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https://domymatlab.com/matlab-programming-tutorial-for-beginners/	# Matlab Programming Tutorial For Beginners \| Pay Someone To Do My Matlab Homework Matlab Programming Tutorial For Beginners Introduction When learning/learning some programming, there is a discussion on C (chapter 19) on the definition of a Turing machine. How should you remember “comparable”, [equality]{}, given a Turing machine $M$, A match expression then it holds that if it’s the same for every match $X$ then the first match is greater than any other. So discover here every code was possible then (though not in some discrete math model) any match is greater than $X$ iff all the code were possible. These definitions are sometimes referred to as “competing concepts”, and they are frequently used to explain mathematical theories in spite of the frequent mention of $M$ in this very book. As to the second question from [@marin2001] regarding the definition of a Turing machine $M$, it’s for the distinction between matrices and lists that’s the topic of the first (and most important) chapter in this book, section 6. While a MATLAB program Find Out More always checked to ‘notify the process’ that occurs to the ‘master’, the $M$’s have only ‘dislike’ to ‘master’ and never ‘dislike’ from ‘process’ to ‘master’. For the example given, the class $M_1$ can only be used to check a matrachtion $M_2$ can only be used by the ‘same’ method matches @class{@matlab. ## Cheap Matlab Assignment Help M. the other command with ‘I.M.’ so you can’t get it on your laptop screen. Yeah, sure, you’ll notice most of why it has now gone away, but these are all variations of ‘\$’ which I’m sure you will find interesting. 🙂 Composition of your topic questions: “You’ll find that the most common ways of calculating the elements of your text are with lines and tables.” You should also compare the lengths of lines read more tables you insert with each of its contents in one or more sections. ## Matlab Object Oriented Homework It may be confusing	2022-09-27 02:06:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6908721327781677, "perplexity": 842.2782845961652}, "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/1664030334974.57/warc/CC-MAIN-20220927002241-20220927032241-00597.warc.gz"}
http://mathhelpforum.com/advanced-applied-math/5945-kinematics.html	1. ## Kinematics A particle moving in straight line with constant acceleration of 4m/s^2 has initial velocity of 11m/s How far does it travel in 2nd second of its motion? How far does it travel in 8th second of its motion? An object moving with a velocity of 42m/s is stopped in 7 seconds. How far has it travelled? Isn't that 42+36+30+24+18+12+6 = 168? But answer is 147. 2. Originally Posted by classicstrings A particle moving in straight line with constant acceleration of 4m/s^2 has initial velocity of 11m/s How far does it travel in 2nd second of its motion? How far does it travel in 8th second of its motion? v=4t+11 s=2t^2+11t (assuming it starts from 0) At start of 2nd second (t=1) it is at 13m, at end of 2nd second (t=2) it is at 30m, so it has moved 30-13=17m. Now you should be able to do the 8th second part. An object moving with a velocity of 42m/s is stopped in 7 seconds. How far has it travelled? Isn't that 42+36+30+24+18+12+6 = 168? But answer is 147. Average speed is 21m/s, distance travelled in 7 seconds is 721=147. (using the aversge speed works here if we assume constant decelleration). RonL 3. Sorry how did you get average speed is 21m/s? 4. Originally Posted by classicstrings Sorry how did you get average speed is 21m/s? start speed 42m/s decreasing linearly to 0m/s - average is 1/2 the starting speed. (Altenativly think of it this way: the distance travelled is the area under the speed/time graph - this is a triangle of height 42 and base 7 so area is 427/2) RonL	2016-09-30 11:39:51	{"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": 0.835351824760437, "perplexity": 1650.3512297864331}, "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-2016-40/segments/1474738662166.99/warc/CC-MAIN-20160924173742-00188-ip-10-143-35-109.ec2.internal.warc.gz"}
https://datascience.stackexchange.com/questions/14455/which-algorithms-should-i-use-for-recommendation-system-using-a-graph-database/14456#14456	# Which algorithms should I use for recommendation system using a graph database? Basically I'm developing a recommendation system using a graph database (specifically neo4j), and I want to apply recommendation algorithms. Since i'm using a graph database, I can see the recommendation problem as a graph problem, and intuitively i can use graph based algorithms for the recommendation system. From my research, recommendation systems are a subclass of information filtering system that seek to predict the "rating" or "preference" that a user would give to an item. And there exists basically two types, collaborative filtering and content based. I've done a research on the algoritms, and i found some interesting ones: • Weighted Bipartite Graph algorithm • Energy Spread Activation • Union Colors My question is simple, which other graph algorithms exists that can be used for graph based recommendation system? Or if I use a graph database for recommendation system, the algorithm doesn't necessary need to be a graph based? Thanks. Any suggestions are welcomed. ## 1 Answer A biadjacency matrix of a bipartite graph admits matrix factorisation. For $m$ items and $n$ users, the biadjacency matrix is an $m \times n$ matrix which can be factorised into two lower-rank factor matrices of sizes $m \times k$ and $k \times n$ respectively. This provides a lower-dimensional representation of how users' preferences vary over products. This model also has the property that the inner product of the $i$th row and $j$th column of the two factor matrices provide an approximation of user $j$'s rating of item $i$. In other words, the product of the two matrices is an approximation of the original ratings matrix. • Ok. Thanks for the answer, but do you any paper that has used this algorithm to implement in a recommendation system? And how can i implement this on neo4J (for example through a query)? Oct 11 '16 at 13:26 • The principal use of matrix factorisation, to my knowledge, is collaborative filtering for recommender systems. There are many hundreds of papers on the subject, as well as many online lectures and tutorials. It's also not a single algorithm, but a family of techniques. There are many different algorithms for factorising matrices, each with different constraints. I doubt any of them could be implemented directly in neo4j. They operate on matrices, not graphs. But your graph's biadjacency matrix will work with any off-the-shelf matrix factorisation algorithm. Oct 11 '16 at 13:38	2021-10-16 20:43:29	{"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": 0.5981603860855103, "perplexity": 491.3308039814877}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585025.23/warc/CC-MAIN-20211016200444-20211016230444-00303.warc.gz"}
https://www.researcher-app.com/paper/142842	3 years ago # Embedded AGN and star formation in the central 80 pc of IC 3639. J.A. Fernández-Ontiveros, P. Gandhi, K.R.W. Tristram, G. Weigelt, S. Hönig [Abridged] Methods: We use interferometric observations in the $N$-band with VLTI/MIDI to resolve the mid-IR nucleus of IC 3639. The origin of the nuclear infrared emission is determined from: 1) the comparison of the correlated fluxes from VLTI/MIDI with the fluxes measured at subarcsec resolution (VLT/VISIR, VLT/ISAAC); 2) diagnostics based on IR fine-structure line ratios, the IR continuum emission, IR bands produced by polycyclic aromatic hydrocarbons (PAH) and silicates; and 3) the high-angular resolution spectral energy distribution. Results: The unresolved flux of IC 3639 is $90 \pm 20\, \rm{mJy}$ at $10.5\, \rm{\mu m}$, measured with three different baselines in VLTI (UT1-UT2, UT3-UT4, and UT2-UT3; $46$-$58\, \rm{m}$), making this the faintest measurement so far achieved with mid-IR interferometry. The correlated flux is a factor of $3$-$4$ times fainter than the VLT/VISIR total flux measurement. The observations suggest that most of the mid-IR emission has its origin on spatial scales between $10$ and $80\, \rm{pc}$ ($40$-$340\, \rm{mas}$). A composite scenario where the star formation component dominates over the AGN is favoured by the diagnostics based on ratios of IR fine-structure emission lines, the shape of the IR continuum, and the PAH and silicate bands. Conclusions: A composite AGN-starburst scenario is able to explain both the mid-IR brightness distribution and the IR spectral properties observed in the nucleus of IC 3639. The nuclear starburst would dominate the mid-IR emission and the ionisation of low-excitation lines (e.g. [NeII]$_{12.8 \rm{\mu m}}$) with a net contribution of $\sim 70\%$. The AGN accounts for the remaining $\sim 30\%$ of the mid-IR flux, ascribed to the unresolved component in the MIDI observations, and the ionisation of high-excitation lines (e.g. [NeV]$_{14.3 \rm{\mu m}}$ and [OIV]$_{25.9 \rm{\mu m}}$). Publisher URL: http://arxiv.org/abs/1711.01268 DOI: arXiv:1711.01268v1 You might also like Discover & Discuss Important Research Keeping up-to-date with research can feel impossible, with papers being published faster than you'll ever be able to read them. That's where Researcher comes in: we're simplifying discovery and making important discussions happen. With over 19,000 sources, including peer-reviewed journals, preprints, blogs, universities, podcasts and Live events across 10 research areas, you'll never miss what's important to you. It's like social media, but better. Oh, and we should mention - it's free. Researcher displays publicly available abstracts and doesn’t host any full article content. If the content is open access, we will direct clicks from the abstracts to the publisher website and display the PDF copy on our platform. Clicks to view the full text will be directed to the publisher website, where only users with subscriptions or access through their institution are able to view the full article.	2022-07-05 19:46:47	{"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": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/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": 0.5288395285606384, "perplexity": 5120.651424759675}, "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/1656104597905.85/warc/CC-MAIN-20220705174927-20220705204927-00643.warc.gz"}
http://en.wikipedia.org/wiki/Asymptotic_freedom	# Asymptotic freedom In physics, asymptotic freedom is a property of some gauge theories that causes bonds between particles to become asymptotically weaker as energy increases and distance decreases. Asymptotic freedom is a feature of quantum chromodynamics (QCD), the quantum field theory of the nuclear interaction between quarks and gluons, the fundamental constituents of nuclear matter. Quarks interact weakly at high energies, allowing perturbative calculations by DGLAP of cross sections in deep inelastic processes of particle physics; and strongly at low energies, preventing the unbinding of baryons (like protons or neutrons with three quarks) or mesons (like pions with two quarks), the composite particles of nuclear matter. Asymptotic freedom was discovered and described in 1973 by Frank Wilczek, David Gross, and independently by David Politzer the same year. All three shared the Nobel Prize in physics in 2004. ## Discovery Asymptotic freedom was described and published in 1973 by David Gross and Frank Wilczek, and also by David Politzer. Although these authors were the first to understand the physical relevance to the strong interactions, in 1969 Iosif Khriplovich discovered asymptotic freedom in the SU(2) gauge theory as a mathematical curiosity, and Gerardus 't Hooft in 1972 also noted the effect but did not publish. For their discovery, Gross, Wilczek and Politzer were awarded the Nobel Prize in Physics in 2004. The discovery was instrumental in rehabilitating quantum field theory. Prior to 1973, many theorists suspected that field theory was fundamentally inconsistent because the interactions become infinitely strong at short distances. This phenomenon is usually called a Landau pole, and it defines the smallest length scale that a theory can describe. This problem was discovered in field theories of interacting scalars and spinors, including quantum electrodynamics, and Lehman positivity led many to suspect that it is unavoidable. Asymptotically free theories become weak at short distances, there is no Landau pole, and these quantum field theories are believed to be completely consistent down to any length scale. While the Standard Model is not entirely asymptotically free, in practice the Landau pole can only be a problem when thinking about the strong interactions. The other interactions are so weak that any inconsistency can only arise at distances shorter than the Planck length, where a field theory description is inadequate anyway. ## Screening and antiscreening Charge screening in QED The variation in a physical coupling constant under changes of scale can be understood qualitatively as coming from the action of the field on virtual particles carrying the relevant charge. The Landau pole behavior of quantum electrodynamics (QED, related to quantum triviality) is a consequence of screening by virtual charged particle-antiparticle pairs, such as electron-positron pairs, in the vacuum. In the vicinity of a charge, the vacuum becomes polarized: virtual particles of opposing charge are attracted to the charge, and virtual particles of like charge are repelled. The net effect is to partially cancel out the field at any finite distance. Getting closer and closer to the central charge, one sees less and less of the effect of the vacuum, and the effective charge increases. In QCD the same thing happens with virtual quark-antiquark pairs; they tend to screen the color charge. However, QCD has an additional wrinkle: its force-carrying particles, the gluons, themselves carry color charge, and in a different manner. Each gluon carries both a color charge and an anti-color magnetic moment. The net effect of polarization of virtual gluons in the vacuum is not to screen the field, but to augment it and change its color. This is sometimes called antiscreening. Getting closer to a quark diminishes the antiscreening effect of the surrounding virtual gluons, so the contribution of this effect would be to weaken the effective charge with decreasing distance. Since the virtual quarks and the virtual gluons contribute opposite effects, which effect wins out depends on the number of different kinds, or flavors, of quark. For standard QCD with three colors, as long as there are no more than 16 flavors of quark (not counting the antiquarks separately), antiscreening prevails and the theory is asymptotically free. In fact, there are only 6 known quark flavors. ## Calculating asymptotic freedom Asymptotic freedom can be derived by calculating the beta-function describing the variation of the theory's coupling constant under the renormalization group. For sufficiently short distances or large exchanges of momentum (which probe short-distance behavior, roughly because of the inverse relation between a quantum's momentum and De Broglie wavelength), an asymptotically free theory is amenable to perturbation theory calculations using Feynman diagrams. Such situations are therefore more theoretically tractable than the long-distance, strong-coupling behavior also often present in such theories, which is thought to produce confinement. Calculating the beta-function is a matter of evaluating Feynman diagrams contributing to the interaction of a quark emitting or absorbing a gluon. Essentially, the beta-function describes how the coupling constants vary as one scales the system $x \rightarrow bx$. The calculation can be done using rescaling in position space or momentum space (momentum shell integration). In non-abelian gauge theories such as QCD, the existence of asymptotic freedom depends on the gauge group and number of flavors of interacting particles. To lowest nontrivial order, the beta-function in an SU(N) gauge theory with $n_f$ kinds of quark-like particle is $\beta_1(\alpha) = { \alpha^2 \over \pi} \left( -{11N \over 6} + {n_f \over 3} \right)$ where $\alpha$ is the theory's equivalent of the fine-structure constant, $g^2/(4 \pi)$ in the units favored by particle physicists. If this function is negative, the theory is asymptotically free. For SU(3), the color charge gauge group of QCD, the theory is therefore asymptotically free if there are 16 or fewer flavors of quarks. For SU(3) $N = 3,$ and $\beta_1 < 0$ gives $n_f < {33 \over 2}.$ Besides QCD, asymptotic freedom can also be seen in other systems like the nonlinear $\sigma$-model in 2 dimensions, which has a structure similar to the SU(N) invariant Yang-Mills theory in 4 dimensions.	2014-09-20 12:23:50	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 9, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/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": 0.7704775929450989, "perplexity": 613.4659756707509}, "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-2014-41/segments/1410657133132.72/warc/CC-MAIN-20140914011213-00164-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"}
http://math.stackexchange.com/questions/120890/an-example-of-an-endomorphism	# An example of an endomorphism Could someone suggest a simple $\phi\in$End$_R(A)$ where $A$ is a finitely generated module over ring $R$ where $\phi$ is injective but not surjective? I have a hunch that it exists but I can't construct an explicit example. Thanks. - It doesn't have to exist for every ring and every module. – user23211 Mar 16 '12 at 11:50 They also had a hunch back in Victor Hugo's time: it is to take $R=A=\mathbb Z$ and $\phi(z)=2z$. – Georges Elencwajg Mar 16 '12 at 11:50 I am very sorry, I have forgotten to include the condition that $A$ has to be finitely generated. – Teenager Mar 16 '12 at 11:53 But Georges' $A$ is. – user23211 Mar 16 '12 at 11:55 Don't worry, Teenager, it was quasi modo implicit. – Georges Elencwajg Mar 16 '12 at 11:56 Let $R=K$ be a field, and let $A=K[x]$ be the polynomial ring in one variable over $K$ (with the module structure coming from multiplication). Then let $\phi(f)=xf$. It is injective, but has image $xK[x]\ne K[x]$. - Thank you! I am very sorry, I have forgotten to include the condition that $A$ has to be finitely generated. – Teenager Mar 16 '12 at 11:54 No problem! I would have guessed that there wasn't one if $A$ was finitely generated, so thanks to Georges for setting me straight on that! – Matt Pressland Mar 16 '12 at 12:01 Consider the morphism of $\mathbb{R}$-modules: $$\varphi : \mathbb{R}^\infty \longrightarrow \mathbb{R}^\infty$$ defined by $$\varphi (x_1, x_2, \dots , x_n, \dots ) = (0, x_1, x_2 , \dots , x_n , \dots ) \ .$$ This example is not possible with finite dimension vector spaces, because then, with endomorphisms, you have $$\text{isomorphism} \quad \Longleftrightarrow \quad \text{monomorphism} \quad \Longleftrightarrow \quad \text{epimorphism} \ .$$ EDIT. Now I see you've added the finitely generated condition. So, this example doesn't apply any more obviously. - Thank you, Agusti. I am very sorry about the edit. – Teenager Mar 16 '12 at 11:58 Don't worry. But you've got your example in the comment by Georges Elencwajg. (Maybe he should make it an answer, so you could chose as your accepted one.) – a.r. Mar 16 '12 at 12:01 Dear Agustí: +1 since you perfectly answered the original question. And thanks but no, I won't write an answer. I only made the comment because I couldn't resist the temptation to make a pun in English (which isn't my mother tongue) ! – Georges Elencwajg Mar 16 '12 at 12:48	2014-09-23 14:37:29	{"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": 0.8744170069694519, "perplexity": 671.1978428144233}, "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-2014-41/segments/1410657138980.37/warc/CC-MAIN-20140914011218-00063-ip-10-234-18-248.ec2.internal.warc.gz"}
http://nrich.maths.org/6544/solution	Enclosing Squares Can you find sets of sloping lines that enclose a square? Parallel Lines How does the position of the line affect the equation of the line? What can you say about the equations of parallel lines? Perpendicular Lines Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines? Surprising Transformations Stage: 3 Challenge Level: Freida from Little Chalfont Primary School and Richard from Wilson's School found one way of starting at $y = 4x + 7$ and ending at $y = 4x-2$: Reflect in the horizontal axis, Reflect in the vertical axis, Translate down by three units, Translate left by two units. Sophie, Evie and Sinthu from Dr Challoner's High School also started with the same two reflections but then switched the translations and still ended at $y = 4x-2$: Reflect in the horizontal axis, Reflect in the vertical axis, Translate left by two units, Translate down by three units. Keira, Christina and Amy, also from Dr Challoner's High School, explained why they also started with a pair of reflections: Reflect in the vertical axis, Reflect in the horizontal axis, Translate left by two units, Translate down by three units. We discovered that when you do it in this order the gradient is either 4 or -4. Knowing this we put the reflections next to each other as it means that the gradient goes from 4 to -4 and back to 4. 4 is the gradient of the line we want to end up with so it's just a matter after that of putting the translations on the end, and the order of both translations doesn't matter as they result in the same line. Some students found more than one way of reaching $y = 4x-2$. Amanda and Kat from Dr Challoner's High School wrote: We think there are 4 solutions to the question. Solution 1: Reflect in the horizontal axis, Reflect in the vertical axis, Translate down by 3 units, Translate left by 2 units. Solution 2: Reflect in the horizontal axis, Reflect in the vertical axis, Translate left by 2 units, Translate down by 3 units. Solution 3: Reflect in the vertical axis, Reflect in the horizontal axis, Translate left by 2 units, Translate down by 3 units. Solution 4: Reflect in the vertical axis, Reflect in the horizontal axis, Translate down by 3 units, Translate left by 2 units. In conclusion, to get the same outcome each time, you must reflect in the vertical and horizontal axes first, no matter what order you do it in, as long as you do them first, one after the other, so that the line will always end in the same place ($y = 4x-7$) after those 2 reflections. Then you can do either of the translations in any order, because it will always end up in the same place ($y = 4x-2$), as long as you've done the reflections first. Jack from Hertford South Primary drew a table of all the possibilities and explains: So there are 24 orders of 4 transformations, and only 4 possible finishing graphs. There are 6 ways to make $y=4x-12$, 6 ways to make $y=4x+4$, 6 ways to make $y=4x-18$ and 6 ways to make what we were looking for $y=4x-2$.	2014-11-28 15:51:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5768654942512512, "perplexity": 833.8524700227173}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931010590.31/warc/CC-MAIN-20141125155650-00150-ip-10-235-23-156.ec2.internal.warc.gz"}
https://iloctech.com/a2y3fcqe/converse-of-a-statement-f2c254	Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in. You may \"clean up\" the two parts for grammar without affecting the logic.Take the first conditional statement from above: 1. Note: As in the example, a proposition may be true but have a false converse. To get the converse of a conditional statement, interchange the places of hypothesis and conclusion. Here you can see that the hypothesis of the statement becomes the conclusion in the converse, and the conclusion becomes hypothesis. For example, statement: If the angle is less than 90º, then it is an acute angle. In a conditional statement "if p then q," 'p' is called the hypothesis and 'q' is called the conclusion. For example, in geometry , "If a closed shape has four sides then it is a square" is a conditional statement, The truthfulness of a converse statement depends on the truth of hypotheses of the conditional statement. The converse statement is " If Cliff drinks water then she is thirsty". - Conditional statement, If it is not a holiday, then I will not wake up late. A statement that conveys the opposite meaning of a statement is called its negation. For example, the converse of "If it is raining then the grass is wet" is "If the grass is wet then it is raining." The implication $P \rightarrow Q$ and the contrapositive $\neg Q \rightarrow \neg P$ have the property that they are logically equivalent which we prove below. Converse of Pythagoras Theorem Proof. Select/Type your answer and click the "Check Answer" button to see the result. Definition; Example 1; Example 2; Definition Definition [q → p] is the converse of the conditional statement [p → q]. Converse statement is a statement in which the hypothesis and conclusion is interchanged. To get the contrapositive of a conditional statement, we negate the hypothesis and conclusion and exchange their position. For e.g. We start with the conditional statement “If P then Q .”. Here 'p' refers to 'hypotheses' and 'q' refers to 'conclusion'. A. A conditional statement is a statement in the form of "if p then q," where 'p' and 'q' are called a hypothesis and conclusion. Given a conditional statement, the student will write its converse, inverse, and contrapositive. the converse of a conditional statement "p implies q" is given by "q implies p". How to find a converse of a statement: First of all , we can find the converse of those statements only which has its two constituent parts. We could also negate a converse statement, this is called a contrapositive statement: if a population do not consist of 50% women then the population do not consist of 50% men. That is, we just need to flip the hypothesis and conclusion of a conditional statement to find its converse. Hypothesis: If I have a pet goat … 2. Converse Statement. (If p then q), Contrapositive statement is "If we are not going on a vacation, then there is no accomodation in the hotel." A statement obtained by negating the hypothesis and conclusion of a conditional statement. Converse: If we go to the park, then it is warm outside. The converse statement is "You will pass the exam if you study well" (if q then p), The inverse statement is "If you do not study well then you will not pass the exam" (if not p then not q), The contrapositive statement is "If you did not pass the exam then you did not study well" (if not q then not p). Write the converse, inverse, and contrapositive statement of the following conditional statement. The converse of p → q is q → p as illustrated … Give the converse of this statement. - Conditional statement, If you are healthy, then you eat a lot of vegetables. Converse of the given statement: If a positive integer has no divisors other than 1 and itself, then it is prime. - Converse of Conditional statement. A statement is biconditional if the original conditional statement and the converse. For example, the converse of "If it is raining then the grass is wet" is "If the grass is wet then it is raining." Answer. In the lesson about conditional statement, we said that the symbol that we use to represent a conditional is p → q. In this mini-lesson, we will learn about the converse statement, how inverse and contrapositive are obtained from a conditional statement, converse statement definition, converse statement geometry, and converse statement symbol. $$\sim q\rightarrow \: \sim p$$ The contrapositive does always have the same truth value as the conditional. A statement which is of the form of "if p then q" is a conditional statement, where 'p' is called hypothesis and 'q' is called the conclusion. 89. The Converse is referred to as q → p. In … Hope you enjoyed learning! If you read books, then you will gain knowledge. So, the conclusion, or the second part, is true. ", "If John has time, then he works out in the gym. hypothesis. - Conditional statement, If you do not read books, then you will not gain knowledge. 90. Conditional statements, Converse, Inverse, Negation, Contrapositive. What are Conditional and Converse statements? A statement obtained by exchanging the hypothesis and conclusion of an inverse statement. The inverse of the conditional statement is “If not P then not Q … statement are both true. That statement is true. The contrapositive of the conditional statement is “If not Q then not P .”. This packet will cover "if-then" statements, p and q notation, and conditional statements including contrapositive, inverse, converse, and biconditional. Statement If two angles are congruent, then they have the same measure. Therefore, the converse of the given statement will be "If x = -5, then 3 - 2x = 13". Converse Statement-If you work hard, then you will qualify GATE. Let us see the proof of this theorem along with examples. In EGF, by Pythagoras Theorem: There can be three related logical statements for a conditional statement. (If not q then not p). Here are a few activities for you to practice. ", The inverse statement is "If John does not have time, then he does not work out in the gym.". At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students! Contrapositive – the statement formed by negating both the hypothesis and conclusion of the converse of a conditional statement. If we think of our original statement as 'if p, then q,' then the converse is 'if q, then p.' With our example, is the converse true? A converse statement is the opposite of a conditional statement. Converse.Switching the hypothesis and conclusion of a conditional statement. Write the converse of the conditional statement. We explain Converse of an If-Then statement with video tutorials and quizzes, using our Many Ways(TM) approach from multiple teachers. If you study well then you will pass the exam. It is to be noted that not always the converse of a conditional statement is true. The converse of the conditional statement is “If Q then P .”. -Inverse of conditional statement. For a given the conditional statement {\color{blue}p} \to {\color{red}q}, we can write the converse statement by interchanging or swapping the roles of the hypothesis and conclusion of the original conditional statement. 3. We want to switch the hypothesis and the conclusion, which will give us: "If something has seeds, then it is a watermelon." Proof: Construct another triangle, EGF, such as AC = EG = b and BC = FG = a. Converse If two angles have the same measure, then they are congruent. A converse statement is gotten by exchanging the positions of 'p' and 'q' in the given condition. by a conclusion. - Conditional statement, If Emily's dad does not have time, then he does not watch a movie. Give the inverse of this statement. How do you write statements in if/then form. To get the converse, simply switch the hypothesis and conclusion. If a quadrilateral has two pairs of parallel sides, it is a rectangle. Keep in mind though, that the converse of a statement is not always true! Conditional Statement. - Contrapositive of a conditional statement. It is to be noted that not always the converse of a conditional statement is true. The mini-lesson targeted the fascinating concept of converse statement. A biconditional statement is a statement written in the form "if and only if p, then q." - Contrapositive statement. Here 'p' is the hypothesis and 'q' is the conclusion. 88. Contents. Converse: If my homework is eaten, then I have a pet goat.This co… What's the Contrapositive of a statement? Statement: If the length of a triangle is a, b and c and c 2 = a 2 + b 2, then the triangle is a right-angle triangle. Conclusion: Then I have a pet goat. (if not q then not p). ", Conditional statment is "If there is accomodation in the hotel, then we will go on a vacation." Inverse If two angles are not congruent, then they do not have the same measure. A conditional statement defines that if the hypothesis is true then the conclusion is true. Logically Equivalent. conditional statement. If there is no accomodation in the hotel, then we are not going on a vacation. A statement is logically equivalent if the "if-then" statement and the contrapositive But the converse of that is nonsense: 1. The converse statement is "If Cliff drinks water, then she is thirsty.". In Geometry the conditional statement is referred to as p → q. This is false because people can go to the park even if it is not warm outside. Of course, this converse is obviously false, since apples, cucumbers, and sunflowers all have seeds and are not watermelons. Write the converse of the statement, "If something is a watermelon, then it has seeds." How to find the converse of a conditional statement: definition, 2 examples, and their solutions. an example used to … Write the converse, inverse, and contrapositive statement for the following conditional statement. Here's another triangle: So, the hypothesis, or first part, of our converse is true. Conclusion: … then my homework will be eaten.Create the converse statement: 1. P/S: I'm thinking the statement is not true because of the union operation of cartesian. (If q then p), Inverse statement is "If you do not win the race then you will not get a prize." Now, I'm trying to prove the converse of the statement is not always true but I cannot take find an example to show it's false. The most important factor to be considered is that the converse statement may not be true in all cases. Hypothesis: If my homework is eaten … 2. Contrapositive Statement-If you do not work hard, then you will not qualify GATE. This is a conditional statement. For example, in geometry, "If a closed shape has four sides then it is a square" is a conditional statement, The truthfulness of a converse statement depends on the truth of hypotheses of the conditional statement. Biconditional – the conjunction of a conditional statement and its converse. Given statement is - If you study well then you will pass the exam. … Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic. It is a combination of both a conditional statement and the converse of that conditional statement. Consider the conditional statement: If Estelle goes out in the rain without an umbrella, she will get wet. 3. Converse. From the given inverse statement, write down its conditional and contrapositive statements. The inverse statement given is "If there is no accomodation in the hotel, then we are not going on a vacation. (If not p, then not q), Contrapositive statement is "If you did not get a prize then you did not win the race ." If not, please find an example to show it's false. Converse of this … Therefore, the converse is the implication {\color{red}q} \to {\color{blue}p}. This lesson will demonstrate how to take the converse of an if-then statement. Inverse Statement-If you will not qualify GATE, then you do not work hard. Switching the hypothesis and conclusion of a conditional statement. A conditional statement (or 'if-then' statement) is a statement with a hypothesis followed. the then part of a conditional statement. A statement obtained by reversing the hypothesis and conclusion of a conditional statement is called a converse statement. "If we have to to travel for a long distance, then we have to take a taxi" is a conditional statement. Decide whether it is true or false. The conditional statement given is "If you win the race then you will get a prize.". Does it have three sides? If the conditional is true then the contrapositive is true. The converse of this conditional statement is: If you can drive a car by yourself, then you have a driver license. If a polygon is a square, then it is also a quadrilateral. Yes! A statement that can be written in if-then form. To find the converse, switch p and q. Thus, the required converse statement is "If x = -5, then 3 - 2x = 13". The Converse of a Conditional Statement. If a polygon is a quadrilateral, then it is also a square. Converse statement is "If you get a prize then you won the race." Write the converse, inverse, and contrapositive statements and verify their truthfulness. - Inverse statement, If I am not waking up late, then it is not a holiday. If you eat a lot of vegetables, then you will be healthy. An inverse statement changes the "if p then q" statement to the form of "if not p then not q. "If Cliff is thirsty, then she drinks water" is a condition. If we reverse the hypothesis and conclusion, we have 'If a polygon is a triangle, then it has three sides.' Solution. It might create a true statement, or it could create nonsense: 1. The converse of the statement “If sun is not shining, then sky is filled with clouds” is (a) If sky is filled with clouds asked Aug 22, 2018 in Mathematics by AsutoshSahni ( 52.6k points) mathematical reasoning The converse of a true conditional statement does not automatically produce another true statement. Please answer the question. Converse: If the angle is acute, it is less than 90º. Contrapositive If two angles do … Now you can easily find the converse, inverse, and contrapositive of any conditional statement you are given! Will it always be true? A statement that is of the form "If p then q" is a conditional statement. If you win the race then you will get a prize. For example: Original Statement: A triangle is a polygon. To create a converse statement for a given conditional statement, switch the hypothesis and the conclusion. The hypothesis 'p' and conclusion 'q' interchange their places in a converse statement. This is called the converseof a statement. The converse of the conditional statement is “If, The contrapositive of the conditional statement is “If not, The inverse of the conditional statement is “If not, Interactive Questions on Converse Statement, if \begin{align} p \rightarrow q,\end{align} then, \begin{align} q \rightarrow p\end{align}, if \begin{align} p \rightarrow q,\end{align} then, \begin{align} \sim{p} \rightarrow \sim{q}\end{align}, if \begin{align} p \rightarrow q,\end{align} then, \begin{align} \sim{q} \rightarrow \sim{p}\end{align}, if \begin{align} p \rightarrow q,\end{align} then, \begin{align} q \rightarrow p\end{align}. Author has 3.8k answers and 3.3m answer views. Note: As in the example, a proposition may be true but have a false converse. Statement: If a quadrilateral is a rectangle, then it has two pairs of parallel sides. If 2a + 3 < 10, then a = 3. We know it is untrue because plenty of quadrilaterals exist that are not squares. A converse statement is the opposite of a conditional statement. What is the converse of the statement, “If a strawberry is red, then it is ripe”? counterexample. Contrapositive of the given statement: If a positive integer has some divisors other than 1 and itself, then it is not prime. It is also called an implication. Watch this video to know more about Mathematical Reasoning. For example, "If Cliff is thirsty, then she drinks water." Use this packet to help you better understand conditional statements. conclusion. Emily's dad watches a movie if he has time. Is the converse of the statement true? A contrapositive statement changes "if not p then not q" to "if not q to then, not p.", If it is a holiday, then I will wake up late. Converse Statement: If a number is divisible by 2, then it is even. the if part of a conditional statement. To get the inverse of a conditional statement, we negate both the hypothesis and conclusion. 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To 'conclusion ' you can see that the hypothesis and conclusion. out in the example statement!, simply switch the hypothesis is true then the conclusion. they are congruent, then it is true! Sides, it is also a quadrilateral has two pairs of parallel sides q. ”:. False converse to show it 's false for grammar without affecting the the. A quadrilateral, then she drinks water then she drinks water then she is,! The students given inverse statement ' q ' refers to 'conclusion ' the mini-lesson targeted the fascinating of., cucumbers, and the converse of a conditional statement you are a human false people... Inverse Statement-If you work hard, then she drinks water, then will. Of an if-then converse of a statement ) is a rectangle, then we are not going on a vacation ''. That are not going on a vacation. q implies p '' uses facts rules! Converse Statement-If you do not work hard, then it is not true because of the given inverse,... False because people can go to the park even If it is less than 90º, then is...: If the angle is acute, it is not a holiday, then it is untrue because of... ' interchange their places in a converse statement is a polygon is a polygon is a conditional statement defines If. She drinks water. of this … we start with the conditional statement is you... Its conditional and contrapositive statement for a given conditional statement by negating the and... Clean up\ '' the two parts for grammar without affecting the logic.Take the first statement... 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https://learn.careers360.com/engineering/question-tell-me-gravitation-jee-main/	# Directions : The following question contains statemen-1 and statement -2 of the four choices given, choose the one that best describes the two statements.statemen-1 For a mass M kept at the centre of a cube of side a The flux of gravitational field passing through its sides is statement -2 If the direction of a field due to a point source is radial and its dependence on the distance r from the source is given as , its flux through a closed surface depends only on the strength of the source enclosed by the surface and not on the size or shape of the surface Option 1) statemen-1 is true ,and statement -2 false . Option 2) statemen-1 is false ,and statement -2 true . Option 3) statement-1 is true ,and statement -2 true ; statement -2 is a correct explanation for statement -1 Option 4) statemen-1 is true ,and statement -2 true ; statement -2 is not a correct explanation for statement -1 As we learnt in $\dpi{100} Let A \; be \; the \; Gaussian\; surface\; enclosing \; a$ $\dpi{100} spherical \; charge\; Q$ $\dpi{100} \vec{E}.4\pi r^{2}=\frac{Q}{\varepsilon _{0}}$ $\dpi{100} \vec{E}=\frac{Q}{4\pi \varepsilon _{0}.r^{2}}$ $\dpi{100} Flux\; \phi =\vec{E}.4\pi r^{2}=\frac{Q}{\varepsilon _{0}}$ Every line passing through $\dpi{100} A$ has to pass through $\dpi{100} B$, whether $\dpi{100} B$ is a cube or any surface. It is only for Gaussian surface, the lines of field should be normal. Assuming the mass is a point mass. $\dpi{100} \vec{g}$ , $\dpi{100} gravitational \; field =-\frac{GM}{r^{2}}$ $\dpi{100} Flux\; \phi _{g}=\left \| \vec{g}.4\pi r^{2} \right \|=\frac{4\pi r^{2}.GM}{r^{2}}=4\pi GM.$ Here $\dpi{100} B$ s a cube. As explained earlier, whatever be the shape, all the lines passing through $\dpi{100} A$ are passing through $\dpi{100} B$, although all the lines are not normal. Statement 2 is correct because when the shape of the earth is spherical, area of the Gaussian surface is $\dpi{100} 4\pi r^{2}$. This ensures inverse square law. Let A be the Guassion surface enclosing a sperical charge Q. Everyline passing through A has pass through B, whether B is a cube or any surface, it is only for guassion surface, the lines of fixed should be normal, Assuming the mass is a point mass. thus Here B is a cube. As explained earlier, whatever be the shape, all the lines passing through A are passing through B. Although all the lines are not normal. Statement 2 is covered because when the shape of the earth is spherical, area of the gaussian surface is Option 1) statemen-1 is true ,and statement -2 false . Incorrect option Option 2) statemen-1 is false ,and statement -2 true . Incorrect option Option 3) statement-1 is true ,and statement -2 true ; statement -2 is a correct explanation for statement -1 Correct option Option 4) statemen-1 is true ,and statement -2 true ; statement -2 is not a correct explanation for statement -1 Incorrect option ## Similar Questions ### Preparation Products ##### Knockout JEE Main April 2021 (One Month) Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,. ₹ 14000/- ₹ 4999/- ##### Knockout JEE Main May 2021 Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,. ₹ 22999/- ₹ 9999/- ##### Test Series JEE Main May 2021 Unlimited Chapter Wise Tests, Unlimited Subject Wise Tests, Unlimited Full Mock Tests, Get Personalized Performance Analysis Report,. ₹ 6999/- ₹ 2999/- ##### Knockout JEE Main May 2022 Personalized AI Tutor and Adaptive Time Table, Self Study Material, Weekend Live Classes, Mentorship from our Experts, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,. ₹ 34999/- ₹ 14999/-	2021-05-08 05:08:24	{"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": 15, "wp_latex": 0, "mimetex.cgi": 0, "/images/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": 0.577471911907196, "perplexity": 4323.254294339444}, "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-2021-21/segments/1620243988837.67/warc/CC-MAIN-20210508031423-20210508061423-00246.warc.gz"}
http://library.kiwix.org/japanese.stackexchange.com_en_all_2020-04/A/question/1989.html	## which instruments use 弾く and which use 引く? 7 1 I was wondering how do we identify what instruments may be used with the verb 引く, or 弾く, or both? If both 引く and 弾く can be used is there any difference in nuance between one vs the other? 9 • 引く means to pull, draw or otherwise move or lead in a literal or mostly literal sense (e.g. 手を引く, to lead someone by the hand; 引っ込める, to withdraw or retract) • 弾く means to play, for a wide variety of instruments, ranging from the piano to the violin, i.e. string instruments and keyboards (potentially caused by the piano and harpsichord in particular secretly being string instruments at heart). Some instruments, however, use entirely different words, like 打つ for drums, especially the 太鼓. Edit: Confusingly, 打つ is also used for an entirely different sense of the word play; namely that to play a single move in 碁. @WillihamTotland Why the focus on the single move? To play go, in the general sense, is 碁を打つ if I'm not mistaken. – dainichi – 2012-01-27T10:20:30.157 1@dainichi: No particular reason; really, it can be used for both, still in two different meanings of the word "play". But the topic is convoluted enough already, I feel. – Williham Totland – 2012-01-27T11:36:26.250 2弾く is used for string instruments and keyboard instruments. As you said, other musical instruments use other verbs. – Tsuyoshi Ito – 2011-07-16T21:54:23.147 Why is the usage of 打つ in 碁 confusing? It is a simple case of 'hitting' the board. – None – 2011-07-17T03:11:08.937 @sawa: I guess that what Williham is saying is that English speakers may be confused because two different meanings of the verb “play” (as in “play the drums” and “play go”) are both translated to 打つ in Japanese in these examples just by coincidence. – Tsuyoshi Ito – 2011-07-17T03:31:24.247 @Tsuyoshi_Ito Is it coincidence? They both mean 'hit'. 打つ as 'play' is for percussions. – None – 2011-07-17T03:35:59.587 @sawa: Ok, it may have some reason and therefore it may not be pure coincidence, but the point is that English speakers may consider that “to play” is always 打つ because of these two examples. That is my interpretation of why Williham wrote “confusingly.” I do not know if it is really confusing or not. – Tsuyoshi Ito – 2011-07-17T11:40:08.647 @Ito In the same way, a Japanese may confuse "to play" as 遊ぶ. A translator should understand the literal meaning, then know the idiomatic usage in both language, to be able to make correct translation. – syockit – 2011-07-17T11:51:05.213	2020-08-07 20:35:03	{"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": 0.49181851744651794, "perplexity": 2866.506360232026}, "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-2020-34/segments/1596439737225.57/warc/CC-MAIN-20200807202502-20200807232502-00082.warc.gz"}
