does not mean that one domino knocks over another domino.[50] • The formal cause is its form, i.e., the arrangement of that matter. It tells us what a thing is, that a thing is determined by the definition, form, pattern, essence, whole, synthesis or archetype. It embraces the account of causes in terms of fundamental principles or general laws, as the whole (i.e., macrostructure) is the cause of its parts, a relationship known as the whole-part causation. Plainly put, the formal cause is the idea in the mind of the sculptor that brings the sculpture into being. A simple example of the formal cause is the mental image or idea that allows an artist, architect, or engineer to create a drawing.[50] • The efficient cause is "the primary source", or that from which the change under consideration proceeds. It identifies 'what makes of what is made and what causes change of what is changed' and so suggests all sorts of agents, nonliving or living, acting as the sources of change or movement or rest. Representing the current understanding of causality as the relation of cause and effect, this covers the modern definitions of "cause" as either the agent or agency or particular events or states of affairs. In the case of two dominoes, when the first is knocked over it causes the second also to fall over.[50] In the case of animals, this agency is a combination of how it develops from the egg, and how its body functions.[52] • The final cause (telos) is its purpose, the reason why a thing exists or is done, including both purposeful and instrumental actions and activities. The final cause is the purpose or function that something is supposed to serve. This covers modern ideas of motivating causes, such as volition.[50] In the case of living things, it implies adaptation to a particular way of life.[52] #### Optics Aristotle describes experiments in optics using a camera obscura in Problems, book 15. The apparatus consisted of a dark chamber with a small aperture that let light in. With it, he saw that whatever shape he made the hole, the sun's image always remained circular. He also noted that increasing the distance between the aperture and the image surface magnified the image.[53] #### Chance and spontaneity According to Aristotle, spontaneity and chance are causes of some things, distinguishable from other types of cause such as simple necessity. Chance as an incidental cause lies in the realm of accidental things, "from what is spontaneous". There is also more a specific kind of chance, which Aristotle names "luck", that only applies to people's moral choices.[54][55] ### Astronomy In astronomy, Aristotle refuted Democritus's claim that the Milky Way was made up of "those stars which are shaded by the earth from the sun's rays," pointing out correctly that if "the size of the sun is greater than that of the earth and the distance of the stars from the earth many times greater than that of the sun, then... the sun shines on all the stars and the earth screens none of them."[56] Aristotle noted that the ground level of the Aeolian islands changed before a volcanic eruption. ### Geology Aristotle was one of the first people to record any geological observations. He stated that geological change was too slow to be observed in one person's lifetime.[57][58] The geologist Charles Lyell noted that Aristotle described such change, including "lakes that had dried up" and "deserts that had become watered by rivers", giving as examples the growth of the Nile delta since the time of Homer, and "the upheaving of one of the Aeolian islands, previous to a volcanic eruption."'[59] ### Biology Among many pioneering zoological observations, Aristotle described the reproductive hectocotyl arm of the octopus (bottom left). #### Empirical research Aristotle was the first person to study biology systematically,[60] and biology forms a large part of his writings. He spent two years observing and describing the zoology of Lesbos and the surrounding seas, including in particular the Pyrrha lagoon in the centre of Lesbos.[61][62] His data in History of Animals, Generation of Animals, Movement of Animals, and Parts of Animals are assembled from his own observations,[63] statements given by people with specialized knowledge such as beekeepers and fishermen, and less accurate accounts provided by travellers from overseas.[64] His apparent emphasis on animals rather than plants is a historical accident: his works on botany have been lost, but two books on plants by his pupil Theophrastus have survived.[65] Aristotle reports on the sea-life visible from observation on Lesbos and the catches of fishermen. He describes the catfish, electric ray, and frogfish in detail, as well as cephalopods such as the octopus and paper nautilus. His description of the hectocotyl arm of cephalopods, used in sexual reproduction, was widely disbelieved until the 19th century.[66] He gives accurate descriptions of the four-chambered fore-stomachs of ruminants,[67] and of the ovoviviparous embryological development of the hound shark.[68] He notes that an animal's structure is well matched to function, so, among birds, the heron, which lives in marshes with soft mud and lives by catching fish, has a long neck and long legs, and a sharp spear-like beak, whereas ducks that swim have short legs and webbed feet.[69] Darwin, too, noted these sorts of differences between similar kinds of animal, but unlike Aristotle used the data to come to the theory of evolution.[70] Aristotle's writings can seem to modern readers close to implying evolution, but while Aristotle was aware that new mutations or hybridizations could occur, he saw these as rare accidents. For Aristotle, accidents, like heat waves in winter, must be considered distinct from natural causes. He was thus critical of Empedocles's materialist theory of a "survival of the fittest" origin of living things and their organs, and ridiculed the idea that accidents could lead to orderly results.[71] To put his views into modern terms, he nowhere says that different species can have a common ancestor, or that one kind can change into another, or that kinds can become extinct.[72] #### Scientific style Aristotle inferred growth laws from his observations on animals, including that brood size decreases with body mass, whereas gestation period increases. He was correct in these predictions, at least for mammals: data are shown for mouse and elephant. Aristotle did not do experiments in the modern sense.[73] He used the ancient Greek term pepeiramenoi to mean observations, or at most investigative procedures like dissection.[74] In Generation of Animals, he finds a fertilized hen's egg of a suitable stage and opens it to see the embryo's heart beating inside.[75][76] Instead, he practiced a different style of science: systematically gathering data, discovering patterns common to whole groups of animals, and inferring possible causal explanations from these.[77][78] This style is common in modern biology when large amounts of data become available in a new field, such as genomics. It does not result in the same certainty as experimental science, but it sets out testable hypotheses and constructs a narrative explanation of what is observed. In this sense, Aristotle's biology is scientific.[77] From the data he collected and documented, Aristotle inferred quite a number of rules relating the life-history features of the live-bearing tetrapods (terrestrial placental mammals) that he studied. Among these correct predictions are the following. Brood size decreases with (adult) body mass, so that an elephant has fewer young (usually just one) per brood than a mouse. Lifespan increases with gestation period, and also with body mass, so that elephants live longer than mice, have a longer period of gestation, and are heavier. As a final example, fecundity decreases with lifespan, so long-lived kinds like elephants have fewer young in total than short-lived kinds like mice.[79] #### Classification of living things Aristotle recorded that the embryo of a dogfish was attached by a cord to a kind of placenta (the yolk sac), like a higher animal; this formed an exception to the linear scale from highest to lowest.[80] Aristotle distinguished about 500 species of animals,[81][82] arranging these in the History of Animals in a graded scale of perfection, a scala naturae, with man at the top. His system had eleven grades of animal, from highest potential to lowest, expressed in their form at birth: the highest gave live birth to hot and wet creatures, the lowest laid cold, dry mineral-like eggs. Animals came above plants, and these in turn were above minerals.[83] see also:[84] He grouped what the modern zoologist would call vertebrates as the hotter "animals with blood", and below them the colder invertebrates as "animals without blood". Those with blood were divided into the live-bearing (mammals), and the egg-laying (birds, reptiles, fish). Those without blood were insects, crustacea (non-shelled – cephalopods, and shelled) and the hard-shelled molluscs (bivalves and gastropods). He recognised that animals did not exactly fit into a linear scale, and noted various exceptions, such as that sharks had a placenta like the tetrapods. To a modern biologist, the explanation, not available to Aristotle, is convergent evolution.