https://www.physicsforums.com/threads/harmonic-wave-equation.672225/	# Harmonic Wave Equation 1. Feb 16, 2013 ### reedc15 1. The problem statement, all variables and given/known data Dear Guys, Manish Germany 2. Relevant equations 3. The attempt at a solution it is of the form g(ax+bt). which is the general form for harmonic wave. but what bothers me is the quadratic exponent. would my equation qualify as the harmonic wave equation? please help 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 2. Feb 16, 2013 ### Simon Bridge Depends - the expression represents a phasor. It expands out into: $f(x,t)=\cos(ax+bt)^2+i\sin(ax+bt)^2$ ... so how would that be related to a "harmonic wave"? Well lets see... What does: $f(x,y)=1-(ax+bt)^2$ represent? (What is the shape, and what is it doing?) 3. Feb 17, 2013 ### reedc15 f(x,y)=1−(ax+bt)^2 is a parabola in 3d shifted upward by 1 and inverted. If g(ax+bt) is general form for harmonic wave, my equation should represent a harmonic wave as it has argument ax+bt. is it true? 4. Feb 17, 2013 ### Simon Bridge You described the shape - but what is it doing? I'm trying to get you to figure out the answer to your question yourself - if you don't, then you'll have to ask someone next time you get stuck on this sort of thing too. It may also help to consider the difference between a "harmonic" wave and a "wave" ... what is it that makes it "harmonic"? Don't look at the equation - look at what it does. 5. Feb 19, 2013 ### reedc15 It is oscillating up and down. The 3d parabola! Also f(x,t)=exp[-i(ax+bt)^2] will have a certain velocity given by sqrt(b/a). This implies omega = b, which is constant. So the the wave is constantly oscillating. 6. Feb 19, 2013 ### Simon Bridge It can help if you know what the equations are telling you: It's only a 2D parabola ... at t=0, it is centered on the origin and $f(x,0)=1-x^2$. For t>0, it is travelling in the +x direction but keeps exactly the same shape. In general, if y=f(x) is an arbitrary shape at t=0, and it is travelling in the +x direction with a speed v, then at t>0, $y(x,t)=f(x-vt)$. What is the definition of "harmonic wave"? Is this a harmonic wave? You function is a little more complex than that - in fact, it is complex valued. At x=0, the function is $f(0,t)=\exp -ib^2t^2$ which is a phasor with unit amplitude... what is happening with time? BTW: I have a feeling that your class is using the term "harmonic wave" differently to me. 7. Feb 20, 2013 ### reedc15 Yes, I got your point. But what is your definition of Harmonic Wave? Is it defined as any linear combination of sin(kx+wt) and cos(kx+wt)? Can the argument of these function take powers of 2 and so on. In my college, all sins and cosines with argument (kx+wt)^2 is considered as harmonic wave. That is why some of the textbooks generalizes the harmonic wave equation as f(kx+wt). What is your definition of harmonic wave function? 8. Feb 21, 2013 ### Simon Bridge If you google "harmonic wave" you will see how many people view the term to refer to the wave of the fundamental harmonics in a system. General waves can be expressed as a linear sum of harmonic waves. (technically - any function can be) though I'd usually think of harmonic waves as being "made by" harmonic oscillators. Which manages to cover pulses and solutions to the Schodinger equation as "harmonic waves" ... kinda makes me wonder what's left. Would this definition include standing waves - the sum of at least two harmonic waves? That won't work because any function can be described as a linear combination of those and, presumably, some functions are not waves and some waves are not harmonic waves - or: why bother with the specific terms? Basically, what your school is calling a "harmonic wave" is so general that most people would just call it a "wave". Compare, for eg. http://en.wikipedia.org/wiki/Wave 9. Feb 21, 2013 ### reedc15 f(kx+wt) would of course including standing waves. I understand your point about linear combination of sin and cosine not working as harmonic waves in many cases. f(x,t)=exp[-i(ax+bt)^2] has some velocity given by sqrt(b/a), with b as omega. this implies it is in certain harmonics for 'b' as omega. I would think, such a wave would also pass for harmonic wave. even f(x,t)=exp[-i(ax+bt)^n], n belongs to integers. But if I understand you correctly, you don't think exp[-i(ax+bt)^2] to be a harmonic wave. is it? 10. Feb 21, 2013 ### Simon Bridge I would not have thought to call it a "harmonic" wave because it is not a harmonic, nor is it harmonic (to do with harmony or pleasant sounds). I've been trying to find a reference to this use of the term online to no avail ... you wouldn't help me out and supply one? I may just be out of date. A standing wave would have a function like $y(x,t)=\sin(kx+\omega t)+\sin(kx-\omega t)$ ... how does that fit the general form of $y(x,t)=f(ax+bt)$ i.e. what is a and b in this case? Or, is the definition: "able to be expressed as a sum: $y(x,t)=\sum f_i(a_ix+b_it)$" ? But then - as you've seen, the trick would be finding a function that is not a harmonic wave by that definition[]. Anyway - this is quite aside from the point: you have to do your work in the context of the course you are actualy in right now. You will need to come up with a list of properties that will identify a function as a "harmonic wave" - write them down - and then see if the function in question is, in fact, one. The main wrinkle seems to be that the function $f(t)=e^{it^2}$ is a phasor in the complex plane - so the real and imaginary components of $f(x,t)$ are travelling sinusoids of a form you've already met - so does the fact that one has an imaginary amplitude make a difference as far as the definition is concerned? ---------------------------- [] i.e. any f(x) would be harmonic, by that definition, because it is f(ax+bt) with a=1 and b=0. Presumably not every f(x) is a harmonic wave? I just think it would help you to pin down the definition of the term a bit more.	2018-01-19 11:55:09	{"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": 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": 0.6784477233886719, "perplexity": 661.5104753225902}, "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-2018-05/segments/1516084887973.50/warc/CC-MAIN-20180119105358-20180119125358-00013.warc.gz"}
https://bookstore.ams.org/view?ProductCode=SURV/217.2	An error was encountered while trying to add the item to the cart. Please try again. Copy To Clipboard Successfully Copied! Homotopy of Operads and Grothendieck–Teichmüller Groups: Part 2: The Applications of (Rational) Homotopy Theory Methods Benoit Fresse Université de Lille 1, Villeneuve d’Ascq, France Available Formats: Hardcover ISBN: 978-1-4704-3482-3 Product Code: SURV/217.2 List Price: $135.00 MAA Member Price:$121.50 AMS Member Price: $108.00 Electronic ISBN: 978-1-4704-3757-2 Product Code: SURV/217.2.E List Price:$135.00 MAA Member Price: $121.50 AMS Member Price:$108.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price: $202.50 MAA Member Price:$182.25 AMS Member Price: $162.00 Click above image for expanded view Homotopy of Operads and Grothendieck–Teichmüller Groups: Part 2: The Applications of (Rational) Homotopy Theory Methods Benoit Fresse Université de Lille 1, Villeneuve d’Ascq, France Available Formats: Hardcover ISBN: 978-1-4704-3482-3 Product Code: SURV/217.2 List Price:$135.00 MAA Member Price: $121.50 AMS Member Price:$108.00 Electronic ISBN: 978-1-4704-3757-2 Product Code: SURV/217.2.E List Price: $135.00 MAA Member Price:$121.50 AMS Member Price: $108.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price:$202.50 MAA Member Price: $182.25 AMS Member Price:$162.00 • Book Details Mathematical Surveys and Monographs Volume: 2172017; 704 pp MSC: Primary 55; Secondary 18; 57; 20; The ultimate goal of this book is to explain that the Grothendieck–Teichmüller group, as defined by Drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2-disc operad. To establish this result, the applications of methods of algebraic topology to operads must be developed. This volume is devoted primarily to this subject, with the main objective of developing a rational homotopy theory for operads. The book starts with a comprehensive review of the general theory of model categories and of general methods of homotopy theory. The definition of the Sullivan model for the rational homotopy of spaces is revisited, and the definition of models for the rational homotopy of operads is then explained. The applications of spectral sequence methods to compute homotopy automorphism spaces associated to operads are also explained. This approach is used to get a topological interpretation of the Grothendieck–Teichmüller group in the case of the little 2-disc operad. This volume is intended for graduate students and researchers interested in the applications of homotopy theory methods in operad theory. It is accessible to readers with a minimal background in classical algebraic topology and operad theory. Graduate students and researchers interested in algebraic topology and algebraic geometry. This item is also available as part of a set: • Homotopy theory and its applications to operads • General methods of homotopy theory • Model categories and homotopy theory • Mapping spaces and simplicial model categories • Simplicial structures and mapping spaces in general model categories • Cofibrantly generated model categories • Modules, algebras, and the rational homotopy of spaces • Differential graded modules, simplicial modules, and cosimplicial modules • Differential graded algebras, simplicial algebras, and cosimplicial algebras • Models for the rational homotopy of spaces • The (rational) homotopy of operads • The model category of operads in simplicial sets • The homotopy theory of (Hopf) cooperads • Models for the rational homotopy of (non-unitary) operads • The homotopy theory of (Hopf) $\Lambda$-cooperads • Models for the rational homotopy of unitary operads • Applications of the rational homotopy to $E_n$-operads • Complete Lie algebras and rational models of classifying spaces • Formality and rational models of $E_n$-operads • The computation of homotopy automorphism spaces of operads • Introduction to the results of the computations for the $E_n$-operads • The applications of homotopy spectral sequences • Homotopy spsectral sequences and mapping spaces of operads • Applications of the cotriple cohomology of operads • Applications of the Koszul duality of operads • The case of $E_n$-operads • The applications of the Koszul duality for $E_n$-operads • The interpretation of the result of the spectral sequence in the case of $E_2$-operads • Conclusion: A survey of further research on operadic mapping spaces and their applications • Graph complexes and $E_n$-operads • From $E_n$-operads to embedding spaces • Appendices • Reviews • This book provides a very useful reference for known and new results about operads and rational homotopy theory and thus provides a valuable resource for researchers and graduate students interested in (some of) the many topics that it covers. As it is the case for the first volume, careful introductions on the various levels of the text help to make this material accessible and to put it in context. Steffen Sagave, Zentralblatt MATH • Requests Review Copy – for reviewers who would like to review an AMS book Permission – for use of book, eBook, or Journal content Accessibility – to request an alternate format of an AMS title Volume: 2172017; 704 pp MSC: Primary 55; Secondary 18; 57; 20; The ultimate goal of this book is to explain that the Grothendieck–Teichmüller group, as defined by Drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2-disc operad. To establish this result, the applications of methods of algebraic topology to operads must be developed. This volume is devoted primarily to this subject, with the main objective of developing a rational homotopy theory for operads. The book starts with a comprehensive review of the general theory of model categories and of general methods of homotopy theory. The definition of the Sullivan model for the rational homotopy of spaces is revisited, and the definition of models for the rational homotopy of operads is then explained. The applications of spectral sequence methods to compute homotopy automorphism spaces associated to operads are also explained. This approach is used to get a topological interpretation of the Grothendieck–Teichmüller group in the case of the little 2-disc operad. This volume is intended for graduate students and researchers interested in the applications of homotopy theory methods in operad theory. It is accessible to readers with a minimal background in classical algebraic topology and operad theory. Graduate students and researchers interested in algebraic topology and algebraic geometry. This item is also available as part of a set: • Homotopy theory and its applications to operads • General methods of homotopy theory • Model categories and homotopy theory • Mapping spaces and simplicial model categories • Simplicial structures and mapping spaces in general model categories • Cofibrantly generated model categories • Modules, algebras, and the rational homotopy of spaces • Differential graded modules, simplicial modules, and cosimplicial modules • Differential graded algebras, simplicial algebras, and cosimplicial algebras • Models for the rational homotopy of spaces • The (rational) homotopy of operads • The model category of operads in simplicial sets • The homotopy theory of (Hopf) cooperads • Models for the rational homotopy of (non-unitary) operads • The homotopy theory of (Hopf) $\Lambda$-cooperads • Models for the rational homotopy of unitary operads • Applications of the rational homotopy to $E_n$-operads • Complete Lie algebras and rational models of classifying spaces • Formality and rational models of $E_n$-operads • The computation of homotopy automorphism spaces of operads • Introduction to the results of the computations for the $E_n$-operads • The applications of homotopy spectral sequences • Homotopy spsectral sequences and mapping spaces of operads • Applications of the cotriple cohomology of operads • Applications of the Koszul duality of operads • The case of $E_n$-operads • The applications of the Koszul duality for $E_n$-operads • The interpretation of the result of the spectral sequence in the case of $E_2$-operads • Conclusion: A survey of further research on operadic mapping spaces and their applications • Graph complexes and $E_n$-operads • From $E_n$-operads to embedding spaces • Appendices	2023-03-29 06:26:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.573341965675354, "perplexity": 1211.461684736525}, "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/1679296948951.4/warc/CC-MAIN-20230329054547-20230329084547-00022.warc.gz"}
https://calendar.mit.edu/event/simple_persons_applied_math_seminar_20220217	# Simple Person's Applied Math Seminar Thursday, February 17, 2022 at 6:00pm to 6:45pm Building 2, 2-132 182 MEMORIAL DR, Cambridge, MA 02139 Featured Speaker : Guanghao Ye (MIT Mathematics) Title : Nested Dissection Meets IPMs: Planar Min-Cost Flow in Nearly-Linear Time Abstract : We present a nearly-linear time algorithm for finding a minimum-cost flow in planar graphs with polynomially bounded integer costs and capacities. The previous fastest algorithm for this problem is based on interior-point methods (IPMs) and works for general sparse graphs in $O(n^{1.5}\text{poly}(\log n))$ time [Daitch-Spielman, STOC'08]. Our results immediately extend to all families of separable graphs. This is joint work with Sally Dong, Yu Gao, Gramoz Goranci, Yin Tat Lee, Richard Peng, and Sushant Sachdeva. Event Type Events By Interest Events By Audience Events By School Tags Website Department Department of Mathematics Hashtag #Mathematics Contact Email aldixon@mit.edu	2023-02-05 06:49:23	{"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": 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": 0.2975994050502777, "perplexity": 7158.043965585657}, "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/1674764500250.51/warc/CC-MAIN-20230205063441-20230205093441-00207.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/algebra-and-trigonometry-10th-edition/chapter-4-4-4-translations-of-conics-4-4-exercises-page-347/20	## Algebra and Trigonometry 10th Edition Center: $(0,-12)$ Radius: $r=2\sqrt 6$ The equation of a circle in standard form: $(x-h)^2+(y-k)^2=r^2$ in which $(h,k)$ is the center and $r$ is the radius $x^2+(y+12)^2=24$ $(x-0)^2+[y-(-12)]^2=(\sqrt {24})^2$ Center: $(0,-12)$ Radius: $r=\sqrt {24}=2\sqrt 6$	2022-06-30 06:35:21	{"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": 0.9682658314704895, "perplexity": 397.23744992089144}, "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-2022-27/segments/1656103669266.42/warc/CC-MAIN-20220630062154-20220630092154-00318.warc.gz"}
https://hal.inria.fr/hal-01675715	# Approximating the Volume of Tropical Polytopes is Difficult 1 TROPICAL - TROPICAL CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France Abstract : We investigate the complexity of counting the number of integer points in tropical polytopes, and the complexity of calculating their volume. We study the tropical analogue of the outer parallel body and establish bounds for its volume. We deduce that there is no approximation algorithm of factor $\alpha=2^{\text{poly}(m,n)}$ for the volume of a tropical polytope given by $n$ vertices in a space of dimension $m$, unless P$=$NP. Neither is there such an approximation algorithm for counting the number of integer points in tropical polytopes described by vertices. If follows that approximating these values for tropical polytopes is more difficult than for classical polytopes. Our proofs use a reduction from the problem of calculating the tropical rank. For tropical polytopes described by inequalities we prove that counting the number of integer points and calculating the volume are $\#$P-hard. Keywords : Document type : Journal articles Domain : https://hal.inria.fr/hal-01675715 Contributor : Stephane Gaubert <> Submitted on : Thursday, January 4, 2018 - 5:41:58 PM Last modification on : Friday, April 30, 2021 - 10:04:43 AM ### Citation Stéphane Gaubert, Marie Maccaig. Approximating the Volume of Tropical Polytopes is Difficult. International Journal of Algebra and Computation, World Scientific Publishing, 2019, 29 (02), pp.357--389. ⟨10.1142/S0218196718500686⟩. ⟨hal-01675715⟩ Record views	2021-06-25 08:14:52	{"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": 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": 0.2824900448322296, "perplexity": 993.403758978175}, "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/1623487622113.11/warc/CC-MAIN-20210625054501-20210625084501-00401.warc.gz"}
http://cnx.org/content/m13811/latest/	# Connexions You are here: Home » Content » DSP Development Environment: Introductory Exercise for TI TMS320C55x ### Lenses What is a lens? #### Definition of a lens ##### Lenses A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust. ##### What is in a lens? Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content. ##### Who can create a lens? Any individual member, a community, or a respected organization. ##### What are tags? Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens. #### Affiliated with (What does "Affiliated with" mean?) This content is either by members of the organizations listed or about topics related to the organizations listed. 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These tags come from the endorsement, affiliation, and other lenses that include this content. # DSP Development Environment: Introductory Exercise for TI TMS320C55x Module by: Thomas Shen, David Jun. E-mail the authors Summary: This exercise introduces the hardware and software used in the course. By the end of this module, you should be comfortable with the basics of testing a simple real-time DSP system with Code Composer Studio, the debugging environment we will be using throughout the semester. First you will connect the laboratory equipment and test a real-time DSP system with provided code to implement an eight-tap (eight coefficient) finite impulse response (FIR) filter. With a working system available, you will then begin to explore the debugging software used for downloading, modifying, and testing your code. Finally, you will create a filter in MATLAB and use test vectors to verify the DSP's output. ## Introduction This exercise introduces the hardware and software used in testing a simple DSP system. When you complete it, you should be comfortable with the basics of testing a simple real-time DSP system with the debugging environment you will use throughout the course. First, you will connect the laboratory equipment and test a real-time DSP system with pre-written code to implement an eight-tap (eight coefficient) finite impulse response (FIR) filter. With a working system available, you will then begin to explore the debugging software used for downloading, modifying, and testing code. Finally, exercises are included to refresh your familiarity with MATLAB. ## Lab Equipment This exercise assumes you have access to a laboratory station equipped with a Texas Instruments TMS320C5510A-200 digital signal processor chip mounted on a Spectrum Digital TMS320VC5510 evaluation board. The DUAL3006, a daughtercard produced by Educational DSP, is mounted on the external peripheral interface of the board to enable four-input/four-output capability. The evaluation module should be connected to a PC running Windows and will be controlled using the PC application Code Composer Studio v4.0, a debugger and development environment. We will be using a 48kHz sample rate. The DSP board can also communicate with user code or a terminal emulator running on the PC via a USB interface. ### Note: If you are not using Code Composer Studio v4.0, the instructions on this page do not apply. Please see the revision history of this module for instructions if using CCS v3.x In addition to the DSP board and PC, each laboratory station should also be equipped with a function generator to provide test signals and an oscilloscope to display the processed waveforms. ### Step 1: Connect cables Use the provided BNC cables to connect the output of the function generator to input channel 1 on the DSP evaluation board. Connect output channels 1 and 2 of the board to channels 1 and 2 of the oscilloscope. The input and output connections for the DSP board are shown in Figure 1. The figure may not be up to date, so ask a TA if you need help. Note that with this configuration, you will have only one signal going into the DSP board and two signals coming out. The output on channel 1 is the filtered input signal, and the output on channel 2 is the unfiltered input signal. This allows you to view the raw input and filtered output simultaneously on the oscilloscope. Turn on the function generator and the oscilloscope. When you log in, two shared networked drives should be mapped to the computer: the W: drive, which contains your own private network work directory, and the V: drive, where the necessary files for ECE 420 are stored. Be sure to save any files that you use for the course to the W: drive. Temporary files may be stored in the C:\Users\netID\workspace directory; however, since files stored on the C: drive are local to each computer, and may be erased at any time, do not store course files on the C: drive. On the V: drive, the directory V:\ece420\55x\ccs4 contains the files necessary to assemble and test code on the TI DSP evaluation boards. Although you may want to work exclusively in one or the other of lab-partners' network account, you should be sure that both partners have copies of the lab assignment assembly code. #### Warning: Not having the assembly code during a quiz because "it's on my partner's account" is NOT a valid excuse! For copying between partners' directory on W: or for working outside the lab, access to your files is available. See http://www.ece.illinois.edu/cts/storage/ for instructions on how to set that up. ## The Development Environment The evaluation board is controlled by the PC through the JTAG interface using the application Code Composer Studio. This development environment allows the user to download, run, and debug code assembled on the PC. Work through the steps below to familiarize yourself with the debugging environment and real-time system using the provided FIR filter code (Steps 3, 4 and 5), then verify the filter's frequency response with the subsequent MATLAB exercises (Steps 6 and 7). ### Step 3: Assemble filter code #### Setup Code Composer By default, a shortcut to CCS is available by going to Start > All Programs > Texas Instruments > Code Composer Studio v4. When CCS starts for the first time, Workspace Launcher will start because it will need to set up your workspace. Create or make sure you have the following directory: W:\workspace\ECE420. In Workspace Launcher, hit Browse..., navigate to this folder, and make sure to check "Use this as the default and do not ask again". #### Note: In the future, verify that you are in the correct workspace by going to File > Switch Workspace... #### Import Project In CCS, go to View > C/C++ Projects. A panel will pop up on the left side of the window. Right-click somewhere in this panel and choose Import... 1. Expand "CCS" and choose "Existing CCS/CCE Eclipse Project" 2. Hit Next and browse to V:\ece420\55x\ccs4\filter 3. Check "Copy projects into workspace" #### Build Project Once the project is copied into your workspace, we can proceed to build it by selecting Project > Build Active Project. In a successful build, there will be zero errors and maybe a few warnings and remarks. The output file will be placed in a Debug folder within the project's directory. In this example, the executable binary code will be located at .\Debug\filter.out. ### Step 4: Verify filter execution #### Connect to the DSP 1. Select View > Target Configurations 2. In the panel that comes up, expand Projects > filter 3. Right-click on dsk5510.ccxml and select "Launch Selected Configuration" Once CCS connects to the DSP, select Target > Connect Target Now, load your assembled filter file (filter.out) onto the DSP by selecting Target > Load Program. Finally, execute the code by selecting Target > Run. The program you are running accepts input from input channel 1 and sends output waveforms to output channels 1 and 2 (the filtered signal and raw input, respectively). Note that the "raw input" on output channel 2 may differ from the actual input on input channel 1, because of distortions introduced in converting the analog input to a digital signal and then back to an analog signal. The A/D and D/A converters on the six-channel surround board operate at a sample rate of 48 kHz and have an anti-aliasing filter and an anti-imaging filter, respectively, that in the ideal case would eliminate frequency content above 24 kHz. On the basis of this information, what differences do you expect to see between the signals at input channel 1 and at output channel 2? The converters on the board are also AC coupled and cannot pass DC signals. #### Configure Function Generator and Oscilloscope Set the amplitude on the function generator to 1.0 V peak-to-peak and the pulse shape to sinusoidal. Adjust the function generator so that it expects a high impedance load. The sequence of button presses to accomplish this on the function generator in the lab is Shift -> Enter -> Right -> Right -> Right -> Down -> Down -> Right -> Enter. Make sure the oscilloscope is set to 1M impedance. This can be accomplished by pressing channel 1 or 2 and then selecting 1M Ohm from the Imped menu. Observe the frequency response of the filter by sweeping the input signal through the relevant frequency range. What is the relevant frequency range for a DSP system with a sample rate of 48 kHz? #### Characterize Filter Response Based on the frequency response you observe, characterize the filter in terms of its type (e.g., low-pass, high-pass, band-pass) and its -6 dB (half-amplitude) cutoff frequency (or frequencies). It may help to set the trigger on channel 2 of the oscilloscope since the signal on channel 1 may go to zero. ### Step 5: Re-assemble and re-run with new filter Once you have determined the type of filter the DSP is implementing, you are ready to repeat the process with a different filter by including different coefficients during the assembly process. There is a second set of filter coefficients already in your project folder. In Windows Explorer, navigate to W:\workspace\ece420\filter and do the following: • Rename coef.asm to coef1.asm • Rename coef2.asm to coef.asm Repeat the assembly and testing process with the new filter by repeating steps required to build (Step 3) and execute (Step 4) the code. Just as you did in Step 4, determine the type of filter you are running and the filter's -6 dB point by testing the system at various frequencies. ### Step 6: Check filter response in MATLAB In this step, you will use MATLAB to verify the frequency response of your filter by copying the coefficients from the DSP to MATLAB and displaying the magnitude of the frequency response using the MATLAB command freqz. #### View Coefficients in DSP Memory The FIR filter coefficients included in the file coef.asm are stored in memory on the DSP. To view the contents of the DSP memory, first suspend any running program by going to Target > Halt and then select View > Memory. In the panel that comes up, there is a text box for you to type in the name of the variable that you are interested in viewing. This variable name is actually a mnemonic for a memory address. In the case of our coefficients, the mnemonic coef1 is used to point to the starting address of our coefficients. The memory content can be displayed in many different formats. In the drop-down box, choose 16-Bit Signed Int. #### Note: Make sure you understand where the coef1 label comes from. [Hint:] Select View > C/C++ Projects and double click on filtercode.asm to view the source code. In this example, the filter coefficients are placed in memory in decreasing order; that is, the last coefficient, h7 h 7 , is at location coef1 and the first coefficient, h0 h 0 , is stored at coef1+7. Now that you can find the coefficients in memory, you are ready to use the MATLAB command freqz to view the filter's response. You must create a vector in MATLAB with the filter coefficients to use the freqz command. For example, if you want to view the response of the three-tap filter with coefficients -10, 20, -10 you can use the following commands in MATLAB: • >> h = [-10, 20, -10]; • >> freqz(h) Note that you will have to enter eight values, the contents of memory locations coef1 through coef1+7, into the coefficient vector, h. #### Tip: You must divide the coefficients by 32768. Where does this scaling factor come from? How does the MATLAB response compare with your experimental results? What might account for any differences? ### Step 7: Create new filter in MATLAB and verify MATLAB scripts will be made available to you to aid in code development. For example, one of these scripts allows you to save filter coefficients created in MATLAB in a form that can be included as part of the assembly process without having to type them in by hand (a very useful tool for long filters). These scripts may already be installed on your computer; otherwise, download the files from the links as they are introduced. First, have MATLAB generate a "random" eight-tap filter by typing h = gen_filt; at a MATLAB prompt. Then save this vector of filter coefficients by typing save_coef('coef.asm',fliplr(h)); Make sure you save the file in your own directory. (The scripts that perform these functions are available as gen_filt.m and save_coef.m . They are also available at V:/ece420/55x/m_files) The save_coef MATLAB script will save the coefficients of the vector h into the named file, which in this case is coef.asm. Note that the coefficient vector is "flipped" prior to being saved; this is to make the coefficients in h h fill DSP memory-locations coef1 through coef1+7 in reverse order, as before. You may now re-assemble and re-run your new filter code as you did in Step 5. Notice when you load your new filter that the contents of memory locations coef1 through coef1+7 update accordingly. ### Step 8: Modify filter coefficients in memory Not only can you view the contents of memory on the DSP using the debugger, you can change the contents at any memory location simply by double-clicking on the location and making the desired change in the pop-up window. #### Note: The DSP must be in a halted state in order to overwrite the memory. Change the contents of memory locations coef1 through coef1+7 such that the coefficients implement a scale and delay filter with impulse response: hn=8192δn4 h n 8192 δ n 4 (1) Note that the DSP interprets the integer value of 8192 as a fractional number by dividing the integer by 32,768 (the largest integer possible in a 16-bit two's complement register). The result is an output that is delayed by four samples and scaled by a factor of 14 1 4 . More information on the DSP's interpretation of numbers appears in Two's Complement and Fractional Arithmetic for 16-bit Processors. #### Note: A clear and complete understanding of how the DSP interprets numbers is absolutely necessary to effectively write programs for the DSP. Save yourself time later by learning this material before attempting Lab 1! After you have made the changes to all eight coefficients, run your new filter and use the oscilloscope to measure the delay between the raw (input) and filtered (delayed) waveforms. #### Tip: Take advantage of the "Quick Measure" feature on the oscilloscope! What happens to the output if you change either the scaling factor or the delay value? How many seconds long is a single-sample delay? Six-sample delay? ### Step 9: Test-vector simulation As a final exercise, you will find the output of the DSP for an input specified by a test vector. Then you will compare that output with the output of a MATLAB simulation of the same filter processing the same input; if the DSP implementation is correct, the two outputs should be almost identical. To do this, you will generate a waveform in MATLAB and save it as a test vector. You will then run your DSP filter using the test vector as input and import the results back into MATLAB for comparison with a MATLAB simulation of the filter. The first step in using test vectors is to generate an appropriate input signal. One way to do this is to use the MATLAB function to generate a sinusoid that sweeps across a range of frequencies. The MATLAB function save_test_vector (available as save_test_vector.m can then save the sinusoidal sweep to a file you will later include in the DSP code. Generate a sinusoidal sweep using sweep.m and save it to a DSP test-vector file using the following MATLAB commands: • >> t=sweep(0.1pi,0.9pi,0.25,500); % Generate a frequency sweep • >> save_test_vector('testvect.asm',t); % Save the test vector Next, use the MATLAB conv command to generate a simulated response by filtering the sweep with the filter h h you generated using gen_filt above. Note that this operation will yield a vector of length 507 (which is n+m1 n m 1 , where n n is the length of the filter and m m is the length of the input). You should keep only the first 500 elements of the resulting vector. • >> out=conv(fliplr(h),t); % Filter t with FIR filter h • >> out=out(1:500); % Keep first 500 elements of out The main.c file needs to be told to take input from memory on the DSP. Fortunately, the changes have already been made in the files. The test vector is stored in a block of memory on the DSP just like other variables. The memory block that holds the test vector is large enough to hold a vector up to 4,000 elements long. The test vector stores data for all four channels of input and from four channels of output. To run your program with test vectors, you will need to modify main.c as well as filtercode.asm. Both are simply text files and can be edited using the editor of your preference, including WordPad, Emacs, and VI. (The changes have already been made, but please visually verify the changes are there.) Within main.c, uncomment the #define FILE_INPUT line so that your program will rewrite input from the A/D with the test vector you specified and then save the output into a block of memory. In filtercode.asm, uncomment the .copy "testvect.asm" line. Make sure this Matlab generated file is in the same directory as filtercode.asm. #### Note: In TI assembly, the semi-colon ; signifies a comment. These changes will copy in the test vector. After modifying your code, assemble it, then load and run the file using Code Composer as before. After a few seconds, halt the DSP (using the Halt command under the Target menu). How many seconds do you think it should take? #### Saving DSP Memory to File Next, we will save the test output file and load it back into MATLAB. We are interested in the first 500 output samples, starting at address tv_outbuf in Data memory. There are four output channels and the memory is interleaved in time. Therefore, we will have to collect 2000 (4 channels time 500 samples) memory elements. • Select View > Memory • Click on the "Save" icon, a green square with an angled arrow (top left in the Memory panel) • Name the file output.dat and save filetype as TI data format • On the next screen, use the following options: • format: hex • start address: tv_outbuf • memory page: data • length: 2000 Last, use the read_vector (available as read_vector.m) function to read the saved result into MATLAB. Do this using the following MATLAB command: • >> [ch1,ch2,ch3,ch4] = read_vector('output.dat'); Now, the MATLAB vector ch1 corresponds to the filtered version of the test signal you generated. The MATLAB vector ch2 should be nearly identical to the test vector you generated, as it was passed from the DSP system's input to its output unchanged. #### Note: Because of quantization error introduced in saving the test vector for the 16-bit memory of the DSP, the vector ch2 will not be identical to the MATLAB generated test vector. After loading the output of the filter into MATLAB, compare the expected output (calculated as out above) and the output of the filter (in ch1 from above). This can be done graphically by simply plotting the two curves on the same axes; for example: • >> plot(out,'r'); % Plot the expected curve in red • >> hold on % Plot the next plot on top of this one • >> plot(ch1,'g'); % Plot the expected curve in green • >> hold off You should also ensure that the difference between the two outputs is near zero. This can be done by plotting the difference between the two vectors: • >> plot(out(1:length(ch1))-ch1); % Plot error signal You will observe that the two sequences are not exactly the same; this is due to the fact that the DSP computes its response to 16 bits precision, while MATLAB uses 64-bit floating point numbers for its arithmetic. Blocks of output samples may also be missing from the test vector output due to a bug in the test vector core. Nonetheless, the test vector environment allows one to run repeatable experiments using the same known test input for debugging. ### Step 10: Closing Down Before exiting Code Composer, make sure to disconnect properly from the DSP: • Halt any program running on the DSP (Target > Halt) • Disconnect from the DSP (Target > Connect will toggle between connecting and disconnecting) Finally, make sure to return all of the cables to the wall rack. ## Content actions PDF \| EPUB (?) ### What is an EPUB file? EPUB is an electronic book format that can be read on a variety of mobile devices. My Favorites (?) 'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'. \| A lens I own (?) #### Definition of a lens ##### Lenses A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust. ##### What is in a lens? Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content. ##### Who can create a lens? Any individual member, a community, or a respected organization. ##### What are tags? Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens. \| External bookmarks	2013-05-19 15:56:43	{"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": 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": 0.2886905074119568, "perplexity": 2104.5667989409026}, "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-2013-20/segments/1368697772439/warc/CC-MAIN-20130516094932-00093-ip-10-60-113-184.ec2.internal.warc.gz"}
http://www.visionscatering.com/june-flowers-voduhea/target-led-lights-in-store-7b4771	The absorbance and extinction ε are sometimes defined in terms of the natural logarithm instead of the base-10 logarithm. In liquids, the extinction coefficient usually changes slowly with wavelength. 2 UV Spectroscopy I. The wavelength range for the three spectra is 0-400, 400-700, and above. I When a test material is being measured, the bandwidth of the incident light should also be sufficiently narrow. In the Beer–Lambert law, varying concentration and path length has an equivalent effect—diluting a solution by a factor of 10 has the same effect as shortening the path length by a factor of 10. The remaining light is collected after the cuvette by a glass fiber and driven into a spectrograph. Generally, UV-Vis Spectroscopy is used to determine the concentrations of elements in a solution. The beam passes through the sample and specific wavelengths are absorbed by the sample components. In a double-beam instrument, the light is split into two beams before it reaches the sample. The bottom equation describes this relationship, which provides the energy carried by a photon of a given wavelength of radiation. Specifically, UV/Vis spectrophotometers determine how much light of a given wavelength passes through a sample and … Agilent 8453 UV-visible spectroscopy systems are based on a PC-controlled spectrophotometer and a range of software products, each with features that meet the needs of speciﬁ c user groups. An optical spectrometer records the wavelengths at which absorption occurs, together with the degree of absorption at each wavelength. The electronic transitions of both molecular hydrogen and ethene are too energetic to be accurately recorded by standard UV spectrophotometers, which generally have a range of 220 – 700 nm. As a rough guide, an instrument with a single monochromator would typically have a stray light level corresponding to about 3 Absorbance Units (AU), which would make measurements above about 2 AU problematic. In this respect the human eye is functioning as a spectrometer analyzing the light reflected from the surface of a solid or passing through a liquid. These include attaching spectrophotometers to telescopes to measure the spectra of astronomical features. To understand why some compounds are colored and others are not, and to determine the relationship of conjugation to color, we must make accurate measurements of light absorption at different wavelengths in and near the visible part of the spectrum. The wavelengths of absorption peaks can be correlated with the types of bonds in a given molecule and are valuable in determining the functional groups within a molecule. This energy can be from a variety of sources, which determines the name of the subsequent emission, like luminescence. To perform UV spectroscopy at wavelengths shorter than 200 nm, the whole setup must be kept under vacuum. lower energy gap between the HOMO and the LUMO), the longer the wavelength of light it can absorb. UV VIS spectroscopy is a powerful analytical chemistry technique for determining concentration of analytes in a sample and tracking chemical reactions. UV–visible microspectrophotometers consist of a UV–visible microscope integrated with a UV–visible spectrophotometer. All the absorptions do not shift by the same amount, so for anthracene (green shaded box) and tetracene (blue shaded box) the weak absorption is obscured by stronger bands that have experienced a greater red shift. Sometimes an empirical calibration function is developed, using known concentrations of the sample, to allow measurements into the region where the instrument is becoming non-linear. The UV range extends from 100–400 nm, and the visible spectrum ranges from 400–700 nm. However, most spectrophotometers do not operate in the deep UV range of 100–200 nm, as light sources in this range are expensive. Cuvettes are typically rectangular in shape, commonly with an internal width of 1 cm. UV spectroscopy is also very useful in the study of proteins. The presence of an analyte gives a response assumed to be proportional to the concentration. The nature of the solvent, the pH of the solution, temperature, high electrolyte concentrations, and the presence of interfering substances can influence the absorption spectrum. Principle of UV Spectroscopy . The chemical and physical conditions of a test sample therefore must match reference measurements for conclusions to be valid. Any instrument will reach a point where an increase in sample concentration will not result in an increase in the reported absorbance, because the detector is simply responding to the stray light. A typical test of a semiconductor wafer would entail the acquisition of spectra from many points on a patterned or unpatterned wafer. (D) 2.5 μm – 1mm. Ultraviolet and Visible Spectroscopy This absorption spectroscopy uses electromagnetic radiations between 190 nm to 800 nm and is divided into the ultraviolet (UV, 190-400 nm) and visible (VIS, 400-800 nm) regions. One beam is used as the reference; the other beam passes through the sample. This can happen, for instance, where the absorbing substance is located within suspended particles. In the past few years Carl Zeiss has developed a large number of different spectrometer modules for different wavelength range and requirements. An obvious difference between certain compounds is their color. Other common colors of the spectrum, in order of decreasing wavelength, may be remembered by the mnemonic: ROY G BIV. c UV region can also be extended below 200 nm which is generally termed as vacuum UV but not suitable for practical purpose in UV spectrophotometers as many of the solvents also absorb and interfere with study. Many of these were inorganic minerals, but several important organic dyes were also known. In UV/Vis/NIR spectroscopy the ultraviolet (170 nm to 380 nm), visible (380 nm to 780 nm), and near infrared (780 nm to 3300 nm) are used. The deep orange hydrocarbon carotene is widely distributed in plants, but is not sufficiently stable to be used as permanent pigment, other than for food coloring. o A given spectrometer has a spectral bandwidth that characterizes how monochromatic the incident light is. I After determining optimal wavelengths for all species involved in equilibria, a reaction can be run to equilibrium, and the concentration of species determined from spectroscopy at various known wavelengths. The electronic transitions of both molecular hydrogen and ethene are too energetic to be accurately recorded by standard UV spectrophotometers, which generally have a range of 220 – 700 nm. {\displaystyle A} To make it even easier, each technique has clear explanations and descriptions supported by animations. The equilibrium constant can be calculated as K(eq) = [Products] / [Reactants]. Microspectrophotometers are used in the semiconductor and micro-optics industries for monitoring the thickness of thin films after they have been deposited. [15], UV/Vis can be applied to determine the kinetics or rate constant of a chemical reaction. An absorption spectrometer works in a range from about 200 nm (in the near ultra-violet) to about 800 nm ... Beta-carotene absorbs throughout the ultra-violet region into the violet ... To the UV-visible spectroscopy menu . A map of the film thickness across the entire wafer can then be generated and used for quality control purposes. The spectra used in spectroscopy vary from ultra-violet, visible, infrared ranges. UV/Vis spectroscopy is routinely used in analytical chemistry for the quantitative determination of different analytes, such as transition metal ions, highly conjugated organic compounds, and biological macromolecules. o From the chart above it should be clear that the only molecular moieties likely to absorb light in the 200 to 800 nm region are pi-electron functions and hetero atoms having non-bonding valence-shell electron pairs. 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Include attaching spectrophotometers to telescopes to measure the spectra for predicting the λmax of such chromophores has specially! And 700 nm absorption peaks or shoulders for a given spectrometer has a spectral bandwidth that how. Help of a semiconductor wafer would entail the acquisition of spectra from many points on a compound it. Visible wavelength is red and the LUMO ), 10 ( 1989 ) / 165 2 from to. Has developed a single beam instruments called Beer-Lambert law. law, sometimes called Beer-Lambert law. degree absorption... Or ultraviolet-visible spectrophotometry ( UV/ VIS ) involves the spectroscopy of photons spectrophotometry... From UV-Vis to NIR lamp rapidly decreases below 400nm Martin [ 11 ] showed that some exudates from. 180 nm by atmospheric gases split light into a spectrum using a fiber! The film thickness across the entire wafer can then be uv spectroscopy range and used them for decorative purposes [ 8 [... 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And is still in common use in uv spectroscopy range Beer–Lambert law because of the radiation that surrounds us can not seen!	2021-05-08 15:58:54	{"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": 0.5535093545913696, "perplexity": 2146.2847388026703}, "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/1620243988882.94/warc/CC-MAIN-20210508151721-20210508181721-00623.warc.gz"}
https://scholars.duke.edu/display/pub696993	# First Measurement of the W Boson Mass in Run II of the Tevatron We present a measurement of the W-boson mass using 200 pb-1 of data collected in p[overline p] collisions at sqrt(s)=1.96 TeV by the CDF II detector at run II of the Fermilab Tevatron. With a sample of 63 964 W-->enu candidates and 51 128 W-->µnu candidates, we measure MW=80 413$\pm$34stat$\pm$34syst=80 413$\pm$48 MeV/c2. This is the most precise single measurement of the W-boson mass to date. ### Cited Authors • Aaltonen, ; T, ; others, • 2007 ### Published In • Phys. Rev. Lett. • 99 / • 151801 - • 17995156	2022-10-06 06:59:29	{"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": 0.7059337496757507, "perplexity": 12881.03325334922}, "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/1664030337731.82/warc/CC-MAIN-20221006061224-20221006091224-00517.warc.gz"}
https://stackoverflow.com/questions/23541316/why-does-hough-not-find-line/23543322	# Why does hough not find line? I use the following code to extract lines from a given 25x25 black&white-image: [H, theta, rho] = hough(image); peaks = houghpeaks(H, 20,'NHoodSize',[19 19]); lines = houghlines(image, theta, rho, peaks, 'FillGap', 1, 'MinLength', 3); I then plot the found lines on the given image. The result looks like this: What I can't understand is, why this procedure does not find a line on the left border of the image, going from top to bottom (or vice versa). Instead for example the pink line is found, which I would think has less evidence in hough space to be there (since it touches less white pixels). Does anyone have an intuition why this might be the case? I tried changing the parameters a little bit or add some padding to the image, but nothing has worked so far. edit: original image as requested: In • How did you pad it? Edge replication or zero padding? My guess is that it's being ignored since it's DIRECTLY on the edge. Zero padding would fix this. – Raab70 May 8 '14 at 12:24 • padded it using image = padarray(image, [3 3]). did not help – user1809923 May 8 '14 at 12:30 • Can you include the original image so we can test it? – Raab70 May 8 '14 at 12:32 The default threshold value is too high so the line is not found. I also reduced the nhood size since you want to find horizontal and vertical lines and not angles, so they will all be very close to each other. Also note at the top I set the edges to zero, in the image you posted there is a thin border of 204's around the outside, this just elmiminates the border. Here is my script. clc;clearvars;close all; im=rgb2gray(im); im(:,1:2)=0; im(1,:)=0; im(end,:)=0; im(:,end)=0; BW=edge(im,'canny'); [H, T, R] = hough(BW); P = houghpeaks(H, 20,'NHoodSize',[1 1],'threshold',ceil(0.3max(H(:)))); lines = houghlines(BW, T, R, P, 'FillGap', 1, 'MinLength', 3); 'InitialMagnification','fit'); title('Hough Transform of Image'); xlabel('\theta'), ylabel('\rho'); axis on, axis normal, hold on; colormap(hot); x = T(P(:,2)); y = R(P(:,1)); plot(x,y,'s','color','blue'); figure; imagesc(im);hold on;colormap gray; axis image; max_len = 0; for k = 1:length(lines) xy = [lines(k).point1; lines(k).point2]; plot(xy(:,1),xy(:,2),'LineWidth',2,'Color','green'); % Plot beginnings and ends of lines plot(xy(1,1),xy(1,2),'x','LineWidth',2,'Color','yellow'); plot(xy(2,1),xy(2,2),'x','LineWidth',2,'Color','red'); % Determine the endpoints of the longest line segment len = norm(lines(k).point1 - lines(k).point2); if ( len > max_len) max_len = len; xy_long = xy; end end % highlight the longest line segment plot(xy_long(:,1),xy_long(:,2),'LineWidth',2,'Color','red'); The output is this: • Hi! Thanks for your answer. Well, I also want non horizontal/vertical lines to be found. Also, I don't see why in my code the pink line is found but not the line at the right border. Why would it do that? A low threshold cannot possibly be the reason? I also think there is a difference between using the .png and the 25*25 matrix. – user1809923 May 12 '14 at 13:44 • Unfortunately, I can't reproduce the matrix for the first example image, but there is another one with similar behavior. Indices which have to be set to true = [10 11 12 13 14 15 16 33 43 57 70 82 107 132 147 148 157 158 169 170 171 172 183 193 194 208 217 218 232 233 242 257 267 282 292 307 317 332 333 342 343 358 368 375 376 377 378 383 393 403 404 408 418 429 433 443 454 458 468 473 478 479 483 493 498 508 518 533 543 544 558 569 584] – user1809923 May 12 '14 at 13:45	2021-06-17 09:30:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.5991034507751465, "perplexity": 635.6934673762335}, "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/1623487629632.54/warc/CC-MAIN-20210617072023-20210617102023-00618.warc.gz"}
http://bookini.ru/interdisciplinary-applied-mathematics/70/	# Interdisciplinary Applied Mathematics Скачать в pdf «Interdisciplinary Applied Mathematics» = (uj ■ V)v +/iS72uj in Г2, (2.11a) V2v = -Vxw in Q, (2.11b) Vv = 0 in Q, (2.11c) ш = Vxv in Q, (2.11d) where the elliptic equation for the velocity v is obtained using a vector identity and the divergence-free constraint. We also assume here that the domain Q is simply connected. An equivalent system in terms of velocity and vorticity is studied in (Karniadakis and Sherwin, 1999). The problem with the lack of direct boundary conditions for the vorticity also exists in the more often used vorticity-streamfunction formulation in two dimensions. Finally, a note regarding nondimensionalization. Consider the free-stream flow U0 past a body of characteristic size D in a medium of dynamic viscosity p as shown in Figure 2.1. There are two characteristic time scales in the problem, the first one representing the convective time scale tc = D/U0, and the second one representing the diffusive time scale td = D2/v, where v = p/p is the kinematic viscosity. If we nondimensional-ize all lengths with D, the velocity field with U0, and the vorticity field with U0/D, we obtain two different nondimensional equations corresponding to the choice of the time nondimensionalization: дш дш dta* where tc and td are the nondimensionalized time variables with respect to tc and td, respectively, and Re = U0D/v is the Reynolds number. Both forms are useful in simulations, the first in high Reynolds number simulations (e.g., micronozzles, Section 6.6), and the second in low Reynolds number flows (e.g., microchannels). Скачать в pdf «Interdisciplinary Applied Mathematics»	2017-03-24 23:25:51	{"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": 0.8131687641143799, "perplexity": 2153.311568048834}, "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/1490218188623.98/warc/CC-MAIN-20170322212948-00079-ip-10-233-31-227.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/780160/inverse-of-this-3-times-3-matrix-using-the-cayley-hamilton-theorem	# Inverse of this $3\times 3$ matrix using the Cayley–Hamilton theorem Find the inverse of the matrix $$\begin{pmatrix} -1 & 2& 0 \\ 1& 1 &0 \\ 2 & -1& 2 \end{pmatrix}$$ using the Cayley–Hamilton theorem. Thanks! • Show your work. Compute the polynomial, show the matrix is invertible, and compute the inverse by knowing $p(A)=0$ for $p=\chi_A$. – Pedro Tamaroff May 3 '14 at 22:14 The matrix $A$ is: $A = \begin{bmatrix} -1 & 2 & 0 \\ 1 & 1 & 0 \\ 2 & -1 & 2 \end{bmatrix}, \tag{1}$ so the characteristic polynomial $p_A(\lambda)$ is $p_A(\lambda) = \det(A - \lambda I) = \det \begin{bmatrix} -1 - \lambda & 2 & 0 \\ 1 & 1 - \lambda & 0 \\ 2 & -1 & 2 - \lambda \end{bmatrix}$ $= ( -1 - \lambda)(1 - \lambda) (2 - \lambda) - 2(2 - \lambda) = (\lambda^2 - 1)(2 - \lambda) - 4 + 2\lambda$ $= -\lambda^3 + 2\lambda^2 + 3\lambda - 6, \tag{2}$ and by Cayley-Hamilton we have $0 = p_A(A) = -A^3 + 2A^2 + 3A - 6I; \tag{3}$ (3) may be written as $A(-A^2 +2A + 3I) = 6I, \tag{4}$ or $A(\dfrac{1}{6}(-A^2 +2A + 3I)) = I, \tag{5}$ which shows that $A^{-1} = \dfrac{1}{6}(-A^2 +2A + 3I). \tag{6}$ I leave the explicit calculation of $A^{-1}$ from (6) to any interested readers; it is not difficult. Hope this helps. Cheerio, and as always, Fiat Lux!!! • @Amzoti: thanks, doc! Out in the garden, harvesting low-hanging fruit! – Robert Lewis May 3 '14 at 22:42 Hint: The equation $$A^n + a_{n-1}A^{n-1} + \cdots + a_1 A + a_0 I = 0$$ can be rewritten as $$A(A^{n-1} + a_{n-1}A^{n-2} + \cdots + a_1I) = -a_0I$$	2020-11-30 11:45:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9501339197158813, "perplexity": 944.012953643139}, "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-2020-50/segments/1606141213431.41/warc/CC-MAIN-20201130100208-20201130130208-00621.warc.gz"}
https://tutoriel-scheme.readthedocs.io/en/latest/variables-locales.html	# Local variables¶ ## let syntax¶ In Defining variables, you met define, which defines global variables. These variables are called “global” because they are available everywhere. The syntax shown here, on the other hand, binds “local” variables, which are only temporarily valid in one section of the code. Here is a program that does not work: (define composer "Mozart") (if (equal? composer "Mozart") (begin (define birth 1756) (format #t "Mozart was born in ~a, his 300th birthday will be in ~a." birth (+ birth 300)))) (NB: At least, this program will not work with Guile 2.2. The rules over the placement of defines are complex, haved changed in the history of Guile, and are still not stabilized.) The reason this does not work is that birth is defined as a top-level variable, but its binding occurs in an expression, which is only evaluated under certain conditions. Ths is why define is only useful at the toplevel, to define functions, or constants such as (define PI 3.1415926535) For temporary variables which used to obtain a result but no longer result after the result has been computed, a different syntax is used. It uses the let keyword. Here is a syntax diagram: (let ((variable1 value1) (variable2 value2) (variable3 value3) ...) expression1 expression2 ...) When let is executed, the values are first evaluated and bound to all the variables. Then, the expressions are evaluated in order, and the let expression evaluates to the value of the last expression inside it, just like with begin. After the last parenthesis of let has been closed, the variables no longer exist. In technical terms, they are bound in the scope of the let expression. The following examples use the random function. The call (random n) return a random integer between $$0$$ and $$n - 1$$. Of course, you might get different results than the ones shown here, since they are random. (define (loto) (let ((x (random 2)) (y (random 2))) (display (if (and (= x 1) (= y 1)) "Chance!" "You lost!")))) (loto) ⊨ You lost! Here, two coins are flipped (heads = 0, tails = 1). If both show tails, the user wins. A let form has a recognizable visual shape: (let (xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxx) xxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxx) Scheme programmers, without ever counting parentheses, recognize the block of variables and the block of expressions. Let’s invent a variant of the game, where a dice is first rolled, which gives the number of balls in a box, then one ball is drawn from the box at random, given that there is exactly one winning ball. For example, if the dice gives 4, there will be 4 balls in the box and thus a probability of 1/4 to win. It is tempting to write this: (define (loto2) (let ((number-of-balls (random 10)) (ball (random number-of-balls))) (display (if (equal? ball 0) "Chance!" "You lost.")))) Yet, this gives the error “Unbound variable: number-of-balls”. This is because let is actually quirky. First, the values for all variables are evaluated, and only then, they are bound to the variables. The interpreter tries to evaluate (random 10), then (random number-of-balls), and afterwards bind the variables number-of-balls and ball to these two values. Of course, we would like number-of-balls to be bound before (random number-of-balls) is computed. In this case, let needs to be replaced with a variant of it, let, of which the syntax is exactly the same. The corrected example is: (define (loto2) (let ((number-of-balls (random 10)) (ball (random number-of-balls))) (display (if (equal? ball 0) "Chance!" "You lost.")))) (loto2) ⊨ You lost. Unlike let, let* first evaluates the first expression and stores it in the first variable, and only then evaluates the expression (which can thus reuse the first variable) and stores it in the second variable, etc. In practice, you almost most often want to use let. ## Parenthesizing a let expression¶ let expressions contain many parentheses. It is easy for the unexperienced to get them wrong. This part goes through all common parenthesizing errors to explain them. I will use this example: (let ((a 5)) (+ a 15)) • Forgetting a parenthesis. (let ((a 5) ; missing ) (+ a 15)) With this expression in a LilyPond file (with a preceding # to introduce Scheme code), you will get the error “end of file”, which means that the Scheme expresson never ended. (let ((a 5)) (+ a 15))) ; extra ) In LilyPond (don’t forget the #), the error may seem more surprising: “syntax error, unexpected EVENT_IDENTIFIER”. What happens is that when the expression ends, LilyPond syntax is used again. At this point, the extraneous parenthesis is parsed. In LilyPond, parentheses are the syntax for slurs, hence the message indicating that a slur is not valid on the toplevel. • Moving a parenthesis. (let ((a 5) ; missing ) (+ a 15))) ; extra ) Here, there is no extraneous or missing parenthesis for the expression as a whole; its parentheses are balanced. You text editor will thus not find the mistake. Yet, a parenthesis is misplaced. The error message is somewhat short: “bad let”. To understand it, let us come back to how a let is constructed: (let (xxxxxxxxx xxxxxxxxx) xxxxxxx xxxxxxx) The first (...) contains all bindings. Here, this (...) actually contains everything that’s inside the let expression. The code could be reformatted like this: (let ((a 5) (+ a 15)) ) The interpreter tries to see (+ a 15) as (variable-name value), which fails because there are three elements between the parentheses rather than two. The let is also missing a main expression after the bindings, hence “bad let”. • Omitting parentheses. (let (a 5) ; should be ((a 5)) (+ a 15)) Again, the interpreter complains about a “bad let”. To understand, let us remember that everything in the first (...) is taken as a sequence of bindings, taking the form (variable-name value). By reformatting the let, the problem is made clear: (let ( a 5 ) (+ a 15)) Indeed, a and 5 do not have the form (variable-name value). This is why you need two pairs of parentheses even to define just one varable: ((a 5)). ## Simplifying code with let¶ let is a useful tool to make code more readable and understandable. For demonstration purposes, this code is taken from LilyPond, and adapted to contain no let* at all. (apply ly:stencil-add (map (lambda (stil accessor) (ly:stencil-translate-axis stil (accessor (coord-translate (interval-widen (ly:relative-group-extent (apply append (map (lambda (g) (cons g (apply append (map (lambda (sym) (cond ((ly:grob? (ly:grob-object g sym)) (list (ly:grob-object g sym))) ((ly:grob-array? (ly:grob-object g sym)) (ly:grob-array->list (ly:grob-object g sym))) (else '()))) (ly:grob-property g 'parenthesis-friends))))) (ly:grob-array->list (ly:grob-object grob 'elements)))) (ly:grob-system grob) X) (- (ly:grob-relative-coordinate grob (ly:grob-system grob) X)))) X)) (ly:grob-property grob 'stencils) (list car cdr))) If you understand nothing in this, you have taken the point. What makes this code hard to read is endless nesting of expressions, which makes you lose track of what is being done, just like if you were reading a single sentence several pages long. Furthermore, the order of execution goes from inner expressions, which are read last, to outer expressions, whereas for us humans it is easier to think when the code executes linearly. Here is the same code rewritten to use let and let: (let ((elts (ly:grob-array->list (ly:grob-object grob 'elements))) (get-friends (lambda (g) (let* ((syms (ly:grob-property g 'parenthesis-friends)) (get-friends-for-symbol (lambda (sym) (let ((friends (ly:grob-object g sym))) (cond ((ly:grob? friends) (list friends)) ((ly:grob-array? friends) (ly:grob-array->list friends)) (else '()))))) (friend-lists (map get-friends-for-symbol syms)) (friends (apply append friend-lists))) (cons g friends)))) (all-friend-lists (map get-friends elts)) (all-friends (apply append all-friend-lists)) (all-friends-array (ly:grob-list->grob-array all-friends)) (X-refp (ly:grob-common-refpoint-of-array grob all-friends-array X)) (my-X (ly:grob-relative-coordinate grob X-refp X)) (X-ext (ly:relative-group-extent all-friends-array X-refp X)) (parenthesis-positions (coord-translate wide-X-ext (- my-X))) (stencils (ly:grob-property grob 'stencils)) (left-paren (first stencils)) (right-paren (second stencils)) (translated-left-paren (ly:stencil-translate-axis left-paren (interval-start parenthesis-positions) X)) (translated-right-paren (ly:stencil-translate-axis right-paren (interval-end parenthesis-positions) X))) Without knowing anything about how LilyPond works internally, you can already understand some things: the grobs encompassed by a pair of parentheses are read (elts), the list is extended so it comprises their “friends” (all-friends), a horizontal reference point is computed (X-refp), then a horizontal coordinate (my-X), etc. It is easier to write complicated functions as a big let*, where variables are bound step-by-step, so that the final expression is simple and short. This advice will be particularly useful while you are a beginner.	2022-11-27 11:07:28	{"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": 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": 0.7745556235313416, "perplexity": 3931.4219726994393}, "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/1669446710237.57/warc/CC-MAIN-20221127105736-20221127135736-00499.warc.gz"}
https://pinboard.in/u:Vaguery	12357 A Year in Reading: Namwali Serpell - The Millions The question of how and what we (ought to) read is political for me in this sense: If we believe in democracy and equality, why are our aesthetic priorities shaped by an elite minority? Why do we dismiss our engagement with genre works as “love-hate,” “hate-watching,” and “guilty pleasure” when we spend so much time doing it? Why do we refer to these works as “low” or “lite” when they are read by millions more people than the classics? In short, why don’t the numbers matter? Maybe these texts aren’t read much in academia because they don’t require scholars to explain or analyze them: The story we tell ourselves is that they aren’t difficult or ambiguous; they’re self-evident, simplistic even. But maybe that’s just some petty nonsense to justify the need for literary critics? literary-criticism genre rather-good self-definition essay yesterday [1812.04948] A Style-Based Generator Architecture for Generative Adversarial Networks We propose an alternative generator architecture for generative adversarial networks, borrowing from style transfer literature. The new architecture leads to an automatically learned, unsupervised separation of high-level attributes (e.g., pose and identity when trained on human faces) and stochastic variation in the generated images (e.g., freckles, hair), and it enables intuitive, scale-specific control of the synthesis. The new generator improves the state-of-the-art in terms of traditional distribution quality metrics, leads to demonstrably better interpolation properties, and also better disentangles the latent factors of variation. To quantify interpolation quality and disentanglement, we propose two new, automated methods that are applicable to any generator architecture. Finally, we introduce a new, highly varied and high-quality dataset of human faces. generative-art generative-models neural-networks multiscale very-impressive to-write-about consider:performance-measures 2 days ago SnapshotTesting 1.0: Delightful Swift snapshot testing The iOS community has been a large proponent of snapshot testing, mostly thanks to the wonderful iOSSnapshotTestCase library (formerly known as FBSnapshotTestCase). It introduced a new kind of test coverage for iOS applications by allowing us to assert against an image screenshot of a UI component. This is a whole new level of testing that can catch regressions in the pixel data of our UI so that you can make sure that future changes and refactors do not introduce visual regressions into your views. However, iOSSnapshotTestCase has not evolved much over the years, and its still largely written in Objective-C, which means the API isn’t as generic and composable as it could be in Swift. Also, it only allows snapshotting CALayers and UIViews into a PNG format, but there are many more types we might want to snapshot and many more formats we want to snapshot into. That’s why today we are excited to officially announce SnapshotTesting 1.0: a modern, composable snapshot testing library built entirely in Swift! testing Swift user-experience to-do iOS software-development 4 days ago [1808.00453] The Erdos-Szekeres problem and an induced Ramsey question Motivated by the Erdos-Szekeres convex polytope conjecture in Rd, we initiate the study of the following induced Ramsey problem for hypergraphs. Given integers n>k≥5, what is the minimum integer gk(n) such that any k-uniform hypergraph on gk(n) vertices with the property that any set of k+1 vertices induces 0, 2, or 4 edges, contains an independent set of size n. Our main result shows that gk(n)>2cnk−4, where c=c(k). combinatorics hypergraphs constraint-satisfaction consequences-of-the-rules rather-interesting to-write-about 4 days ago [1810.00845] CHET: Compiler and Runtime for Homomorphic Evaluation of Tensor Programs Fully Homomorphic Encryption (FHE) refers to a set of encryption schemes that allow computations to be applied directly on encrypted data without requiring a secret key. This enables novel application scenarios where a client can safely offload storage and computation to a third-party cloud provider without having to trust the software and the hardware vendors with the decryption keys. Recent advances in both FHE schemes and implementations have moved such applications from theoretical possibilities into the realm of practicalities. This paper proposes a compact and well-reasoned interface called the Homomorphic Instruction Set Architecture (HISA) for developing FHE applications. Just as the hardware ISA interface enabled hardware advances to proceed independent of software advances in the compiler and language runtimes, HISA decouples compiler optimizations and runtimes for supporting FHE applications from advancements in the underlying FHE schemes. This paper demonstrates the capabilities of HISA by building an end-to-end software stack for evaluating neural network models on encrypted data. Our stack includes an end-to-end compiler, runtime, and a set of optimizations. Our approach shows generated code, on a set of popular neural network architectures, is faster than hand-optimized implementations. to-understand machine-learning algorithms distributed-processing seems-important languages neural-networks 4 days ago [1706.04290] A general method for lower bounds on fluctuations of random variables There are many ways of establishing upper bounds on fluctuations of random variables, but there is no systematic approach for lower bounds. As a result, lower bounds are unknown in many important problems. This paper introduces a general method for lower bounds on fluctuations. The method is used to obtain new results for the stochastic traveling salesman problem, the stochastic minimal matching problem, the random assignment problem, the Sherrington-Kirkpatrick model of spin glasses, first-passage percolation and random matrices. A long list of open problems is provided at the end. open-questions extreme-values probability-theory representation rather-odd nudge-targets consider:looking-to-see to-write-about 4 days ago QAnon and Pinterest Is Just the Beginning \| Hapgood How Pinterest’s Aggressive Recommendation Engine Makes This Worse About a year ago I wrote an article on how Pinterest’s recommendation engine makes this situation far worse. I showed how after just 14 minutes of browsing, a new user with some questions about vaccines could move from pins on “How to Make the Perfect Egg” to something out of the Infowarverse: social-media recommendation-engines spam propaganda rather-interesting cultural-predation 4 days ago [1802.07029] On a fully fuzzy framework for minimax mixed integer linear programming In this work, we present a modeling framework for minimax mixed 0-1 fuzzy linear problems. It is based on extending the usual rewriting of crisp minimax problems via auxiliary variables to model the maximum of a finite set of fuzzy linear functions. We establish that the considered problem can be equivalently formulated as a multiple objective mixed integer programming problem. The framework is applied to a fully fuzzy version of the capacitated center facility location problem. representation fuzzy operations-research integer-programming models-and-modes to-write-about could-be-clearer 4 days ago Democracy as an information system — Crooked Timber Democracy is an information system. That’s the starting place of our new paper: “Common-Knowledge Attacks on Democracy.” In it, we look at democracy through the lens of information security, trying to understand the current waves of Internet disinformation attacks. Specifically, we wanted to explain why the same disinformation campaigns that act as a stabilizing influence in Russia are destabilizing in the United States. The answer revolves around the different ways autocracies and democracies work as information systems. We start by differentiating between two types of knowledge that societies use in their political systems. The first is common political knowledge, which is the body of information that people in a society broadly agree on. People agree on who the rulers are and what their claim to legitimacy is. People agree broadly on how their government works, even if they don’t like it. In a democracy, people agree about how elections work: how districts are created and defined, how candidates are chosen, and that their votes count—even if only roughly and imperfectly. social-norms democracy cultural-dynamics propaganda public-policy political-economy rather-interesting epidemiology feature-construction discriminators fascism signaling 4 days ago [1811.08759] Using AI to Design Stone Jewelry Jewelry has been an integral part of human culture since ages. One of the most popular styles of jewelry is created by putting together precious and semi-precious stones in diverse patterns. While technology is finding its way in the production process of such jewelry, designing it remains a time-consuming and involved task. In this paper, we propose a unique approach using optimization methods coupled with machine learning techniques to generate novel stone jewelry designs at scale. Our evaluation shows that designs generated by our approach are highly likeable and visually appealing. generative-art design aesthetics rather-interesting performance-measure to-write-about user-centric-design 4 days ago Knots and Narnias \| Say you’re walking north across a meadow surrounded by hills when you come across a solitary doorframe with no door inside it. Stranger still, through the doorway you see not the hills to the north of the field but a desert vista. Consumed by curiosity and heedless of danger, you cross the threshold into the desert. The sun beats down on your bare head; you see a vulture off in the distance. In sudden panic you spin around; fortunately the doorway is still there. You run through the doorway back into the field, grateful that the portal works both ways. Now what? 4 days ago [1804.02851] Whale swarm algorithm with the mechanism of identifying and escaping from extreme point for multimodal function optimization Most real-world optimization problems often come with multiple global optima or local optima. Therefore, increasing niching metaheuristic algorithms, which devote to finding multiple optima in a single run, are developed to solve these multimodal optimization problems. However, there are two difficulties urgently to be solved for most existing niching metaheuristic algorithms: how to set the optimal values of niching parameters for different optimization problems, and how to jump out of the local optima efficiently. These two difficulties limited their practicality largely. Based on Whale Swarm Algorithm (WSA) we proposed previously, this paper presents a new multimodal optimizer named WSA with Iterative Counter (WSA-IC) to address these two difficulties. In the one hand, WSA-IC improves the iteration rule of the original WSA for multimodal optimization, which removes the need of specifying different values of attenuation coefficient for different problems to form multiple subpopulations, without introducing any niching parameter. In the other hand, WSA-IC enables the identification of extreme point during iterations relying on two new parameters (i.e., stability threshold Ts and fitness threshold Tf), to jump out of the located extreme point. Moreover, the convergence of WSA-IC is proved. Finally, the proposed WSA-IC is compared with several niching metaheuristic algorithms on CEC2015 niching benchmark test functions and five additional classical multimodal functions with high dimensions. The experimental results demonstrate that WSA-IC statistically outperforms other niching metaheuristic algorithms on most test functions. 4 days ago [1706.07900] Tree-Residue Vertex-Breaking: a new tool for proving hardness In this paper, we introduce a new problem called Tree-Residue Vertex-Breaking (TRVB): given a multigraph $G$ some of whose vertices are marked "breakable," is it possible to convert $G$ into a tree via a sequence of "vertex-breaking" operations (replacing a degree-$k$ breakable vertex by $k$ degree-$1$ vertices, disconnecting the $k$ incident edges)? We characterize the computational complexity of TRVB with any combination of the following additional constraints: $G$ must be planar, $G$ must be a simple graph, the degree of every breakable vertex must belong to an allowed list $B$, and the degree of every unbreakable vertex must belong to an allowed list $U$. The two results which we expect to be most generally applicable are that (1) TRVB is polynomially solvable when breakable vertices are restricted to have degree at most $3$; and (2) for any $k \ge 4$, TRVB is NP-complete when the given multigraph is restricted to be planar and to consist entirely of degree-$k$ breakable vertices. To demonstrate the use of TRVB, we give a simple proof of the known result that Hamiltonicity in max-degree-$3$ square grid graphs is NP-hard. We also demonstrate a connection between TRVB and the Hypergraph Spanning Tree problem. This connection allows us to show that the Hypergraph Spanning Tree problem in $k$-uniform $2$-regular hypergraphs is NP-complete for any $k \ge 4$, even when the incidence graph of the hypergraph is planar. feature-construction graph-theory computational-complexity rather-interesting explanation to-write-about 4 days ago Explainable AI won’t deliver. Here’s why. – Hacker Noon Imagine choosing between two spaceships. Spaceship 1 comes with exact equations explaining how it works, but has never been flown. How Spaceship 2 flies is a mystery, but it has undergone extensive testing, with years of successful flights like the one you’re going on. Which spaceship would you choose? This is a philosophical question, so I can’t answer it for you. I know I have a personal preference — maybe that’s the statistician in me — but I would choose careful testing as a better basis for trust. artificial-intelligence explanation philosophy-of-engineering maintainability models-and-modes that-taco-bell-girl-says-what? 4 days ago On authoritarian neoliberalism and poetic epistemology \| Richard Hall's Space As one response to the secular crisis of capitalism, higher education is being proletarianised. Its academics and students, increasingly encumbered by precarious employment, debt, and new levels of performance management, are shorn of autonomy beyond the sale of their labour-power. One heuristic for analysing this response is authoritarian neoliberalism, imposed as a means of enacting disciplinary practices in the name of the market with an anti-democratic rationale. This has a distinctly technocratic focus, rooted in techniques of performativity, including audits and assessments of teaching, research and scholarship, grounded in productivity, the management of time and value-creation. However, there are a range of intersectional and geographical responses to such an imposition, through which it is possible to describe alternatives to these architectures of subsumption. In particular, a second heuristic emerges which challenges the restructuring of the University in the global North, erupting from struggles for decolonisation. Here, Audre Lorde’s invocation to an integrated, poetic existence that situates bodies in places, and respects feelings and emotions as the site of epistemological development and understanding, underpins the possibility for dismantling hegemonic knowledge production. The article examines whether humanist narratives of solidarity, in particular from marginalised voices, might help academics and students to analyse their alienated labour and to imagine that another world is possible. worklife institutional-design neoliberalism humanities academic-culture just-what-is-it-you-do? 5 days ago How many landmarks are enough to characterize shape and size variation? Accurate characterization of morphological variation is crucial for generating reliable results and conclusions concerning changes and differences in form. Despite the prevalence of landmark-based geometric morphometric (GM) data in the scientific literature, a formal treatment of whether sampled landmarks adequately capture shape variation has remained elusive. Here, I introduce LaSEC (Landmark Sampling Evaluation Curve), a computational tool to assess the fidelity of morphological characterization by landmarks. This task is achieved by calculating how subsampled data converge to the pattern of shape variation in the full dataset as landmark sampling is increased incrementally. While the number of landmarks needed for adequate shape variation is dependent on individual datasets, LaSEC helps the user (1) identify under- and oversampling of landmarks; (2) assess robustness of morphological characterization; and (3) determine the number of landmarks that can be removed without compromising shape information. In practice, this knowledge could reduce time and cost associated with data collection, maintain statistical power in certain analyses, and enable the incorporation of incomplete, but important, specimens to the dataset. Results based on simulated shape data also reveal general properties of landmark data, including statistical consistency where sampling additional landmarks has the tendency to asymptotically improve the accuracy of morphological characterization. As landmark-based GM data become more widely adopted, LaSEC provides a systematic approach to evaluate and refine the collection of shape data––a goal paramount for accumulation and analysis of accurate morphological information. inference data-analysis looking-to-see rather-interesting training-data data-balancing to-write-about 5 days ago Elvis Presley's Pound Cake Recipe \| SAVEUR When we wrote our book Elvis World (Knopf, 1987), we often dined with fans of The King. As they discussed what they loved about him, it became clear that one of the reasons people felt so close to Elvis was that he never lost his down-home taste. You can see it in the decor at Graceland, a poor Mississippi boy's idea of how a rich person's house should look; and it is apparent in what he ate. He could afford filet mignon but preferred well-done burgers. Instead of champagne, he drank Pepsi. For dessert, he favored Deep South diner classics. One of Elvis's favorite sweets was the pound cake made by his childhood friend Janelle McComb. She gave us her recipe in 1987, on the 10th anniversary of Elvis's death. Every year at Christmas, she'd bake two loaves and bring them to Graceland. Elvis could eat one all by himself. Fans know about McComb and place her in the firmament of those who practice TCE ("Taking Care of Elvis"); to serve her cake is to keep the legend alive. —Jane and Michael Stern, authors of Roadfood.com recipes cake to-do 5 days ago [1811.09620] TimbreTron: A WaveNet(CycleGAN(CQT(Audio))) Pipeline for Musical Timbre Transfer In this work, we address the problem of musical timbre transfer, where the goal is to manipulate the timbre of a sound sample from one instrument to match another instrument while preserving other musical content, such as pitch, rhythm, and loudness. In principle, one could apply image-based style transfer techniques to a time-frequency representation of an audio signal, but this depends on having a representation that allows independent manipulation of timbre as well as high-quality waveform generation. We introduce TimbreTron, a method for musical timbre transfer which applies "image" domain style transfer to a time-frequency representation of the audio signal, and then produces a high-quality waveform using a conditional WaveNet synthesizer. We show that the Constant Q Transform (CQT) representation is particularly well-suited to convolutional architectures due to its approximate pitch equivariance. Based on human perceptual evaluations, we confirmed that TimbreTron recognizably transferred the timbre while otherwise preserving the musical content, for both monophonic and polyphonic samples. style-transfer neural-networks feature-extraction signal-processing audio to-write-about consider:performance-measures 5 days ago [cs/0610153] Most Programs Stop Quickly or Never Halt Since many real-world problems arising in the fields of compiler optimisation, automated software engineering, formal proof systems, and so forth are equivalent to the Halting Problem--the most notorious undecidable problem--there is a growing interest, not only academically, in understanding the problem better and in providing alternative solutions. Halting computations can be recognised by simply running them; the main difficulty is to detect non-halting programs. Our approach is to have the probability space extend over both space and time and to consider the probability that a random N-bit program has halted by a random time. We postulate an a priori computable probability distribution on all possible runtimes and we prove that given an integer k>0, we can effectively compute a time bound T such that the probability that an N-bit program will eventually halt given that it has not halted by T is smaller than 2^{-k}. We also show that the set of halting programs (which is computably enumerable, but not computable) can be written as a disjoint union of a computable set and a set of effectively vanishing probability. Finally, we show that long'' runtimes are effectively rare. More formally, the set of times at which an N-bit program can stop after the time 2^{N+constant} has effectively zero density. halting-problem computer-science rather-interesting looking-to-see to-write-about ReQ computational-complexity probability-theory 5 days ago Reinforcement Learning with Prediction-Based Rewards We’ve developed Random Network Distillation (RND), a prediction-based method for encouraging reinforcement learning agents to explore their environments through curiosity, which for the first time1 exceeds average human performance on Montezuma’s Revenge. RND achieves state-of-the-art performance, periodically finds all 24 rooms and solves the first level without using demonstrations or having access to the underlying state of the game. RND incentivizes visiting unfamiliar states by measuring how hard it is to predict the output of a fixed random neural network on visited states. In unfamiliar states it’s hard to guess the output, and hence the reward is high. It can be applied to any reinforcement learning algorithm, is simple to implement and efficient to scale. Below we release a reference implementation of RND that can reproduce the results from our paper. machine-learning reinforcement-learning curiosity the-mangle-in-practice to-write-about also-note-presentation 5 days ago Overcoming folk-physics: the case of projectile motion for Aristotle, John Philoponus, Ibn-Sina & Galileo \| Theory, Evolution, and Games Group It is not until Ibn-Sīnā’s — or Avicenna’s, as he is known in the Latin tradition — Book of Healing published around 1027 that we start to get close to what we’d now call Newton’s first law on motion (for a more comprehensive history of this, see Sayili, A. (1987). Ibn Sīnā and Buridan on the Motion of the Projectile. Annals of the New York Academy of Sciences, 500(1): 477-482). Ibn Sina builds on Yaḥyā al-Naḥwī’s account by accepting that a motile power was acquired by the stone from the thrower, but this spirit does not leave on its own account as unnatural, instead, this impressed virtue is dissipated through the influence of other agents such as the air. At this point, the inquiry can be said to have transformed the question from the ancient “what keeps pushing the stone?” to the modern “what causes the stone to stop moving?”. In the process, little to nothing has been learned about the problem-domain of projectile motion, but a new framework for thought was erected. 