[85] He believed that purposive final causes guided all natural processes; this teleological view justified his observed data as an expression of formal design.[86] Aristotle's Scala naturae (highest to lowest) Group Examples (given by Aristotle) Blood Legs Souls (Rational, Sensitive, Vegetative) Qualities (HotCold, WetDry) Man Man with blood 2 legs R, S, V Hot, Wet Live-bearing tetrapods Cat, hare with blood 4 legs S, V Hot, Wet Cetaceans Dolphin, whale with blood none S, V Hot, Wet Birds Bee-eater, nightjar with blood 2 legs S, V Hot, Wet, except Dry eggs Egg-laying tetrapods Chameleon, crocodile with blood 4 legs S, V Cold, Wet except scales, eggs Snakes Water snake, Ottoman viper with blood none S, V Cold, Wet except scales, eggs Egg-laying fishes Sea bass, parrotfish with blood none S, V Cold, Wet, including eggs (Among the egg-laying fishes): placental selachians Shark, skate with blood none S, V Cold, Wet, but placenta like tetrapods Crustaceans Shrimp, crab without many legs S, V Cold, Wet except shell Cephalopods Squid, octopus without tentacles S, V Cold, Wet Hard-shelled animals Cockle, trumpet snail without none S, V Cold, Dry (mineral shell) Larva-bearing insects Ant, cicada without 6 legs S, V Cold, Dry Spontaneously-generating Sponges, worms without none S, V Cold, Wet or Dry, from earth Plants Fig without none V Cold, Dry Minerals Iron without none none Cold, Dry ### Psychology #### Soul Aristotle proposed a three-part structure for souls of plants, animals, and humans, making humans unique in having all three types of soul. Aristotle's psychology, given in his treatise On the Soul (peri psychēs), posits three kinds of soul ("psyches"): the vegetative soul, the sensitive soul, and the rational soul. Humans have a rational soul. The human soul incorporates the powers of the other kinds: Like the vegetative soul it can grow and nourish itself; like the sensitive soul it can experience sensations and move locally. The unique part of the human, rational soul is its ability to receive forms of other things and to compare them using the nous (intellect) and logos (reason).[87] For Aristotle, the soul is the form of a living being. Because all beings are composites of form and matter, the form of living beings is that which endows them with what is specific to living beings, e.g. the ability to initiate movement (or in the case of plants, growth and chemical transformations, which Aristotle considers types of movement).[15] In contrast to earlier philosophers, but in accordance with the Egyptians, he placed the rational soul in the heart, rather than the brain.[88] Notable is Aristotle's division of sensation and thought, which generally differed from the concepts of previous philosophers, with the exception of Alcmaeon.[89] #### Memory According to Aristotle in On the Soul, memory is the ability to hold a perceived experience in the mind and to distinguish between the internal "appearance" and an occurrence in the past.[90] In other words, a memory is a mental picture (phantasm) that can be recovered. Aristotle believed an impression is left on a semi-fluid bodily organ that undergoes several changes in order to make a memory. A memory occurs when stimuli such as sights or sounds are so complex that the nervous system cannot receive all the impressions at once. These changes are the same as those involved in the operations of sensation, Aristotelian 'common sense', and thinking.[91][92] Aristotle uses the term 'memory' for the actual retaining of an experience in the impression that can develop from sensation, and for the intellectual anxiety that comes with the impression because it is formed at a particular time and processing specific contents. Memory is of the past, prediction is of the future, and sensation is of the present. Retrieval of impressions cannot be performed suddenly. A transitional channel is needed and located in our past experiences, both for our previous experience and present experience.[93] Because Aristotle believes people receive all kinds of sense perceptions and perceive them as impressions, people are continually weaving together new impressions of experiences. To search for these impressions, people search the memory itself.[94] Within the memory, if one experience is offered instead of a specific memory, that person will reject this experience until they find what they are looking for. Recollection occurs when one retrieved experience naturally follows another. If the chain of "images" is needed, one memory will stimulate the next. When people recall experiences, they stimulate certain previous experiences until they reach the one that is needed.[95] Recollection is thus the self-directed activity of retrieving the information stored in a memory impression.[96] Only humans can remember impressions of intellectual activity, such as numbers and words. Animals that have perception of time can retrieve memories of their past observations. Remembering involves only perception of the things remembered and of the time passed.[97] Senses, perception, memory, dreams, action in Aristotle's psychology. Impressions are stored in the sensorium (the heart), linked by his laws of association (similarity, contrast, and contiguity). Aristotle believed the chain of thought, which ends in recollection of certain impressions, was connected systematically in relationships such as similarity, contrast, and contiguity, described in his laws of association. Aristotle believed that past experiences are hidden within the mind. A force operates to awaken the hidden material to bring up the actual experience. According to Aristotle, association is the power innate in a mental state, which operates upon the unexpressed remains of former experiences, allowing them to rise and be recalled.[98][99] #### Dreams Aristotle describes sleep in On Sleep and Wakefulness.[100] Sleep takes place as a result of overuse of the senses[101] or of digestion,[100] so it is vital to the body.[101] While a person is asleep, the critical activities, which include thinking, sensing, recalling and remembering, do not function as they do during wakefulness. Since a person cannot sense during sleep they can not have desire, which is the result of sensation. However, the senses are able to work during sleep,[101] albeit differently,[100] unless they are weary.[101] Dreams do not involve actually sensing a stimulus. In dreams, sensation is still involved, but in an altered manner.[101] Aristotle explains that when a person stares at a moving stimulus such as the waves in a body of water, and then look away, the next thing they look at appears to have a wavelike motion. When a person perceives a stimulus and the stimulus is no longer the focus of their attention, it leaves an impression.[100] When the body is awake and the senses are functioning properly, a person constantly encounters new stimuli to sense and so the impressions of previously perceived stimuli are ignored.[101] However, during sleep the impressions made throughout the day are noticed as there are no new distracting sensory experiences.[100] So, dreams result from these lasting impressions. Since impressions are all that are left and not the exact stimuli, dreams do not resemble the actual waking experience.[102] During sleep, a person is in an altered state of mind. Aristotle compares a sleeping person to a person who is overtaken by strong feelings toward a stimulus. For example, a person who has a strong infatuation with someone may begin to think they see that person everywhere because they are so overtaken by their feelings. Since a person sleeping is in a suggestible state and unable to make judgements, they become easily deceived by what appears in their dreams, like the infatuated person.[100] This leads the person to believe the dream is real, even when the dreams are absurd in nature.[100] In De Anima iii 3, Aristotle ascribes the ability to create, to store, and to recall images in the absence of perception to the faculty of imagination, phantasia.[15] One component of Aristotle's theory of dreams disagrees with previously held beliefs. He claimed that dreams are not foretelling and not sent by a divine being. Aristotle reasoned naturalistically that instances in which dreams do resemble future events are simply coincidences.[103] Aristotle claimed that a dream is first established by the fact that the person is asleep when they experience it. If a person had an image appear for a moment after waking up or if they see something in the dark it is not considered a dream because they were awake when it occurred. Secondly, any sensory experience that is perceived while a person is asleep does not qualify as part of a dream. For example, if, while a person is sleeping, a door shuts and in their dream they hear a door is shut, this sensory experience is not part of the dream. Lastly, the images of dreams must be a result of lasting impressions of waking sensory experiences.