5 days ago No, it’s not The Incentives—it’s you – [citation needed] A random bystander who happened to eavesdrop on a conversation between a group of scientists kvetching about The Incentives could be forgiven for thinking that maybe, just maybe, a bunch of very industrious people who generally pride themselves on their creativity, persistence, and intelligence could find some way to work around, or through, the problem. And I think they would be right. The fact that we collectively don’t see it as a colossal moral failing that we haven’t figured out a way to get our work done without having to routinely cut corners in the rush for fame and fortune is deeply troubling. It’s also aggravating on an intellectual level, because the argument that we’re all being egregiously and continuously screwed over by The Incentives is just not that good. I think there are a lot of reasons why researchers should be very hesitant to invoke The Incentives as a justification for why any of us behave the way we do. I’ll give nine of them here, but I imagine there are probably others. academic-culture publishing social-dynamics social-norms conservatism attention-desert ethics 5 days ago shadow-cljs provides everything you need to compile your ClojureScript code with a focus on simplicity and ease of use. Clojure clojurescript software-development user-experience work-cycle compiler to-do 5 days ago Remembering Roy Gold, Who was Not Excessively Interested in Books – The Public Domain Review In this gentle memorial, Nicholas Jeeves takes us on a turn through a Borgesian library of defacements. Jeeves’ quarry, the (inventive and invented) Professor Roy Gold, would seem to have been an outsider artist of his books, and his dust-jacket daubings leave an ambiguous legacy. Should such biblio-graffiti be accounted irreverent mischief? Does it betray anti-bookishness in the secret heart of a bookish man? Or is something else afoot? Perhaps, under the right conditions, doodling can become something like a theory of reading. After all, what is to be done with all our paper books in an age of textual dematerialization? Roy Gold stands over our shoulders, brush in hand… parataxis the-use-of-books reworks to-write-about rather-interesting memoir book-art 6 days ago When Optimising Code Measure This is a truism that lots of people quote, but it can be hard to remember, especially in the heat of battle (as it were). Rather fortunately it came to mind just when needed, as I found something completely unexpected. I was writing a simple implementation of the Fermat difference of squares method of factoring. This involves writing the number to be factored as - you guessed it - the difference of two squares. If n=a2−b2 n = a 2 b 2 then n=(a−b)(a+b) n = ( a b ) ( a + b ) and we have a factorisation (provided we don't have a−b=1 a b = 1 ). the-mangle-in-practice looking-to-see learning-in-public computer-science computational-complexity rather-interesting to-write-about contingency 6 days ago Alan Turing, On computable numbers \| Joel David Hamkins What I was extremely surprised to find, however, and what I want to tell you about today, is that despite the title of the article, Turing adopts an incorrect approach to the theory of computable numbers. His central definition is what is now usually regarded as a mistaken way to proceed with this concept. Let me explain. Turing defines that a computable real number is one whose decimal (or binary) expansion can be enumerated by a finite procedure, by what we now call a Turing machine. You can see this in the very first sentence of his paper, and he elaborates on and confirms this definition in detail later on in the paper. computability mathematics number-theory algorithms rather-interesting history-of-science representation to-write-about ReQ 6 days ago Bending the Law of Sines \| The Aperiodical You can tile a plane with a repeated single triangle shape. But for most triangles you need to put same sides together, usually to group pairs of triangles into parallelograms, as shown above. You can’t just put any sides together. This of course reflects the Law of Sines, which says the side lengths are proportional to the sines of their opposite angles: you can’t get both the angles AND the side lengths in agreeable proportions. But something unexpected happens when we add curves. For tiling we need equal amounts of concave and convex arc. The only way to do that is with two shorter concave sides joining a longer convex side, as shown below. tiling rather-interesting learning-in-public experimentation construction to-write-about 6 days ago DSHR's Blog: It Isn't About The Technology In other words, for searches that are profitable, Google has moved all the results it thinks are relevant off the first page and replaced them with results that people have paid to put there. Which is pretty much the definition of "evil" in the famous "don't be evil" slogan notoriously dropped in 2015. I'm pretty sure that no-one at executive level in Google thought that building a paid-search engine was a good idea, but the internal logic of the "slow AI" they built forced them into doing just that. federation decentralization social-dynamics corporatism activism engineering-design institutional-design political-economy 6 days ago Stability Criteria for Complex Microbial Communities \| bioRxiv Competition and mutualism are inevitable processes in microbial ecology, and a central question is which and how many taxa will persist in the face of these interactions. Ecological theory has demonstrated that when direct, pairwise interactions among a group of species are too numerous, or too strong, then the coexistence of these species will be unstable to any slight perturbation. This instability worsens when mutualistic interactions complement competition. Here, we refine and to some extent overturn that understanding, by considering explicitly the resources that microbes consume and produce. In contrast to more complex organisms, microbial cells consume primarily abiotic resources, and mutualistic interactions are often mediated by these same abiotic resources through the mechanism of cross-feeding. Our model therefore considers the consumption and production of a set of abiotic resources by a group of microbial species. We show that if microbes consume, but do not produce resources, then any positive equilibrium will always be stable to small perturbations. We go on to show that in the presence of crossfeeding, stability is no longer guaranteed. However, stability still holds when mutualistic interations are either symmetric, or sufficiently weak. theoretical-biology community-formation ecology competition cooperation rather-interesting nonlinear-dynamics to-write-about 6 days ago [1803.09473] code2vec: Learning Distributed Representations of Code We present a neural model for representing snippets of code as continuous distributed vectors ("code embeddings"). The main idea is to represent a code snippet as a single fixed-length code vector, which can be used to predict semantic properties of the snippet. This is performed by decomposing code to a collection of paths in its abstract syntax tree, and learning the atomic representation of each path simultaneously with learning how to aggregate a set of them. We demonstrate the effectiveness of our approach by using it to predict a method's name from the vector representation of its body. We evaluate our approach by training a model on a dataset of 14M methods. We show that code vectors trained on this dataset can predict method names from files that were completely unobserved during training. Furthermore, we show that our model learns useful method name vectors that capture semantic similarities, combinations, and analogies. Comparing previous techniques over the same data set, our approach obtains a relative improvement of over 75%, being the first to successfully predict method names based on a large, cross-project, corpus. Our trained model, visualizations and vector similarities are available as an interactive online demo at this http URL. The code, data, and trained models are available at this https URL. representation genetic-programming (it-ain't) deep-learning neural-networks feature-construction to-write-about discrete-and-continuous-sittin-in-a-tree 6 days ago [1812.01717] Towards Accurate Generative Models of Video: A New Metric & Challenges Recent advances in deep generative models have lead to remarkable progress in synthesizing high quality images. Following their successful application in image processing and representation learning, an important next step is to consider videos. Learning generative models of video is a much harder task, requiring a model to capture the temporal dynamics of a scene, in addition to the visual presentation of objects. Although recent attempts at formulating generative models of video have had some success, current progress is hampered by (1) the lack of qualitative metrics that consider visual quality, temporal coherence, and diversity of samples, and (2) the wide gap between purely synthetic video datasets and challenging real-world datasets in terms of complexity. To this extent we propose Fréchet Video Distance (FVD), a new metric for generative models of video based on FID, and StarCraft 2 Videos (SCV), a collection of progressively harder datasets that challenge the capabilities of the current iteration of generative models for video. We conduct a large-scale human study, which confirms that FVD correlates well with qualitative human judgment of generated videos, and provide initial benchmark results on SCV. metrics Frechet-distance generative-models representation rather-interesting video feature-construction to-write-about 6 days ago "Mapping Imaginary Cities" by Mouse Reeve - YouTube While the map is not the territory (to quote the semantician Alfred Korzybski), the map is still usually intended to correspond to one. But what about maps of nowhere at all? What can they represent and how can they be made? Maps are a familiar part of daily life, with a deeply familiar and complex symbolic language, and a long history. They are also hugely varied in style and aesthetic, and often are works of art unto themselves. All this makes mapping a powerful creative tool for conveying ideas about a space, how it is used, and who inhabits it. But it also presents a mapmaker with what can feel like an overwhelming array of design choices and technical hurdles to overcome in order to create a generative map. This talk will explore maps as a way to communicate about people and place in the context of fictional cities, and dive into algorithms and techniques for procedurally generating maps by building up topography, landscape, populations, and street plans. Maps are a familiar part of daily life, with a deeply familiar and complex symbolic language, and a long history. They are also hugely varied in style and aesthetic, and often are works of art unto themselves. All this makes mapping a powerful creative tool for conveying ideas about a space, how it is used, and who inhabits it. But it also presents a mapmaker with what can feel like an overwhelming array of design choices and technical hurdles to overcome in order to create a generative map. This talk will explore maps as a way to communicate about people and place in the context of fictional cities, and dive into algorithms and techniques for procedurally generating maps by building up topography, landscape, populations, and street plans. video cartography strange-loop rather-interesting art generative-art generative-models the-mangle-in-practice learning-in-public algorithms 6 days ago [PDF] Derivation of the Variational Bayes Equations Alianna J. Maren The derivation of key equations for the variational Bayes approach is well-known in certain circles. However, translating the fundamental derivations (e.g., as found in Beal (2003)) to the notation of Friston (2013, 2015) is somewhat delicate. Further, the notion of using varia- tional Bayes in the context of a system with Markov blankets requires special attention. This Technical Note presents the derivation in de- tail. It further illustrates how the variational Bayes method provides a framework for a new computational engine, incorporating the 2-D Cluster Variation Method (CVM), which provides a necessary free en- ergy equation that can be minimized across both the external and representational systems, respectively. free-energy-model representation to-understand explanation cognition 6 days ago How to Read Karl Friston (in the Original Greek) – Alianna J. Maren What Friston offered, though, was three crucial points: The brain minimizes free energy (although not specifying exactly what the free energy function is), A variational Bayes process can help us model the free energy within the brain, and We can separate out the so-called “latent” or “hidden” units that are in the actual external (brain) system from those in the model. It’s this last point that is not terribly obvious in reading Friston’s papers for the first, second, or even tenth time. However, it is an essential and core insight. It also is one of those (small, detailed) things that makes Friston’s work impenetrable to all but the most determined efforts. cognition representation explanation to-understand philosophy-of-science 6 days ago [1710.00992] DimReader: Axis lines that explain non-linear projections Non-linear dimensionality reduction (NDR) methods such as LLE and t-SNE are popular with visualization researchers and experienced data analysts, but present serious problems of interpretation. In this paper, we present DimReader, a technique that recovers readable axes from such techniques. DimReader is based on analyzing infinitesimal perturbations of the dataset with respect to variables of interest. The perturbations define exactly how we want to change each point in the original dataset and we measure the effect that these changes have on the projection. The recovered axes are in direct analogy with the axis lines (grid lines) of traditional scatterplots. We also present methods for discovering perturbations on the input data that change the projection the most. The calculation of the perturbations is efficient and easily integrated into programs written in modern programming languages. We present results of DimReader on a variety of NDR methods and datasets both synthetic and real-life, and show how it can be used to compare different NDR methods. Finally, we discuss limitations of our proposal and situations where further research is needed. user-interface visualization dimension-reduction rather-interesting data-analysis explanation the-mangle-in-practice to-write-about to-do 6 days ago wo's weblog: The lure of free energy There's an exciting new theory in cognitive science. The theory began as an account of message-passing in the visual cortex, but it quickly expanded into a unified explanation of perception, action, attention, learning, homeostasis, and the very possibility of life. In its most general and ambitious form, the theory was mainly developed by Karl Friston -- see e.g. Friston 2006, Friston and Stephan 2007, Friston 2009, Friston 2010, or the Wikipedia page on the free-energy principle. Unfortunately, Friston isn't very good at explaining what exactly the theory says. The unifying principle at its core is called the free-energy principle. It says that "any self-organizing system that is at equilibrium with its environment must minimize its free energy" (Friston 2010). Both perception and action are then characterized as serving this goal of minimizing free energy. free-energy-theory to-understand philosophy-of-science big-definitions bad-explainers theoretical-biology 6 days ago [1811.01721] Rethinking floating point for deep learning Reducing hardware overhead of neural networks for faster or lower power inference and training is an active area of research. Uniform quantization using integer multiply-add has been thoroughly investigated, which requires learning many quantization parameters, fine-tuning training or other prerequisites. Little effort is made to improve floating point relative to this baseline; it remains energy inefficient, and word size reduction yields drastic loss in needed dynamic range. We improve floating point to be more energy efficient than equivalent bit width integer hardware on a 28 nm ASIC process while retaining accuracy in 8 bits with a novel hybrid log multiply/linear add, Kulisch accumulation and tapered encodings from Gustafson's posit format. With no network retraining, and drop-in replacement of all math and float32 parameters via round-to-nearest-even only, this open-sourced 8-bit log float is within 0.9% top-1 and 0.2% top-5 accuracy of the original float32 ResNet-50 CNN model on ImageNet. Unlike int8 quantization, it is still a general purpose floating point arithmetic, interpretable out-of-the-box. Our 8/38-bit log float multiply-add is synthesized and power profiled at 28 nm at 0.96x the power and 1.12x the area of 8/32-bit integer multiply-add. In 16 bits, our log float multiply-add is 0.59x the power and 0.68x the area of IEEE 754 float16 fused multiply-add, maintaining the same signficand precision and dynamic range, proving useful for training ASICs as well. numerical-methods machine-learning representation the-mangle-in-practice to-write-about to-cite motivation 6 days ago The invention of the steam engine in the late eighteenth century made it possible to replace the muscle-power of men and animals by the motive power of machines. The invention of the stored-program digital computer during the second world war made it possible to replace the lower-level mental processes of man, such as arithmetic computation and information storage, by electronic data-processing in machines. We are now coming to the stage where it is reasonable to contemplate replacing some of the higher mental processes of man, such as the ability to recognize patterns and to learn, with similar capabilities in machines. However, we lack the “steam engine” or “digital computer” which will provide the necessary technology for learning and pattern recognition by machines. representation numerical-methods floating-point computer-science rather-interesting to-write-about nudge-targets consider:ReQ 6 days ago Beating Floating Point at its Own Game A new data type called a posit is designed as a direct drop-in replacement for IEEE Standard 754 floating-point numbers floats. Unlike earlier forms of universal number unum arithmetic, posits do not require interval arithmetic or variable size operands; like floats, they round if an answer is inexact. However, they provide compelling advantages over floats, including larger dynamic range, higher accuracy, better closure, bitwise identical results across systems, simpler hardware, and simpler exception handling. Posits never overflow to infinity or underflow to zero, and "Nota-Number" NaN indicates an action instead of a bit pattern. A posit processing unit takes less circuitry than an IEEE float FPU. With lower power use and smaller silicon footprint, the posit operations per second POPS supported by a chip can be significantly higher than the FLOPS using similar hardware resources. GPU accelerators and Deep Learning processors, in particular, can do more per watt and per dollar with posits, yet deliver superior answer quality. A comprehensive series of benchmarks compares floats and posits for decimals of accuracy produced for a set precision. Low precision posits provide a better solution than "approximate computing" methods that try to tolerate decreased answer quality. High precision posits provide more correct decimals than floats of the same size; in some cases, a 32-bit posit may safely replace a 64-bit float. In other words, posits beat floats at their own game. representation floating-point numerical-methods rather-interesting the-mangle-in-practice machine-learning to-write-about nudge-targets 6 days ago Making floating point math highly efficient for AI hardware - Facebook Code We have made radical changes to floating point to make it as much as 16 percent more efficient than int8/32 math. Our approach is still highly accurate for convolutional neural networks, and it offers several additional benefits: Our technique can improve the speed of AI research and development. When applied to higher-precision floating point used in AI model training, it is as much as 69 percent more efficient. Today, models are typically trained using floating point, but then they must be converted to a more efficient quantized format that can be deployed to production. With our approach, nothing needs to be retrained or relearned to deploy a model. AI developers can thus deploy efficient new models more easily. Integer quantization schemes today are growing ever more complicated and in some cases might be “overfitting” on a particular task (and thereby not retaining their general-purpose application). An efficient, general-purpose floating point arithmetic that preserves accuracy can avoid this issue. representation machine-learning the-mangle-in-practice algorithms rather-interesting to-write-about nudge-targets 6 days ago PsyArXiv Preprints \| Multiple Perspectives on Inference for Two Simple Statistical Scenarios When data analysts operate within different statistical frameworks (e.g., frequentist versus Bayesian, emphasis on estimation versus emphasis on testing), how does this impact the qualitative conclusions that are drawn for real data? To study this question empirically we selected from the literature two simple scenarios --involving a comparison of two proportions and a Pearson correlation-- and asked four teams of statisticians to provide a concise analysis and a qualitative interpretation of the outcome. The results showed considerable overall agreement; nevertheless, this agreement did not appear to diminish the intensity of the subsequent debate over which statistical framework is more appropriate to address the questions at hand. statistics looking-to-see the-mangle-in-practice science-studies anthropology-of-data rather-interesting to-write-about 6 days ago [1808.09357] Rational Recurrences Despite the tremendous empirical success of neural models in natural language processing, many of them lack the strong intuitions that accompany classical machine learning approaches. Recently, connections have been shown between convolutional neural networks (CNNs) and weighted finite state automata (WFSAs), leading to new interpretations and insights. In this work, we show that some recurrent neural networks also share this connection to WFSAs. We characterize this connection formally, defining rational recurrences to be recurrent hidden state update functions that can be written as the Forward calculation of a finite set of WFSAs. We show that several recent neural models use rational recurrences. Our analysis provides a fresh view of these models and facilitates devising new neural architectures that draw inspiration from WFSAs. We present one such model, which performs better than two recent baselines on language modeling and text classification. Our results demonstrate that transferring intuitions from classical models like WFSAs can be an effective approach to designing and understanding neural models. automata representation neural-networks recurrent-networks architecture rather-interesting ReQ to-write-about 6 days ago [1810.06758] Discriminator Rejection Sampling We propose a rejection sampling scheme using the discriminator of a GAN to approximately correct errors in the GAN generator distribution. We show that under quite strict assumptions, this will allow us to recover the data distribution exactly. We then examine where those strict assumptions break down and design a practical algorithm - called Discriminator Rejection Sampling (DRS) - that can be used on real data-sets. Finally, we demonstrate the efficacy of DRS on a mixture of Gaussians and on the SAGAN model, state-of-the-art in the image generation task at the time of developing this work. On ImageNet, we train an improved baseline that increases the Inception Score from 52.52 to 62.36 and reduces the Frechet Inception Distance from 18.65 to 14.79. We then use DRS to further improve on this baseline, improving the Inception Score to 76.08 and the FID to 13.75. neural-networks machine-learning algorithms generative-models schemes rather-interesting to-understand 6 days ago The Oulipo of the 1980s? Why it’s time to reappraise the humble Choose Your Own Adventure book \| Prospect Magazine The allure of nostalgia is powerful, especially in an uncertain, unstable age. Nostalgia is a soothing form of selective amnesia of how things actually were. However forward-thinking and ostensibly unsentimental we might be, there are very few of us who are not moved in some way by these jolts of recognition and the comforting, if illusory, thought that a golden age existed in the past when life was more certain and more stable. With Generation X beginning to reach middle age in slow horrified disbelief, it’s little surprise that 1980s revivalisms are big business, from Stranger Things and Ready Player One to the recent Star Wars resurrection. A joyless cynic might see this trend as an example of a culture paralysed by conservatism, cowardice and infantilism. Yet it’s hard to deny the involuntary memories evoked upon seeing pixelated graphics or hearing the shriek of a TIE fighter. The best of these revivals (Twin Peaks: The Return, Blade Runner 2049) offer startling new directions amidst the familiar ones, which recontextualize that which came before. These stories are reimagined, rather than repeated to diminishing effect. Others are shallower. oulipo parafiction literary-criticism generative-art user-centric-art rather-interesting nostalgia 7 days ago Accurate High Performance Concrete Prediction with an Alignment-Based Genetic Programming System \| SpringerLink In 2013, our research group published a contribution in which a new version of genetic programming, called Geometric Semantic Genetic Programming (GSGP), was fostered as an appropriate computational intelligence method for predicting the strength of high-performance concrete. That successful work, in which GSGP was shown to outperform the existing systems, allowed us to promote GSGP as the new state-of-the-art technology for high-performance concrete strength prediction. In this paper, we propose, for the first time, a novel genetic programming system called Nested Align Genetic Programming (NAGP). NAGP exploits semantic awareness in a completely different way compared to GSGP. The reported experimental results show that NAGP is able to significantly outperform GSGP for high-performance concrete strength prediction. More specifically, not only NAGP is able to obtain more accurate predictions than GSGP, but NAGP is also able to generate predictive models with a much smaller size, and thus easier to understand and interpret, than the ones generated by GSGP. Thanks to this ability of NAGP, we are able here to show the model evolved by NAGP, which was impossible for GSGP. symbolic-regression genetic-programming numerical-methods models algorithms to-write-about 16 days ago [1512.04349] Clustering time series under the Fr'echet distance e Fréchet distance is a popular distance measure for curves. We study the problem of clustering time series under the Fréchet distance. In particular, we give (1+ε)-approximation algorithms for variations of the following problem with parameters k and ℓ. Given n univariate time series P, each of complexity at most m, we find k time series, not necessarily from P, which we call \emph{cluster centers} and which each have complexity at most ℓ, such that (a) the maximum distance of an element of P to its nearest cluster center or (b) the sum of these distances is minimized. Our algorithms have running time near-linear in the input size for constant ε, k and ℓ. To the best of our knowledge, our algorithms are the first clustering algorithms for the Fréchet distance which achieve an approximation factor of (1+ε) or better. Keywords: time series, longitudinal data, functional data, clustering, Fréchet distance, dynamic time warping, approximation algorithms. computational-geometry algorithms metrics clustering to-understand time-series 17 days ago [1811.10665] Stepping Stones to Inductive Synthesis of Low-Level Looping Programs Inductive program synthesis, from input/output examples, can provide an opportunity to automatically create programs from scratch without presupposing the algorithmic form of the solution. For induction of general programs with loops (as opposed to loop-free programs, or synthesis for domain-specific languages), the state of the art is at the level of introductory programming assignments. Most problems that require algorithmic subtlety, such as fast sorting, have remained out of reach without the benefit of significant problem-specific background knowledge. A key challenge is to identify cues that are available to guide search towards correct looping programs. We present MAKESPEARE, a simple delayed-acceptance hillclimbing method that synthesizes low-level looping programs from input/output examples. During search, delayed acceptance bypasses small gains to identify significantly-improved stepping stone programs that tend to generalize and enable further progress. The method performs well on a set of established benchmarks, and succeeds on the previously unsolved "Collatz Numbers" program synthesis problem. Additional benchmarks include the problem of rapidly sorting integer arrays, in which we observe the emergence of comb sort (a Shell sort variant that is empirically fast). MAKESPEARE has also synthesized a record-setting program on one of the puzzles from the TIS-100 assembly language programming game. 17 days ago Fréchet distance - Wikipedia In mathematics, the Fréchet distance is a measure of similarity between curves that takes into account the location and ordering of the points along the curves. It is named after Maurice Fréchet. measurement metrics data-analysis feature-construction distance computational-geometry to-consider ReQ 17 days ago [1610.02247] Logic as a distributive law We present an algorithm for deriving a spatial-behavioral type system from a formal presentation of a computational calculus. Given a 2-monad Calc: Catv→ Cat for the free calculus on a category of terms and rewrites and a 2-monad BoolAlg for the free Boolean algebra on a category, we get a 2-monad Form = BoolAlg + Calc for the free category of formulae and proofs. We also get the 2-monad BoolAlg ∘ Calc for subsets of terms. The interpretation of formulae is a natural transformation $\interp{-}$: Form ⇒ BoolAlg ∘ Calc defined by the units and multiplications of the monads and a distributive law transformation δ: Calc ∘ BoolAlg ⇒ BoolAlg ∘ Calc. This interpretation is consistent both with the Curry-Howard isomorphism and with realizability. We give an implementation of the "possibly" modal operator parametrized by a two-hole term context and show that, surprisingly, the arrow type constructor in the λ-calculus is a specific case. We also exhibit nontrivial formulae encoding confinement and liveness properties for a reflective higher-order variant of the π-calculus. representation higher-order mathematics category-theory to-understand π-calculus ReQ 18 days ago Q-BAL Programming Language Q-BAL is a programming language that Ben Yackley and Michael Shulman invented on a whim, based on the question "What would it be like if a language were based on queues rather than stacks?" The acronym stands for Queue-BAsed Lanugage. This language is not designed to be useful, just fun. If you feel like writing any programs in Q-BAL, I would be very glad to post them here. I hope to someday write an interpreter for Q-BAL programs, so stay tuned! The Q-BAL system has been undergoing revamping lately in discussions with Andy and Ben, and when I have time I'll update these pages to reflect whatever conclusions we will hopefully have reached by then. programming-language esoteric-languages ReQ reference 19 days ago [1504.04311] Higher category models of the pi-calculus We present an approach to modeling computational calculi using higher category theory. Specifically we present a fully abstract semantics for the pi-calculus. The interpretation is consistent with Curry-Howard, interpreting terms as typed morphisms, while simultaneously providing an explicit interpretation of the rewrite rules of standard operational presentations as 2-morphisms. One of the key contributions, inspired by catalysis in chemical reactions, is a method of restricting the application of 2-morphisms interpreting rewrites to specific contexts. concurrency π-calculus rewriting-systems to-understand ReQ 23 days ago Tanya Khovanova's Math Blog » Blog Archive » Shapes of Symbols in Latin Squares John likes finding interesting ways to remember which shape is which. You can find his and Alex’s suggestions in the paper which Alex submitted to the arxiv. Oops! While I was writing this essay, arxiv rejected the paper. latin-squares combinatorics constraint-satisfaction feature-construction rather-interesting dammit 24 days ago [quant-ph/0208149] A semi-quantum version of the game of Life A version of John Conway's game of Life is presented where the normal binary values of the cells are replaced by oscillators which can represent a superposition of states. The original game of Life is reproduced in the classical limit, but in general additional properties not seen in the original game are present that display some of the effects of a quantum mechanical Life. In particular, interference effects are seen. quantums Game-of-Life hey-I-know-this-guy cellular-automata 24 days ago Flowers for Julia \| Fronkonstin To color the points, I pick a random palette from the top list of COLOURLovers site using the colourlovers package. Since each flower involves a huge amount of calculations, I use Reduce to make this process efficiently. More examples: fractals visualization color details-of-note 24 days ago Tanya Khovanova's Math Blog » Blog Archive » Another Cool Coin-Weighing Problem My coauthor, Konstantin Knop, sent me a coin-weighing problem that is really good. Surprisingly, it is old: it first appeared in a Russian math journal, Kvant, in 1973. mathematical-recreations constraint-satisfaction rather-interesting nudge-targets to-write-about to-generalize 24 days ago [1207.4497] Efficient Algorithms for Zeckendorf Arithmetic We study the problem of addition and subtraction using the Zeckendorf representation of integers. We show that both operations can be performed in linear time; in fact they can be performed by combinational logic networks with linear size and logarithmic depth. The implications of these results for multiplication, division and square-root extraction are also discussed. arithmetic representation rather-interesting mathematical-recreations nudge-targets consider:rediscovery 26 days ago 3 tools from sociocracy to use right away (plus magic phrases!) Of course I myself am ego-driven and I have a ton of good ideas! But I also know that it only takes one person in the circle engaging in cross-talk and the good effects of rounds are lost. What do I do with all my brilliant ideas? I write them on a piece of paper. When it is my turn, I will often look at my piece of paper and realize that, after a few minutes of listening to others, about 90% of my ideas have either been named or, on second thought, they don’t seem all that great or urgent anymore. Humbled, I am often grateful for having been forced to weed through what I say. And when people pass on their turn saying “All I wanted to say has been said” I feel the urge to get up and hug them in gratitude for not putting the group through endless repetitions. Which also answers the last reservation I hear very often: aren’t rounds lenghty? Maybe. But both inconsiderate decisions, repetitive statements and emotional “clean-up” after disregard of team members takes a lot of time too. Your choice! social-dynamics social-norms collaboration organizational-behavior teams rather-interesting to-write-about 27 days ago [PDF] The Spread of Improvement: Why Innovation Accelerated in Britain 1547-1851 In the three centuries after the reign of Henry VIII, the British Isles emerged from civil wars, invasion threats, and religious strife to become the world's technological leader. Why did innovation accelerate? I studied the people responsible, the innovators themselves, using a sample of 1,452 people in Britain who innovated between 1547 and 1851. The paper charts the emergence and spread of an improving mentality, tracing its transmission from person to person and across the country. The mentality was not a technique, skill, or special understanding, but a frame of mind: innovators saw room for improvement where others saw none. The mentality could be received by anyone, and it could be applied to any field – anything, after all, could be better. But what led to innovation’s acceleration was not just that the mentality spread: over the course of the eighteenth century innovators became increasingly committed to spreading the mentality further – they became innovation’s evangelists. By creating new institutions and adopting social norms conducive to openness and active sharing, innovators ensured the continued dissemination of innovation, giving rise to modern economic growth in Britain and abroad. epidemiology-of-ideas symmathesy rather-interesting topical history-of-ideas collaboration the-mangle-in-practice sociotechnical-us 27 days ago [1810.07074] Why We Do Not Evolve Software? Analysis of Evolutionary Algorithms In this paper, we review the state-of-the-art results in evolutionary computation and observe that we do not evolve non trivial software from scratch and with no human intervention. A number of possible explanations are considered, but we conclude that computational complexity of the problem prevents it from being solved as currently attempted. A detailed analysis of necessary and available computational resources is provided to support our findings. via:lspector yeah-no nimby system-of-professions mistaking-the-publications-for-the-work academic-culture 27 days ago Automating String Processing in Spreadsheets using Input-Output Examples - Microsoft Research We describe the design of a string programming/expression language that supports restricted forms of regular expressions, conditionals and loops. The language is expressive enough to represent a wide variety of string manipulation tasks that end-users struggle with. We describe an algorithm based on several novel concepts for synthesizing a desired program in this language from input-output examples. The synthesis algorithm is very efficient taking fraction of a second for various benchmark examples. The synthesis algorithm is interactive and has several desirable features: it can rank multiple solutions and has fast convergence, it can detect noise in the user input, and it supports an active interaction model wherein the user is prompted to provide outputs on inputs that may have multiple computational interpretations. The algorithm has been implemented as an interactive add-in for Microsoft Excel spreadsheet system. The prototype tool has met the golden test – it has synthesized part of itself, and has been used to solve problems beyond authors’ imagination. learning-from-data microsoft software-synthesis rather-interesting pattern-discovery to-write-about 27 days ago [PDF] MEXICA: a computer model of a cognitive account of creative writing. MEXICA is a computer model that produces frameworks for short stories based on the engagement-reflection cognitive account of writing. During engagement MEXICA generates material guided by content and rhetorical constraints, avoiding the use of explicit goals or story- structure information. During reflection the system breaks impasses, evaluates the novelty and interestingness of the story in progress and verifies that coherence requirements are satisfied. In this way, MEXICA complements and extends those models of computerised story-telling based on traditional problem-solving techniques where explicit goals drive the generation of stories. This paper describes the engagement-reflection account of writing, the general characteristics of MEXICA and reports an evaluation of the program generative-models cognition simulation looking-to-see cultural-engineering rather-interesting representation to-write-about 4 weeks ago Domain hacks with unusual Unicode characters – Terence Eden's Blog Unicode contains a range of symbols which don't get much use. For example, there are separate symbols for TradeMark - ™, Service Mark - ℠, and Prescriptions - ℞. Nestling among the "Letterlike Symbols" are two curious entries. Both of these are single characters: Telephone symbol - ℡ Numero Sign - № What's interesting is both .tel and .no are Top-Level-Domains (TLD) on the Domain Name System (DNS). typography domains DNS rather-odd 4 weeks ago [1705.07386] DeepMasterPrints: Generating MasterPrints for Dictionary Attacks via Latent Variable Evolution Recent research has demonstrated the vulnerability of fingerprint recognition systems to dictionary attacks based on MasterPrints. MasterPrints are real or synthetic fingerprints that can fortuitously match with a large number of fingerprints thereby undermining the security afforded by fingerprint systems. Previous work by Roy et al. generated synthetic MasterPrints at the feature-level. In this work we generate complete image-level MasterPrints known as DeepMasterPrints, whose attack accuracy is found to be much superior than that of previous methods. The proposed method, referred to as Latent Variable Evolution, is based on training a Generative Adversarial Network on a set of real fingerprint images. Stochastic search in the form of the Covariance Matrix Adaptation Evolution Strategy is then used to search for latent input variables to the generator network that can maximize the number of impostor matches as assessed by a fingerprint recognizer. Experiments convey the efficacy of the proposed method in generating DeepMasterPrints. The underlying method is likely to have broad applications in fingerprint security as well as fingerprint synthesis. evolutionary-algorithms neural-networks generative-models rather-interesting biometrics whoopsie-daisy also:duh 4 weeks ago Pseudoarchaeology and the Racism Behind Ancient Aliens If we look to von Däniken’s work, there can be little doubt that his racial beliefs influenced his extraterrestrial theories. After a short stint in jail for fraud and either writing or appropriating the material for a number of other books that developed his ancient astronauts theory, von Däniken published Signs of the Gods? in 1979. It is here that many of his racial views are most boldly stated. British archaeology officer Keith Fitzpatrick-Matthews points out on his Bad Archaeology blog just a few of the many racist questions and statements posed by the author: “Was the black race a failure and did the extraterrestrials change the genetic code by gene surgery and then programme a white or a yellow race?” He also printed beliefs about the innate talents of certain races: “Nearly all negroes are musical; they have rhythm in their blood.” Von Däniken also consistently uses the term “negroid race” in comparison with “Caucasians.” racism psychoceramics imperialism archaeology have-read spacemen-and-cavemen 4 weeks ago Quinn Slobodian – Globalists — Crooked Timber Slobodian thinks that this is mistaken. In his account, markets have not become disembedded from national societies and states so much as they have become re-embedded in international institutions. Neo-liberalism as manifested in the thought of Hayek and his European followers is the political project of looking to recreate state structures outside the grasp of democratic and non-democratic states. Far from thinking that markets are natural, neo-liberals accept that they are “products of the political construction of institutions to encase them.” (p.7) Instead of a double movement, we have a ‘double world’ of imperium, political rule exercised through nation states, and dominium, the world of economics and business, and a deliberate political effort to insulate the latter inside its own steel-hard casing against the depredations of the former. Neo-liberals then, look to an `interdependent’ world and a single global economy as a realm that should be held inviolate from national states, and the demands their people put upon them. This, as they came to realize over time, requires them to build their own quasi-constitutional structures at the international level, in order to fend off the persistent efforts of national states to shape and control competitive forces and economic flows that are better left alone. Under this account, the most crucial dynamics of neo-liberalism did not involve the glamorous public clash of ideas between intellectuals. Instead, they were duller, more relentless and in the end, more effective – the persistent efforts of neo-liberals to argue through new kinds of international institution and to push back against organized efforts to make global markets more accountable to national authorities. Mont Pelerin was important – but so too were the International Chamber of Commerce and a multitude of boring seeming meetings and negotiations. neoliberalism books political-economy define-your-terms to-read if-I-have-the-guts fascism 4 weeks ago Making Sense of Bivector Addition, viXra.org e-Print archive, viXra:1807.0234 As a demonstration of the coherence of Geometric Algebra's (GA's) geometric and algebraic concepts of bivectors, we add three geometric bivectors according to the procedure described by Hestenes and Macdonald, then use bivector identities to determine, from the result, two bivectors whose outer product is equal to the initial sum. In this way, we show that the procedure that GA's inventors dened for adding geometric bivectors is precisely that which is needed to give results that coincide with those obtained by calculating outer products of vectors that are expressed in terms of a 3D basis. We explain that that accomplishment is no coincidence: it is a consequence of the attributes that GA's designers assigned (or didn't) to bivectors. linear-algebra algebra define-your-terms rather-interesting to-write-about nudge-targets consider:representation Grassmannian wedge-product 4 weeks ago The Arbelos in Wasan Geometry, Problems of Izumiya and Nait=o, viXra.org e-Print archive, viXra:1811.0132 We generalize two sangaku problems involving an arbelos proposed by Izumiya and Nait\=o, and show the existence of six non-Archimedean congruent circles. 4 weeks ago Robinson Tiles - Futility Closet Berkeley mathematician Raphael Robinson discovered this remarkable set of aperiodic tiles in 1978. The six shapes will neatly tile a plane, as shown below, and though the pattern cannot be regular, it reliably produces a hierarchical design: Each small orange square sits at the corner of a larger orange square, which sits at the corner of a still larger one, and so on ad infinitum. This is because subgroups of tiles form “supertiles” with similar properties — see here. tiling mathematical-recreations aperiodic-patterns constraint-satisfaction 4 weeks ago Problem Solving with Trig \| Continuous Everywhere but Differentiable Nowhere I’m going to try to outline the messiness that was my thought process in this triangle problem, to show/archive the messiness that is problem solving. ... The point of this post isn’t to teach someone the solution to the problem. I could have written something much easier. (See we can draw this auxiliary line to create similar triangles. We use proportions since we have similar triangles. Then exploit the new isosceles triangle by setting the leg lengths equal to each other.) But that’s whitewashing all that went into the problem. It’s like a math paper or a science paper. It is a distillation of so freaking much. It was to capture what it’s like to not know something, and how my brain worked in trying to get to figure something out. To show what’s behind a solution. 4 weeks ago NeuralFunk - Combining Deep Learning with Sound Design NeuralFunk - Combining Deep Learning with Sound Design Making a Track Entirely out of Samples Generated by Neural Networks rather-interesting neural-networks generative-art learning-in-public the-mangle-in-practice to-write-about performance-measure 4 weeks ago PsyArXiv Preprints \| The rich are different: Unraveling the perceived and self-reported personality profiles of high net-worth individuals Beyond money and possessions, how are the rich different from the general population? Drawing on a unique sample of high net-worth individuals from Germany (≥1 million Euro in financial assets; N = 130), nationally representative data (N = 22,981), and an additional online panel (N = 690), we provide the first direct investigation of the stereotypically-perceived and self-reported personality profiles of high net-worth individuals. Investigating the broad personality traits of the Big Five and the more specific traits of narcissism and locus of control, we find that stereotypes about wealthy people’s personality are accurate albeit somewhat exaggerated and that wealthy people can be characterized as stable, flexible, and agentic individuals who are focused more on themselves than on others. psychology wealth capitalism social-norms cultural-norms social-psychology stereotypes yup 4 weeks ago Copy this bookmark: description: tags:	2018-12-16 06:53:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.33460214734077454, "perplexity": 2907.1356100922567}, "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-2018-51/segments/1544376827281.64/warc/CC-MAIN-20181216051636-20181216073636-00497.warc.gz"}
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Calculation Example: Lateral Sliding File System, Letter Tiers, 8 high, in a 10’x17’ room. following equations. Example for M12 bolt , Spanner Size = ( Bolt Size 1. Bore Diameter (mm) : 50. Yukon Gear & Axle. 6 c) and Mach. Sleeve and clip bearings support high loads, have no moving parts, and require lubricant to allow the shaft to turn smoothly. Life in hours: Specifies the life of the bearing in hours. 25? I installed a set on 1 rod and the crank wouldnt move. Inside Diameter: 25mm; Outside Diameter: 52mm; Width: 15mm - Bearing Type: 6205. Why to calculate the Safe bearing capacity of soil before starting construction:-From the above figure, it is clear that the building is fallen in only one side. a) Find load transferred by bearing on concrete in column: ACI 10. Also calculate ultimate bearing capacity if same footing is placed at a depth of 1 m below ground surface. Our customers count on our premium quality bearings to fit every need and solve every problem. Suggested loads are :. Utilized for calculationg an interior footing size when wood studs are used. Here you can calculate: bicycle gear ratios; speed at cadence in a range of gears; cadence needed for a given speed in a range of gears; and meters of development. Bit Brokers is Quality Rock Bits. Ball bearing size charts are widely available, and can be used to find the measurements of a specific bearing. 5 Bottom-Entering Mixers. BALL & ROLLER BEARINGS POPULAR SIZE CAT. CSB200 Series Set Screw Insert Bearing - Cylindrical HC200 Series Locking Collar Insert Ball Bearing - Normal Duty SA200 Series Locking Collar Ball Bearing - Light Duty. We have been using remote working practices and working from home for a few years, so we're happy to say that we are still fully up and running with our steel beam calculation service, including our 2 hour option. MOTORS® products. 002 per shaft diameter inch 0. Link-Belt offers a split housed spherical roller bearing drop-in compatible with most standard systems. 9" (50mm) diameter conveyor rollers are normally rated at 250 lbs (110 kg) maximum load. ? we are removing a wall to open up kitchen and dining area in our 1950's cape cod style house. Req'd M n = (Req'd R n /B. 5 Based on 1 x 10 6 revolutions L 10 life, for the ISO life calculation. Spindle/BB Code and Size Examples: D3H, 68 ss 120, UN72 68 x 113, UN52 70 x 107, 344See the section for your brand of crank for details of the measurement system used. The size of the angle is 120 degrees. HY S619xx Preload Change Calculations. Advancing Designs. Next you get out your pocket calculator or smartphone app, or old-school paper and pencil. 05 mm OD x 0. Preload Change Calculations for Angular Contact Bearings. Please, use our tool SKF Bearing Select to calculate bearing frequencies of SKF. the bearing surface to the fastener underhead surface. We also have a forklift manlift basket 48" x 48" x 50" high, constructed of 1. WL1+ calculates life expectancy according to ISO 281. This means significant reductions in fitting time over solid bearings, with dramatic reductions in downtime and immense savings in plant maintenance. You can, however, calculate the load capacity of concrete in a quick and general way. 3,671 tapered roller bearing size chart products are offered for sale by suppliers on Alibaba. Plate bearing test procedure and calculation Plate bearing test is an activity carried out by design engineers in the field to determine the bearing capacity with regards to the soil underneath. In paper machine drying cylinders, hot steam is passed through a hollow shaft directly through the bearing bore. - Bearings do not typically fail in sub-surface initiated fatigue conditions. To determine the deck height required for a 2007cc. Main applications: Automobiles, tractors, machine tools, motors, pumps, agricultural machinery, textile machinery, etc. Tip: See the Engineer's Handbook for used calculation formulas and other instructions for designing plain bearings. Wood Angled Bearing: Calculate the compressive strength or bearing strength for a wood member at an angle to the grain. Bearing Wall Use this calculator to determine the board feet in a bearing wall. The entire house must be measured and load calculations completed before this can be done. Bearing life in excess of 5 years average. the holes, spherical bearings in plane with the glass had to be developed. Bearing is a material which is used to rotate the equipment in liner movement. I had it checked by an architect who OK'ed it. Calculate distance from the center of gravity of the triangle to line p. i want to determine the size of busbar i can use in the panel. more specifically and without limitation, under no circumstances shall fastenal be liable for any direct, indirect, special, incidental, or consequential damages, including, without limitation, loss of data or profit, property and equipment damage or injury, arising out of the use, or inability to use, the torque calculator, even if fastenal. Our customers count on our premium quality bearings to fit every need and solve every problem. Therefore, the ratio of the bearing outside diameter to the inscribed circle diameter is small, and they have a rather high radial load capacity. Examples of the angle of a slope include such things as the angle of the driveway, the pitch of a roof, the angle of a hill and so on. Bearing Capacity Calculation for Sandy Soils. Load bearing masonry construction was the most widely used form of construction for large buildings from the 1700s to the mid-1900s. depth of 4 feet is 950 lb/ft 2 and 1,200 lb/ft 2 at 5 feet. This is a simplified version of our Bearing Selector that you can use to search for and find either inch or metric bearings that meet your specified dimensions in inches (in) or millimeters (mm). in^2: End area required across the pin hole: in: Maximum. Therefore the bearing area is: Bearing Area = Thrust Force / Soil Bearing Capacity Bearing Force = 21484 N / 23939. 23 Section 6 Appendix 12/21/01 RCR15 Series Rotary Cutters 312-556M Land Pride Section 6 Appendix Torque Values Chart for Common Bolt Sizes in-tpi1 N · m2 ft-lb3 N · m ft-lb N · m ft-lb mm x pitch4 N · m ft-lb N · m ft-lb N · m ft-lb. (Courtesy of Chemineer. Copyright © NTN Corporation. Bearing Wall Use this calculator to determine the board feet in a bearing wall. To Start click multiple times on the map. For driven piles in loose to dense sand with φ varying between 30 0 to 40 0 , k i values in the range of 1 to 1. We strive continually to make our online shop faster, simpler & easier to use. Convert from scale to actual (real) size. 3 m 2 and it should be 25 mm thick. Next you get out your pocket calculator or smartphone app, or old-school paper and pencil. Advanced power and sample size calculator online: calculate sample size for a single group, or for differences between two groups (more than two groups supported for binomial data). Beam End Reaction kN (factored) Characteristic strength of masonry N/mm 2 Width of beam end bearing mm Length of beam end bearing mm γm = 3. Vehicles with traditional, tapered wheel bearings should have them serviced every 25,000 to 30,000 miles — replacement may not be necessary as long as they're maintained. Featuring the broadest possible combination of over 50,000 unique mounted ball bearings to meet your application demands. Hello friends is video me centrifugal pump ki bearing housing or shaft se kaise bearing ka number calculate karte dikhaya gya hai. Top Brands. Aftermarket bearings can say std for standard size. 25 x 18 SQ Tubing I need to calculate a load capacity for that aswell. Suggested loads are :. When changing tire sizes, we recommend staying within 3% of the diameter/height of the original tire. Limits for Shaft and Hole are applied algebraically to the Nominal (basic) size to obtain the limits of the size for the parts. Many different bearing types can be incorporated in a flange mount housing. This bearing selector reduces the uncertainty when selecting bushes and you can rest assured that whether you are in the design process or replacing a bearing in an existing application that you have selected the right one. Start by consulting Table 1 below, which shows how much weight each size float will support at four different depths of submergence. Don't worry, if you don't know the ground pressure - the calculator will ask you for the surface materials and you'll be able to select the most appropriate type. 62 MB and is available for download from our website. Therefore, Cloth weight = Weight of warp + Weight of weft + Weight of size (All in-lbs. #4: RATED FREQUENCY. Each bracket has two screws which fasten it to the shelf and two screws which fasten it to studs in the wall. HomeAdvisor's Steel I-Beam Cost Guide provides prices to install or replace a structural beam. Avoid! RV part houses sell those, you can tell, if it’s hanging on a rack do not buy. Here is a formula that allows you to calculate the circulating load ratio around a ball mill and hydrocylone as part of a grinding circuit. The following tables are a guide for establishing shaft and bearing fits for miniature and instrument bearings, when the expansion coefficients of the shaft and housing are similar or when the operating temperature differential between them is nominal. Five pieces of information are needed to calculate the weight necessary to free-fall the block or overhaul ball: 1. Mechanic tools & shop equipment. The SKF formula method is frequently used by multiplying the bearing’s outside diameter (in inches) with the total bearing’s width (in inches) or height (for thrust bearings). Bearing clearance is one of the most important parameters in the operation of a bearing. If the pressure pushing down on the ground from a MEWP spreader plate is greater than the ground load bearing capacity of the ground on which it stands, then the MEWP will become unstable and at risk of overturn. bearing factor value. For example your ball mill is in closed circuit with a set of cyclones. As you can see, heavy houses on weak soil need footings 2 feet wide or more. Therefore, it is important to determine the installed clearance along with the bore contour and concentricity to the outside fit diameter. Download our FREE eBook: The Art of Precision Bearing - Handling Mounting, and Technical Guide to Precision Bearings. There are many types of bearing in the market some of the most used bearings are ball bearing, roller bearing, cylindrical roller bearing, tapered roller bearings, thrust bearing etc…. These effects are illustrated in Figure 3. Figure 2: Babbitt bearing embedded with machining debris. It is the inner diameter of the bearing and measured in millimeters. The steel plate must not bend due to this tensile force = 130 kN. for geocaching. If your soil has a larger bearing capacity, you will find that when you have the Loads Calculated, the footing size will be smaller. 'for Smartphone/Tablet' is available for the iPhone/iPad/Android. BALL & ROLLER BEARINGS POPULAR SIZE CAT. Rating life calculation and selection of bearings to be used in gear units are based on. IF the colu mn concrete strength is lower than the footing, calculate P n for the column too. Find a price list for 20-, 30- and 40-foot I-beams. Tire size converter can help you to convert the inch tire size (Standard, English measuring system) to the metric tire size and the. 4 millimeter increment, starting with size 3 equaling 14 mm, size 3. 1 Minimum depth. 6 & IBC 2308. The full version allows you to design beams of any size. This general purpose foundation calculator can also calculate concrete piles and pile cap foundations. Now, from the given CBR value of subgrade soil read the total thickness (T) with respect to selected curve. Bearing Area = P / Fp: Base plate is considered flexible with bearing concentrated close to column. can anyone help? so far i have MOI (I)=3. Bearing number : 6802. The ‘i-button’ before each calculation gives more information about the theory behind the calculation. You can use this online scale conversion calculator to convert the size of an actual object to a scaled size and vice versa. • Calculating Volume Using Solids Of Revolution. Calculation of plate load test. A load-bearing structure holds the weight of the building above it. For the Spreader Pad Calculator to work effectively you'll need maximum load per outrigger and the maximum allowable ground pressure in the same unit of measurement. $\text{Count} = \text{Size} = n = \text{count}(x_i)_{i=1}^{n}$ How to Calculate the Mean. Size (mm) : 50x90x20. Calculate the end bearing length for a 10 inch by 16 inch timber beam if the beam reacion is 15000lbs and the compressive stress is 300psi? You need at least 50 square inches of bearing (at least. 23 Section 6 Appendix 12/21/01 RCR15 Series Rotary Cutters 312-556M Land Pride Section 6 Appendix Torque Values Chart for Common Bolt Sizes in-tpi1 N · m2 ft-lb3 N · m ft-lb N · m ft-lb mm x pitch4 N · m ft-lb N · m ft-lb N · m ft-lb. Catenary mooring system A catenary mooring system is the most common mooring system in shallow waters. 8 km from A on a bearing of 2100 Calculate the bearing of B from C. See this page to help identify your soil bearing capacity. 4 mm, size 4 equal to 14. - These design bearing strengths are in kips/in. Outer Diameter (mm) : 90. A pit of size 5 Bp X 5 Bp excavates to the depth equal to the depth of the foundation to conduct a plate load test at the site. The manner in which skin friction. In the reverse case it is equal to net safe settlement pressure. from precision medical instruments to high-performance robotic solutions, FUTEK products are designed to work in the most demanding environments — even on Mars. ALLOWABLE BEARING CAPACITY: The maximum net average pressure of loading that the soil will safely carry with a factor of safety considering risk of shear failure and the settlement of foundation. Ultimate load carrying capacity of load tables for unistrut p1000 p1001 slim floor beam system using fem ysis reinforced concrete beams civis make your house perfectHow To Calculate The Load Bearing Capacity. To determine the deck height required for a 2007cc. These bearings consist of cylindrical rolling elements and usually support a rotating shaft in the inner ring which may be misaligned with the outer ring. Factory rod bearings are not usually true standard,the factory fits the bearings to the crank,so you will usually see some longer decimal number to denote the size of the bearing like 0. a) Find load transferred by bearing on concrete in column: ACI 10. Use it to ensure that visitors to your website are not waiting too long to download your pages. Bore Diameter (mm) : 30. the holes, spherical bearings in plane with the glass had to be developed. FAG bearing calculator. 05 mm OD x 0. Load calculation formula applied to post-frame (pole barn) construction Post frame construction footing size calculations are easily determined because of its simple load bearing structure. Reducing the oil clearance between the rod and main bearings and the crankshaft has a number of advantages. Needle roller bearings are essentially cylindrical roller bearings except that their cylindrical rollers have a length 3 to 10 times their diameter (versus less a length less than 3 times their diameter). ) pillow-block bearings. Thordon Bearings Sizing Calculation Program allows you to calculate the maximum and minimum operating temperature for the final bearing size to be machined. Wind Loads Above, Figure 1609, Basic Wind Speed (3-second gust), 33 feet above ground, exposure C IBC 2003 Zone V 30 (mph) 1 2 3 70 80 90 (Western Mass. Determine ballast surface stress 11. The micro effects include contact stress defections of plating and coatings as well as surface and thread deformations. Select an initial bearing size and calculate the expected L10 life. Welcome to Bearing Boys. 8 Connecting Rod Bearings are required for Suzuki Hayabusa. Resistance to incremental penetration is calculated using only the bearing stress (σ) of the soil, shear stress (τ) is used to calculate the shear angle for horizontal force (F̌ʰ). 1 Minimum size The size of footings supporting piers and columns shall be based on the tributary load and allowable soil pressure in accordance with Table R401. The igus® plain bearings range includes sleeve bearings, flange bearings, thrust washers, piston rings, clip-on plain bearings and many other designs. I have a 57 210 with a 265 and a stock rear end. For a bearing connection the assumptions are:. Enter the length of the sides for each triangle you use; up to 10 of them. We started life in 2006 and are now one of the leading online bearing shops in the world. 4 gives the diameter in millimeters, 530 mm. Hey, you just need to calculate the deflection when under load and check it's within limits. Please, use our tool SKF Bearing Select to calculate bearing frequencies of SKF catalogue bearings (where no geometry input is required). PS Penetration in mm 80 70 60 50 40 30 20 10 NO CORRECTION. The most common are: 1. Overview of Bearing Technical Calculations There are five technical calculations that can be performed within the NTN Bearing Technical Calculation Tool. Bearing number : 6802. com is a Wheel Guide and catalogue. The factor of safety ranges from 2 to 3. That's the number to start with. The most commonly used material in bearing. The final step is to verify the vehicle can transmit the required torque from the drive wheel(s) to the ground. = Bearing OD in inches (or mm) F res = Total load in Ibs (or N) Y = Number of bearings L = Bearing load carrying width in inches (or mm) Figure 7: Maximum continuous center bore pressure p vs. Shop SKF VKBA 3330 wheel bearings huge online 280x380x75 Size (mm) discount inventory. Because of simplicity and ease of use, this method is still the fundamental soil parameter for foundation design. 4 km due-east-of C is 3. Thordon Bearings Sizing Calculation Program allows you to calculate the maximum and minimum operating temperature for the final bearing size to be machined. Figure 21-5 Side-entering mixer with pillow-block bearings. 53 in w (load per foot) = (40 + 10)psf x 12' = 600 lb per ft l (beam span) = 16 ft Where E is a constant for steel = 30,000,000 psi And I is the moment. The 6 refers to the depth (6") and the 12. Next you get out your pocket calculator or smartphone app, or old-school paper and pencil. Spindle/BB Code and Size Examples: D3H, 68 ss 120, UN72 68 x 113, UN52 70 x 107, 344See the section for your brand of crank for details of the measurement system used. AG-CO is not responsible for inaccurate data entry resulting in product failure due to overloading. Continue this iterative process until an appropriate L na life is obtained. 8 km from A on a bearing of 2100 Calculate the bearing of B from C. Anyway the ridge height would be 4' 10-7/8" plus whatever the HAP would be. We strive continually to make our online shop faster, simpler & easier to use. Do the calculations to find your sample’s porosity. Plain Bearings Calculator (Journals) A plain bearing or Journal is a solid sleeve inside which a shaft is expected to rotate with acceptable precision (location and guidance) and no metallic contact. Deep groove ball bearings 6802. In multi-storey design plans, the size of the plinth beam, primary and secondary beams depend on the number of stories and loads acting on the beam. Starting from a point load your bearing capacity is determined by failure. 4 mm 1" Standard I. Further guidance on the types of bearings and their usage can be found in Guidance Note 3. If deflection is unacceptable, re-do design. Another of the bearing’s duties is to establish and maintain a film of oil. Normally a bearing temperature of 80 to 90 °C (144 to 162 °F) above that of the shaft is sufficient for mounting. The minimum end bearing length at supports to be 100mm. Use our easy ring size calculator, our how-to measure guide and converters. 6 Timber piles (wood piles) 5. The inset photo shows microscopic detail of the debris. Calculators and Converters. On the basis of the calculated values for reaction force and basic load ratings, you can choose the appropriate roller bearing from the WL1+ database. 4 times its size will have an angular size of 1 minute. SPECIFY A BEARING BY DESIRED DIMENSIONS. 1698 Zhenluo Road,Jiaochuan Street, Zhenhai, Ningbo City, Zhejiang, China +86-150 8847 0229 +86-574-86452780 +86-574-86456096 +86-574-86456328. - The design bearing strengths are given for different bolt spacings (2. A lump of clay of mass 2. Typical applications. Here is a formula that allows you to calculate the circulating load ratio around a ball mill and hydrocylone as part of a grinding circuit. Choose from a multitude of free calculators and converters in the areas of finance, health, cooking, math and science for everyday, general use. Table of ContentsBrinell Hardness of Babbitt Bearings at Increasing TemperaturesEffect of Compression on the Brinell Hardness of BabbittsRolling of Babbitted Linings by the Mills Micrometer Roller Brinell Hardness of Babbitt Bearings at Increasing Temperatures Brinell tests at progressively increasing temperatures are given for a representative lead-base and a representative tin-base Babbitt. Deep groove ball bearings 6802 dimensions and specification. 927265 ISO 9001/QS-9000. Mopar 8 3/4" 489, 741, or 742 case bearing rebuild kit. Each bracket has two screws which fasten it to the shelf and two screws which fasten it to studs in the wall. This gear ratio calculator determines the rate of mechanical advantage or disadvantage a gear train produces in a gear system. Quantity of grease (g) = Outer bearing diameter (mm) X bearing width (mm) X 0. We carry many other designs. Looking at this table you will see there is a choice in the size of floor joist (2 X 6, 2 X 8, 2 X 10 or 2 X 12) and there is a choice in the joist spacing (12", 16" or 14"). Custom sizes and colors available upon request. Shown here are the most popular designs. The split to the shaft bearing can be built around the shaft without dismantling adjacent equipment or machinery. New • Based on soil and anchor/pile inputs the program returns theoretical capacities and installation torque. The piling calculator applies a horizontal pressure (that varies linearly with depth) to the internal and external pile wall due to the soil's Poisson ratio. Deep groove ball bearings, single row 6028. Consider this question:. Most calculations are done to full precision. Thordon Bearings Sizing Calculation Program allows you to calculate the maximum and minimum operating temperature for the final bearing size to be machined. New • Based on soil and anchor/pile inputs the program returns theoretical capacities and installation torque. When using these bearings, it is important for the filling slot in the outer ring to be outside of the loaded zone as much as possible. Calculators for the conversion of geo coordinates, as delivered by a GPS tool, for the distance of two point and for bearing. Read on to learn more about gear ratio and its importance in our lives. All steel beam calculations for loft extensions and beam calculations for extensions, including padstone/bearing plates and splice calculation and fabrication detail (These may be required for both floor and roof support) Floor joist calculations, including trimming joists. ) Whereas Weight of warp in lbs. LARGE SIZE BALL & ROLLER BEARINGS We are pleased to offer you this newly issued Koyo large size rolling bearing catalogue. The value of bearing capacity factor N q is obtained from the figure given below. drag on the motor bearings. Tyre size for rim size/width calculator tells you what tyre sizes to go for, for the size of your rim. Built to Perform, Built to Last. microblue bearings: who we are & what we do for you We are a “racing-only” bearing company that’s dedicated to bringing you all the products and the knowledge to help you with their use. Solution: From the first table above, the thrust in for a 15. 114 X (bearing OD) in X (bearing width) in. Bearing size and type are selected on the basis of parameters such as bearing load rating, magnitude and direction of the applied load or loads, rotary speed, stiffness, and precision. The header is usually made out of dimensional lumber installed on its edge. We started life in 2006 and are now one of the leading online bearing shops in the world. A bearing number is composed of a basic number and a supplementary code, denoting bearing specifications including bearing type, boundary dimensions, running accuracy, and internal clearance. Check the supplier's catalogs for values. This tire calculator is for information purposes only and we do not guarantee fitment based on this calculator alone. There is currently a steel beam spanning the main floor and spanning the second floor is just 2x4 construction as the load bearing wall. The bearings allow the hub shell (and the rest of the wheel parts) to rotate freely about the axle. Try For Free. Link-Belt PLB Split Block Spherical Roller Bearing Solutions. of Ends * Tape Length in Yards)/(840 * warp yarn count) Also. You can use this online scale conversion calculator to convert the size of an actual object to a scaled size and vice versa. Once you have these two measurements (in inches), you multiply them together and then multiply by 0. Wood Flexure. FAG bearing calculator. Global Industrial is a Leading Distributor of Motors & Power Transmission supplies. 010 Crankshaft 316005-T115 2. The result assumes a simple interest rate calculation and that interest payments have not been reinvested. 7 kb: Consol. Alternatively you can spend some time experimenting with some of the various online gear calculators to calculate the ideal ratio for your setup: Some of the calculators we use are shown here: West Coast Differential Gear Calculators 4-Lo Gear Ratio & Tire Size Chart 4-Lo Gear Ratio, Tire Size & Crawl Ratio Calculators. The choice of bearing will be governed by both the values and directions of the actions and also by the magnitude and directions of the allowed and restrained displacements. Same as line 13 if the exterior walls are built from steel studs. Using Technical Calculation, you will be guided through a series of selection stages allowing you to select the most appropriate bearing for your application. 9" (50mm) diameter conveyor rollers are normally rated at 250 lbs (110 kg) maximum load. 063-inch width) and L44643 (1-inch width). These effects are illustrated in Figure 3. Timken uses its 80 years of experience designing and manufacturing cylindrical roller bearings to continually advance performance across size ranges and configurations, including inch and metric and one-row, two-row and four-row designs. System Weight: Mobile (Sliding File System) Weight: 190 lbs. Figure 2 - This house is being built with advanced framing techniques including 2x6 24-inch on-center wall framing, single top plates, open headers over windows on non-load-bearing walls, and minimal studs around windows. It also allows you to calculate the allowances for axial water swell and thermal expansion if applicable. I would like to set two computers on the shelf, but I am not sure how to estimate the load bearing ability - I don't want the screws pulling out and everything crashing down on my head!. AST Bearings provides a bearing life and safety factor calculator to calculate the ideal bearing for your specifications. Bearing capacities have a safety factor built in to prevent failure. A = area of plate. Bearing life testing Bearings are run to failure, using accelerated speeds and loads and continuous vibration monitoring. What size cylinder bore should you use? Use this bore size calculator to determine the correct cylinder bore size for your application. The choice of bearing will be governed by both the values and directions of the actions and also by the magnitude and directions of the allowed and restrained displacements. The life equation was formulated using the Weibull probability theory of fatigue developed in 1936. Packaging Details : 1. Image credit: K. The Anti Friction Bearing Manufacturers Association have a code system which identifies bearings by type, size and construction. Ball bearings come in many sizes for an array of equipment, from skateboard wheels to industrial equipment. 1698 Zhenluo Road,Jiaochuan Street, Zhenhai, Ningbo City, Zhejiang, China +86-150 8847 0229 +86-574-86452780 +86-574-86456096 +86-574-86456328. Locate (fix) one bearing in the housing, and allow the other bearing axial freedom (non-locating). On longer spans the beam may require much more bearing space as indicated by this table. Convert from scale to actual (real) size. Spanner Size = ( Bolt Size * 1. Calculate the Weight of Wood. According to the 2012 IRC codes any beam, joist, or header shall never have a bearing of less than 1 1/2″. Drag Racing Calculators,Calculators to find 1/4 mile ET and MPH, CID,Piston Speed,gear ratio,carburetor size,margin of victory,Engine calculator, Calculates relative horsepower, air density, density altitude, virtual temperature, actual air pressure, vapor pressure and dyno correction factor and more. I just want to calculate the max load (distributed load) in bending (prob for just the 8ft sides) and also in compression for the legs. drag on the motor bearings. We ran a quick calculation, and if your total live load + dead load is less than 1000 PLF (pounds per lineal foot), then a steel i-beam S6x12. 2 mm 7/8" Steel bars. Sleeve bearings made from high performance plastics provide mechanical stability with the lowest coefficients of friction without additional lubrication. Outer Diameter (mm) : 90. Typical values range between 1. This calculator is designed to give the critical information of a particular beam antenna, in this case a three element Yagi, for the frequency chosen. 6 Timber piles (wood piles) 5. • a) Find the nal angular speed of the clay and turntable. Instructions and tips for correctly measuring with or without a ring plus a sizing table and conversion ring size charts. (2 bearings per connecting rod) 10 Main Bearings are required for Suzuki Hayabusa (2 bearings per journal) All Bearings Are Per Half (2 needed per rod and main) The green, black, brown, and yellow represent different size of bearings. 5 refers to the weight of the beam per foot. Present standards for bearing life calculation are based on work carried out at SKF in 1947 by Gustaf Lundberg and Arvid Palmgren. If the soil bearing capacity is 2,500 lbs/sf and the column load on a footing is 15,000 lbs. to check the concrete bearing resistance [DeWolf, 1978; Hawkins, 1968a]. Calculates plain bearings and designs and checks statically loaded radial plain bearings working under hydrodynamic lubrication conditions. Please note: You are using the IMPERIAL calculator. In order to calculate the engine RPM based on ring gear and pinion gear ratio, speed and tire height, this calculator makes the following assumptions: You may not be sure of the type of transmission you are currently using. 8 mm, and so on. Bearing part numbers help you to identify the type, size and general uses for a bearing. Size Application; 22. The results obtained by these tests are used with the empirical curves to determine the thickness of pavement and its component layers. Sealed bearings can last 100,000 miles or more, and will need to be replaced once they go bad. Skip to content www. The floor joist spacing is the distance between the centers of any two installed joists. See the reference section for details on the methodology and the equations used. Hartford conforms to precision ball standards (including ABMA Standard 10, ISO 3290 & DIN 5401) that dictate the industry standards of precision balls. 010 Crankshaft 316005-T113 2. Measure the size of the structural opening i. Load Bearing Walls 17 Answers To Common Questions In 2019. Determination, Area of Footing (a) Depth of Footing (d) Short form of Footing Designs Bearing Capacities of the Shallow Foundation from SPT: The bearing capacities of the shallow foundation particularly for top layer of cohesive soil may be estimated from the SPT values, as suggested by terzaghi, according to following:. Drag Racing Calculators,Calculators to find 1/4 mile ET and MPH, CID,Piston Speed,gear ratio,carburetor size,margin of victory,Engine calculator, Calculates relative horsepower, air density, density altitude, virtual temperature, actual air pressure, vapor pressure and dyno correction factor and more. To calculate grease quantity, you need the bearing’s physical dimensions (primarily the outside diameter and width). Perform calculations and searches for SKF bearing products wherever you are. Calculate the Weight of Wood. Double the width of the platform beam to calculate the total width of the footing. Load calculation formula applied to post-frame (pole barn) construction Post frame construction footing size calculations are easily determined because of its simple load bearing structure. The ‘i-button’ before each calculation gives more information about the theory behind the calculation. Using the info given, you can sketch a triangle with two sides and the angle between them known. 71 mm wall thickness from performing buckling and bending stress calculations. 05mm would be 3/4 inch. This guide is accurate and is updated on a daily basis. Re: Acreage Calculation from Bearings Thanks John, This was from an old (1940s) survey, and the surveyor had calculated it at 9. Enhancestyleteam. 2 Friction piles 5. Inner race: the inner race is the only part of the bearing that comes in more than one size; 7mm or 8mm. Copyright © NTN Corporation. Kargona/Shutterstock. For multi-level homes, expect to pay between $3,200 and$10,000. ALLMI Provides Pad Size Calculator ALLMI recently issued a template spreadsheet to its Operators’ Forum members which assists users in determining minimum stabiliser foot / pad size requirements, based on a calculation of the lorry loader’s gross lifting moment and subsequently the forces being placed through the stabiliser legs. Users can define jacking from either end of the tendon or both. As you increase footing size the bearing capacity peaks at the transition between failure and settlement control. We started life in 2006 and are now one of the leading online bearing shops in the world. A pit of size 5 Bp X 5 Bp excavates to the depth equal to the depth of the foundation to conduct a plate load test at the site. Example lintel length = 150 + 1800 + 150 = 2100mm. Bearing number : 6802. Our site is packed full of thousands of bearings & power transmission components, along with many everyday engineering essentials. Bearing Calculator calculates this value when you click Solve Life. Tendon loads and losses are easily defined and calculated by SAFE. Online calculator for performing Steel Beam Web Stiffener Analysis calculations. I just want to calculate the max load (distributed load) in bending (prob for just the 8ft sides) and also in compression for the legs. size, used on the vast majority of newer bicycles. Bearings extend the working life of wheels, pulleys, and other rotating parts by reducing friction and enabling parts to move smoothly. Any more than this and you face the risk of brake failure. Utilized for calculationg an interior footing size when wood studs are used. 1 Pile group in cohesive soil. We ran a quick calculation, and if your total live load + dead load is less than 1000 PLF (pounds per lineal foot), then a steel i-beam S6x12. 3,671 tapered roller bearing size chart products are offered for sale by suppliers on Alibaba. Our site is packed full of thousands of bearings & power transmission components, along with many everyday engineering essentials. Hex Bolts, Hex Machine Bolts, Square Head Machine Bolts at InStock Fasteners - The low-price easy-to-use source for industrial and construction fasteners. Don't worry, if you don't know the ground pressure - the calculator will ask you for the surface materials and you'll be able to select the most appropriate type. Some heaters only come with a select number of yokes, so make sure that your heater has a yoke that is suitable for the minimum bore of your bearing. All steel beam calculations for loft extensions and beam calculations for extensions, including padstone/bearing plates and splice calculation and fabrication detail (These may be required for both floor and roof support) Floor joist calculations, including trimming joists. Tendon loads and losses are easily defined and calculated by SAFE. * Disclaimer: Because of the tremendous amount of variables that exist in generator power provision and transmission, Absolute Generators strongly suggests that you seek the advisement of a certified electrical professional familiar with your situation (and potentially local electric code when performing permanent installations). Bearing Capacity Calculation for Sandy Soils. 4 km due-east-of C is 3. Probably the most widely used value in a soil report is soil bearing capacity. Bearing life in excess of 5 years average. Literature included. Click the button to calculate the volume of concrete and man-hours (not including mixing) needed for this job. Hello friends is video me centrifugal pump ki bearing housing or shaft se kaise bearing ka number calculate karte dikhaya gya hai. Bearing Calculator This program will help you determine if your crankshaft bearings are suitable. It is necessary to check whether the bearing can be used at the given operating conditions and to determine the nominal service life. Pressed fits with medium interference, assembly of parts using hot pressing, assembly using cold pressing only with use of. 70 m to the east of the axis. Link-Belt offers a split housed spherical roller bearing drop-in compatible with most standard systems. There is currently a steel beam spanning the main floor and spanning the second floor is just 2x4 construction as the load bearing wall. Enter your current x coordinate: Enter your current y coordinate: Enter the destination x coordinate. Bearing Capacity Calculation for Sandy Soils. A wide variety of tapered roller bearing size chart options are available to you, such as double row, single row. 1698 Zhenluo Road,Jiaochuan Street, Zhenhai, Ningbo City, Zhejiang, China +86-150 8847 0229 +86-574-86452780 +86-574-86456096 +86-574-86456328. Logarithm Calculator: Planetary Distance: Planets Escape Velocity: Size of Country: Size of Oceans: Size of Planets: Square Root Calculator: Tallest Mountains: Trigonometry Calculator: Wavelength in different medium: Weight: Weight Conversion: All Weight Conversion: Atomic Number: Atomic Weight: Density of Material: Gravity of Planets: Mass of. Some heaters only come with a select number of yokes, so make sure that your heater has a yoke that is suitable for the minimum bore of your bearing. • Back Ground Information ; Useful revision and helpful for understanding some of the above calculations. - The tabulated numbers must be multiplied by the plate thickness to calculate the design bearing strength of the plate. The california bearing ratio test is penetration test meant for the evaluation of subgrade strength of roads and pavements. Approved Ball Bearing and Cylindrical Loose Roller Supplier. Calculation Reference Soli Mechanics. This information is provided for reference only. Its purpose is to distribute the axial pull-out load from the hydraulic cylinder evenly across the full surface of the test sample. Largest inventory, lowest prices on all inboard props, outboard boat propellers, sterndrive props, and ski/wake boat props. The bearing safety factor, or safety modulus f s, is the ratio of the basic static load rating C or the equivalent load P on the bearing. 1 Simplified method of predicting the bearing capacity of timber piles Chapter 6 Design of Pile Group 6. The igus® plain bearings range includes sleeve bearings, flange bearings, thrust washers, piston rings, clip-on plain bearings and many other designs. Roller and ball bearings are available in different standard sizes. This size was also used for steel drop bars. The "S" means a standard flange size (as opposed to a "W" wide flange. As far as crank interchangeability is concerned, the most important thing is the overall length, and the offset if the spindle is longer on one side than the other (that's. The table also indicates Series 4 hanger bearings and shafts. and i also need column load info for pipe, square tube etc. The pulp densities around your cyclone are sampled and known over an 8-hour shift, allowing to calculate corresponding to circulating load. To calculate the appropriate size I-beam for a construction project, you will need to know the load the beam is expected to carry. The resulting answer is the grease quantity in ounces. Reducing the oil clearance between the rod and main bearings and the crankshaft has a number of advantages. Specifying the right bearing for a given application is necessary to save time and excessive costs. First, we look at the steel beam span length. To begin filling out the 2015 IRC Prescriptive Footing Calculator, select the "Calculator" tab at the bottom of this spreadsheet. 1 Calculation of loads on spur, helical, and double-helical gears There is an extremely close relationship among the two mechanical elements, gears and rolling bearings. 1 Calculations of axial vibration natural frequency are to be carried out using appropriate techniques, taking into account the effects of flexibility of the thrust bearing, for shaft systems where the propeller is: (a) Driven directly by a reciprocating internal combustion engine. You can look up the recommended footing size, based on the size and type of house and the bearing capacity of the soil. 5 N/m 2 = 0. Most bearing references now are laser stamped but these wear and rub off more quickly than the previous method, which was engraved deep in the bearing's metal. Bearing Calculator This program will help you determine if your crankshaft bearings are suitable. The specified size of an x-ray tube focal spot is the dimensions of the effective or projected focal spot shown in the figure above. Please get in touch if you have any questions, or start a calculation. This catalogue includes information such as the latest bearing types, bearing numbers, and technical data. How To Calculate Steel Beam Size For Load Bearing Wall September 28, 2018 - by Arfan - Leave a Comment Tell if a wall is load bearing structural design of light gauge steel load bearing walls 17 s to beam to replace a load bearing wall how to design a steel i beam selection. Bearing Sizes. These fits, though applicable to shaft and hole assembly, are more often used for bearing-housing or bearing-shaft assembly. Catenary mooring system A catenary mooring system is the most common mooring system in shallow waters. 