[102] ## Practical philosophy Aristotle's practical philosophy covers areas such as ethics, politics, economics, and rhetoric.[42] Virtues and their accompanying vices[4] Too little Virtuous mean Too much Humbleness High-mindedness Vainglory Lack of purpose Right ambition Over-ambition Spiritlessness Good temper Irascibility Rudeness Civility Obsequiousness Cowardice Courage Rashness Insensibility Self-control Intemperance Sarcasm Sincerity Boastfulness Boorishness Wit Buffoonery Shamelessness Modesty Shyness Callousness Just resentment Spitefulness Pettiness Generosity Vulgarity Meanness Liberality Wastefulness ### Just war theory Aristotelian just war theory is not well regarded in the present day, especially his view that warfare was justified to enslave "natural slaves". In Aristotelian philosophy, the abolition of what he considers "natural slavery" would undermine civic freedom. The pursuit of freedom is inseparable from pursuing mastery over "those who deserve to be slaves". According to The Cambridge Companion to Aristotle's Politics the targets of this aggressive warfare were non-Greeks, noting Aristotle's view that "our poets say 'it is proper for Greeks to rule non-Greeks'".[104] Aristotle generally has a favourable opinion of war, extolling it as a chance for virtue and writing that "the leisure that accompanies peace" tends to make people "arrogant". War to "avoid becoming enslaved to others" is justified as self-defense. He writes that war "compels people to be just and temperate", however, in order to be just "war must be chosen for the sake of peace" (with the exception of wars of aggression discussed above).[104] ### Ethics Aristotle considered ethics to be a practical rather than theoretical study, i.e., one aimed at becoming good and doing good rather than knowing for its own sake. He wrote several treatises on ethics, including most notably, the Nicomachean Ethics.[105] Aristotle taught that virtue has to do with the proper function (ergon) of a thing. An eye is only a good eye in so much as it can see, because the proper function of an eye is sight. Aristotle reasoned that humans must have a function specific to humans, and that this function must be an activity of the psuchē (soul) in accordance with reason (logos). Aristotle identified such an optimum activity (the virtuous mean, between the accompanying vices of excess or deficiency[4]) of the soul as the aim of all human deliberate action, eudaimonia, generally translated as "happiness" or sometimes "well being". To have the potential of ever being happy in this way necessarily requires a good character (ēthikē aretē), often translated as moral or ethical virtue or excellence.[106] Aristotle taught that to achieve a virtuous and potentially happy character requires a first stage of having the fortune to be habituated not deliberately, but by teachers, and experience, leading to a later stage in which one consciously chooses to do the best things. When the best people come to live life this way their practical wisdom (phronesis) and their intellect (nous) can develop with each other towards the highest possible human virtue, the wisdom of an accomplished theoretical or speculative thinker, or in other words, a philosopher.[107] ### Politics In addition to his works on ethics, which address the individual, Aristotle addressed the city in his work titled Politics. Aristotle considered the city to be a natural community. Moreover, he considered the city to be prior in importance to the family which in turn is prior to the individual, "for the whole must of necessity be prior to the part".[108] He famously stated that "man is by nature a political animal" and argued that humanity's defining factor among others in the animal kingdom is its rationality.[109] Aristotle conceived of politics as being like an organism rather than like a machine, and as a collection of parts none of which can exist without the others. Aristotle's conception of the city is organic, and he is considered one of the first to conceive of the city in this manner.[110] Aristotle's classifications of political constitutions The common modern understanding of a political community as a modern state is quite different from Aristotle's understanding. Although he was aware of the existence and potential of larger empires, the natural community according to Aristotle was the city (polis) which functions as a political "community" or "partnership" (koinōnia). The aim of the city is not just to avoid injustice or for economic stability, but rather to allow at least some citizens the possibility to live a good life, and to perform beautiful acts: "The political partnership must be regarded, therefore, as being for the sake of noble actions, not for the sake of living together." This is distinguished from modern approaches, beginning with social contract theory, according to which individuals leave the state of nature because of "fear of violent death" or its "inconveniences."[L] In Protrepticus, the character 'Aristotle' states:[111] For we all agree that the most excellent man should rule, i.e., the supreme by nature, and that the law rules and alone is authoritative; but the law is a kind of intelligence, i.e. a discourse based on intelligence. And again, what standard do we have, what criterion of good things, that is more precise than the intelligent man? For all that this man will choose, if the choice is based on his knowledge, are good things and their contraries are bad. And since everybody chooses most of all what conforms to their own proper dispositions (a just man choosing to live justly, a man with bravery to live bravely, likewise a self-controlled man to live with self-control), it is clear that the intelligent man will choose most of all to be intelligent; for this is the function of that capacity. Hence it's evident that, according to the most authoritative judgment, intelligence is supreme among goods.[111] ### Economics Aristotle made substantial contributions to economic thought, especially to thought in the Middle Ages.[112] In Politics, Aristotle addresses the city, property, and trade. His response to criticisms of private property, in Lionel Robbins's view, anticipated later proponents of private property among philosophers and economists, as it related to the overall utility of social arrangements.[112] Aristotle believed that although communal arrangements may seem beneficial to society, and that although private property is often blamed for social strife, such evils in fact come from human nature. In Politics, Aristotle offers one of the earliest accounts of the origin of money.[112] Money came into use because people became dependent on one another, importing what they needed and exporting the surplus. For the sake of convenience, people then agreed to deal in something that is intrinsically useful and easily applicable, such as iron or silver.[113] Aristotle's discussions on retail and interest was a major influence on economic thought in the Middle Ages. He had a low opinion of retail, believing that contrary to using money to procure things one needs in managing the household, retail trade seeks to make a profit. It thus uses goods as a means to an end, rather than as an end unto itself. He believed that retail trade was in this way unnatural. Similarly, Aristotle considered making a profit through interest unnatural, as it makes a gain out of the money itself, and not from its use.[113] Aristotle gave a summary of the function of money that was perhaps remarkably precocious for his time. He wrote that because it is impossible to determine the value of every good through a count of the number of other goods it is worth, the necessity arises of a single universal standard of measurement. Money thus allows for the association of different goods and makes them "commensurable".[113] He goes on to state that money is also useful for future exchange, making it a sort of security. That is, "if we do not want a thing now, we shall be able to get it when we do want it".[113] ### Rhetoric and poetics The Blind Oedipus Commending his Children to the Gods (1784) by Bénigne Gagneraux. In his Poetics, Aristotle uses the tragedy Oedipus Tyrannus by Sophocles as an example of how the perfect tragedy should be structured, with a generally good protagonist who starts the play prosperous, but loses everything through some hamartia (fault).[114] Aristotle's Rhetoric proposes that a speaker can use three basic kinds of appeals to persuade his audience: ethos (an appeal to the speaker's character), pathos (an appeal to the audience's emotion), and logos (an appeal to logical reasoning).[115] He also categorizes rhetoric into three genres: epideictic (ceremonial speeches dealing with praise or blame), forensic (judicial speeches over guilt or innocence), and deliberative (speeches calling on an audience to make a decision on an issue).[116] Aristotle also outlines two kinds of rhetorical proofs: enthymeme (proof by syllogism) and paradeigma (proof by example).[117] Aristotle writes in his Poetics that epic poetry, tragedy, comedy, dithyrambic poetry, painting, sculpture, music, and dance are all fundamentally acts of mimesis ("imitation"), each varying in imitation by medium, object, and manner.[118][119] He applies the term mimesis both as a property of a work of art and also as the product of the artist's intention[118] and contends that the audience's realisation of the mimesis is vital to understanding the work itself.[118] Aristotle states that mimesis is a natural instinct of humanity that separates humans from animals[118][120] and that all human artistry "follows the pattern of nature".[118] Because of this, Aristotle believed that each of the mimetic arts possesses what Stephen Halliwell calls "highly structured procedures for the achievement of their purposes."[118] For example, music imitates with the media of rhythm and harmony, whereas dance imitates with rhythm alone, and poetry with language. The forms also differ in their object of imitation. Comedy, for instance, is a dramatic imitation of men worse than average; whereas tragedy imitates men slightly better than average. Lastly, the forms differ in their manner of imitation – through narrative or character, through change or no change, and through drama or no drama.[121] While it is believed that Aristotle's Poetics originally comprised two books – one on comedy and one on tragedy – only the portion that focuses on tragedy has survived. Aristotle taught that tragedy is composed of six elements: plot-structure, character, style, thought, spectacle, and lyric poetry.[122] The characters in a tragedy are merely a means of driving the story; and the plot, not the characters, is the chief focus of tragedy. Tragedy is the imitation of action arousing pity and fear, and is meant to effect the catharsis of those same emotions. Aristotle concludes Poetics with a discussion on which, if either, is superior: epic or tragic mimesis. He suggests that because tragedy possesses all the attributes of an epic, possibly possesses additional attributes such as spectacle and music, is more unified, and achieves the aim of its mimesis in shorter scope, it can be considered superior to epic.[123] Aristotle was a keen systematic collector of riddles, folklore, and proverbs; he and his school had a special interest in the riddles of the Delphic Oracle and studied the fables of Aesop.[124] ### Views on women Aristotle's analysis of procreation describes an active, ensouling masculine element bringing life to an inert, passive female element. On this ground, proponents of feminist metaphysics have accused Aristotle of misogyny[125] and sexism.[126] However, Aristotle gave equal weight to women's happiness as he did to men's, and commented in his Rhetoric that the things that lead to happiness need to be in women as well as men.[M] ## Influence More than 2300 years after his death, Aristotle remains one of the most influential people who ever lived.[128][129] He contributed to almost every field of human knowledge then in existence, and he was the founder of many new fields. According to the philosopher Bryan Magee, "it is doubtful whether any human being has ever known as much as he did".[130] Among countless other achievements, Aristotle was the founder of formal logic,[131] pioneered the study of zoology, and left every future scientist and philosopher in his debt through his contributions to the scientific method.[132][133][134] Taneli Kukkonen, writing in The Classical Tradition, observes that his achievement in founding two sciences is unmatched, and his reach in influencing "every branch of intellectual enterprise" including Western ethical and political theory, theology, rhetoric and literary analysis is equally long. As a result, Kukkonen argues, any analysis of reality today "will almost certainly carry Aristotelian overtones ... evidence of an exceptionally forceful mind."[134] Jonathan Barnes wrote that "an account of Aristotle's intellectual afterlife would be little less than a history of European thought".[135] ### On his successor, Theophrastus Frontispiece to a 1644 version of Theophrastus's Historia Plantarum, originally written around 300 BC Aristotle's pupil and successor, Theophrastus, wrote the History of Plants, a pioneering work in botany. Some of his technical terms remain in use, such as carpel from carpos, fruit, and pericarp, from pericarpion, seed chamber.[136] Theophrastus was much less concerned with formal causes than Aristotle was, instead pragmatically describing how plants functioned.[137][138] ### On later Greek philosophers The immediate influence of Aristotle's work was felt as the Lyceum grew into the Peripatetic school. Aristotle's notable students included Aristoxenus, Dicaearchus, Demetrius of Phalerum, Eudemos of Rhodes, Harpalus, Hephaestion, Mnason of Phocis, Nicomachus, and Theophrastus. Aristotle's influence over Alexander the Great is seen in the latter's bringing with him on his expedition a host of zoologists, botanists, and researchers. He had also learned a great deal about Persian customs and traditions from his teacher. Although his respect for Aristotle was diminished as his travels made it clear that much of Aristotle's geography was clearly wrong, when the old philosopher released his works to the public, Alexander complained "Thou hast not done well to publish thy acroamatic doctrines; for in what shall I surpass other men if those doctrines wherein I have been trained are to be all men's common property?"[139] ### On Hellenistic science After Theophrastus, the Lyceum failed to produce any original work. Though interest in Aristotle's ideas survived, they were generally taken unquestioningly.[140] It is not until the age of Alexandria under the Ptolemies that advances in biology can be again found. The first medical teacher at Alexandria, Herophilus of Chalcedon, corrected Aristotle, placing intelligence in the brain, and connected the nervous system to motion and sensation. Herophilus also distinguished between veins and arteries, noting that the latter pulse while the former do not.[141] Though a few ancient atomists such as Lucretius challenged the teleological viewpoint of Aristotelian ideas about life, teleology (and after the rise of Christianity, natural theology) would remain central to biological thought essentially until the 18th and 19th centuries. Ernst Mayr states that there was "nothing of any real consequence in biology after Lucretius and Galen until the Renaissance."[142] ### On Byzantine scholars Greek Christian scribes played a crucial role in the preservation of Aristotle by copying all the extant Greek language manuscripts of the corpus. The first Greek Christians to comment extensively on Aristotle were Philoponus, Elias, and David in the sixth century, and Stephen of Alexandria in the early seventh century.[143] John Philoponus stands out for having attempted a fundamental critique of Aristotle's views on the eternity of the world, movement, and other elements of Aristotelian thought.[144] Philoponus questioned Aristotle's teaching of physics, noting its flaws and introducing the theory of impetus to explain his observations.[145] After a hiatus of several centuries, formal commentary by Eustratius and Michael of Ephesus reappeared in the late eleventh and early twelfth centuries, apparently sponsored by Anna Comnena.[146] ### On the medieval Islamic world Islamic portrayal of Aristotle, c. 1220 Aristotle was one of the most revered Western thinkers in early Islamic theology. Most of the still extant works of Aristotle,[147] as well as a number of the original Greek commentaries, were translated into Arabic and studied by Muslim philosophers, scientists and scholars. Averroes, Avicenna and Alpharabius, who wrote on Aristotle in great depth, also influenced Thomas Aquinas and other Western Christian scholastic philosophers. Alkindus greatly admired Aristotle's philosophy,[148] and Averroes spoke of Aristotle as the "exemplar" for all future philosophers.[149] Medieval Muslim scholars regularly described Aristotle as the "First Teacher".[147] The title "teacher" was first given to Aristotle by Muslim scholars, and was later used by Western philosophers (as in the famous poem of Dante) who were influenced by the tradition of Islamic philosophy.[150] ### On medieval Europe With the loss of the study of ancient Greek in the early medieval Latin West, Aristotle was practically unknown there from c. AD 600 to c. 1100 except through the Latin translation of the Organon made by Boethius. In the twelfth and thirteenth centuries, interest in Aristotle revived and Latin Christians had translations made, both from Arabic translations, such as those by Gerard of Cremona,[152] and from the original Greek, such as those by James of Venice and William of Moerbeke. After the Scholastic Thomas Aquinas wrote his Summa Theologica, working from Moerbeke's translations and calling Aristotle "The Philosopher",[153] the demand for Aristotle's writings grew, and the Greek manuscripts returned to the West, stimulating a revival of Aristotelianism in Europe that continued into the Renaissance.