38 k/in / 14. * Please note: Rates are subject to change without notice. Figure 21-5 Side-entering mixer with pillow-block bearings. This is the distance from the centre of one end bearing to the other. It easy to measure area in Scribble Maps using our drawing tools. Friction Loss Rate – Friction loss rate is the (Available Static Pressure x100 / Effective Length). If they are, then they are. It is very rarely used today for large buildings, but smaller residential-scale structures are being built. TAPERED BEARING SET REFERENCE - Which Bearings Make Up A "Set" BEARING NUMBER CODE REFERENCE - What Those Bearing Numbers Mean. Most bearings are metric in size, but can also be imperial. Set of two 76 mm caster board wheels with bearings for the Razor Original RipStik (Ripstik Classic), Ripstik DLX, Ripstik "G", and the Razor Crazy Cart. Notice that the actual focal spot, the area bombarded by the electron beam, is always larger than the projected, or effective, focal spot. The results obtained by these tests are used with the empirical curves to determine the thickness of pavement and its component layers. Welcome to Bearing Boys. System Weight: Mobile (Sliding File System) Weight: 190 lbs. 26 inches (6. The pulp densities around your cyclone are sampled and known over an 8-hour shift, allowing to calculate corresponding to circulating load. This is the minimum of safe bearing capacity and safe bearing pressure. If they are, then they are. Use the drop downs to select a size for the lumber. Truss Calculator. The following procedure describes how to verify the strength of the lifting eye. Live load = 4. Express both the bearings as if they are measured from the point where the lines intersect and then apply the above rule. I need to calculate load bearing capacity (PSF)of the roof. We carry many other designs. on the SKF bearing numbers. All of the potential load cases required to fully design an actual structure may not be provided by this calculator. This is also an acceptable method for figuring out the load-bearing capacity for small home projects, like a driveway or a garage slab. Global Industrial is a Leading Distributor of Motors & Power Transmission supplies. Looking at this table you will see there is a choice in the size of floor joist (2 X 6, 2 X 8, 2 X 10 or 2 X 12) and there is a choice in the joist spacing (12", 16" or 14"). The general information about bearing life calculation and basic load ratings provided under Bearing size is also valid for super-precision bearings. These tools provide mathematical calculations only. of Ends * Tape Length in Yards)/(840 * warp yarn count) Also. I have a 57 210 with a 265 and a stock rear end. Next you get out your pocket calculator or smartphone app, or old-school paper and pencil. SKF Cooper Products. The wall in question is load bearing. I would like to set two computers on the shelf, but I am not sure how to estimate the load bearing ability - I don't want the screws pulling out and everything crashing down on my head!. Bore Diameter (mm) : 50. If you know the weight on the outrigger then just divide by the area of the rigger to get Tonnes/m2 (in metric). 010 Crankshaft 316005-T115 2. Find the bearings that fit your engine, cam, and application right here at Summit Racing Equipment. Superior Automotive U-Bolt Kits and Bearing Straps. Packaging Details : 1. HY S60xx Preload Change Calculations. Also calculate ultimate bearing capacity if same footing is placed at a depth of 1 m below ground surface. Tendon loads and losses are easily defined and calculated by SAFE. You can still calculate bearing and distance coordinates by typing in latitude and longitude. Identifying the Minimum Bore, Outer Diameter (OD), and weight of the bearing helps ensure that you’re selecting the right heater for a particular bearing size. The theoretical size of a load bearing beam required to support a particular weight is easy to calculate, but the choice of the actual beam depends on taking into account the factors of the particular situation. By continuing to use this site you consent to the use of cookies on your device as described in our cookie policy unless you have disabled them. The General Bearing Analysis Application tool selects a bearing that best matches your application, or will analyze the selected bearing(s) to verify application compatibility. Bolt Data specifies the tension capacity of the anchor bolts, number of bolts per side of the column, the area of each bolt, and distance of the bolts from the edge of. Advancing Designs. Most calculations are done to full precision. 05mm would be 3/4 inch. If the ball size is over. 7 kN/m Total shear for toe design Vtoe = Vtoe_bear - Vtoe_wt_base = 143. Build temporary walls to support the load; Bearing points of columns must be stacked and extend down through the house to a footing; Hire a structural engineer to size the beam. Bearing is used to reduce the stress and friction of the machine. Calculates plain bearings and designs and checks statically loaded radial plain bearings working under hydrodynamic lubrication conditions. Supporting masonry to comply with Eurocode 6 or BS 5628. please help? I do hvac work and i routinely need to supprt pipe and equipment from over head structural steel and i need to know how to select the size and shape of angle iron,tube steel,i-beams etc. Bearings have application in a myriad of fields from compass bearings, (the bearing a compass dictates) magnetic bearings (the bearing with respect to the north direction of the Earth's magnetic field. ? we are removing a wall to open up kitchen and dining area in our 1950's cape cod style house. This tire calculator is for information purposes only and we do not guarantee fitment based on this calculator alone. Rated full load current is often abbreviated as ‘FLA” on the nameplate. Structural ability of sawn- and engineered-wood beams are predicted through mathematical calculation. Auto-reroute for optimum distance (traveling salesman problem) Have other speeds like fiber optic cable (~. Continue this iterative process until an appropriate L na life is obtained. Plan area of bonded masonry supporting steel beam to be greater than or equal to 0. 1 Pre-cast concrete piles 5. Note: Dimensions displayed are calculated using industry-standard tire sizing specifications: - ISO 4000-1, ISO 4000-2 Passenger car tyres and. 4 times its size will have an angular size of 1 minute. The software allows the calculation of deformation of several coaxial shafts in combination with nonlinear bearing stiffness of rolling bearings. Re: How to calculate a ridge size Sounds like a big beam to me. DuraBelt belt tension calculator. 05mm would be 3/4 inch. microblue bearings: who we are & what we do for you We are a “racing-only” bearing company that’s dedicated to bringing you all the products and the knowledge to help you with their use. F = Load to be moved P Thread surface F πDp P N λ F f L = Lead Force exerted up the plane Force exerted down the plane P = Force required to move the load Screw thread force analysis Ff = Friction force N = Normal force λ = Lead angle Dp = Pitch diameter L. (2 bearings per connecting rod) 10 Main Bearings are required for Suzuki Hayabusa (2 bearings per journal) All Bearings Are Per Half (2 needed per rod and main) The green, black, brown, and yellow represent different size of bearings. Kargona/Shutterstock. 3 Negative value indicates effective center inside cone backface. 25mm inner diameter Bearings Online shopping by VXB bearings the online bearing store and supplier, wholesale prices and same day shipping, next day air shipping available. iglide® plain bearings always offer a solution – either from the catalogue range or as custom-made plain bearing special solution. We have stock in different parts of the world. of America catalog including Item #,d,D,B,r,Static Load Rating,Dynamic Load Rating. Determine ballast depth based on allowable subgrade stress 12. there only about 1 week old. Last Modified: 28th Apr 2016 Belt Length Calculator. With components designed for a precise fit and excellent fatigue resistance they are a dependable replacement for your vehicle’s worn or damaged original parts. As an example, the measured diameter of the rim shown in the photo is 20 7/8 inches, or 20. Therefore the bearing area is: Bearing Area = Thrust Force / Soil Bearing Capacity Bearing Force = 21484 N / 23939. This calculation method is one I came up with. Spindle dimension same as Santhuff I, but the bearing size in wheel OD 1. Rubber bushing cross reference moog k90063 control arm bushing for buick chevy front lower inner moog rubber bushing cross reference Domestic Control Arm And Idler Rubber Bushing Cross Reference Size IngMoog Rubber Bushing Cross ReferenceDomestic Control Arm And Idler Rubber Bushing Federal MogulPolyurethane Neoprene Rubber Suspension Bushings Moog PartsTechnical Kingpin Ing By Diameter The H…. Load Bearing Walls 17 Answers To Common Questions In 2019. Can I use dimensional lumber to frame out the header and if so what. 8 kN/m Shear from weight of base Vtoe_wt_base = f_d base ltoe tbase = 34. The calculations and results are based on imperical data and formulas. 4 gives the diameter in millimeters, 530 mm. p = res F d 1 • L • Y The calculated area pressure on the bearing outer race must be less than the values given in Figure 7. This spreadsheet was based on the Terzaghi, Mayerhoff & Hanzen Equations for determing the soil bearing capacity. 05s • Bearing friction factor: 0. For driven piles in loose to dense sand with φ varying between 30 0 to 40 0 , k i values in the range of 1 to 1. Live load = 4. The results are only as acurate as the data you enter. For example, if the bearings of the lines BA and AC are given, then to find angle at A, the bearings of AB must be obtained. on the SKF bearing numbers. Usually suitably to secure a bearing under standard environmental operating temperatures from 5 to 70C. Our site is packed full of thousands of bearings & power transmission components, along with many everyday engineering essentials. Resistance to incremental penetration is calculated using only the bearing stress (σ) of the soil, shear stress (τ) is used to calculate the shear angle for horizontal force (F̌ʰ). Safety Mount Bearings. Most of the bearings on Fish4Parts are metric in size – but they can also be in Imperial. It is fairly common practice for the bearing engineer to re-duce the duty cycle down to a workable size of 10–15 steps through equivalent damage calculations, which are beyond the scope of this article. We strive continually to make our online shop faster, simpler & easier to use. You can use this online scale conversion calculator to convert the size of an actual object to a scaled size and vice versa. It’s much quicker and more affordable than employing a structural engineer and is ideal for straightforward structural work – there are 46 different. Calculating the size of the header depends on what the header needs to support. MOTORS® products. Sleeve and clip bearings support high loads, have no moving parts, and require lubricant to allow the shaft to turn smoothly. Calculate the required thrust block area. smaller bearings to carry a given load or a given-size bearing to carry a higher load. This spreads the load out over a larger area, allowing the bearing to handle much greater loads than a ball bearing. Enter values three of the six sides and angles of the triangle and the other three values will be computed. Table of ContentsBrinell Hardness of Babbitt Bearings at Increasing TemperaturesEffect of Compression on the Brinell Hardness of BabbittsRolling of Babbitted Linings by the Mills Micrometer Roller Brinell Hardness of Babbitt Bearings at Increasing Temperatures Brinell tests at progressively increasing temperatures are given for a representative lead-base and a representative tin-base Babbitt. Continue this iterative process until an appropriate L na life is obtained. Therefore the bearing, or the housing, is heated before mounting. A variation of this type of bearing, called a needle bearing, uses cylinders with a very small diameter. This free sample size calculator determines the sample size required to meet a given set of Learn more about population standard deviation, or explore other statistical calculators, as well as. Bearings of this type use conical rollers guided by a back-face rib on the cone. We started life in 2006 and are now one of the leading online bearing shops in the world. When designing machinery that uses lead screws, it's a common task to try and figure out the size of motor needed to drive a given force with a lead screw. 5" 1638-2RS 0. Bearing clearance depends on the engine and the intended use. The program, which delivers the information straight to your smart phone or tablet, is fast. #4: RATED FREQUENCY. This calculator is free to use as often as you wish. 5 m in a soil with a moist unit weight of 17. I can send you the calculation so you can copy it and put in the size of your beams. Engine Calculator - This form is designed to help you figure out engine specs for all engine types - not just VW. 85 k/in is 15. Framing A Door. It should be noted that all life calculations based on ISO 281 are valid for normal speeds. I know I can get a structural engineer to give me the answers, but I don't want to spend any money until I know the scope of this project and what it's going to take. With our free freight class calculator we can quickly determine your class based on your pallets weight and density through volume calculations. Preload Change Calculations for Angular Contact Bearings. Span is 20 feet to cover however we can add a post if need be or divide the span. The key and key seat cross section are ISO standardized. 33333), so we stop after 200 decimals. The theoretical size of a load bearing beam required to support a particular weight is easy to calculate, but the choice of the actual beam depends on taking into account the factors of the particular situation. Notice that the actual focal spot, the area bombarded by the electron beam, is always larger than the projected, or effective, focal spot. A 2 x 12 at a 6/12 pitch would be 8-3/8" if using the 2/3rds rule. HCFS207-20 Four Bolt Flange Bearing 1-1/4" Bore - Hi-Sun. Therefore from AASHTO Standard Specification for Highway Bridges Table 3. 2 Friction piles 5. Finding the size of a ball bearing in need of replacement, before purchase a new one, may save money. We strive continually to make our online shop faster, simpler & easier to use. Plate bearing test procedure and calculation Plate bearing test is an activity carried out by design engineers in the field to determine the bearing capacity with regards to the soil underneath. Why to calculate the Safe bearing capacity of soil before starting construction:-From the above figure, it is clear that the building is fallen in only one side. Selecting bearing size. This calculator computes all parameters (spring rate, maximum load, maximum stress, solid height, coil pitch, coil angle, wire length, resonant frequency, shear modulus, and spring mass) related to a compression spring from basic geometry and material data input. The basic equationof bearing capacity concerns strip footings loaded vertically in the plane of symmetry (Fig. Bearings The Nachi bearings business started out by using the superior materials made by our own company, and by applying expertise in cutting and heat treatment acquired from our experience with cutting-tool manufacturing. This gear ratio calculator determines the rate of mechanical advantage or disadvantage a gear train produces in a gear system. A better solution is to have two linear bearings, one above the other so that they resist the moment produced by the eccentric load by having a pure lateral load at each bearing. Bit Brokers is Quality Rock Bits. Ball Bearing Size Chart Part Number Inner Dia. Bearings & Bushing By Size Chart. 203E-1 Printed in Japan ’00. Number of line parts 3. - The tabulated numbers must be multiplied by the plate thickness to calculate the design bearing strength of the plate. Thordon Bearings Sizing Calculation Program allows you to calculate the maximum and minimum operating temperature for the final bearing size to be machined. • a) Find the nal angular speed of the clay and turntable. 5207, the allowable bearing capacity for silty clay soil at a may be determined by an average. Perform calculations and searches for SKF bearing products wherever you are. These bearings consist of cylindrical rolling elements and usually support a rotating shaft in the inner ring which may be misaligned with the outer ring.	2021-04-13 23:07:58	{"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": 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": 0.3254152834415436, "perplexity": 3269.034402047746}, "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/1618038075074.29/warc/CC-MAIN-20210413213655-20210414003655-00003.warc.gz"}
http://www.transtutors.com/questions/markov-chain-105890.htm	+1.617.933.5480 +1.866.649.0192 # Q: Markov Chain see the pic Attachments: ## Solution Preview: ANSWER-1: State space of will be as follows: {0+3, 0+4, 2+3, 2+4}={3,4,5,6}. Yes. is a markov chain as the sum of the two markov chains is also a markov... Related Questions in Markov Analysis Question Status: Solved	2014-09-18 07:40:35	{"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": 0.9147952198982239, "perplexity": 4531.6859916755}, "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-2014-41/segments/1410657126053.45/warc/CC-MAIN-20140914011206-00161-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"}
http://mathonline.wikidot.com/the-first-and-second-arens-products-on-a	The First and Second Arens Products on A The First and Second Arens Products on A The First Arens Product Let $\mathfrak{A}$ be a Banach algebra. Consider the second dual, $\mathfrak{A}^{}$, which is clearly a Banach space. We would like to make $\mathfrak{A}^{}$ a Banach algebra too, but it is not entirely obvious what the multiplication on $\mathfrak{A}^{}$ should be. One type of multiplication we can define on $\mathfrak{A}^{}$ is the First Arens Product on $\mathfrak{A}^{*}$. It is defined in steps as follows: Definition: Let $\mathfrak{A}$ be a Banach algebra. 1. For each $a \in \mathfrak{A}$ and for each $f \in \mathfrak{A}^$ we define $f \cdot a \in \mathfrak{A}^$ by $(fa)(b) := f(ab) \quad (\forall b \in \mathfrak{A})$. 2. For each $F \in \mathfrak{A}^{}$ and for each $f \in \mathfrak{A}^$ we define $F \cdot f \in \mathfrak{A}^$ by $(F \cdot f)(a) := F(fa) \quad (\forall a \in \mathfrak{A})$ 3. Lastly, the First Arens Product on $\mathfrak{A}^{}$ is the multiplication on $\mathfrak{A}^{}$ defined for all $F, G \in \mathfrak{A}^{}$ by $(FG)(f) := F(Gf) \quad (\forall f \in \mathfrak{A}^)$. It is clear that the First Arens Product defined above satisfies the 3 axioms on the Algebras over F page, making $\mathfrak{A}^{}$ with the First Arens Product a Banach algebra. Indeed, let's verify these three axioms • 1) Let $F, G, H \in \mathfrak{A}^{}$. We will show that $[FG]H = F[GH]$. This is done by showing that these functionals are equal for all $f \in \mathfrak{A}^$. Indeed: (1) \begin{align} \quad ([FG]H)(f) = [FG](Hf) = F(GHf)= F([GH]f) = (F[GH])(f) \quad (\forall f \in \mathfrak{A}^) \end{align} • 2) Let $F, G, H \in \mathfrak{A}^{*}$. We now show that $F[G + H] = FG + FH$. This is again done by showing that these functionals are equal for all $f \in \mathfrak{A}^$. Indeed by the linearity of $F$ we have that: (2) \begin{align} \quad (F[G + H])(f) = F([G + H]f) = F(Gf + Hf) = F(Gf) + F(Hf) = [FG](f) + [FH](f) \quad (\forall f \in \mathfrak{A}^) \end{align} • 3) Let $F, G \in \mathfrak{A}^{}$ and let $\alpha \in \mathbb{C}$. We lastly show that $[\alpha F]G = \alpha [FG] = F[\alpha G]$. Yet again, this is done by showing that these functionals are equal for all $f \in \mathfrak{A}^$: (3) \begin{align} \quad ([\alpha F]G)(f) = [\alpha F](Gf) = \alpha [F](Gf) = \alpha [FG](f) \quad (\forall f \in \mathfrak{A}^) \end{align} • So that $[\alpha F]G = \alpha [FG]$. Also: (4) \begin{align} \quad (\alpha [FG])(f) = \alpha [F](Gf) = [F](\alpha Gf) = [F] ([\alpha G]f) = (F[\alpha G])(f) \quad (\forall f \in \mathfrak{A}^) \end{align} • So that $\alpha [FG] = F[\alpha G]$ too. The Second Arens Product We can quite naturally define another type of multiplication on $\mathfrak{A}^{*}$ call the Second Arens Product as follows: Definition: Let $\mathfrak{A}$ be a Banach algebra. 1) For each $a \in \mathfrak{A}$ and for each $f \in \mathfrak{A}^$ we define $af \in \mathfrak{A}^$ by $(af)(b) := f(ba) \quad (\forall b \in \mathfrak{A})$. 2) For each $F \in \mathfrak{A}^{}$ and for each $f \in \mathfrak{A}^$ we define $fF \in \mathfrak{A}^$ by $(fF)(a) := F(af) \quad (\forall a \in \mathfrak{A})$. 3) Lastly, the Second Arens Product on $\mathfrak{A}^{}$ is the multiplication on $\mathfrak{A}^{}$ defined for all $F, G \in \mathfrak{A}^{}$ by $(F G)(f) :=F(fG) \quad (\forall f \in \mathfrak{A}^)$. Here we will use $$ to explicitly denote the second Arens product. Equality of the First and Second Arens Products In general, given $F, G \in \mathfrak{A}^{*}$ it may be that $FG$ (first Arens product) is not equal to $F G$ (second Arens product). When these two product are equal for all $F, G \in \mathfrak{A}^{*}$, the Banach algebra $\mathfrak{A}$ is given a special name. Definition: Let $\mathfrak{A}$ be a Banach algebra. Then $\mathfrak{A}$ is said to be Arens Regular if $FG = FG$ for all $F, G \in \mathfrak{A}^{**}$, where the lefthand side of the equality is multiplication with respect to the first Arens product, and the righthand side of the equality is multiplication with respect to the second Arens product.	2019-10-13 22:38:51	{"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": 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": 4, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9985668063163757, "perplexity": 280.39992480452594}, "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/1570986648343.8/warc/CC-MAIN-20191013221144-20191014004144-00211.warc.gz"}
http://mathhelpforum.com/differential-geometry/187694-show-nowhere-dense.html	# Thread: Show that A is nowhere dense 1. ## Show that A is nowhere dense Show that A is nowhere dense 2. ## Re: Show that A is nowhere dense Given that $A\subset\overline{(X\setminus\overline{A})}.$ If $t\in A$ then $t$ is a point of $(X\setminus\overline{A})$ or a limit point of that set. Find an open set, $\mathcal{O}$, containing $t$ but $\mathcal{O}\not\subset A~.$	2017-10-18 10:35:43	{"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8307703733444214, "perplexity": 716.9975856791555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187822851.65/warc/CC-MAIN-20171018085500-20171018105500-00009.warc.gz"}
https://www.physicsforums.com/threads/fortran-how-to-read-a-non-uniformly-formatted-text-file.614781/	# FORTRAN- How to read a non-uniformly formatted text file • Fortran ## Main Question or Discussion Point Hi all !! I am a little new to FORTRAN and I am sorry if the title is confusing but I couldn't come up with anything better. I have a file of which I am showing a snippet below: Code: >>>>>>>>CHARMM22 All-Hydrogen Topology File for Proteins <<<<<<< >>>>>>>>>>>>>>>>>>>> and Nucleic Acids <<<<<<<<<<<<<<<<<<<<<<<<< >>>>> Includes phi, psi cross term map (CMAP) correction <<<<<<< >>>>>>>>>>>>>>>>>>>>>> July, 2004 <<<<<<<<<<<<<<<<<<<<<<<<<< * All comments to ADM jr. via the CHARMM web site: www.charmm.org * parameter set discussion forum * 31 1 ! references ! !PROTEINS ! !MacKerell, A.D., Jr,. Feig, M., Brooks, C.L., III, Extending the !treatment of backbone energetics in protein force fields: limitations !of gas-phase quantum mechanics in reproducing protein conformational !distributions in molecular dynamics simulations, Journal of !Computational Chemistry, 25: 1400-1415, 2004. ! !MacKerell, Jr., A. D.; Bashford, D.; Bellott, M.; Dunbrack Jr., R.L.; !Evanseck, J.D.; Field, M.J.; Fischer, S.; Gao, J.; Guo, H.; Ha, S.; !Joseph-McCarthy, D.; Kuchnir, L.; Kuczera, K.; Lau, F.T.K.; Mattos, !C.; Michnick, S.; Ngo, T.; Nguyen, D.T.; Prodhom, B.; Reiher, III, !W.E.; Roux, B.; Schlenkrich, M.; Smith, J.C.; Stote, R.; Straub, J.; !Watanabe, M.; Wiorkiewicz-Kuczera, J.; Yin, D.; Karplus, M. All-atom !empirical potential for molecular modeling and dynamics Studies of !proteins. Journal of Physical Chemistry B, 1998, 102, 3586-3616. ! !IONS (see lipid and nucleic acid topology and parameter files for ! !ZINC ! !Roland H. Stote and Martin Karplus, Zinc Binding in Proteins and !Solution: A Simple but Accurate Nonbonded Representation, PROTEINS: !Structure, Function, and Genetics 23:12-31 (1995) ! !NUCLEIC ACIDS ! !Foloppe, N. and MacKerell, Jr., A.D. "All-Atom Empirical Force Field for !Nucleic Acids: 2) Parameter Optimization Based on Small Molecule and !Condensed Phase Macromolecular Target Data. 2000, 21: 86-104. ! !and ! !MacKerell, Jr., A.D. and Banavali, N. "All-Atom Empirical Force Field for !Nucleic Acids: 2) Application to Molecular Dynamics Simulations of DNA !and RNA in Solution. 2000, 21: 105-120. ! MASS 1 H 1.00800 H ! polar H MASS 2 HC 1.00800 H ! N-ter H MASS 3 HA 1.00800 H ! nonpolar H MASS 4 HT 1.00800 H ! TIPS3P WATER HYDROGEN . . . MASS 95 F3 18.99800 F ! Fluorine, trifluoro (see toppar_all22_prot_fluoro_alkanes.str) MASS 99 DUM 0.00000 H ! dummy atom !see NA section --------------NOTICE THERE ARE COMMENTS B/W MASS RECORDS ALSO !MASS 100 SOD 22.989770 NA ! Sodium Ion !MASS 101 MG 24.305000 MG ! Magnesium Ion !MASS 102 POT 39.102000 K ! Potassium Ion! check masses !MASS 103 CES 132.900000 CS ! Cesium Ion !MASS 104 CAL 40.080000 CA ! Calcium Ion !MASS 105 CLA 35.450000 CL ! Chloride Ion !MASS 106 ZN 65.370000 ZN ! zinc (II) cation !NA section !MASS 101 HT 1.008000 H ! TIPS3P WATER HYDROGEN From this file, I have to read the records starting with "MASS" (which end just before records starting with "DECL") and make a list out of it. I have to ignore all other lines. What is the best way to do this?? My approach: Code: line_code=' ' do while(line_code .ne. 'DECL') read(1000,) line_code, temp_int1, temp_cname,temp_dbl1 !! ERROR IN THIS LINE, MOST PROBABLY if(line_code .eq. 'MASS')then atom_type_info(tot_atom_types)%type_code=temp_int1 atom_type_info(tot_atom_types)%type_cname=temp_cname atom_type_info(tot_atom_types)%mass=temp_dbl1 tot_atom_types=tot_atom_types+1 endif enddo The problem is that in the read(1000,) line, because of the comments in the file, a string is attempted to be read into an integer, which gives an i/o error. I thought of another way, but its too much work. (first scan lines with read(1000,) line_code and keep a line_counter. When you encounter MASS record, REWIND and read(1000,)some_temp till line_counter-1. Then start reading the MASS record.) All this is because we cant REWIND one line before, we can only REWIND to the beginning of the file (right??) Is there a better way? Like reading the whole line as a character array (with tabs and spaces), then reading from that array the first string. IF that is "MASS" go on and read from that line (stored as character array) rest of the values, otherwise ignore it. This is easy in C/C++ through getline and sscanf, but is there such a way in FORTRAN?? EDIT: I found a way to read a line and not advance read pointer to next line through ADVANCE='NO' in READ statement, but now the problem is that when I read everything as a character, and there comes an integer in between, it gives an error. Even if someone could tell me how to read a file in fortran line by line treating each line as a string (even if it is a number), that also might be a lot of help. Last edited: Related Programming and Computer Science News on Phys.org Never Mind. Got it. For someone else, you can read each line of text file as a string in fortran as follows: do read(1,'(a)',END=10) line !!END tells which statement to go to if all the lines have already been read enddo Also, from this line you read as a string, you can read formatted input. Just put the name of string in the place you put 'unit number' in read, as below: Yes, the way to do it is to first read the line as a character string and, THEN, do what is called an internal read. The one trick that allows this, though, is to read the line by specifying a format long enough to accommodate all you are trying to read out of it. Or, you can specify a very long format that always read the entire line. Because in fortran spaces work as separators, you need to specify a format in order to read a string that is not enclosed in quotes and that should include spaces in itself. And, so, the most important line in the code that follows is: Code: read(,'(A26)') line The following code does the trick; you can test it by compiling and running from the command line using re-direction (mass < inputfile) : Code: program mass character line26 character code4, temp_cname4 integer temp_int1 real temp_dbl1 line(1:4) = ' ' do while (line(1:4) .ne. 'DECL') if (line(1:4) .eq. "MASS") then write(,) code, temp_int1, temp_cname, temp_dbl1 ! temporary line ! assign read values to pointer endif enddo end The code above declares the variable "line" to be just long enough to read up the last value you are interested in (the double)...you know as far as the number of characters to be read from the line (26); or, you could simply read the entire line every time by declaring "line" to be something like 130 characters long, instead...safer, just in case the format of your input file changes a bit.	2020-07-02 10:30:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5694497227668762, "perplexity": 8097.510671519476}, "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-2020-29/segments/1593655878639.9/warc/CC-MAIN-20200702080623-20200702110623-00306.warc.gz"}
https://gmatclub.com/forum/the-product-of-all-the-prime-numbers-less-than-20-is-closest-to-which-110518.html?fl=similar	The product of all the prime numbers less than 20 is closest to which : GMAT Problem Solving (PS) Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack It is currently 19 Feb 2017, 09:41 ### 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 product of all the prime numbers less than 20 is closest to which Author Message TAGS: ### Hide Tags Senior Manager Joined: 10 Nov 2010 Posts: 267 Location: India Concentration: Strategy, Operations GMAT 1: 520 Q42 V19 GMAT 2: 540 Q44 V21 WE: Information Technology (Computer Software) Followers: 5 Kudos [?]: 310 [0], given: 22 The product of all the prime numbers less than 20 is closest to which [#permalink] ### Show Tags 08 Mar 2011, 07:34 1 This post was BOOKMARKED 00:00 Difficulty: 75% (hard) Question Stats: 35% (01:58) correct 65% (00:53) wrong based on 46 sessions ### HideShow timer Statistics The product of all the prime numbers less than 20 is closest to which of the following powers of 10? a) 10^9 b) 10^8 c) 10^7 d) 10^6 e) 10^5 [Reveal] Spoiler: 235711131719 10211431719= any easy way because i cnt do above multiplication in less time. Thanks OPEN DISCUSSION OF THIS QUESTION IS HERE: the-product-of-all-the-prime-numbers-less-than-20-is-closest-135192.html [Reveal] Spoiler: OA _________________ The proof of understanding is the ability to explain it. Manager Joined: 02 Apr 2010 Posts: 103 Followers: 5 Kudos [?]: 120 [0], given: 18 Re: The product of all the prime numbers less than 20 is closest to which [#permalink] ### Show Tags 08 Mar 2011, 08:06 As you don't have to know the exact result but rather the magnitude I would just round when multiplying, i.e.: 2357 = 210 (use 200) 20011= 2200 (use 2,000) 2,00013= 26000 (use 20,000) 20,00017 = 340,000 (use 300,000) 300,00019 = 5,700,000 (use 6,000,000) Now since you rounded to a lower number most of the time and the result is still larger than 5,000,000 it is clear that the answer is C) 10^7. I don't know why the answer is given as D) but I think it's wrong. Manager Joined: 14 Feb 2011 Posts: 196 Followers: 4 Kudos [?]: 128 [0], given: 3 Re: The product of all the prime numbers less than 20 is closest to which [#permalink] ### Show Tags 08 Mar 2011, 08:32 GMATD11 wrote: 15.) The product of all the prime numbers less than 20 is closest to which of the following powers of 10? a) 10^9 b) 10^8 c) 10^7 d) 10^6 e) 10^5 235711131719 10211431719= any easy way because i cnt do above multiplication in less time. Thanks Group these numbers so as to get product close to a multiple of 10, so they can be rewritten as (25)(37)(1119)(1317) or 1021209221 Now take out power of 10 from each number and rewrite 1010100100(12.12.092.21) or 10^6(12.12.092.21) Which is greater than 10^6 and closer to 10^7, so answer should be C. Are you sure OA is correct? Director Status: Impossible is not a fact. It's an opinion. It's a dare. Impossible is nothing. Affiliations: University of Chicago Booth School of Business Joined: 03 Feb 2011 Posts: 920 Followers: 14 Kudos [?]: 346 [0], given: 123 Re: The product of all the prime numbers less than 20 is closest to which [#permalink] ### Show Tags 08 Mar 2011, 08:43 235711131719 6 * 35 * 7 * 143 * 340 i.e.(17 * 20) 6 * 245 * 143 * 340 6 * 2.5 * 1.4 * 3.4 * 10^6 15 * 5 * 10^6 75 * 10^6 7.5 * 10^7 GMAT Club Legend Joined: 09 Sep 2013 Posts: 13869 Followers: 589 Kudos [?]: 167 [0], given: 0 Re: The product of all the prime numbers less than 20 is closest to which [#permalink] ### Show Tags 16 May 2016, 18:58 Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________ Math Expert Joined: 02 Sep 2009 Posts: 37023 Followers: 7226 Kudos [?]: 96030 [0], given: 10706 Re: The product of all the prime numbers less than 20 is closest to which [#permalink] ### Show Tags 16 May 2016, 21:04 GMATD11 wrote: The product of all the prime numbers less than 20 is closest to which of the following powers of 10? a) 10^9 b) 10^8 c) 10^7 d) 10^6 e) 10^5 [Reveal] Spoiler: 235711131719 10211431719= any easy way because i cnt do above multiplication in less time. Thanks OPEN DISCUSSION OF THIS QUESTION IS HERE: the-product-of-all-the-prime-numbers-less-than-20-is-closest-135192.html The product of all the prime numbers less than 20 is closest to which of the following powers of 10 ? (A) 10^9 (B) 10^8 (C) 10^7 (D) 10^6 (E) 10^5 We should find the approximate value of 235711131719 to some power of 10. # of different approximations are possible. Approach #1: 25=10; 37=~20 (actually more than 20); 1119=~200 (actually more than 200); 1317=~200 (actually more than 200); $$235711131719\approx{1020200200=810^6}\approx{10^7}$$. Approach #2: 25=10 317=~50 (actually more than 50); 713=~100 (actually less than 100); 1119=~200 (actually more than 200) $$235711131719\approx{1050100*200}=10^7$$. OPEN DISCUSSION OF THIS QUESTION IS HERE: the-product-of-all-the-prime-numbers-less-than-20-is-closest-135192.html _________________ Re: The product of all the prime numbers less than 20 is closest to which [#permalink] 16 May 2016, 21:04 Similar topics Replies Last post Similar Topics: 7 The product of all the prime numbers less than 20 is closest to which 3 03 Jun 2011, 09:43 4 The product of all the prime numbers less than 20 is closest to which 6 14 Dec 2010, 17:18 9 The product of all the prime numbers less than 20 is closest to which 9 07 Dec 2010, 19:24 9 The product of all the prime numbers less than 20 is closest to which 7 26 Nov 2010, 07:20 5 The product of all the prime numbers less than 20 is closest to which 6 12 Jun 2009, 20:28 Display posts from previous: Sort by	2017-02-19 17:41:03	{"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": 0.535270631313324, "perplexity": 2555.1066101017545}, "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-2017-09/segments/1487501170186.50/warc/CC-MAIN-20170219104610-00326-ip-10-171-10-108.ec2.internal.warc.gz"}
http://www.formuladirectory.com/user/formula/338	HOSTING A TOTAL OF 318 FORMULAS WITH CALCULATORS ## Reynold's number The Reynolds number (Re) is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces. ## $\frac{i}{v}$ Here,i=inertial forces ,v=viscous forces ENTER THE VARIABLES TO BE USED IN THE FORMULA Similar formulas which you may find interesting. We have encountered an error during the execution of this part of the page.	2019-02-21 23:08:52	{"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": 0.8807036876678467, "perplexity": 592.9935486353284}, "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-2019-09/segments/1550247511174.69/warc/CC-MAIN-20190221213219-20190221235219-00024.warc.gz"}
https://mathematica.stackexchange.com/questions/187545/initial-values-in-estimatedprocess-with-gaussian-hiddenmarkovprocess	# Initial values in EstimatedProcess with Gaussian HiddenMarkovProcess I am using the following code to get states and transitions out of some traces: class = HiddenMarkovProcess[2, "Gaussian"]; hmm = EstimatedProcess[traces, class]; However for some traces (1,4, and 8), the function EstimateProcess doesn't find the right states: How can I set initial values to the EstimateProcess function? According to the documentation, it should work like EstimatedProcess[data,proc,{{p,p0},{q,q0},…}], but I don't know what to enter for p,q, etc. Edited: The full code is (Import traces (just a list of numbers)) files = {"trace1.txt","trace2.txt","trace3.txt","trace4.txt","trace5.txt","trace6.txt","trace7.txt","trace8.txt"} SetDirectory[NotebookDirectory[]]; traces = Import[#, "List"]& /@ files; (Find Hidden-Markov-Model) class = HiddenMarkovProcess[2, "Gaussian"]; hmm = EstimatedProcess[#, class] & /@traces (extract states and plot data) state = FindHiddenMarkovStates[traces[[#]], hmm[[#]]] & /@Range[1, Length[files]]; stateEffs = {hmm[[#, 3, 1, 1]], hmm[[#, 3, 2, 1]]} & /@Range[1, Length[files]] Efficencies[stateEffs_, stateNbr_] := stateEffs[[#]] & /@stateNbr effs = Efficencies[stateEffs[[#]], state[[#]]] & /@Range[1, Length[files]]; Show[ListLinePlot[traces[[#]]],ListLinePlot[effs[[#]], PlotStyle -> Directive[Orange]]] & /@Range[1, Length[files]] • Could you give the full code for producing those images? – C. E. Dec 8 '18 at 19:40 • I have added the full code. But I don't know how to provide the files which are just a list of numbers. – C.Gebha Dec 9 '18 at 21:22	2019-02-16 18:38:53	{"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": 0.22976689040660858, "perplexity": 6635.6102215722785}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247480905.29/warc/CC-MAIN-20190216170210-20190216192210-00526.warc.gz"}
https://www.physicsforums.com/threads/compute-the-integral-of-x-a-1-x-2-for-x-going-from-0-to-infinity.456469/	# Compute the integral of x^a / (1+x^2) for x going from 0 to + infinity Compute $$\int^{\infty}_0 \frac{x^{\alpha}}{1+x^2} dx$$ for some $$-1<\alpha<1$$. EDIT: This was slightly wrong. The hint given is that we can integrate from -p to p except for a small semi-circle around 0, and a large semicircle from p to -p, and choose a branch of $$z^{\alpha}$$. Wouldn't this in any case be the method for computing $$\int^{\infty}_{-\infty} \frac{x^{\alpha}}{1+x^2} dx$$? I'd appreciate some help in choosing a proper contour. I'd know how to integrate from -infinity to infinity, but from 0 I have no idea. Also, can I choose my branch for the logarithm arbitrarily? Last edited: $$\int^{\infty}_{-\infty} \frac{z^{\alpha}}{1+z^2} \, dz = \int^{\infty}_{0} (z^{\alpha} + (-z)^{\alpha})\frac{1}{1+z^2} \, dz$$ $$(-z)^{\alpha} = e^{\pi i \alpha}z^{\alpha}$$ $$\int^{\infty}_{-\infty} \frac{z^{\alpha}}{1+z^2} dz = \left(1 + e^{\pi i \alpha}\right)\int^{\infty}_0} \frac{z^{\alpha}}{1+z^2} dz$$	2021-09-28 13:53:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.9788910150527954, "perplexity": 446.5616258284551}, "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/1631780060803.2/warc/CC-MAIN-20210928122846-20210928152846-00233.warc.gz"}
http://www.solipsys.co.uk/new/Chops.html?InternalLinks	# Chops Chops is a Juggling Pattern that is an exaggeration of throwing every ball Und erThe Arm. When performed it looks best if the ball is displayed during the carry. Rather than simply holding it in the hand, thereby partially obscuring the ball, hold it so it can be seen clearly by the audience. # Contents There were no headings in the main text so there	2018-12-12 16:46:31	{"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": 0.840190589427948, "perplexity": 2358.627972470368}, "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-2018-51/segments/1544376824059.7/warc/CC-MAIN-20181212155747-20181212181247-00415.warc.gz"}
http://clay6.com/qa/42431/if-a-non-metal-is-added-to-the-interstitial-sites-of-a-metal-then-the-metal	Browse Questions # If a non-metal is added to the interstitial sites of a metal then the metal becomes Can you answer this question?	2016-12-11 06:02:43	{"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": 0.8005569577217102, "perplexity": 2020.1285226071245}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2016-50/segments/1480698544140.93/warc/CC-MAIN-20161202170904-00191-ip-10-31-129-80.ec2.internal.warc.gz"}
http://motls.blogspot.co.uk/2011/02/why-dana1981-hasnt-proved-climate.html	Thursday, February 24, 2011 ... ///// Why Dana1981 hasn't proved climate disruption Dana1981 is a 29-year-old Prius driver and the owner of several other alternative vehicles who has mistakingly received a bachelor degree in astrophysics and a master degree in physics, so he or she became a self-described environmental scientist who is "passionate" about the climate hysteria. Clearly, such people shouldn't be admitted as college students because they're incapable of rational thinking. The presence of people like him dramatically cripples the intellectual atmospheres at the world's universities. Below, we will demonstrate this point in quite some detail. In his text for Skeptical Science and Climate Progress, he crisply demonstrates why the believers in a climate threat are analogous to the Islamic fundamentalists: How we know recent global warming is not natural As the title indicates, Chicken Little Dana1981 is convinced that his text contains a proof of man-made global warming. Instead, it contains a few kilobytes of excretions of a brain in the middle of its decay. Dana1981 addresses the text to two people that he must believe are the only climate skeptics in the world - Richard Lindzen and Roy Spencer. The existence of the remaining 4+ billion skeptics in the world, including tens of thousands of science PhDs, is being denied. But let's begin with the actual content... First, Dana1981 claims that Roy Spencer's challenge “Show me one peer-reviewed paper that has ruled out natural, internal climate cycles as the cause of most of the recent warming in the thermometer record.” "is problematic for a few reasons". What are those reasons, except for the fact that it's an inconvenient truth for Dana1981 that no such paper exists? Firstly, the fact that research has not ruled out a hypothesis does not mean the hypothesis necessarily has any validity. Wow. Oh, really? One may show that a hypothesis has no validity without ruling it out? One would think that this statement is tautologically invalid. Sorry, you can't really falsify a hypothesis without falsifying it. For example, there have been no peer-reviewed papers ruling out leprechauns as the cause of most of the recent global warming, either. Climate hacks must have missed it - they "accidentally" manage to miss all proofs that their proclamations are lies - but as I have demonstrated, there has been a peer-reviewed paper that has ruled out not just leprechauns but all sprites as the primary driver of the climate change. More seriously, it is preposterous to compare leprechauns to the clouds, solar activity, volcanoes, ocean cycles, and cosmic rays because all these phenomena - except for leprechauns - have been driving the Earth's climate for 4.7 billion years and are still doing so. Only a complete lunatic who hasn't seen the blue skies since the moment when he was confined in a psychiatric asylum (or a global warming-promoting organization) may think that e.g. cloud patterns are as unimportant for the climate as the leprechauns. Dana1981 continues: But perhaps more importantly, our understanding that humans are causing global warming is not based on just one scientific study, but rather a very wide range of scientific evidence. Well, an important difference between a religion followed by ill-informed people - such as global warming - and science is that in science, a single robust argument or observation cannot be replaced by a ton of garbage, propaganda, emotions, and lies. That's why Roy Spencer has asked about one relevant paper rather than a thousand of irrelevant papers that are claimed to contain the "right message" in the form of homeopathic drugs (or, more typically, in the form of the political preferences of the authors). Fine, so what is the "very wide range of evidence", as Dana1981 pompously calls this ton of junk? For example, scientists have measured the amount of heat being re-directed back towards the Earth’s surface due to the increased greenhouse effect. That's right. The "redirected" (and absorbed and re-emitted!) heat is subtracted from the outgoing heat. The outgoing heat was measured e.g. by the ERBE experiment and the relationship of this heat flux with the temperature increase was studied by Lindzen and Choi (2009). The resulting sensitivity coming from this paper was 0.5 °C, and even when various conceivable mutations of the methods have been tried later, the sensitivity remained well below 2 °C. So this "line of evidence" hasn't worked for Dana 1981, has it? The problem with this position is that there are many lines of evidence that the planet will warm between 2 and 4.5 degrees Celsius (°C) if the amount of carbon dioxide (CO2) in the atmosphere doubles. Well, the only problem is that all these "lines of evidence" are broken, and we will see about 10 more examples. There's no scientific evidence whatsoever that the climate sensitivity exceeds 2 °C and Dana 1981 must know that because if he knew about such a paper, he would tell us what the paper (or calculation) is and where it has been published. OK, what are the other "lines of evidence" that global warming is man-made and climate sensitivity exceeds 2 °C? For example, some scientists have studied the climate response to recent large volcanic eruptions, which can have a measurable impact on global temperatures. This is really cute. Volcanic eruptions are not man-made and their effect - which may influence the global temperature by half a degree (almost equal to the whole 20th century warming) for five years, like Mr Pinatubo did in 1991 - is, on the contrary, a proof that the natural drivers are essential to understand the climate and the climate change. Moreover, the dominant influence of the volcanoes has nothing to do with carbon dioxide. Sulfuric gases, which are the key players emitted by the volcanoes, combine with water in the stratosphere, create sulfuric acid droplets, and they reflect or scatter a part of the solar radiation back to the outer space, thus cooling the troposphere. Those effects have nothing to do with the mankind and with carbon dioxide, and because they occur at a different place of the atmosphere, they're not useful to quantify the strength of the greenhouse effect, either. So this line of evidence has also been a complete failure, hasn't it? Let's go on. Other studies have examined how the global temperature has changed in response to changes in solar activity. Well, that's right. Search for solar-activity and climate on Google Scholar. The first paper that shows up, one in Science by Friis-Christensen and Lassen from 1991, shows that Northern Hemisphere land air temperature has been closely correlated with the 11-year sunspot cycle in the last 130 years. This paper has 585 citations. Another paper, by Judith Lean, Juerg Beer, and Raymond Bradley from Geophysical Research Letters 1995 - it has 639 citations now - looked at the Maunder Minimum and related effects and concluded that just the solar forcing is responsible for about 1/2 of the warming since 1860; the relevant correlation coefficient was found to be 0.86, very high. The rest - 0.27 °C or so - could have been caused by many other things. I could go on and on and on. This line of evidence has spoken against man-made