[154] These thinkers blended Aristotelian philosophy with Christianity, bringing the thought of Ancient Greece into the Middle Ages. Scholars such as Boethius, Peter Abelard, and John Buridan worked on Aristotelian logic.[155] The medieval English poet Chaucer describes his student as being happy by having at his beddes heed Twenty bookes, clad in blak or reed, Of aristotle and his philosophie,[156] A cautionary medieval tale held that Aristotle advised his pupil Alexander to avoid the king's seductive mistress, Phyllis, but was himself captivated by her, and allowed her to ride him. Phyllis had secretly told Alexander what to expect, and he witnessed Phyllis proving that a woman's charms could overcome even the greatest philosopher's male intellect. Artists such as Hans Baldung produced a series of illustrations of the popular theme.[157][151] The Italian poet Dante says of Aristotle in The Divine Comedy: Dante L'Inferno, Canto IV. 131–135 Translation Hell vidi 'l maestro di color che sanno seder tra filosofica famiglia. Tutti lo miran, tutti onor li fanno: quivi vid'ïo Socrate e Platone che 'nnanzi a li altri più presso li stanno; I saw the Master there of those who know, Amid the philosophic family, By all admired, and by all reverenced; There Plato too I saw, and Socrates, Who stood beside him closer than the rest. ### On Early Modern scientists William Harvey's De Motu Cordis, 1628, showed that the blood circulated, contrary to classical era thinking. In the Early Modern period, scientists such as William Harvey in England and Galileo Galilei in Italy reacted against the theories of Aristotle and other classical era thinkers like Galen, establishing new theories based to some degree on observation and experiment. Harvey demonstrated the circulation of the blood, establishing that the heart functioned as a pump rather than being the seat of the soul and the controller of the body's heat, as Aristotle thought.[158] Galileo used more doubtful arguments to displace Aristotle's physics, proposing that bodies all fall at the same speed whatever their weight.[159] ### On 19th-century thinkers The 19th-century German philosopher Friedrich Nietzsche has been said to have taken nearly all of his political philosophy from Aristotle.[160] Aristotle rigidly separated action from production, and argued for the deserved subservience of some people ("natural slaves"), and the natural superiority (virtue, arete) of others. It was Martin Heidegger, not Nietzsche, who elaborated a new interpretation of Aristotle, intended to warrant his deconstruction of scholastic and philosophical tradition.[161] The English mathematician George Boole fully accepted Aristotle's logic, but decided "to go under, over, and beyond" it with his system of algebraic logic in his 1854 book The Laws of Thought. This gives logic a mathematical foundation with equations, enables it to solve equations as well as check validity, and allows it to handle a wider class of problems by expanding propositions of any number of terms, not just two.[162] ### Modern rejection and rehabilitation "That most enduring of romantic images, Aristotle tutoring the future conqueror Alexander".[134] Illustration by Charles Laplante [fr], 1866 During the 20th century, Aristotle's work was widely criticized. The philosopher Bertrand Russell argued that "almost every serious intellectual advance has had to begin with an attack on some Aristotelian doctrine". Russell called Aristotle's ethics "repulsive", and labelled his logic "as definitely antiquated as Ptolemaic astronomy". Russell stated that these errors made it difficult to do historical justice to Aristotle, until one remembered what an advance he made upon all of his predecessors.[5] The Dutch historian of science Eduard Jan Dijksterhuis wrote that Aristotle and his predecessors showed the difficulty of science by "proceed[ing] so readily to frame a theory of such a general character" on limited evidence from their senses.[163] In 1985, the biologist Peter Medawar could still state in "pure seventeenth century"[164] tones that Aristotle had assembled "a strange and generally speaking rather tiresome farrago of hearsay, imperfect observation, wishful thinking and credulity amounting to downright gullibility".[164][165] By the start of the 21st century, however, Aristotle was taken more seriously: Kukkonen noted that "In the best 20th-century scholarship Aristotle comes alive as a thinker wrestling with the full weight of the Greek philosophical tradition."[134] Ayn Rand accredited Aristotle as "the greatest philosopher in history" and cited him as a major influence on her thinking.[166] More recently, Alasdair MacIntyre has attempted to reform what he calls the Aristotelian tradition in a way that is anti-elitist and capable of disputing the claims of both liberals and Nietzscheans.[167] Kukkonen observed, too, that "that most enduring of romantic images, Aristotle tutoring the future conqueror Alexander" remained current, as in the 2004 film Alexander, while the "firm rules" of Aristotle's theory of drama have ensured a role for the Poetics in Hollywood.[134] Biologists continue to be interested in Aristotle's thinking. Armand Marie Leroi has reconstructed Aristotle's biology,[168] while Niko Tinbergen's four questions, based on Aristotle's four causes, are used to analyse animal behaviour; they examine function, phylogeny, mechanism, and ontogeny.[169][170] ## Surviving works ### Corpus Aristotelicum First page of a 1566 edition of the Nicomachean Ethics in Greek and Latin The works of Aristotle that have survived from antiquity through medieval manuscript transmission are collected in the Corpus Aristotelicum. These texts, as opposed to Aristotle's lost works, are technical philosophical treatises from within Aristotle's school. Reference to them is made according to the organization of Immanuel Bekker's Royal Prussian Academy edition (Aristotelis Opera edidit Academia Regia Borussica, Berlin, 1831–1870), which in turn is based on ancient classifications of these works.[171] ### Loss and preservation Aristotle wrote his works on papyrus scrolls, the common writing medium of that era.[N] His writings are divisible into two groups: the "exoteric", intended for the public, and the "esoteric", for use within the Lyceum school.[173][O][174] Aristotle's "lost" works stray considerably in characterization from the surviving Aristotelian corpus. Whereas the lost works appear to have been originally written with a view to subsequent publication, the surviving works mostly resemble lecture notes not intended for publication.[175][173] Cicero's description of Aristotle's literary style as "a river of gold" must have applied to the published works, not the surviving notes.[P] A major question in the history of Aristotle's works is how the exoteric writings were all lost, and how the ones we now possess came to us.[177] The consensus is that Andronicus of Rhodes collected the esoteric works of Aristotle's school which existed in the form of smaller, separate works, distinguished them from those of Theophrastus and other Peripatetics, edited them, and finally compiled them into the more cohesive, larger works as they are known today.[178][179] ## Legacy ### Depictions Paintings Aristotle has been depicted by major artists including Lucas Cranach the Elder,[180] Justus van Gent, Raphael, Paolo Veronese, Jusepe de Ribera,[181] Rembrandt,[182] and Francesco Hayez over the centuries. Among the best-known is Raphael's fresco The School of Athens, in the Vatican's Apostolic Palace, where the figures of Plato and Aristotle are central to the image, at the architectural vanishing point, reflecting their importance.[183] Rembrandt's Aristotle with a Bust of Homer, too, is a celebrated work, showing the knowing philosopher and the blind Homer from an earlier age: as the art critic Jonathan Jones writes, "this painting will remain one of the greatest and most mysterious in the world, ensnaring us in its musty, glowing, pitch-black, terrible knowledge of time."[184][185] Sculptures ### Eponyms The Aristotle Mountains in Antarctica are named after Aristotle. He was the first person known to conjecture, in his book Meteorology, the existence of a landmass in the southern high-latitude region and called it Antarctica.[186] Aristoteles is a crater on the Moon bearing the classical form of Aristotle's name.[187] ## References ### Notes 1. ^ That these dates (the first half of the Olympiad year 384/383 BC, and in 322 shortly before the death of Demosthenes) are correct was shown by August Boeckh (Kleine Schriften VI 195); for further discussion, see Felix Jacoby on FGrHist 244 F 38. Ingemar Düring, Aristotle in the Ancient Biographical Tradition, Göteborg, 1957,p. 253 2. ^ See Shields 2012, pp. 3–16; Düring 1957 covers ancient biographies of Aristotle. 3. ^ This type of syllogism, with all three terms in 'a', is known by the traditional (medieval) mnemonic Barbara.[27] 4. ^ M is the Middle (here, Men), S is the Subject (Greeks), P is the Predicate (mortal).