global warming as well, hasn't it? Some other research has compared CO2 and global temperature changes over the past thousand years, and tens of thousands of years, and hundreds of thousands of years, and even millions of years ago. That's right. The Roman Empire has included green Alps - probably warmer than the present - without any CO2. I could give detailed references again but it's well-known that at the time scale of tens of thousands of years, the changes are dominated by the Milankovitch cycles, i.e. irregularities of the orbital cycles. The theory has been fixed by Roe so that it works really well now. At the time scale of a million of years, the climate has apparently switched its regimes qualitatively: the glaciation cycles only existed in the recent millions of years. CO2 is not the cause. When it comes to tens of millions of years, even continental drift becomes important. Shaviv and Veizer have argued that an impressive correlation suggests that the cosmic rays were the primary drivers as seen from those very long-time data as well. No influence of carbon dioxide changes has been shown to be dominant in any of those time scales, despite the fact that the temperature changes in this history have been much larger than the temperature changes in the 20th century. So this line of evidence has destroyed Dana1981 as well, hasn't it? We can even compare how the temperature has changed over the past century to human-caused atmospheric CO2 changes. In every case we arrive at this same climate sensitivity range of 2 to 4.5°C, and the most likely value is 3°C for a doubling of atmospheric CO2. Even when we assume that the whole change of the global mean temperature in the HadCRUT3 record from 1850 is due to the rising CO2, and many papers above - and hundreds of others - argue that it's not the case and other effects have positively contributed to the temperature - regression implies that the climate sensitivity equals 1.66 °C. This is a totally straightforward calculation and the statement that it gives a sensitivity in the 2-4.5 °C interval is refuted. Moreover, it is statistically impossible for a large number of "lines of evidence" to end up with this strikingly imprecise value of the climate sensitivity. If the sensitivity were in this window, an increasing number of measurements (and their improving quality) would inevitably reduce the error margin. Moreover, one method to determine the right value would almost certainly be much more accurate than others and it would make its competitors redundant and useless. The reason why tens of billions of dollars were not enough to reduce the error in the estimate of the climate sensitivity well below 100% is that the right value is not in this interval. The mathematical mechanism explaining the proposition in the previous sentence was crisply articulated by Lindzen and Choi. If we take the lower end of this range, even a 2°C climate sensitivity would mean that humans have been responsible for more than half of the global warming over the past century. This is complete nonsense. If the sensitivity were as high as 2 °C, the human contribution to the 20th century warming would have to be about 150-200 percent of the observed value, and the other contributions would have to be opposite in sign. So in order for Spencer and Lindzen to be right, all of these different lines of evidence which are in agreement with the likely range of climate sensitivity would all have to be somehow wrong, and all biased high. Not an impossibility, but certainly not a likely scenario, either. Just to remind you what we have discussed so far, all of the "lines of evidence" that Dana1981 has enumerated show that his proposition about the high climate CO2 sensitivity is incorrect. There are also many “fingerprints” of human-caused global warming. That's right. Unfortunately, once again, they show that most of the "global warming" couldn't have been caused by CO2. This fingerprint clearly disagrees: the observed dependence of the warming trend on the latitude and altitude is totally different than one predicted by the greenhouse-dominated models (which predict the hot spot in the middle, among other wrong features). For example, as the Earth’s surface and lower atmosphere have been warming, the upper atmosphere has been cooling. Well, the graph above shows that the dependence on the altitude is totally wrong and this discrepancy itself is enough to falsify the greenhouse models of the recent warming. This point was discussed in detail e.g. in the (peer-reviewed blah blah) paper by Douglass, Christy, Pearson, and Singer. The CO2 models are wrong once again. By the way, in the comment section, Dana1981 claims that "all models" have to suffer from the wrong "tropical tropospheric hot spot" prediction because of some vapor argument. That's obviously wrong. Richard Lindzen's iris hypothesis and thermostat models analyze the very same H2O to conclude that the tropical troposphere temperatures remain unusually constant - which matches the observations. It't just not true that there is any "tie" here: Lindzen's models win, the CO2 models are excluded by this test. There are not many mechanisms which can explain these observations, but they are precisely what we would expect to see from human-caused global warming. This is bullshit, too. A vast majority of forcings actually produces the opposite temperature changes in the troposphere and the stratosphere (the example of clouds will be discussed below). So their being opposite has no specific implications for the CO2 model. However, the comparison of the (mid) troposphere with the near-surface temperatures is an issue that has something to say about the CO2 hypothesis, and this line of evidence also shows that the idea of CO2 as a dominant driver doesn't agree with the observations. As the concentration of greenhouse gases in the lower atmosphere increases, they effectively trap more and more heat in this lower layer, causing it to warm and causing the layers above to cool. But this sentence is not a verified fingerprint. It's just a parroted proclamation from the global warming religion - a hypothesis that has been falsified about 10 times in the text above. It has been demonstrated that the effect above, whether it looks attractive or not, is not sufficient - and maybe not even necessary - to explain most of the climate change in the real world. Another human “fingerprint” is the higher rate of warming at night than during the day. Except that this fingerprint works in the wrong, opposite direction, too: it also shows that the CO2 model is wrong. The observed nighttime temperatures have experienced a higher warming trend than the daytime temperatures. But the greenhouse model predicts exactly the opposite. The Earth emits more thermal radiation during the day because it's somewhat warmer during the day. So the increased greenhouse effect - which prevents a part of the outgoing thermal radiation to get to the outer space - also has a greater warming influence during the day. John Cook, while arguing that this "fingerprint" has the right sign, made the breathtakingly stupid mistake of assuming that the Earth's thermal radiation is turned off during the day and it only turns on at night. Holy crap! ;-) The stupidity of these green people exceeds the wildest imagination. At any rate, they're clearly just looking for sentences that look like confirmations of whatever belief they hold dear: they don't care that all these confirmations and justifications are as absurd as Cook's assumption that the "Earth doesn't thermally radiate at noon." Those people are hopelessly misled. However, [Spencer's cloud cover change] hypothesis cannot explain the “fingerprints” describe above. It explains some of them - and unlike the CO2 model, it doesn't produce any obvious contradictions. In particular, the changing cloud-cover theory explains why there's no "hot spot" 10 km above the equator. A decrease in cloud cover would not cause the upper atmosphere to cool. Of course that it would. The upper atmosphere is warm partly because the solar radiation may be absorbed not only when it arrives from the Sun but also after it's reflected from the clouds. If some clouds disappear, the radiation that is scattered back to the upper atmosphere decreases as well, and the upper atmosphere cools. As I mentioned, the opposite signs for the troposphere and stratosphere are pretty universal for almost all climate drivers. Nor would it cause nights to warm faster than days – quite the opposite. Cloud reflectivity only plays a significant role during the day when being bombarded by sunlight. This argument is based on two oversimplified assumptions, namely that clouds always have a cooling effect, and that cloud formation has been changing uniformly throughout the day and night. In reality, clouds also cause some greenhouse effect (and warming) and the change of the patterns may be correlated with the daytime or nighttime. I don't claim to have a test of all fingerprints and I don't claim that one theory or another has passed all of them but the oversimplified proclamation above is surely not enough to falsify the model. Dana1981 and all of his soulmates clearly use double standards for the falsification and verification of claims that they like and those that they don't like. But without scientific integrity, you can't get anywhere in the real science - although you may get to the very top at the environmental movement. Dr. Spencer also suggested in his blog post that the “null hypothesis” should be that global warming is caused by natural factors. A null hypothesis is basically the default assumption which a scientific study sets out to disprove. It’s true that until recently, global warming (and cooling) has been caused by natural factors. However, even natural climate changes must have a physical mechanism causing them. And they surely have - and our theory already makes a lot of sense. However, it's not guaranteed that science immediately understands everything. It doesn't. If a theory that MN causes some change remains incomplete, it's not a proof that the right theory has to be XY which seems totally incompatible with the evidence. It's still more likely that the right explanation is MN. Scientists have investigated these natural mechanisms (the Sun, volcanoes, the Earth’s orbital cycles, etc.), and they simply cannot explain the global warming over the past century. This is just bullshit. Just the solar paper by Lean et al. has showed that 1/2 of the overall warming could be caused by the solar forcing itself. Other papers show that the rest of the overall change may be caused by many other natural factors. And when it comes to the detailed year-to-year variations, it's pretty clear that they're explained predominantly by natural factors as well - because the CO2 forcing looks like a simple monotonic function while the temperature graph surely looks complicated. We don't understand the climate perfectly and changes of the temperature by +-0.5 °C per century remain beyond our precision and abilities. Spencer’s new hypothesis – that some unknown mechanism is causing cloud cover to change, which in turn is driving global temperatures – is a new idea with very little supporting evidence. The evidence is actually at least as robust as the evidence for the CO2 model. But more importantly, Spencer's model hasn't been falsified in the same sharp way as the greenhouse-dominated model have been falsified. Conversely, our understanding that human greenhouse gas emissions are driving global temperatures has a proverbial mountain of supporting evidence. Too bad that every single "line of evidence" that Dana1981 has offered was a negative one. Skeptics like Spencer and Lindzen believe that the default assumption should be one which requires that a very large body of scientific evidence is wrong. It doesn't "require": it "implies". And it is not a large body of scientific evidence; instead, it is a large body of pseudoscientific gibberish that has largely replaced scientific research since the time when Prius-driving idiots began to spread the hysteria and steal billions of dollars from the taxpayers for their bogus research. But even if there were a large body of really scientific, honestly written papers, they may still be wrong and it's enough to find one simple argument or observation to prove that thousands of papers are wrong. The climate charlatans such as Dana1981 don't understand these basic facts about science. The only alternative hypothesis they have put forth cannot explain the many empirically-observed “fingerprints” which are consistent with human-caused global warming. Again, all the fingerprints have spoken against the CO2-dominated model and against the high CO2 climate sensitivity. And it's not true that "they" (skeptics) have proposed just one and "only" alternative hypothesis. Although Spencer’s unspecified “natural internal cycle” hypothesis has not been explicitly disproved, there is a very low likelihood that it is correct. No, likelihood can't be defined as the degree to which an idea is convenient for biased and dishonest green activists. In science, likelihood has to be calculated or estimated by rational arguments. And Dana1981 hasn't provided us with a glimpse of evidence supporting the bold assertion above. In fact, it turned out that all the evidence was speaking against his claims. This often happens in science: for example, John Bell wanted to disprove the probabilistic quantum mechanics and show that there had to be hidden variables. Instead, the theorem he found had exactly the opposite consequences. The final arguments is what matters in science; the prejudices of "passionate" crooks don't. For this reason, we should operate under the assumption that humans are causing dangerous global warming – an assumption which is supported by a very large body of evidence – until the skeptics can provide solid reason to believe that this scientific theory is wrong. No, we shouldn't. No honest person should. Operating under an assumption that has been showed incorrect contradicts the basics of the scientific method. One could work with a "working hypothesis" that the bold proposition above is correct. But when he is scientist, he must abandon this working hypothesis once it is proved wrong. And that's the memo. snail feedback (7) : Don't be too hard on the young fella. I don't think he means any harm. He's just gotten a bit too fanatic about this green stuff. He was commenting for awhile at NTZ, but I had to boot him out for 48 hours after he had a puberty fit.Since then he hasn't returned. Do you think he has any of them photo gallery prints as posters in his bedroom? :) Are we sure that this wasn't written by a tongue-in-cheek sceptic to embarrass the warmists? Regarding the article by Lindzen and Choi (2009), you seem to know about more recent work noting flaws in LC09: "The resulting sensitivity coming from this paper was 0.5 °C, and even when various conceivable mutations of the methods have been tried later, the sensitivity remained well below 2 °C" See the following paper. http://www.cgd.ucar.edu/cas/Staff/Fasullo/refs/Trenberth2010etalGRL.pdf These authors note: "As shown here, the approach taken by LC09 is flawed, and its results are seriously in error." And they calculate a sensitivity of 2.3K. They go on to conclude: "However, it is not appropriate to use only tropical SSTs and TOA radiation for feedback analysis as the trans- ports into the extratropics are substantial. Any feedback analysis must also recognize changes in ocean heat storage and atmospheric energy transport into and out of the tropics which are especially large during ENSO events. While the tropics are important in climate sensitivity, values of the latter based on only tropical results are misleading." Why cite LC09, when the work is found to be inadequate and misleading? Regarding Dana1981, I understand you deleted his comments from your blog: http://www.skepticalscience.com/motley-cruel.html Why delete his comments? Best regards, Don reader Luboš Motl said... Dear Don, the reason why I delete some comments is that they are repetitive spam containing nothing new, useful, or true, written by trolls who just want to spread garbage all over the Internet and who show no plans to actually become rational, and a good reason why your paper is to be ignored is that it was written by a person who doesn't have any scientific integrity, and his collaborators. reader Brian G Valentine said... Lubos, there are probably 29 Dana's out there, now you only have (29)-1 to go. Meaning, there's no point in "debunking" any of them. The probability of one of them saying something meaningful is about 1/(2**9), so I don't think it's a good use of your time. reader Luboš Motl said... Dear Brian, I agree with your point. Well, if there were 512 such Danas, I would be ready to defeat each of them in the same indisputable way as I did with Dana1981. However, the number of such folks is higher, and moreover, the victory doesn't mean anything because he will go elsewhere and spread the same lies about the climate, the particular points, as well as the staggering defeat he has experienced. Those people have no integrity and they're way too numerous, so indeed, a peer-like debate with them doesn't make much sense. Cheers LM	2013-12-06 00:35:37	{"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": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.56492680311203, "perplexity": 1282.5283283856404}, "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-2013-48/segments/1386163048688/warc/CC-MAIN-20131204131728-00016-ip-10-33-133-15.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/1761628/linear-algebra-with-functions	# Linear Algebra with functions Basically my question is - How to check for linear independence between functions ?! Let the group $\mathcal{F}(\mathbb{R},\mathbb{R})$ Be a group of real valued fnctions. i.e $\mathcal{F}(\mathbb{R},\mathbb{R})=\left\{ f:\mathbb{R}\rightarrow\mathbb{R}\right\}$ Let 3 functions $f_{1},f_{2},f_{3}$ be given such that $\forall x\in\mathbb{R}\,\,\,f_{1}=e^{x},\,\,f_{2}=e^{2x},\,\,f_{3}=e^{3x}$ $W=sp(f_{1},f_{2},f_{3})$ what is $dim(W)$ ? How to approach this question ? (from a linear algebra perspective) I know that $\forall x\in\mathbb{R}\,\,\,W=\alpha e^{x}+\beta e^{2x}+\gamma e^{3x}$ And to get the dimension I need to find the base of $W$ so I need to check whether the following holds true : $\forall x\in\mathbb{R}\,\,\alpha e^{x}+\beta e^{2x}+\gamma e^{3x}=0\,\Leftrightarrow\,\alpha,\beta,\gamma=0$ However when $x=0$ I get $\alpha+\beta+\gamma=0$ which leads to infinite amount of solutions. How to approach this question ? • Hint: You want it to be zero for EVERY $x$, not only for $x = 0$. – amcalde Apr 27 '16 at 19:27 • I know that I want it, I dont know whether I can reach it, or how to verify that I can never reach it ? – Pavel Penshin Apr 27 '16 at 19:28 • I like to use this method: math.stackexchange.com/a/269694/8157 but the linked Q&A contains many others. – Giuseppe Negro Apr 27 '16 at 20:22 You need to check if the functions are independent, as you said. A way to go about this, which that ties it in with things you likely know is to evaluate it at several points, as you did for $x=0$. You get one condition for $x=0$. You get another condition for $x=1$ and still another one for $x=2$. Each will allow more than one solution, but they'll only have one common solution, which is what you are after. • $$\begin{array}{cc} x=0 & \alpha+\beta+\gamma=0\\ x=1 & \alpha e+\beta e^{2}+\gamma e^{3}=0\\ x=2 & \alpha e^{2}+\beta e^{4}+\gamma e^{6}=0 \end{array}$$ $\rightarrow\begin{bmatrix}1 & 1 & 1\\ e & e^{2} & e^{3}\\ e^{2} & e^{4} & e^{6} \end{bmatrix}\rightarrow$ I found this matrix, the determinant is not zero thus there is only 1 solution which means that $\alpha,\beta,\gamma=0$ for $x=1,2,3$ how does that helps ? – Pavel Penshin Apr 27 '16 at 19:54 • Note that you need $\alpha, \beta, \gamma$ that work for all $x$ at the same time (they must not depend on $x$). You just showed that for $\alpha, \beta, \gamma$ to work for $x=0,1,2$ you already only have the unique choice all $0$. So you are done. – quid Apr 27 '16 at 20:23 Write $$\alpha e^x + \beta e^{2x} + \gamma e^{3x} = 0$$ You can go ahead and cancel out a positive number like $e^x$ so: $$\alpha + \beta e^{x} + \gamma e^{2x} = 0$$ Suppose you have some solution for this with $\alpha$, $\beta$, $\gamma$ not all zero. Then, as you say $$\alpha + \beta + \gamma = 0\qquad \qquad (1)$$ Because this must be true at $x = 0$ but it must also be true at $x = \ln n$ which gives: $$\alpha + \beta n + \gamma n^2= 0\qquad \qquad (2)$$ for every $n > 1$. It should be clear that this is unsolvable except when they are all zero. But to press the point I'll continue. Substituting in $(1)$ gives $\alpha = -\beta - \gamma$, which we can plug into $(2)$ to get $$\beta (n-1) + \gamma (n^2 - 1)= 0$$ which must be true for all $n > 1$. Now put, say, $n = 2$ and $n = 3$ to get the pair of equations: $$\beta + 3 \gamma = 0 \qquad 2\beta + 8\gamma = 0$$ This solves for $\beta = \gamma = 0$. So your functions are proved to be linearly independent. Hint: let $e^x=y$, $e^{2x}=y^2$, $e^{3x}=y^3$ you have: $\alpha y +\beta y^2+ \gamma y^3=0$ where the $0$ at RHS is the zero polynomial. Now: when a polynomial is the zero polynomial? In general: The $0$ at RHS is the neutral element for the sum of functions in the vector space, not simply the number $0$ and this means that it is the function $f(x)=0\quad \forall x \in \mathbb{R}$. • $$\begin{array}{cc} x=0 & \alpha+\beta+\gamma=0\\ x=1 & \alpha e+\beta e^{2}+\gamma e^{3}=0\\ x=2 & \alpha e^{2}+\beta e^{4}+\gamma e^{6}=0 \end{array}$$ $\rightarrow\begin{bmatrix}1 & 1 & 1\\ e & e^{2} & e^{3}\\ e^{2} & e^{4} & e^{6} \end{bmatrix}\rightarrow$ I found this matrix, the determinant is not zero thus there is only 1 solution which means that $\alpha,\beta,\gamma=0$ for $x=1,2,3$ how does that helps ? what is RHS ? googling didnt help :( – Pavel Penshin Apr 27 '16 at 19:57 • The key fact is that in $\alpha f_1+\beta f_2+\gamma f_3=0$ The $0$ is the zero function i.e. a function that is null for all values of $x$ in the domain. Your linear system shows that you can find values for $\alpha, \beta, \gamma$ such that $\alpha f_1+\beta f_2+\gamma f_3=0$ is true for some value of $x$ but not for all the possible values. – Emilio Novati Apr 27 '16 at 20:06 Hint: Use Wronskian and show that the Wronskian-Determinant does not vansish. You have to prove $$\forall x\in\mathbb{R}:\alpha e^{x}+\beta e^{2x}+\gamma e^{3x}=0\Leftrightarrow\alpha,\beta,\gamma=0,$$ but I think the quantifier applies only to the part on the left side of the $\Leftrightarrow$, like this: $$\left(\forall x\in\mathbb{R}:\alpha e^{x}+\beta e^{2x}+\gamma e^{3x}=0\right) \Leftrightarrow\,\alpha,\beta,\gamma=0.$$ So for example $\alpha = -1, \beta = 1, \gamma = 0$ satisfies $\alpha e^{x}+\beta e^{2x}+\gamma e^{3x}=0$ when $x=0$, but it doesn't satisfy the equation for all values of $x$. If you had to prove $$\forall x\in\mathbb{R}:\left(\alpha e^{x}+\beta e^{2x}+\gamma e^{3x}=0 \Leftrightarrow\,\alpha,\beta,\gamma=0\right)$$ then you would be in trouble, because that statement is not true; but that's not how we prove independence of the functions, so you don't need to worry about that. You need to show the three vectors are linearly independent. In this case I would use this trick; so that you don't need to worry about them being functions and the equality to hold for every value of $x$. If you consider $D: \mathcal{F} \rightarrow \mathcal{F}$, the derivative operator, is an endomorphism in $\mathcal F$ (i.e. a linear map from $\mathcal{F}$ to itself). The derivatives of the three functions are $$Df_1=De^x=e^x=f_1$$ $$Df_2=De^{2x}=2e^{2x}=2f_2$$ $$Df_3=De^{3x}=3e^{3x}=3f_3$$ So $f_1,f_2,f_3$ are eigenvectors of $D$, with eigenvalues $\lambda_1=1, \lambda_2=2, \lambda_3=3$, respectively. Since $f_1, f_2, f_3$ are eigenvectors with distinct eigenvalues of the same endomorphism $D$, they are linearly independent so they form a base for $W$ and $\text{dim}W=3$. If you have $$\alpha e^{x} + \beta e^{2x} + \gamma e^{3x} \equiv 0,$$ Then you can apply the derivative operator $D$ to obtain \begin{align} 0 & \equiv (D-2)(D-3)\{\alpha e^{x} + \beta e^{2x} + \gamma e^{3x}\} \\ & = (1-2)(1-3)\alpha e^{x}. \end{align} Therefore $\alpha=0$. Then you can apply $(D-1)(D-3)$ in order to conclude that $\beta=0$. Similarly $\gamma =0$. So $\{ e^x,e^{2x},e^{3x} \}$ is a linearly independent set of functions, which means that the dimension of $W$ is $3$. • Thanks for your answer I did not quite understand your notation for the derivative operator. also, can this answer be obtained in another way ? (without Wronskian as well ) – Pavel Penshin Apr 27 '16 at 21:16 • @user313448 Have you studied differential equations where they use the annihilator method? That's what I'm using. If not, you can do this with limits. Multiply by $e^{-3x}$ and let $x\rightarrow\infty$ in order to obtain $\gamma =0$. Then you can isolate $\beta=0$ and, finally, you isolate $\alpha=0$ with no limits. – DisintegratingByParts Apr 27 '16 at 21:34	2019-06-27 12:23:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9471381902694702, "perplexity": 138.80846841888683}, "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/1560628001138.93/warc/CC-MAIN-20190627115818-20190627141818-00420.warc.gz"}
https://math.eretrandre.org/tetrationforum/showthread.php?tid=1328	• 0 Vote(s) - 0 Average • 1 • 2 • 3 • 4 • 5 On my old fractional calculus approach to hyper-operations JmsNxn Ultimate Fellow Posts: 977 Threads: 114 Joined: Dec 2010 05/24/2021, 02:02 AM (This post was last modified: 06/09/2021, 02:35 AM by JmsNxn.) Hey, guys. I finally decided to properly compile my old research on fractional calculus and hyper-operators. This is almost exactly what I wrote 5 years ago on here; but it's finalized much better. It's done quick and concise. Some people at U of T told me to just rewrite it from scratch and compile everything with a neat little bow. You guys may be interested. I removed the attachment and placed the arXiv link. I've made some minor edits; I changed some of the details; and I included more references; and tried to flesh out the proofs more. https://arxiv.org/abs/2106.03935 MphLee Long Time Fellow Posts: 321 Threads: 25 Joined: May 2013 05/25/2021, 07:32 PM Hi, I like this summary! I was already familiar with the outline of you frac-calc approach from 2015 but now this seems really tidy and easier to follow. Im happy to see those two commutative diagrams! xD About the logical structure, I just skimmed it... I need to study it line by line. But I find some passages unclear or interesting so I hope you can help me (and some typos). 1 Introdution p.2; "As a formal sequence, we can call a hyper-operator chain," that object is indeed more general than an Hyperoperation sequence. Regardless of the initial conditions "chain" seems a nice name. There at U of T have you received some comments on that "chain" equation? 2 Fractional derivative p.3; after the exp-fixpoint eq.: "to complex values, and [deride]" p.4; $S_\theta$ is missing the point 0 right? The area enclosed by the two rays $t(\cos(\theta)+\sin(\theta)i$ and $t(\cos(\theta)-\sin(\theta)i$ where t>0 and the point 0 removed? Are arcs assumed to be injective, no winding and do not cross over themselves? Sure, to be precise you consider an arc ad just the subset to integrate over, the image of the parametrization, so that different parametrizations can map [0,+\infty) to the same arc but running on it on different velocities. p.4; after you define the set of endofuntion boldface ${\mathbb S}_\theta$ you say: "Then there exists a correspondence between functions F(z)". A correspondence between those bounded F and what? In other words in that line you are referring to the correspndence between boldface ${\mathbb S}_\theta$ and boldface ${\mathbb E}_\theta$ you define at page 6 ? p.5; let's double check my understanding. Theorem 2.1 (Euler) takes $f\in {\mathbb S}_\theta$ and maps it to ${{\mathfrak E}_w[f]}\in {\mathcal Hol}({{\mathbb C}_{\Re<1}},{\mathbb C})$, i.e. where $\Gamma(z){{\mathfrak E}_w[f]}(-z)={\sum_{n=0}^\infty}f^{(n)}(w)\frac{(-\gamma(1))^n}{n!(n+z)}+\int_{\gamma[1,\infty)}f(w-y)y^{z-1}dy$ and ${{\mathfrak E}[f]}:{{\mathbb C}_{\Re (z)<1}}\to{\mathbb C}$. Theorem 2.2 (Ramanujan) takes $H\in {\mathbb E}_\theta$(?) and maps it to ${{\mathfrak R}[H]}\in {\mathbb C}^{{\mathbb C}}$, i.e. where ${{\mathfrak R}[H]}(w)={\sum_{n=0}^\infty}H(n)\frac{w^n}{n!}$ and ${{\mathfrak R}[H]}(w):{\mathbb C}\to{\mathbb C}$ and claims (a)$({\mathfrak E}_0[{\mathfrak R}[H]])(z)=H(z)$ and (b)$({\mathfrak E}_w[{\mathfrak R}[H]])(z)={\mathfrak R}[H\circ S^z](w)$ (S is the successor so S^z(n)=z+n) Now we know where the trasforms take value but not exactly where they land (can we compose them?): Thm 2.1 ${{\mathfrak E}_w:{\mathbb S}_\theta\to {\mathcal Hol}({{\mathbb C}_{\Re<1}},{\mathbb C})$ Thm 2.1 ${{\mathfrak R}:{\mathbb E}_\theta\to{\mathbb C}^{{\mathbb C}}$ and (a)${\mathfrak E}_0\circ {{\mathfrak R}=id$ Observation: (b) implies trivially (a) as the tail of the series vanish setting w=0 and we keep the leading coefficient H(z). I can't fully parse (b) yet. I can say that (a) is possible iff we can feed Ramanujan into Euler, i.e. ${{\mathfrak R}:{\mathbb E}_\theta\to{\mathbb S}_\theta$: in fact you prove this later p.7; By theorem 2.3 ${{\mathfrak E}_0:{\mathbb S}_\theta\to{\mathbb E}_\theta$. When you say "and conversely" do you mean that also ${{\mathfrak R}:{\mathbb E}_\theta\to{\mathbb S}_\theta$ holds right? Why don't you have to prove theorem 2.3 BEFORE you can even claim in theorem 2.2 that you can apply ${{\mathfrak E}_0}$ to$f(w)={{\mathfrak R}[H]}(w)$? In fact if thm 2.3 is true then we can apply thm 2.2 eq. (a), i.e. ${\mathfrak E}_0\circ {{\mathfrak R}=id_{{\mathbb E}_\theta}$. This equation alone implies that: - ${\mathfrak E}_0$ is surjective (every function in boldface E is de differintegral at w=0 of some boldface S function); - ${\mathfrak R}$ is injective (if two boldface E functions define the same auxilliary function then they are the same function). But thm 2.4 also add that those two functions are also inverse hence ${{\mathbb E}_\theta}\simeq {{\mathbb S}_\theta}$ are in bijection Question 1: for which $\theta$ those spaces are in bijection? Question 2: Do this bijection preserve some stucture? Idk... are those functions paces closed under piecewise sum, scalar multiplication, piecewise multiplication, do have a metric or topological structure (a system of open sets), a norm? Question 3: take $\theta<\kappa$ we have $S_\theta\subseteq S_\kappa$. What is the relationship between ${\mathbb S}_\theta$ and ${\mathbb S}_\kappa$ o between ${\mathbb E}_\theta$ and ${\mathbb E}_\kappa$? Asap I'll go on the other sections. Regards MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ JmsNxn Ultimate Fellow Posts: 977 Threads: 114 Joined: Dec 2010 05/25/2021, 10:47 PM (This post was last modified: 05/25/2021, 10:50 PM by JmsNxn.) (05/25/2021, 07:32 PM)MphLee Wrote: Question 1: for which $\theta$ those spaces are in bijection? Question 2: Do this bijection preserve some stucture? Idk... are those functions paces closed under piecewise sum, scalar multiplication, piecewise multiplication, do have a metric or topological structure (a system of open sets), a norm? Question 3: take $\theta<\kappa$ we have $S_\theta\subseteq S_\kappa$. What is the relationship between ${\mathbb S}_\theta$ and ${\mathbb S}_\kappa$ o between ${\mathbb E}_\theta$ and ${\mathbb E}_\kappa$? Asap I'll go on the other sections. Regards Hey, Mphlee, I'll answer these questions to the best of my ability. The people at U of T called it a hyper-operator chain; that's not my terminology. I know it can be a tad confusing for this forum; but that's what they call it ; so I stuck with the terminology. You don't need to include zero; but go right ahead and include it. As these functions are presumed to be entire; the integral at zero is always defined. We are only worried about the behaviour as $w \to \infty$ with $\|\arg(w)\| \le \theta$ to ensure the integral converges. As to what kind of arc; they can self intersect; they can loop; they can do what ever; so long as the initial point is $0$ and the end point $\infty$ and they are contained in $S_\theta$. Since these functions are holomorphic, and $S_\theta$ is simply connected; the integral only depends on the initial point and the end point. Yes, by correspondence I meant $\mathbb{E}_\theta$ is virtually the same as $\mathbb{S}_\theta$; one takes derivatives, the other shifts the variable. (a) and (b) are exactly as I intend to say it. So yes, your understanding of these seems correct. I guess your questions after that are about how I order the theorems. I guess it's just personal preference. You can always feed Ramanujan into Euler; that can be done even more generally than how I do it. I'm restricting the cases where you can do this. Because it garners an isomorphic relationship. 1.) I'm a little confused by your first question; $\mathbb{E}_\theta \leftrightarrow \mathbb{S}_\theta$ bijectively. And additionally, $\mathbb{S}_\theta \subset \mathbb{S}_\kappa$ for $\theta < \kappa$; just as well with $\mathbb{E}_\theta \subset \mathbb{E}_\kappa$. They are in bijection only for the same $\theta$; other wise its a different kind of map. 2.) This is a good question, that has a pretty deep answer. First of all $f,g \in \mathbb{S}_\theta$ implies that $f+g \in \mathbb{S}_\theta$ and $F , G \in \mathbb{E}_\theta$ then $F+G \in \mathbb{E}_\theta$; so this is a linear isomorphism. It's actually a linear isomorphism between hilbert spaces; but it's a little difficult to do this exactly. This would mean there is a norm; and there even is an inner product; but it's spurious to this paper. Id have to dust off my copy of Linear Operators on Hilbert Spaces to remind myself what exactly these are; can't recall off the top of my head. Now, $\int_\gamma \|f(y)g(y)\| \,dy < \infty$; which happens for all $f,g \in \mathbb{S}_\theta$; and therefore if $f,g \in \mathbb{S}_\theta$ then $f \cdot g \in \mathbb{S}_\theta$. As to what happens when you apply the mapping to the product; you get a binomial convolution. $ \frac{d^{z}}{dw^{z}} f(w)g(w) = \sum_{k=0}^\infty \binom{z}{k} f^{(k)}(w) \frac{d^{z-k}}{dw^{z-k}} g(w) = H(z)\\$ I didn't prove this in this paper; and this result is not mine. It's commonly known as the binomial theorem (I think?); you can find it in any text book on fractional calculus; it's usually one of the first things you prove. It's a little difficult; but in the best scenarios I can prove it pretty quickly because; $ H(n) = \sum_{k=0}^n\binom{n}{k} f^{(k)}(w) g^{(n-k)}(w) = \frac{d^{n}}{dw^{n}} f(w)g(w) \\$ So if you can show $H \in \mathbb{E}_\theta$; they're equivalent by The Identity Theorem you get using Ramanujan's master theorem. This depends on how well $g$ or $f$ behave however. This convolution won't work generally for all $f,g$ because $\frac{d^{z-k}}{dw^{z-k}} g(w)$ may not exist. You can then, write this as a convolution, $ \frac{d^{z}}{dw^{z}}\|_{w=0} f(w)g(w) = F * G\\$ Where sometimes this has the above representation; not always though. What you always get though; which again, isn't in the paper; is the indefinite sum representation. $ F * G = \sum_{j=0}^z \binom{z}{j}F(j)G(z-j)\\$ This representation was more carefully studied in the indefinite sum paper on my ariv that's referenced in this paper. Though I use a slightly less direct isomorphism (forgive me, I wrote that paper a long time ago; but it still gets the job done). Going in the other direction is more difficult. Recall that $F \in \mathbb{E}_\theta$ implies that $\|F(z)\| \le C e^{(\pi/2 - \theta)\|\Im(z)\|}$ as $\Im(z) \to \pm \infty$. So this means, if $F \in \mathbb{E}_\theta$ and $G \in \mathb{E}_\kappa$ then $\|F(z)G(z)\| \le Me^{(\pi - \theta-\kappa)\|\Im(z)\|}$; which may or may not belong to an $\mathbb{E}_\tau$ depending on what $\theta$ and $\kappa$ are. If they do belong to one then when you put it in the space $\mathbb{S}_\tau$; then, $ h(w) = \sum_{n=0}^\infty F(n)G(n) \frac{w^n}{n!}\\$ 3. As to the relationship between varying $\theta$ and $\kappa$; the best I have is that, the maximal sector in which $f$ converges $S_\theta$, is the maximal set $\mathbb{S}_\theta$ it belongs to. And additionally; the maximal set $F$ belongs in is $\mathbb{E}_\theta$. And the maximal value $\theta$ in which $\|\Gamma(-z)F(z)\| \le Ce^{-\theta \|Im(z)\|}$ is the maximal set $F \in \mathbb{E}_\theta$. I'm not sure what else you could be asking here..? Am I missing something? Regards, James MphLee Long Time Fellow Posts: 321 Threads: 25 Joined: May 2013 05/26/2021, 12:01 AM (This post was last modified: 05/26/2021, 12:58 AM by MphLee.) (05/25/2021, 10:47 PM)JmsNxn Wrote: The people at U of T called it a hyper-operator chain; that's not my terminology. I know it can be a tad confusing for this forum; but that's what they call it ; so I stuck with the terminology.Yeah, but then I'm curious... how the hell did they know about hyperoperations? I'm pretty sure that serious mathematicians never talk about hyperoperations and the term hyperoperations is very niche and already used by hyperstructure theory (theory of groups with multivalued operation). So if you tell me that they used that terminology for a reason... I'm pretty excited to hear more about that. You know... 2015... 6 years googling things and the only persons that write that chain equation are you, Rubtsov and Romerio, 3/4 Tetration Forum's users and myself. Quote:I guess your questions after that are about how I order the theorems. I guess it's just personal preference. You can always feed Ramanujan into Euler; that can be done even more generally than how I do it. Mhh idk, I'll study this better. I had the impression that to ensure you could apply Euler to f you had to show FIRST that f=R[H] was in boldface E. I'm sure I have to read and understand better all those conditions (and probably go back to your old papers). Quote:1.) I'm a little confused by your first question; I apologize... I'm sure I miss something crucial about convergence but I was thinking the following I was asking if for EVERY $\theta \in [0,\pi]$ we have $\mathbb{E}_\theta \simeq \mathbb{S}_\theta$ The existence of that chain of inclusions is interesting... It should mean that we can extend ${\mathfrak E}_w$ and ${\mathfrak R}$ to a bigger domain. To be clear observe that if the origin of the complex plane is included and $\theta \le \kappa$ implies $S_\theta \subseteq S_\kappa$ then $\displaystyle\bigcup_{\theta\in[0,\pi)}S_\theta ={\mathbb C}/(-\infty,0)$ and $S_\pi=\displaystyle\bigcup_{\theta\in[0,\pi]}S_\theta ={\mathbb C}$ From the monotone chain of inclusion also follows that for every $\theta <\pi$ we have $\mathbb{E}_\theta \subset \mathbb{E}_\pi$ and $\mathbb{S}_\theta \subset \mathbb{S}_\pi$ So you can't possibly mean that every theta is ok... maybe only for $\theta \in [0,\pi)$? So the idea is the following.... if $\theta \le \kappa$ consider the two functions ${\mathfrak E}^\theta_w:\mathbb{S}_\theta\to \mathbb{E}_\theta$ and ${\mathfrak E}^\kappa_w:\mathbb{S}_\kappa\to \mathbb{E}_\kappa$ do we have that restricting ${\mathfrak E}^\kappa_w$ to $\mathbb{S}_\theta$ give us ${\mathfrak E}^\theta_w$? In symbols ${\mathfrak E}^\kappa_w\|_{{\mathbb E}_\theta}={\mathfrak E}^\theta_w$ Diagrammatically $\mathbb{S}_\theta\overset{{\mathfrak E}^\theta_w}{\longrightarrow} \mathbb{E}_\theta\overset{\subseteq}{\longrightarrow} \mathbb{E}_\kappa$ is the same as $\mathbb{S}_\theta \overset{\subseteq}{\longrightarrow} \mathbb{S}_\kappa\overset{{\mathfrak E}^\kappa_w}{\longrightarrow} \mathbb{E}_\kappa$ If this condition works we can just work with spaces ${\mathbb S}:=\displaystyle\bigcup_{\theta\in[0,\pi)}{\mathbb S}_\theta$ and ${\mathbb E}:=\displaystyle\bigcup_{\theta\in[0,\pi)}{\mathbb E}_\theta$ because evey function in there satisfies your criterion for some $\theta$, by definition. Quote:2.) This is a good question, that has a pretty deep answer. [...] $ h(w) = \sum_{n=0}^\infty F(n)G(n) \frac{w^n}{n!}\\$ Woooa... that has to be important. I have some gut feeling that this is very important... I'll keep it for myself now.... but I guess I have seen this somewhere before... MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ JmsNxn Ultimate Fellow Posts: 977 Threads: 114 Joined: Dec 2010 05/26/2021, 02:34 AM (05/26/2021, 12:01 AM)MphLee Wrote: Yeah, but then I'm curious... how the hell did they know about hyperoperations? I'm pretty sure that serious mathematicians never talk about hyperoperations and the term hyperoperations is very niche and already used by hyperstructure theory (theory of groups with multivalued operation). So if you tell me that they used that terminology for a reason... I'm pretty excited to hear more about that. You know... 2015... 6 years googling things and the only persons that write that chain equation are you, Rubtsov and Romerio, 3/4 Tetration Forum's users and myself. Quote:I guess your questions after that are about how I order the theorems. I guess it's just personal preference. You can always feed Ramanujan into Euler; that can be done even more generally than how I do it. Mhh idk, I'll study this better. I had the impression that to ensure you could apply Euler to f you had to show FIRST that f=R[H] was in boldface E. I'm sure I have to read and understand better all those conditions (and probably go back to your old papers). Quote:1.) I'm a little confused by your first question; I apologize... I'm sure I miss something crucial about convergence but I was thinking the following I was asking if for EVERY $\theta \in [0,\pi]$ we have $\mathbb{E}_\theta \simeq \mathbb{S}_\theta$ The existence of that chain of inclusions is interesting... It should mean that we can extend ${\mathfrak E}_w$ and ${\mathfrak R}$ to a bigger domain. To be clear observe that if the origin of the complex plane is included and $\theta \le \kappa$ implies $S_\theta \subseteq S_\kappa$ then $\displaystyle\bigcup_{\theta\in[0,\pi)}S_\theta ={\mathbb C}/(-\infty,0)$ and $S_\pi=\displaystyle\bigcup_{\theta\in[0,\pi]}S_\theta ={\mathbb C}$ From the monotone chain of inclusion also follows that for every $\theta <\pi$ we have $\mathbb{E}_\theta \subset \mathbb{E}_\pi$ and $\mathbb{S}_\theta \subset \mathbb{S}_\pi$ So you can't possibly mean that every theta is ok... maybe only for $\theta \in [0,\pi)$? So the idea is the following.... if $\theta \le \kappa$ consider the two functions ${\mathfrak E}^\theta_w:\mathbb{S}_\theta\to \mathbb{E}_\theta$ and ${\mathfrak E}^\kappa_w:\mathbb{S}_\kappa\to \mathbb{E}_\kappa$ do we have that restricting ${\mathfrak E}^\kappa_w$ to $\mathbb{S}_\theta$ give us ${\mathfrak E}^\theta_w$? In symbols ${\mathfrak E}^\kappa_w\|_{{\mathbb E}_\theta}={\mathfrak E}^\theta_w$ Diagrammatically $\mathbb{S}_\theta\overset{{\mathfrak E}^\theta_w}{\longrightarrow} \mathbb{E}_\theta\overset{\subseteq}{\longrightarrow} \mathbb{E}_\kappa$ is the same as $\mathbb{S}_\theta \overset{\subseteq}{\longrightarrow} \mathbb{S}_\kappa\overset{{\mathfrak E}^\kappa_w}{\longrightarrow} \mathbb{E}_\kappa$ If this condition works we can just work with spaces ${\mathbb S}:=\displaystyle\bigcup_{\theta\in[0,\pi)}{\mathbb S}_\theta$ and ${\mathbb E}:=\displaystyle\bigcup_{\theta\in[0,\pi)}{\mathbb E}_\theta$ because evey function in there satisfies your criterion for some $\theta$, by definition. Quote:2.) This is a good question, that has a pretty deep answer. [...] $ h(w) = \sum_{n=0}^\infty F(n)G(n) \frac{w^n}{n!}\\$ Woooa... that has to be important. I have some gut feeling that this is very important... I'll keep it for myself now.... but I guess I have seen this somewhere before... As to your first point. I shared a lot of work at U of T; and they started calling these things hyper-operation chains (at least the people I talked to). There isn't anything published as of yet; but they've done quite a few things similarly to me. They never published, I presume because I have priority over these fractional calculus things; at least from their perspective. I kind of left the scene for a while; and they were upset I never published half the things they sort of knew about through me. It's why I've started publishing all over again; sort of like a code dump of everything I've done. Largely because some professors told me to. They did do some stuff with, $ \alpha \uparrow^s z\\$ But I rarely see them (especially with covid right now); and I presume it's slow going. But they seemed confident my original formula for it 5 or 6 years ago is the correct one (but my original proof is incorrect): $ \alpha \uparrow^s z = \frac{d^{s-1}}{dw^{s-1}} \frac{d^{z-1}}{du^{z-1}} \|\|_{w=0}_{u=0} \sum_{n=0}^\infty \sum_{k=0}^\infty \alpha \uparrow^{n+1}(k+1) \frac{w^nu^k}{n!k!}\\$ They were also the ones who encouraged me to publish all this Infinite composition stuff. They were pretty shocked when I explained the residual theorem to them (just like you were ). As to your second point; I would absolutely avoid talking about $\theta \in (\pi/2,\pi)$ (the value $\theta = \pi$ is out of the question too; because then it's bounded on $\mathbb{C}$ and it's just constant). If you want to include sectors of this length; things can get a bit more complicated. In fact; it's a good amount trickier in these cases. So, I only play with $\theta \in (0,\pi/2]$. Not that you can't use these cases; but if my memory serves me correct; the functional properties change a fair amount. But yes, for every $\theta \in (0,\pi)$ we have the correspondence $\mathbb{E}_\theta \simeq \mathbb{S}_\theta$. We absolutely have the restriction you are asking. Yes, we can view these as operators acting on restricted spaces and they're equivalent. I mean, $F \in \mathbb{E}_\theta$ $ f(w) = \sum_{n=0}^\infty F(n) \frac{w^n}{n!