[27] 5. ^ The first equation can be read as 'It is not true that there exists an x such that x is a man and that x is not mortal.'[28] 6. ^ Rhett Allain notes that Newton's First Law is "essentially a direct reply to Aristotle, that the natural state is not to change motion.[44] 7. ^ Leonard Susskind comments that Aristotle had clearly never gone ice skating or he would have seen that it takes force to stop an object.[46] 8. ^ For heavenly bodies like the Sun, Moon, and stars, the observed motions are "to a very good approximation" circular around the Earth's centre, (for example, the apparent rotation of the sky because of the rotation of the Earth, and the rotation of the moon around the Earth) as Aristotle stated.[47] 9. ^ Drabkin quotes numerous passages from Physics and On the Heavens (De Caelo) which state Aristotle's laws of motion.[45] 10. ^ Drabkin agrees that density is treated quantitatively in this passage, but without a sharp definition of density as weight per unit volume.[45] 11. ^ Philoponus and Galileo correctly objected that for the transient phase (still increasing in speed) with heavy objects falling a short distance, the law does not apply: Galileo used balls on a short incline to show this. Rovelli notes that "Two heavy balls with the same shape and different weight do fall at different speeds from an aeroplane, confirming Aristotle's theory, not Galileo's."[47] 12. ^ For a different reading of social and economic processes in the Nicomachean Ethics and Politics see Polanyi, Karl (1957) "Aristotle Discovers the Economy" in Primitive, Archaic and Modern Economies: Essays of Karl Polanyi ed. G. Dalton, Boston 1971, 78–115. 13. ^ "Where, as among the Lacedaemonians, the state of women is bad, almost half of human life is spoilt."[127] 14. ^ "When the Roman dictator Sulla invaded Athens in 86 BC, he brought back to Rome a fantastic prize – Aristotle's library. Books then were papyrus rolls, from 10 to 20 feet long, and since Aristotle's death in 322 BC, worms and damp had done their worst. The rolls needed repairing, and the texts clarifying and copying on to new papyrus (imported from Egypt – Moses' bulrushes). The man in Rome who put Aristotle's library in order was a Greek scholar, Tyrannio."[172] 15. ^ Aristotle: Nicomachean Ethics 1102a26–27. Aristotle himself never uses the term "esoteric" or "acroamatic". For other passages where Aristotle speaks of exōterikoi logoi, see W.D. Ross, Aristotle's Metaphysics (1953), vol. 2 pp= 408–10. Ross defends an interpretation according to which the phrase, at least in Aristotle's own works, usually refers generally to "discussions not peculiar to the Peripatetic school", rather than to specific works of Aristotle's own. 16. ^ "veniet flumen orationis aureum fundens Aristoteles", (Google translation: "Aristotle will come pouring forth a golden stream of eloquence").[176] 17. ^ Compare the medieval tale of Phyllis and Alexander above. ### Citations 1. ^ Kantor 1963, p. 116. 2. ^ 3. ^ 4. ^ a b c d 5. ^ a b c d 6. ^ Barnes 1995, p. 9. 7. ^ 8. ^ McLeisch 1999, p. 5. 9. ^ 10. ^ Hall 2018, p. 14. 11. ^ Anagnostopoulos 2013, p. 4. 12. ^ Blits 1999, pp. 58–63. 13. ^ 14. ^ Aristotle 1984, pp. Introduction. 15. ^ a b c 16. ^ a b Green 1991, pp. 58–59. 17. ^ Smith 2007, p. 88. 18. ^ Green 1991, p. 460. 19. ^ Filonik 2013, pp. 72–73. 20. ^ Jones 1980, p. 216. 21. ^ Gigon 2017, p. 41. 22. ^ Düring 1957, p. T44a-e. 23. ^ Haase 1992, p. 3862. 24. ^ Degnan 1994, pp. 81–89. 25. ^ Corcoran 2009, pp. 1–20. 26. ^ Kant 1787, pp. Preface. 27. ^ a b c 28. ^ 29. ^ Pickover 2009, p. 52. 30. ^ 31. ^ 32. ^ Prior Analytics, pp. 24b18–20. 33. ^ 34. ^ a b c 35. ^ 36. ^ Aristotle 1999, p. 111. 37. ^ Metaphysics, p. VIII 1043a 10–30. 38. Cohen 2016. 39. ^ Lloyd 1968, pp. 43–47. 40. ^ Metaphysics, p. IX 1050a 5–10. 41. ^ Metaphysics, p. VIII 1045a–b. 42. ^ a b c d 43. ^ a b Lloyd 1968, pp. 133–39, 166–69. 44. ^ a b 45. Drabkin 1938, pp. 60–84. 46. Rovelli 2015, pp. 23–40. 47. ^ Carteron 1923, pp. 1–32 and passim. 48. ^ Leroi 2015, pp. 88–90. 49. Lloyd 1996, pp. 96–100, 106–07. 50. ^ Hankinson 1998, p. 159. 51. ^ a b Leroi 2015, pp. 91–92, 369–73. 52. ^ 53. ^ Physics, p. 2.6. 54. ^ Miller 1973, pp. 204–13. 55. ^ Meteorology, p. 1. 8. 56. ^ Moore 1956, p. 13. 57. ^ Meteorology, p. Book 1, Part 14. 58. ^ Lyell 1832, p. 17. 59. ^ Leroi 2015, p. 7. 60. ^ Leroi 2015, p. 14. 61. ^ Thompson 1910, p. Prefatory Note. 62. ^ "Darwin's Ghosts, By Rebecca Stott". independent.co.uk. 2 June 2012. Retrieved 19 June 2012. 63. ^ Leroi 2015, pp. 196, 248. 64. ^ Day 2013, pp. 5805–16. 65. ^ Leroi 2015, pp. 66–74, 137. 66. ^ Leroi 2015, pp. 118–19. 67. ^ Leroi 2015, p. 73. 68. ^ Leroi 2015, pp. 135–36. 69. ^ Leroi 2015, p. 206. 70. ^ Sedley 2007, p. 189. 71. ^ Leroi 2015, p. 273. 72. ^ Taylor 1922, p. 42. 73. ^ Leroi 2015, pp. 361–65. 74. ^ 75. ^ Leroi 2015, pp. 197–200. 76. ^ a b Leroi 2015, pp. 365–68. 77. ^ Taylor 1922, p. 49. 78. ^ Leroi 2015, p. 408. 79. ^ Leroi 2015, pp. 72–74. 80. ^ Bergstrom & Dugatkin 2012, p. 35. 81. ^ Rhodes 1974, p. 7. 82. ^ Mayr 1982, pp. 201–02. 83. ^ 84. ^ Leroi 2015, pp. 111–19. 85. ^ Mason 1979, pp. 43–44. 86. ^ Leroi 2015, pp. 156–63. 87. ^ Mason 1979, p. 45. 88. ^ Guthrie 2010, p. 348. 89. ^ Bloch 2007, p. 12. 90. ^ Bloch 2007, p. 61. 91. ^ Carruthers 2007, p. 16. 92. ^ Bloch 2007, p. 25. 93. ^ Warren 1921, p. 30. 94. ^ Warren 1921, p. 25. 95. ^ Carruthers 2007, p. 19. 96. ^ Warren 1921, p. 296. 97. ^ Warren 1921, p. 259. 98. ^ Sorabji 2006, p. 54. 99. Holowchak 1996, pp. 405–23. 100. Shute 1941, pp. 115–18. 101. ^ a b Modrak 2009, pp. 169–81. 102. ^ Webb 1990, pp. 174–84. 103. ^ a b Deslauriers & Destrée 2013, pp. 157-162. 104. ^ 105. ^ Nicomachean Ethics Book I. See for example chapter 7. 106. ^ Nicomachean Ethics, p. Book VI. 107. ^ Politics, pp. 1253a19–24. 108. ^ Aristotle 2009, pp. 320–21. 109. ^ Ebenstein & Ebenstein 2002, p. 59. 110. ^ a b Hutchinson & Johnson 2015, p. 22. 111. ^ a b c Robbins 2000, pp. 20–24. 112. ^ a b c d Aristotle 1948, pp. 16–28. 113. ^ Kaufmann 1968, pp. 56–60. 114. ^ Garver 1994, pp. 109–10. 115. ^ Rorty 1996, pp. 3–7. 116. ^ Grimaldi 1998, p. 71. 117. Halliwell 2002, pp. 152–59. 118. ^ Poetics, p. I 1447a. 119. ^ Poetics, p. IV. 120. ^ Poetics, p. III. 121. ^ Poetics, p. VI. 122. ^ Poetics, p. XXVI. 123. ^ Aesop 1998, pp. Introduction, xi–xii. 124. ^ 125. ^ Morsink 1979, pp. 83–112. 126. ^ Rhetoric, p. Book I, Chapter 5. 127. ^ Leroi 2015, p. 8. 128. ^ 129. ^ Magee 2010, p. 34. 130. ^ Guthrie 1990, p. 156. 131. ^ 132. ^ Durant 2006, p. 92. 133. Kukkonen 2010, pp. 70–77. 134. ^ Barnes 1982, p. 86. 135. ^ Hooker 1831, p. 219. 136. ^ Mayr 1982, pp. 90–91. 137. ^ Mason 1979, p. 46. 138. ^ Plutarch 1919, p. Part 1, 7:7. 139. ^ Annas 2001, p. 252. 140. ^ Mason 1979, p. 56. 141. ^ Mayr 1985, pp. 90–94. 142. ^ Sorabji 1990, pp. 20, 28, 35–36. 143. ^ Sorabji 1990, pp. 233–74. 144. ^ Lindberg 1992, p. 162. 145. ^ Sorabji 1990, pp. 20–21, 28–29, 393–406, 407–08. 146. ^ a b 147. ^ 148. ^ Averroes 1953, p. III, 2, 43. 149. ^ Nasr 1996, pp. 59–60. 150. ^ a b 151. ^ 152. ^ 153. ^ 154. ^ 155. ^ Allen & Fisher 2011, p. 17. 156. ^ 157. ^ Aird 2011, pp. 118–29. 158. ^ 159. ^ Durant 2006, p. 86. 160. ^ Sikka 1997, p. 265. 161. ^ 162. ^ Dijksterhuis 1969, p. 72. 163. ^ a b Leroi 2015, p. 353. 164. ^ Medawar & Medawar 1984, p. 28. 165. ^ 166. ^ Knight 2007, pp. passim. 167. ^ 168. ^ MacDougall-Shackleton 2011, pp. 2076–85. 169. ^ 170. ^ 171. ^ 172. ^ a b Barnes 1995, p. 12. 173. ^ House 1956, p. 35. 174. ^ Irwin & Fine 1996, pp. xi–xii. 175. ^ 176. ^ Barnes & Griffin 1999, pp. 1–69. 177. ^ Anagnostopoulos 2013, p. 16. 178. ^ Barnes 1995, pp. 10–15. 179. ^ 180. ^ 181. ^ 182. ^ 183. ^ 184. ^ 185. ^ 186. ^	2023-02-05 14:16:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "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": 0.4537208080291748, "perplexity": 6011.559047973056}, "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/1674764500255.78/warc/CC-MAIN-20230205130241-20230205160241-00786.warc.gz"}
https://physics.stackexchange.com/questions/352630/are-these-two-square-root-non-square-root-worldline-actions-equivalent-at-quan?noredirect=1	# Are these two square root & non-square root worldline actions equivalent at quantum level? (Kleinert's method) The relativistic particle action is (in its more natural form) $$A=McS=Mc\int_{\lambda_a}^{\lambda_b}d\lambda\sqrt{x'^2(\lambda)}.\tag{19.12}$$ That action doesn't lend itself easily to a calculation of the corresponding path integral, so (as done in many papers and textbooks) we can use the alternative form that is more suitable since it is quadratic: $$\bar{A}=\int_{\lambda_a}^{\lambda_b}d\lambda \left[\frac{Mc}{2h(\lambda)}x'^2(\lambda)+h(\lambda)\frac{Mc}{2}\right].\tag{19.10}$$ Although the $\bar{A}$ describes the same classical physics as the original action (19.12), I wonder that it may lead to a different quantum physics. The textbooks give some arguments for equivalence but they don't actually "calculate" the integral, which seems imposible due to the square root. Kleinert's book1 "Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and ..." is the only one I have found (if you know another reference dealing with the equivalence issue please tell me), page 1424, Appendix 19A. It seems that he got $$(x_b\|x_a)=N''\int\frac{d^D k}{(2\pi)^D}\frac{1}{k^2+M_Rc^2/\hbar^2}e^{ik}(x_b-x_a)\tag{19A.15b}$$ with $$z \equiv \nu\log\nu , \qquad \nu\equiv \frac{D+1}{\bar{\epsilon}}\lambda_C, \qquad \bar{\epsilon} = \frac{S}{N+1},\tag{19A.13}$$ $$M_R=M(1+z)^{1/2} .\tag{19A.16}$$ So, in principle, they are different, similar but different. So, my questions are: 1. What is happening here? I mean with the mass, it looks like regularization or renormalization. 2. Is it necessary to calculate both path integrals in order to show they are equivalent? 3. Has the path integral been calculated correctly (by Kleinert) without cheating nor using any hocus-pocus process? Or what has he done? 1 If you don't have he book, please read this draft http://users.physik.fu-berlin.de/~kleinert/b5/psfiles/pthic19.pdf, page 1436. • Non-Gaussian path integrals are hocus-pocus by definition, or more precisely, by absence of definition :) You might find interesting Polyakov's treatment of this in his "Gauge fields and strings", chapter on random surfaces. He shows that there's a hocus-pocus which kinda justifies saying that any path integral with a field which enters the Lagrangian without derivatives is equivalent to the one for the reduced action. – Prof. Legolasov Aug 19 '17 at 5:03 • @SolenodonParadoxus what do you mean by the reduced action? and how can a lagrangia have no derivatives? – Pathy Aug 19 '17 at 5:41 • in your question the Lagrangian has no derivatives of $h$ in it. Reduced action is what you end up with when you solve the algebraic (classical) equation of motion for the field $h$ (or generally, the field which enters without derivatives) and plug this back in the action. For the relativistic particle this gives the square-root action. – Prof. Legolasov Aug 19 '17 at 8:21 • @SolenodonParadoxus yes off course, h is just like a lagrange multiplier, it is not a true degree of freedom. – Pathy Aug 19 '17 at 14:51 • Which is exactly my point. So Polyakov's handwaving is a more general claim, a special case of which is that your two actions give the same quantum dynamics. – Prof. Legolasov Aug 20 '17 at 8:42 1. Let us start by summarizing Kleinert's results for a relativistic massive point$^1$ particle: • On one hand, Kleinert finds no mass-renormalization (19.30) for the Hamiltonian phase space path integral (19.16) with a non-square root action (19.10). • On the other hand, Kleinert finds mass-renormalization (19A.15b) with for the Lagrangian path integral (19A.2) with a square root action (19.12). 2. Note that there doesn't seem to be a truly independent definition of the path integral (19A.2) with a square root action (19.12). Even Kleinert starts his derivation by immediately going to a non-square root integral representation (19A.3). 3. The reason why the results (19.30) & (19A15b) are different is not an issue of square root vs. non-square root action per se. Rather it is an issue of the Lagrangian vs. the Hamiltonian path integral formulation. There is no canonical choice of path integral measure within the Lagrangian formulation. (Recall e.g. the famous Feynman fudge factor.) The Hamiltonian formulation is more fundamental in this respect, cf. e.g. this Phys.SE post. In the Lagrangian path integral (19A.2), Kleinert (from the onset) omits the pertinent path integral measure factor (which can only be deduced in the Hamiltonian formulation). Hence there is no reason the results should agree. In turn, omitting the correct path integral measure factor introduces renormalization, even in point mechanics. 4. If one performs the Dirac-Bergmann analysis of both the Lagrangian square root action (19.12) and non-square root action (19.10) (which both possesses a world-line (WL) reparametrization gauge symmetry), and accounts for the pertinent first-class constraints, etc, one eventually ends up with one and the same Hamiltonian action formulation, cf. e.g. this, this & this Phys.SE posts and links therein. In conclusion, since the Lagrangian square root action (19.12) and non-square root action (19.10) have identical Hamiltonian action, then any path integral quantization scheme (that is consistent with the Hamiltonian formulation) must be identical as well. References: 1. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edition; Chapter 19 (pdf). -- $^1$ For the analogous Polyakov vs. Nambu-Goto question in string theory, see e.g. this Phys.SE post.	2021-04-16 12:26:23	{"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": 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": 0.9170677661895752, "perplexity": 453.3199145014575}, "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-2021-17/segments/1618038056325.1/warc/CC-MAIN-20210416100222-20210416130222-00085.warc.gz"}
https://www.molpro.net/manual/doku.php?id=region&do=	# Region The REGION program allows a correlated method to be applied to a target region within a molecule, where it is restricted to a subset of spatially localised orbitals, and the rest of the valence electrons left uncorrelated and only treated at the mean-field level. The correlated region can be embedded within an SCF environment (that of the reference wave function), or within a lower level correlated method using a simple subtractive approach (see section Multi-layer embedding). Multi-layer embedding can also be achieved using the same subtractive approach. Firstly, molecular orbitals are localised using the intrinsic bonding orbital (IBO) method, described in chapter intrinsic basis bonding analysis (IAO/IBO). Occupied orbitals to be embedded are either chosen directly by the user, or the user chooses a set of atoms, and the orbitals sat on these atoms are embedded (see section selecting the Embedded Orbitals). The occupied orbitals are reordered putting the embedded orbitals at the top, followed by the other valence orbitals which are placed into the core. The embedded and active/open orbital spaces are then made pseudo-canonical. The virtual orbital space can also be reduced and localised to the embedding region (see section truncating the virtual space). This can only be utilised by the FCI program within Molpro (and some third-party programs interfaced to Molpro), because the number of molecular orbitals no longer equals the number of basis functions. However, regional local coupled cluster can still be performed by the LCC program, using its own REGION directive (see section correlating subsets of electrons (REGION) for details). The molecular orbitals are controlled using the directives: START,$<$record.file$>$ SAVE,$<$record.file$>$ Molecular orbitals are read from the dump record given after START, and saved to the dump record given after SAVE The initial occupancies of each orbital space are controlled using the usual directives: CORE,$<$integer$>$ CLOSED,$<$integer$>$ OCC,$<$integer$>$ WF,charge=$<$integer$>$,sym=$<$integer$>$,spin=$<$integer$>$ The correct number of core orbitals must be known to the REGION program (either through inheriting it from the previous calculation, or through explicit declaration by the user). The core orbitals are kept separate from the inactive orbital space when performing the IBO localisation. If the definition of the core is changed, these two spaces will mix, leading to a slightly different set of embedded IBOs. The IBOs to be embedded can be specified directly with the ORBITALS option, either using their index, or as a range of indices e.g., {REGION,ORBITALS=[10-15,17,21]} Alternatively, the orbitals can be found automatically by choosing a set of target atoms with the ATOMS option. Orbitals exerting an IBO partial charge of at least THRESH_REGION (default=0.2 a.u.) on any of the target atoms, are included in the embedding region. Atoms may be specified using their names in the geometry specification, index, or using ranges of indices e.g. {REGION,ATOMS=[H10-H15,C17,N21]} or {REGION,ATOMS=[H,17,N2]} Further control of the ATOMS option is achieved using the ORB_SELECT option, which controls how strictly the target atoms inherit their orbitals. When ORB_SELECT=INCLUSIVE (the default), an orbital is embedded if any of the atoms it sits on are included in the target atoms. When ORB_SELECT=EXCLUSIVE, an orbital is only embedded if all the atoms it sits on are included in the target atoms. For example, in the following molecule: H H \ / C1=C2 / \ H H The command {REGION,ATOMS=[C1,C2],ORB_SELECT=EXCLUSIVE} selects only the carbon-carbon sigma and pi bonds, whereas the command {REGION,ATOMS=[C1,C2],ORB_SELECT=INCLUSIVE} selects all the bonds. For both the ORBTIALS and ATOMS options, the active orbital space (in a CASSCF wave function) or the open orbitals (in a single-reference wave function) are always included in the embedding region. Therefore using an empty ORBITALS or ATOMS option will embed only the active orbitals e.g. {REGION,ORBTIALS=[]} {REGION,ATOMS=[]} Core orbitals (as defined by the CORE variable) are exuded from the embedding region. Finally the embedded orbital space is made pseudo-canonical by diagonalising the embedded-embedded block of the Fock matrix, and similarly the active/open orbitals. This can be disabled by setting the SEMI_CAN=false. The FCI program, and some third-party programs interfaced to Molpro, can handle a reduction in the number of virtual orbitals. This procedure is called the deleted virtual approximation, and can significantly reduce the cost of correlated calculations. When the option FULL_VIRT=false, a truncated virtual space localized to the embedding region is constructed from projected atomic orbitals (PAOs). By default FULL_VIRT=true and the full canonical virtual space from the previous SCF calculation is retained. PAOs located on a set of ‘host atoms’ are transformed by means of a singular value decomposition of their overlap matrix. Virtual functions with an eigenvalue below THRESH_REDUN (default=$1.0\times10^{-8}$) are considered redundant and deleted. PAOs on the other, environment, atoms are also discarded. Host atoms are those sat ‘underneath’ the embedded occupied orbitals. When the embedded orbitals exert an IBO partial charge of at least THRESH_REGION (default 0.2 a.u.) on an atom, then it is classified as a host atom. Further host atoms may be manually included by using the HOST_ATOMS option. This allows the virtual domain to be extended. The truncated virtual space is made pseudo-canonical by diagonalising the virtual-virtual block of the Fock matrix. This can be disabled by setting the SEMI_CAN=false. Treating the environment with a second correlated method (that is usually less expensive) is done with a simple subtractive embedding procedure, defined as $$E = E_{low}(A+B) - E^{reg}_{low}(A) + E^{reg}_{high}(A).$$ The argument $A$ denotes the central target region, and $B$ the environment, and the subscripts denote the accuracy (and cost) of the correlated methods. $E^{reg}_{low}(A)$ is an energy calculation performed using the REGION program, with a lower level correlated method on the target region, for example MP2. Similarly $E^{reg}_{high}(A)$ is performed with a higher level correlated method on the same target region, for example CISD. Multiple embedding layers can be achieved through repeatedly applying the above equation. • OBRITALS=$<$string$>$ Example: ‘[6,7,8,9-12]’. Indices of IBOs to be embedded. • ATOMS=$<$string$>$ Example: ‘[C1,C2,H3,H4-H6]’. Orbitals exerting an IBO partial charge of at least THRESH_REGION on these atoms are embedded. • ORB_SELECT=$<$string$>$ Either ‘INCLUSIVE’ or ‘EXCLUSIVE’. Controls how strictly embedded orbitals are selected, when using the ATOMS option. Default=INCLUSIVE. • HOST_ATOMS=$<$string$>$ Example: ‘[C4,C5,H6-H10]’. List of extra host atoms, added to those chosen automatically. Only relevant when FULL_VIRT=false. • THRESH_REGION=$<$double$>$ IBO partial charge threshold. Determines which orbitals are sat on those atoms specified with the ATOMS option, as well as host atoms underneath embedded orbitals. Default=0.2 atomic units. • THRESH_REDUN=$<$double$>$ Eigenvalue threshold. Determines redundant virtual functions when FULL_VIRT=false. Default=$1.0\times10^{-8}$. • SEMI_CAN=$<$logical$>$ Control pseudo-canonicalisation of the embedded orbital spaces. Default=true. • FULL_VIRT=$<$logical$>$ When true, the full canonical virtual space from the previous SCF calculation is retained. When false, a truncated virtual space is constructed. Default=true. • IBO_AO_TYPE=$<$string$>$ Minimal basis set used by the IBO program. Default is that of the IBO program (MINAO-PP). • PLOT_PAOS=$<$logical$>$ Writes the PAOs to the dump record, so they can be visualized by an orbital viewing program. Default=false. • PRINT_MAP=$<$logical$>$ Prints a mapping of the rearranged IBO indices to the original IBO indices. Default=true. • REGION_TYPE=$<$integer$>$ When set to 0, the atomic core and environment valence orbitals are totally deleted. This allows the user to plot the embedded orbitals only, though cannot be used for viable calculations. When set to 1, the atomic core and environment orbitals are placed into the core. These orbitals must be used for actual calculations. Default=1. This is an example of embedding MRCI inside CASPT2 for Butane. memory,100,M gprint,orbitals,civector nosym;noextra ANGSTROM Geometry={ C1 ,, 0.0000 ,0.7647+y ,0.0000 C2 ,, 0.0000 ,-0.7647-y, 0.0000 C3 ,, -1.4029, 1.3695+y,0.0000 C4 ,, 1.4029 ,-1.3695-y, 0.0000 H5 ,, 0.5526 ,1.1231+y ,0.8740 H6 ,, 0.5526 ,1.1231+y ,-0.8740 H7 ,, -0.5526, -1.1231-y,0.8740 H8 ,, -0.5526, -1.1231-y,-0.8740 H9 ,, -1.3684, 2.4598+y,0.0000 H10,, 1.3684 ,-2.4598-y,0.0000 H11,, -1.9674, 1.0558+y,-0.8807 H12,, -1.9674, 1.0558+y,0.8807 H13,, 1.9674 ,-1.0558-y,-0.8807 H14,, 1.9674 ,-1.0558-y,0.8807} y=1.0 basis=vdz {multi;closed,16;occ,18;wf,charge=0,sym=1,spin=0} {rs2c} e1=energy {region,orbitals='[7-10]'; core,4;closed,16;occ,18;save,2200.2;wf,charge=0,sym=1,spin=0} {put,molden,p1.molden;nosort} {rs2c;orbital,2200.2} e2=energy {ci;orbital,2200.2} e3=energy e=e1-e2+e3	2022-09-30 10:40:35	{"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": 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": 0.8299554586410522, "perplexity": 3491.4697629882335}, "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/1664030335448.34/warc/CC-MAIN-20220930082656-20220930112656-00341.warc.gz"}
http://nora.nerc.ac.uk/515423/	nerc.ac.uk The energy expenditure of free-ranging black-browed albatrosses Bevan, R. M.; Butler, P. J.; Woakes, A. J.; Prince, P. A.. 1995 The energy expenditure of free-ranging black-browed albatrosses. Philosophical Transactions of the Royal Society B: Biological Sciences, 350 (1332). 119-131. 10.1098/rstb.1995.0146 Full text not available from this repository. (Request a copy) Abstract/Summary As heart rate (f<latex>$_{\text{H}}$</latex>) can be used to determine the energy expenditure of black-browed albatrosses (Diomedea melanophrys) (Bevan et al. 1994), data loggers - recording f<latex>$_{\text{H}}$</latex> and abdominal temperature (T<latex>$_{\text{ab}}$</latex>) - were implanted into free-ranging black-browed albatrosses breeding at South Georgia. Five birds also had salt water switches (sws) attached to one leg to record when the birds were on the water, and two others had satellite transmitters attached to their back to determine the birds' position at sea. The birds were released into their natural environment and recaptured, on average, 23 days later when the data loggers were removed. The f<latex>$_{\text{H}}$</latex> data were then converted into estimates of energy expenditure (EE) using a previously derived equation. The mean EE during incubation and brooding were 2.22 and 2.42 W kg<latex>$^{-1}$</latex>, respectively. When the birds were foraging at sea, EE increased to between 4.63 and 5.80 W kg<latex>$^{-1}$</latex>, depending on the phase of the reproductive cycle. As the birds spent approximately the same length of time at the nest and at sea during incubation and brooding, the overall mean EE during these phases were 3.63 and 3.54 W kg<latex>$^{-1}$</latex> respectively. These rates are significantly lower than that during the chick-rearing phase when a high level of foraging EE is maintained almost continuously. By combining information from the sws with the f<latex>$_{\text{H}}$</latex> data, it was possible to determine the EE of the birds when on the water (5.77 W kg<latex>$^{-1}$</latex>) and when flying (6.21 W kg<latex>$^{-1}$</latex>). These values are approximately twice the estimated basal metabolic rate (BMR) for the species. The energy costs of flight are half previous values, estimated using the doubly labelled water technique, because of the previous assumption that birds on the water have an EE equivalent to BMR. When the birds were on the nest, T<latex>$_{\text{ab}}$</latex> was 39.3 <latex>$\pm$</latex> 0.4 degrees C and this changed very little with time. However, when they were at sea, T<latex>$_{\text{ab}}$</latex> showed large variations, depending on the behaviour of the bird. Information from the sws indicated that all large drops (> 0.5 degrees C) in T<latex>$_{\text{ab}}$</latex> occurred when the birds were on water. The mean minimum value reached was 32.5 <latex>$\pm$</latex> 2.0 degrees C. It is likely that ingestion of prey or water are the major causes of this decrease. This is the first study to have used f<latex>$_{\text{H}}$</latex> extensively to determine the EE of a free-ranging marine bird. The advantages of using this technique are that data can be obtained over long durations with high resolution, permitting the EE of different activities to be estimated. Item Type: Publication - Article 10.1098/rstb.1995.0146 BAS Programmes > Pre 2000 programme 0962-8436 06 Dec 2016 14:58 +0 (UTC) http://nora.nerc.ac.uk/id/eprint/515423	2017-08-16 21:51:18	{"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": 0.5774644017219543, "perplexity": 3942.505185440728}, "config": {"markdown_headings": false, "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-2017-34/segments/1502886102663.36/warc/CC-MAIN-20170816212248-20170816232248-00477.warc.gz"}