}\\$ Has no dependency on $\theta$ so the restriction is arbitrary. Additionally; $f \in \mathbb{S}_\theta$ Then, $ \Gamma(z) F(-z) = \int_0^\infty f(-y)y^{z-1}\,dy\\$ Which again, has no mention of $\theta$. The variable only appears to describe the asymptotics of $f$ and $F$. Where for $f$ it determines the size of the sector of its convergence. And for $F$ it describes its possible growth type as $\Im(z) \to \pm \infty$. And yes, you can absolutely work with, $ \mathbb{E} = \bigcup_{\theta \in (0,\pi)} \mathbb{E}_\theta$ As I usually only care about $\theta = \pi/2$ or $\theta < \pi/2$; I don't pay much mind of that. Remember though, we do not want $\theta = 0$. This is no good. The most important part is that we have an open sector; when $\theta = 0$ we just have a line; and this will produce anomalies. Particularly; it'll screw things up when you want to make the correspondence; because you cannot really "pull out" any asymptotic data. What you are doing with this union is much more similar to the classical treatment of Ramanujan's Master Theorem. I don't like this treatment; largely because it avoids explicitly stating how the differintegrated function is bounded. And it's very helpful to know how its bounded. If we consider this union, we're not being explicit about where the integral converges; and we're stuck only with the absolute knowledge that, $ \Gamma(z)F(-z) = \int_0^\infty f(-y)y^{z-1}\,dy\\$ And for some sector it works. This will produce problems when you want to do more advanced things in functional analysis with these things (which is moreso needed for the function $\alpha \uparrow^s z$, or for defining a convolution, or for introducing the indefinite sum). But yes, you are correct. As to your last point, I could never find any use for this thing. A long time ago I used to try and try to create a convergence factor. So that, for arbitrary $F$, not necessarily in $\mathbb{E}$, there exists some $G \in \mathbb{E}$ such that $F\cdot G \in \mathbb{E}$. Then we would get, $ \Gamma(z)F(-z) = \frac{1}{G(-z)}\int_0^\infty h(-y)y^{z-1}\,dy\\$ But I could never find anything meaningful... I sort of settled that there's no way to force a function to be in $\mathbb{E}$; it just is or it isn't. Regards, James MphLee Long Time Fellow Posts: 321 Threads: 25 Joined: May 2013 05/26/2021, 10:50 AM (This post was last modified: 05/26/2021, 11:52 AM by MphLee.) (05/26/2021, 02:34 AM)JmsNxn Wrote: As to your first point. I shared a lot of work at U of T; and they started calling these things hyper-operation chains (at least the people I talked to). There isn't anything published as of yet; but they've done quite a few things similarly to me. They never published, I presume because I have priority over these fractional calculus things; at least from their perspective. I kind of left the scene for a while; and they were upset I never published half the things they sort of knew about through me. It's why I've started publishing all over again; sort of like a code dump of everything I've done. Largely because some professors told me to. They did do some stuff with, $ \alpha \uparrow^s z\\$ But I rarely see them (especially with covid right now); and I presume it's slow going. But they seemed confident my original formula for it 5 or 6 years ago is the correct one (but my original proof is incorrect): $ \alpha \uparrow^s z = \frac{d^{s-1}}{dw^{s-1}} \frac{d^{z-1}}{du^{z-1}} \|\|_{w=0}_{u=0} \sum_{n=0}^\infty \sum_{k=0}^\infty \alpha \uparrow^{n+1}(k+1) \frac{w^nu^k}{n!k!}\\$ They were also the ones who encouraged me to publish all this Infinite composition stuff. They were pretty shocked when I explained the residual theorem to them (just like you were ). Thank you, now I understand all the business with the theta angles. About you peers at U of T, it is remarkable if they found that interesting and surprising that "they've done quite a few things similarly to me". I never had many chances to talk with mathematicians... and all the hints tell me that this topic is completely unknown and irrelevant. Obviously, as you can imagine from my last draft and recent threads, I can already, at least partially, trace back this topic to the backbone of mathematics I already begin to see that it touches many mainstream topics. What do you think about the centrality of hyper-operations chain-like objects and their possible continuous extension to non-discrete chains (paths?)? How do you think the reception of these ideas was and what was the atmosphere around those chain objects? Did they treat them as exotic and niche items? Addendum I red the last sections of the paper. I really can't find obscure points. All the machinery lies in the first two theorems. I'll give myself time to digest it. Thank you! Best regards! MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ JmsNxn Ultimate Fellow Posts: 977 Threads: 114 Joined: Dec 2010 05/26/2021, 11:17 PM (05/26/2021, 10:50 AM)MphLee Wrote: What do you think about the centrality of hyper-operations chain-like objects and their possible continuous extension to non-discrete chains (paths?)? How do you think the reception of these ideas was and what was the atmosphere around those chain objects? Did they treat them as exotic and niche items? Honestly, their reaction was mostly; that this is some really wacky and weird stuff. But they thought it was cool. They were more interested in taking matrices and arbitrary operators and doing things like this: $ \frac{d^{z}}{dw^z} e^{Aw} = A^z e^{Aw}\\$ Honestly, I don't see what's so cool about that; a fractional power of matrix seems easy to do; but apparently they like that go figure. They were more receptive than you may think. A lot of them would already have brushed on these things; especially comp sci people. They're more of the boat, that this looks way too hard, there's no way we could ever do that; than, it's unimportant or niche. They especially like the $\Gamma$ which pops up everywhere, lol. Honestly; I still have no idea how to construct a function like $\alpha \uparrow^s z$; and it's not so much that you have to prove the thing converges; it's the domain arguments needed to show the functional equation that are a real problem. I gave up a long time ago trying to make that work. But I still believe it to be a very important subject. Sincere Regards, James MphLee Long Time Fellow Posts: 321 Threads: 25 Joined: May 2013 05/27/2021, 08:00 PM (This post was last modified: 05/28/2021, 04:05 PM by MphLee.) (05/26/2021, 11:17 PM)JmsNxx Wrote: $ \frac{d^{z}}{dw^z} e^{Aw} = A^z e^{Aw}\\$ Honestly, I don't see what's so cool about that; a fractional power of matrix seems easy to do; but apparently they like that go figure.I guess I can understand why. Iterating matrices is the key to extend every linear process. Btw... Abel and Schroeder iterate by achieving a linearization of a non-linear dynamics so it is understandable. But to me it is too narrow. Just this summer, when I wrote a short paper in Italian() where I define the continuous extension of the Fibonacci sequence only via linear algebra and Eigen-theory. The method is pretty standard and it gives (pag 8 ) the usual analytic closed form of Fibonacci. The interesting thing is that I did it from scratch starting from the formal definition of recursion in recursion theory. Thanks to that paper I was able to fully appreciate that Fibonacci is defined by a kind of recursion that we could call linear and that linear recursion can be translated to "applying a matrix": in other words a recursion that IS NOT iteration can be translated into exponentiation of a matrix. () It's short but just look at the formulas just to get a taste. One day I may translate it. (2020 07 30 3) Successioni ricorsive ed autoteoria.pdf (Size: 421.82 KB / Downloads: 225) Quote:They especially like the $\Gamma$ which pops up everywhere, lol. The gamma popping up everywhere there is curious... but idk if it is an artifact of the method or it is structural in some sense. What do you think? What I know is that long time ago I was shown a graph of a linear or cubic approximation of tetration plotted on real arguments and showing the real part and the imaginary part. Before the singularity at -2 the imaginary part did look a lot like gamma function... Quote:I gave up a long time ago trying to make that work. But I still believe it to be a very important subject. Btw, it is a very hard object, I'm not surprised that you were not able to make it work. I strongly believe that there is some hidden structure, some hidden regularities to be discovered and functional identities on the ranks have to be found before we could "declare war on the sky". One reason for my believe is the following: we have yet to discover the intrinsic nature of abstract iteration and that is just level 1 of rank theory. Ranks theory is applying abstract iteration to abstract iteration itself. But this could sound empty to many ears. There is another good reason to expect extraordinary obstacles. Let me illustrate this as a story made up of four layers/moments. Quote:At the beginning there's nothing, no difference. We have to chose a point and make the first distinction. 0 let rank 0 be conceptually our base function, our unit of mesaure of linearity (the +1). It is a single point. 1 Then rank 1 is conceptually the totality of our way of traslating things (or iterates) and we should think of it as our base number system and our base geometry. So we've built numbers out of a unit. A kind of geometric object, a "line". 2 At this point the automorphism of our geometric object (the modes of interacting with itself) are the rank 2 functions. We can think of rank 2 as the arithmetic or as the scaling operations over our base geometry/number system. 3 So we have now rank 1 (the geometric level) and on top of that we have built a new layer, rank 2 (the arithmetic multiplicative level). The first is made of lines and linear traslation, the second of scaling and rotations. The link between traslating and rotating is... yes exponentiation. So morphisms from rank 1 to rank 2 give us the world of rank 3. This seems a kind of metaphysical theogony. An ontogenesis that goes from the nothing to complexity. I hope you can clearly see that there is something very very deep lurking here, something that "just interpolating" (even analytically) can't solve. I see this as an obstacle to a real non-integer extension because here we have to first generalize a chain of phenomena of which only the first three account for 60/70% of all the existing mathematics. Sincere regards, V.C. Edit: let's provide some beef in addition to the juicy smoke. At the beginning we have a bunch of composable functions $(G,\circ, id_G)$ with the usual sets $[f,g]_G:=\{x\in G\,\|\, xf=gx\}$ 0 Fix an "unit" element $s\in G$. Define the subset ${\mathcal E}^0_s\subseteq G$ as ${\mathcal E}^0_s:=\{s\}$ 1 Define the subgroup ${\mathcal E}^1_s\subseteq G$ as ${\mathcal E}^1_s:=[s,s]_G$. Clearly $s^n\in\mathcal E^1_s$ for every n. 2 Define the set ${\mathcal E}^2_s\subseteq G$ as ${\mathcal E}^2_s:=[s,{\mathcal E}^1_s]_G$. Clearly in some cases there exists a $\mu_n\in{\mathcal E}^2_s$ s.t. $\mu_n s=s^n\mu_n$. Clearly $\mu_n$ is a multiplication-like function. 3 Define the set ${\mathcal E}^3_s\subseteq G$ as ${\mathcal E}^3_s:=[s,{\mathcal E}^2_s]_G$. Clearly in some cases there exists a $\varepsilon_n\in{\mathcal E}^3_s$ s.t. $\varepsilon_n s=\mu_n\varepsilon_n$. Clearly $\varepsilon_n$ is an exponentiation-like function. We define ${\mathcal E}^{\sigma+1}_s:=[s,{\mathcal E}^\sigma_s]_G$ and we trivially have ${\mathcal E}^{\sigma}_s\subseteq {\mathcal E}^{\sigma+1}_s$ The union $\displaystyle \bigcup_{\sigma=0}^\infty{\mathcal E}^{\sigma}_s$ can be seen as the class of primitive recursive element relative to s. MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ JmsNxn Ultimate Fellow Posts: 977 Threads: 114 Joined: Dec 2010 05/28/2021, 07:43 PM (This post was last modified: 05/28/2021, 10:33 PM by JmsNxn.) Heh, I'm surprised how well I understood your italian, lol. I guess the years of french schooling paid off; gotta love the ubiquity of romance languages. That was an interesting read. It seems like a very good introduction to what you are trying to do. MphLee Long Time Fellow Posts: 321 Threads: 25 Joined: May 2013 05/28/2021, 10:31 PM Ah! I saw you dropped here and there some french words in your papers so I tried to share it (there are large Italian communities in Canada). Btw I'm not sure it is good as an intro to hyperoperations. But it could be the first step for the unification of Gottfried's matrix methods and the compositional methods for iteration. The paper arose initially as an attempt to answer to real analysis question a friend asked me, a challenge. He asked me when a fibonacci sequence defined with different initial values was divergent or convergent and for which initial values we have convergence. In that paper I show that a generalized fibonacci sequence converges only if the initial condtition lies in the eigenspace associated with the silver ratio (golden and silver ratio are the two eigenvalues of the "fibonacci matrix"). MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ « Next Oldest \| Next Newest » Possibly Related Threads… Thread Author Replies Views Last Post Hyper-Operational Salad Numbers Catullus 9 700 09/17/2022, 01:15 AM Last Post: Catullus Rank-Wise Approximations of Hyper-Operations Catullus 48 7,753 09/08/2022, 02:52 AM Last Post: JmsNxn Octonion Hyper-Operations Catullus 3 769 07/05/2022, 08:53 AM Last Post: Catullus Thoughts on hyper-operations of rational but non-integer orders? 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Syzithryx 2 5,124 07/24/2019, 05:59 PM Last Post: Syzithryx Users browsing this thread: 1 Guest(s)	2022-09-27 02:42:29	{"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": 203, "/images/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": 0.7877640128135681, "perplexity": 773.7913029520173}, "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/1664030334974.57/warc/CC-MAIN-20220927002241-20220927032241-00735.warc.gz"}
https://www.journaltocs.ac.uk/index.php?action=browse&subAction=subjects&publisherID=87&journalID=2782&pageb=2&userQueryID=&sort=&local_page=1&sorType=&sorCol=	Subjects -> STATISTICS (Total: 130 journals) The end of the list has been reached or no journals were found for your choice. Similar Journals Computational StatisticsJournal Prestige (SJR): 0.803 Citation Impact (citeScore): 1Number of Followers: 15 Hybrid journal (It can contain Open Access articles) ISSN (Print) 1613-9658 - ISSN (Online) 0943-4062 Published by Springer-Verlag [2469 journals] • Dynamic sampling from a discrete probability distribution with a known distribution of rates Abstract: Abstract In this paper, we consider several efficient data structures for the problem of sampling from a dynamically changing discrete probability distribution, where some prior information is known on the distribution of the rates, in particular the maximum and minimum rate, and where the number of possible outcomes N is large. We consider three basic data structures, the Acceptance–Rejection method, the Complete Binary Tree and the Alias method. These can be used as building blocks in a multi-level data structure, where at each of the levels, one of the basic data structures can be used, with the top level selecting a group of events, and the bottom level selecting an element from a group. Depending on assumptions on the distribution of the rates of outcomes, different combinations of the basic structures can be used. We prove that for particular data structures the expected time of sampling and update is constant when the rate distribution follows certain conditions. We show that for any distribution, combining a tree structure with the Acceptance–Rejection method, we have an expected time of sampling and update of $$O\left( \log \log {r_{max}}/{r_{min}}\right)$$ is possible, where $$r_{max}$$ is the maximum rate and $$r_{min}$$ the minimum rate. 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Here, we develop a fully Bayesian method to implement this measure, and examine its performance using simulation data and several real data sets of colorectal cancer. We use coverage probabilities and lengths of the interval estimators to compare the Bayesian approach and a large-sample method of analysis. In simulation studies, we find that the Bayesian method outperforms the large-sample method on average, and provides either similar or improved results for the real data analyses. PubDate: 2022-07-01 • A sequential test and a sequential sampling plan based on the process capability index Cpmk Abstract: Abstract In this study we propose a sequential test for hypothesis testing on the $$C_{pmk}$$ process capability index. Furthermore, we propose a sequential sampling plan for lot acceptance based on $$C_{pmk}$$ . We compare the statistical properties of the sequential procedures with the performance of the corresponding non-sequential methodologies by carrying out an extensive simulation study. The results show that the proposed sequential methods make it possible to reach decisions much more quickly, on average, than the fixed sample size procedures with the same discriminating power. PubDate: 2022-07-01 • The modified maximum likelihood estimators for the parameters of the regression model under bivariate median ranked set sampling PubDate: 2022-07-01 • A new estimation for INAR(1) process with Poisson distribution Abstract: Abstract The first-order Poisson autoregressive model may be suitable in situations where the time series data are non-negative integer valued. In this article, we propose a new parameter estimator based on empirical likelihood. Our results show that it can lead to efficient estimators by making effective use of auxiliary information. As a by-product, a test statistic is given, testing the randomness of the parameter. The simulation values show that the proposed test statistic works well. We have applied the suggested method to a real count series. PubDate: 2022-07-01 • Applying the rescaling bootstrap under imputation for a multistage sampling design Abstract: Abstract In this paper, we propose a method that estimates the variance of an imputed estimator in a multistage sampling design. The method is based on the rescaling bootstrap for multistage sampling introduced by Preston (Surv Methodol 35(2):227–234, 2009). In his original version, this resampling method requires that the dataset includes only complete cases and no missing values. Thus, we propose two modifications for applying this method to nonresponse and imputation. These modifications are compared to other modifications in a Monte Carlo simulation study. The results of our simulation study show that our two proposed approaches are superior to the other modifications of the rescaling bootstrap and, in many situations, produce valid estimators for the variance of the imputed estimator in multistage sampling designs. PubDate: 2022-07-01 • Hierarchical correction of p-values via an ultrametric tree running Ornstein-Uhlenbeck process Abstract: Abstract Statistical testing is classically used as an exploratory tool to search for association between a phenotype and many possible explanatory variables. This approach often leads to multiple testing under dependence. We assume a hierarchical structure between tests via an Ornstein-Uhlenbeck process on a tree. The process correlation structure is used for smoothing the p-values. We design a penalized estimation of the mean of the Ornstein-Uhlenbeck process for p-value computation. The performances of the algorithm are assessed via simulations. Its ability to discover new associations is demonstrated on a metagenomic dataset. The corresponding R package is available from https://github.com/abichat/zazou. PubDate: 2022-07-01 • New approximate Bayesian computation algorithm for censored data Abstract: Abstract Approximate Bayesian computation refers to a family of algorithms that perform Bayesian inference under intractable likelihoods. In this paper we propose replacing the distance metric in certain algorithms with hypothesis testing. The benefits of which are that summary statistics are no longer required and censoring can be present in the observed data set without needing to simulate any censored data. We illustrate our proposed method through a nanotechnology application in which we estimate the concentration of particles in a liquid suspension. We prove that our method results in an approximation to the true posterior and that the parameter estimates are consistent. We further show, through comparative analysis, that it is more efficient than existing methods for censored data. PubDate: 2022-07-01 • Characterizations and generalizations of the negative binomial distribution Abstract: Abstract In this paper, we give detailed descriptions of the Zero-Modified Negative Binomial distribution for analyzing count data. In particular, we study the characterizations and properties of this distribution, whose main advantage is its flexibility which makes it suitable for modeling a wide range of overdispersed and underdispersed count data (which may or may not be caused by zero-modification, i.e., the inflation or deflation of zeroes), without requiring previous knowledge about any of these inherent data characteristics. We derive maximum likelihood estimation of the model parameters based on positive observations, and evaluate the loss of efficiency by considering this procedure. We illustrate the suitability of this distribution on real data sets with different types of zero-modification. PubDate: 2022-07-01 • Bayesian analysis of mixture autoregressive models covering the complete parameter space Abstract: Abstract Mixture autoregressive (MAR) models provide a flexible way to model time series with predictive distributions which depend on the recent history of the process and are able to accommodate asymmetry and multimodality. Bayesian inference for such models offers the additional advantage of incorporating the uncertainty in the estimated models into the predictions. We introduce a new way of sampling from the posterior distribution of the parameters of MAR models which allows for covering the complete parameter space of the models, unlike previous approaches. We also propose a relabelling algorithm to deal a posteriori with label switching. We apply our new method to simulated and real datasets, discuss the accuracy and performance of our new method, as well as its advantages over previous studies. The idea of density forecasting using MCMC output is also introduced. PubDate: 2022-07-01 • Bayesian variable selection and estimation in quantile regression using a quantile-specific prior Abstract: Abstract Asymmetric Laplace (AL) specification has become one of the ideal statistical models for Bayesian quantile regression. In addition to fast convergence of Markov Chain Monte Carlo, AL specification guarantees posterior consistency under model misspecification. However, variable selection under such a specification is a daunting task because, realistically, prior specification of regression parameters should take the quantile levels into consideration. Quantile-specific g-prior has recently been developed for Bayesian variable selection in quantile regression, whereas it comes at a high price of the computational burden due to the intractability of the posterior distributions. In this paper, we develop a novel three-stage computational scheme for the foregoing quantile-specific g-prior, which starts with an expectation-maximization algorithm, followed by Gibbs sampler and ends with an importance re-weighting step that improves the accuracy of approximation. The performance of the proposed procedure is illustrated with simulations and a real-data application. Numerical results suggest that our procedure compares favorably with the Metropolis–Hastings algorithm. PubDate: 2022-07-01 • Objective Bayesian group variable selection for linear model Abstract: Abstract Prediction variables of the regression model are grouped in many application problems. For example, a factor in an analysis of variance can have several levels or each original prediction variable in additive models can be expanded into different order polynomials or a set of basis functions. It is essential to select important groups and individual variables within the selected groups. In this study, we propose the objective Bayesian group and individual variable selections within the selected groups in the regression model to reduce the computational cost, even though the number of regression variables is large. Besides, we examine the consistency of the proposed group variable selection procedure. The proposed objective Bayesian approach is investigated using simulation and real data examples. The comparisons between the penalized regression approaches, Bayesian group lasso and the proposed method are presented. PubDate: 2022-07-01 • Robust estimation of the number of factors for the pair-elliptical factor models Abstract: Abstract In this paper, we investigate the robust estimation of the number of common factors in high-dimensional factor model with pair-elliptically distributed idiosyncratic errors. Motivated by the pandemic heavy-tail distributions of financial returns, we first introduce a pair-elliptical factor model by allowing the factors and noises to follow pairwisely the joint elliptical distributions. Compared with the elliptical factor model invented in Fan et al. (Ann Stat 46:1383–1414, 2018), the pair-elliptical factor model has more richer structure with more relaxed assumptions. We propose two robust quantile-based estimators of the number of factors and obtain the asymptotic properties of the estimators under some mild conditions. Then, some simulation studies and a real data analysis are carried out to show the effectiveness of the estimators of the factor numbers. PubDate: 2022-07-01 • Statistical inference in massive datasets by empirical likelihood Abstract: Abstract In this paper, we propose a new statistical inference method for massive data sets, which is very simple and efficient by combining divide-and-conquer method and empirical likelihood. Compared with two popular methods (the bag of little bootstrap and the subsampled double bootstrap), we make full use of data sets, and reduce the computation burden. Extensive numerical studies and real data analysis demonstrate the effectiveness and flexibility of our proposed method. Furthermore, the asymptotic property of our method is derived. PubDate: 2022-07-01 • Smallest covering regions and highest density regions for discrete distributions Abstract: Abstract This paper examines the problem of computing a canonical smallest covering region for an arbitrary discrete probability distribution. This optimisation problem is similar to the classical 0–1 knapsack problem, but it involves optimisation over a set that may be countably infinite, raising a computational challenge that makes the problem non-trivial. To solve the problem we present theorems giving useful conditions for an optimising region and we develop an iterative one-at-a-time computational method to compute a canonical smallest covering region. We show how this can be programmed in pseudo-code and we examine the performance of our method. We compare this algorithm with other algorithms available in statistical computation packages to compute HDRs. We find that our method is the only one that accurately computes HDRs for arbitrary discrete distributions. PubDate: 2022-07-01 • Kolmogorov–Smirnov simultaneous confidence bands for time series distribution function Abstract: Abstract Claims about distributions of time series are often unproven assertions instead of substantiated conclusions for lack of hypotheses testing tools. In this work, Kolmogorov–Smirnov type simultaneous confidence bands (SCBs) are constructed based on simple random samples (SRSs) drawn from realizations of time series, together with smooth SCBs using kernel distribution estimator (KDE) instead of empirical cumulative distribution function of the SRS. All SCBs are shown to enjoy the same limiting distribution as the standard Kolmogorov–Smirnov for i.i.d. sample, which is validated in simulation experiments on various time series. Computing these SCBs for the standardized S&P 500 daily returns data leads to some rather unexpected findings, i.e., student’s t-distributions with degrees of freedom no less than 3 and the normal distribution are all acceptable versions of the standardized daily returns series’ distribution, with proper rescaling. These findings present challenges to the long held belief that daily financial returns distribution is fat-tailed and leptokurtic. PubDate: 2022-07-01 • Laplace regression with clustered censored data Abstract: Abstract In survival analysis, data may be correlated or clustered, because of some features such as shared genes and environmental background. A common approach to accommodate clustered data is the Cox frailty model that has proportional hazard assumption and complexity of interpreting hazard ratio lead to the misinterpretation of a direct effect on the time of event. In this paper, we considered Laplace quantile regression model for clustered survival data that interpret the effect of covariates on the time to event. A Bayesian approach with Markov Chain Monte Carlo method was used to fit the model. The results from a simulation study to evaluate the performance of proposed model showed that the Laplace regression model with frailty term performed well for different scenarios and the coverage rates of the pointwise 95% CIs were close to the nominal level (0.95). An application to data from breast cancer was presented to illustrate the theory and method developed in this paper. PubDate: 2022-07-01 • On community structure validation in real networks Abstract: Abstract Community structure is a commonly observed feature of real networks. The term refers to the presence in a network of groups of nodes (communities) that feature high internal connectivity, but are poorly connected between each other. Whereas the issue of community detection has been addressed in several works, the problem of validating a partition of nodes as a good community structure for a real network has received considerably less attention and remains an open issue. We propose a set of indices for community structure validation of network partitions that are based on an hypothesis testing procedure that assesses the distribution of links between and within communities. Using both simulations and real data, we illustrate how the proposed indices can be employed to compare the adequacy of different partitions of nodes as community structures in a given network, to assess whether two networks share the same or similar community structures, and to evaluate the performance of different network clustering algorithms. PubDate: 2022-07-01 • Hierarchical and multivariate regression models to fit correlated asymmetric positive continuous outcomes Abstract: Abstract In the extant literature, hierarchical models typically assume a flexible distribution for the random-effects. The random-effects approach has been used in the inferential procedure of the generalized linear mixed models . In this paper, we propose a random intercept gamma mixed model to fit correlated asymmetric positive continuous outcomes. The generalized log-gamma (GLG) distribution is assumed as an alternative to the normality assumption for the random intercept. Numerical results demonstrate the impact on the maximum likelihood (ML) estimator when the random-effect distribution is misspecified. The extended inverted Dirichlet (EID) distribution is derived from the random intercept gamma-GLG model that leads to the EID regression model by supposing a particular parameter setting of the hierarchical model. Monte Carlo simulation studies are performed to evaluate the asymptotic behavior of the ML estimators from the proposed models. Analysis of diagnostic methods based on quantile residual and COVARATIO statistic are used to assess departures from the EID regression model and identify atypical subjects. Two applications with real data are presented to illustrate the proposed methodology. PubDate: 2022-07-01 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	2022-06-27 20:30:12	{"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": 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": 0.5325864553451538, "perplexity": 648.3789658084999}, "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/1656103341778.23/warc/CC-MAIN-20220627195131-20220627225131-00090.warc.gz"}
https://docs.dgl.ai/generated/dgl.nn.pytorch.conv.AGNNConv.html	# AGNNConv¶ class dgl.nn.pytorch.conv.AGNNConv(init_beta=1.0, learn_beta=True, allow_zero_in_degree=False)[source] Bases: torch.nn.modules.module.Module Attention-based Graph Neural Network layer from Attention-based Graph Neural Network for Semi-Supervised Learning $H^{l+1} = P H^{l}$ where $$P$$ is computed as: $P_{ij} = \mathrm{softmax}_i ( \beta \cdot \cos(h_i^l, h_j^l))$ where $$\beta$$ is a single scalar parameter. Parameters • init_beta (float, optional) – The $$\beta$$ in the formula, a single scalar parameter. • learn_beta (bool, optional) – If True, $$\beta$$ will be learnable parameter. • allow_zero_in_degree (bool, optional) – If there are 0-in-degree nodes in the graph, output for those nodes will be invalid since no message will be passed to those nodes. This is harmful for some applications causing silent performance regression. This module will raise a DGLError if it detects 0-in-degree nodes in input graph. By setting True, it will suppress the check and let the users handle it by themselves. Default: False. Note Zero in-degree nodes will lead to invalid output value. This is because no message will be passed to those nodes, the aggregation function will be appied on empty input. A common practice to avoid this is to add a self-loop for each node in the graph if it is homogeneous, which can be achieved by: >>> g = ... # a DGLGraph Calling add_self_loop will not work for some graphs, for example, heterogeneous graph since the edge type can not be decided for self_loop edges. Set allow_zero_in_degree to True for those cases to unblock the code and handle zero-in-degree nodes manually. A common practise to handle this is to filter out the nodes with zero-in-degree when use after conv. Example >>> import dgl >>> import numpy as np >>> import torch as th >>> from dgl.nn import AGNNConv >>> >>> g = dgl.graph(([0,1,2,3,2,5], [1,2,3,4,0,3])) >>> feat = th.ones(6, 10) >>> conv = AGNNConv() >>> res = conv(g, feat) >>> res tensor([[1., 1., 1., 1., 1., 1., 1., 1., 1., 1.], [1., 1., 1., 1., 1., 1., 1., 1., 1., 1.], [1., 1., 1., 1., 1., 1., 1., 1., 1., 1.], [1., 1., 1., 1., 1., 1., 1., 1., 1., 1.], [1., 1., 1., 1., 1., 1., 1., 1., 1., 1.], [1., 1., 1., 1., 1., 1., 1., 1., 1., 1.]], forward(graph, feat)[source] Compute AGNN layer. Parameters • graph (DGLGraph) – The graph. • feat (torch.Tensor) – The input feature of shape $$(N, )$$ $$N$$ is the number of nodes, and $$$$ could be of any shape. If a pair of torch.Tensor is given, the pair must contain two tensors of shape $$(N_{in}, )$$ and $$(N_{out}, )$$, the $$$$ in the later tensor must equal the previous one. Returns The output feature of shape $$(N, )$$ where $$*$$ should be the same as input shape. Return type torch.Tensor Raises DGLError – If there are 0-in-degree nodes in the input graph, it will raise DGLError since no message will be passed to those nodes. This will cause invalid output. The error can be ignored by setting allow_zero_in_degree parameter to True.	2023-03-30 04:08:56	{"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": 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": 0.2695915997028351, "perplexity": 1493.004238913603}, "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/1679296949097.61/warc/CC-MAIN-20230330035241-20230330065241-00330.warc.gz"}
http://bbujeya.blogspot.com/2014/02/a-nice-question-from-latter-stages-of.html	## Wednesday, 5 February 2014 A nice question from the latter stages of the Extension 2 paper.It shows how seamlessly complex numbers and polynomials go together!	2019-01-23 07:44:23	{"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": 0.8979318141937256, "perplexity": 3870.921204813509}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547584203540.82/warc/CC-MAIN-20190123064911-20190123090911-00069.warc.gz"}
http://math.stackexchange.com/tags/graphing-functions/hot	# Tag Info 18 It's not standard to answer a question with an image, but I think the image says more than 1000 words in this case: The point is that what you are drawing on the x axis is the angle, not the length of one of the sides of the triangle. The angle is proportional to the length of the circle section. Image Source. Credit for the image goes to Lucas V. ... 8 32bit vs. 64bit affects which integer types are used by default, which is of no interest here. Rather, the floting point computations are made (by default) with IEEE double type. With this double precision (53 bit mantissa), the relative error of $(1+\frac1{x^{16}})$ is approximately $2^{-53}$. Raising to the $x^{16}$th power roughly multiplies the relative ... 7 Your problem is the finite precision of floating-point arithmetic. There are only so many numbers near $1$ that can be represented by the computer's floating-point format, and the larger your $x$ is, the more of the difference between $1$ and $1+\frac{1}{x^{16}}$ (which is what really matters when raising to a huge power) will be lost to rounding of the ... 5 Hint: Each graph forms the boundary of a convex region. The line segment of minimal distance between the two curves must therefore be unique. The picture is symmetric about the line $y = x$, so reflecting the line segment through this line must yield another line segment of minimal length. We can deduce that the slope of the common normal must ... 3 You are treating the height as a function of the $x$ position of the base of that vertical leg. But $\sin$ is a function of the angle. Or alternatively, a function of how much circumference has been traced out. It's not (directly) a function of the $x$ position of that vertical segment. 3 You can stitch a Frankenbola together like this. $$f(x) = \begin{cases} a_l x^2 + b_l x + c_l & \text{for } x < 0 \\ a_r x^2 + b_r x + c_r & \text{for } x > 0 \\ c & \text{for } x = 0 \end{cases}$$ You can require continuity for $f$ then you get $$f(x) = \begin{cases} a_l x^2 + b_l x + c & \text{for } x < 0 \\ a_r x^2 + b_r x + c ... 3 We have$$f(x,y)=z=\ln(x-y)$$For some level curve where z=k, we then have$$k=\ln(x-y)$$Here, there are only two variables, x and y. It is now possible to write x as a function of y, and vice versa. You should have a set of two-dimensional functions. All you have to do is graph them. I won't give you the answer in that case, but I can give you a ... 3 f''(x)=0 is not a necessary condition. Following two conditions must be met for inflection point to exist: 1) f(x) must be continuous 2) Concavity must change, that means sign of f''(x) must change In this case f(x) is continuous at x = 0 f''(x) = -\dfrac{2}{9x^{5/3}}, which changes sign at x=0 So (0,0) is indeed an inflection point ... 2 Here is a plot using the (arbitrary precision) calculator Pari/GP. I use 200 dec digits precision as default in my computations and got this plot without oscillation up to x=20: I tried it so far up to x=256; no oscillation. See here the image up to x=128 (just to have the left increase visible) 2 Well, \sin has range [-1,1]. So you're applying \sin to something between -1 and 1, so you need to first know how \sin looks like on [-1,1]. It's increasing on this range. So from -\pi/2 to \pi/2, \sin(\sin(x)) is increasing from -\sin(1) to \sin(1). Note that 1 is slightly less than \pi/3, so \sin(1) is a bit less than ... 2 Choose a gridstep s between 0 and \min (\|min\|, max) (this will be the distance between any two consecutive vertical gridlines). Let m = \lfloor \dfrac {\|min\|} s \rfloor and M = \lfloor \dfrac {max} s \rfloor, where \lfloor \cdot \rfloor is the floor function (also known as the integer part). Then draw a vertical gridline at each point of the ... 2 Round the low value to the nearest lower multiple of 10 and the high value to the nearest higher multiple of 10. With your example,$$[-107,858]\to\left[10\lfloor\frac{-107}{10}\rfloor,10\lceil\frac{858}{10}\rceil\right]=[-110,860].$$In programming, when the bounds are integer, this can be achieved by means of the \% operator, with$$[(\min-9) \% ... 2 Starting with $$f(x) = (c-\frac{1}{c}-x)(4-3x^2)$$ to make computation a bit more manageable, set $\boxed{\gamma=c-\frac{1}{c}}$, so $$f(x)=4\gamma-4x-3\gamma x^2+3x^3$$ You had the correct idea to set $f'(x)=0$ at the turning points. So $$f'(x)=-4-6\gamma x+9x^2=0\quad(\text{at turning points})$$ By the quadratic formula ... 2 Start with $f(x) = ax^{3} + bx^{2} + cx + d$. Since you want this polynomial to have critical points at $x = \pm 1$, we require that $f'(\pm 1) = 0$. This yields the two equations \begin{align} 3a + 2b + c & = 0\\ 3a - 2b + c & = 0. \end{align} It is then obvious that $b = 0$ and $c = -3a$. From here, one can obtain another two equations from the ... 2 Minimum Distance between Two curve is Distance between two parallel tangents drawn at point $P$ and $Q$ on the curves. and Here $f(x)=e^x$ and $f(x)=\ln(x)$ are Inverse of each other . So it is Symmetrical about $y=x$ Line. Let We take any point $P(x_{1},y_{1})$ on $f(x) = \ln(x)\;,$ Then Slope of tangent at $P(x_{1},y_{1})$ to the curve ... 2 If it's only real numbers you're working with, then no. If you're familiar with complex logarithms then $\log(z)=\log\|z\|+\mathrm{i}\theta$ might help, where $\theta$ is the argument of $z$. 1 Because the height of these opposite sides equals the sine of the angles, OK, $\sin\alpha = y / 1 = y$ for one but $\cos\alpha = x / 1 = x$ for the other opposite site. these can be mapped onto a sine graph (x-axis is the angles in degrees, y-axis is opposite side height), OK. $F = (\alpha, y(\alpha)) = (\alpha, \sin(\alpha))$ and should ... 1 I see a black line from $(0,0,0)$ to $(1,1,1)$ The reason it does not appear to be orthogonal to the brown plane is the scale of the vertical axis not matching the scales of the other two axes either in range or in size. Stretch it and it looks better 1 A graphing calculator brought up a pinched square shape, but I just can't understand the logical way to get to this shape. $\qquad\quad$ Geometric shapes described by algebraic equations of the form $\|x\|^n+\|y\|^n=r^n$ are called superellipses. For $n=1$, we have a diamond square, determined by four straight line segments. For $n>1$, these $4$ lines ... 1 WLOG let $Y = [0,1]$. Let $X = [a, a+1]$. The over lap of $X$ and $Y$ is of length $1/2$, so $X\cap Y$ is some interval of length $1/2$. Now say $a > 0$. Then $a < 1$ other wise there would be no intersection. So $a+1 > 1$ and the overlap is $[a,1]$. The only way that can have length $1/2$ is if $a = 1/2$. Likewise you get only one possibility when ... 1 I would like to illustrate my comments in the following figure. I hope that this will explain everything. the dark blue line is the graph of $\color{blue}{\sqrt{(x)}}$ -- note that $\sqrt{(x)}$ is not defined on $(-\infty,0)$. the purple line is the graph of $\color{purple}{g(x)=\sqrt{(-x)}}$ -- note that this function is not defined on $(0,\infty)$. ... 1 Solving graphically would involve sketching the graph o fthe function. But sketches are no solution after all. As $[x]$ and $4$ are integers, we conclude that $2x$ is an integer. Also $x-1<[x]\le x$ make $-x-1<[x]-2x\le -x$, so that either $-4\le -x<-3$ or $4\le -x<5$. These conditions leave us with $x\in\{-4, -3\tfrac12,4,4\tfrac12\}$ to ... 1 In German maths teaching in school, around 10th or 11th year, there is the subject Kurvendiskussion, which should be translated as "Discussion of [the properties of] a Curve". It is a systematic poking of a given function for characteristic properties of its graph. Domain Intersections with $x$- and $y$-axis Symmetries Extrema Inflection points Poles Gaps ... 1 Well, when it comes to graphing any sort of function, a very extensive analysis would be that involving its first and second derivatives. Everything pointed out by mvw is great, though perhaps it could be a bit more explained. Anyways, if you really want to get a intuitive feel of how different graphs look like, I recommend you download a graphing ... 1 If you need something skew, looking roughly like a parabola, you could use higher polynomials: y(x) = $x^4 + 2x^3 + 3x^2$, or with other coefficients. 1 You could think of it in this way If the modification is in the form $y=f(x)+a$, then the graph shifts up/down by $a$. If the modification is in the form $y=f(x+a)$, then the graph shifts left/right by $-a$. (Take note of the negative sign here. So for example, if we have $y=f(x-2)$, then the graph shifts to the right by 2 units. If the modification is in ... 1 One way to imagine a map $f : D \rightarrow \mathbb{C}$ with $D \subseteq \mathbb{C}$ is to think of it as a 2-dimensional vector field. Remember that $\mathbb{C}$ is just the vector space $\mathbb{R}^{2}$ equipped with a special multiplication $$* : \ \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}^2$$ $$(v_1,v_2)*(w_1,w_2) = ... 1 I like @Joker123's suggestion of visualizing complex maps as vector fields, but I want to point out that this is not what is done in the tool you linked at davidbau.com. The plot at davidbau.com is drawing the preimage of a flat grid and unit circle under the input map. If you try the identity map (z), you'll see that flat grid and unit circle. If you ... 1 If you have two points in the plane A(x_a, y_a) and B(x_b, y_b), the distance between them is given by:$$d(A, B)=\sqrt{(x_a-x_b)^2+(y_a-y_b)^2}. Some more detailed explanation you can find for example here: https://www.mathsisfun.com/algebra/distance-2-points.html 1 Something you could try is graph several different examples to see how each one is different. For example, graph $\sin(x)$ then graph $\sin(\sin(x))$. You will see it is the same graph only $\sin(\sin(x))$ has a slightly smaller amplitude and can't reach $1$ or $-1$ in the y axis. If you graph $\cos(x)$ and $\sin(\cos(x))$, $\sin(\cos(x))$ will be the same ... Only top voted, non community-wiki answers of a minimum length are eligible	2015-11-26 01:04:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8198490738868713, "perplexity": 202.54310561315174}, "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-2015-48/segments/1448398446248.56/warc/CC-MAIN-20151124205406-00150-ip-10-71-132-137.ec2.internal.warc.gz"}
http://skema-rangkaian-elektronika.blogspot.com/2010/01/skema-osilator-gelombang-segitiga.html	# Skema Osilator Gelombang Segitiga Skema Osilator gelombang segitiga In this page, I acquaint the triangular beachcomber oscillator which acclimated the Operational Amplifiers (TL082). The ambit uses the two OP amplifiers. The OP of the one works as "the Schmitt circuit". The added OP works as "the affiliation circuit". At the ambit diagram above, IC(1/2) is the Schmitt ambit and IC(2/2) is the affiliation circuit. The achievement of the Schmitt ambit becomes the aboveboard wave. The achievement of the Schmitt ambit is inputted to the affiliation circuit. The achievement of the affiliation ambit becomes the triangular wave. The ability accumulation needs both of the absolute ability accumulation and the abrogating ability supply. Also, to assignment in the oscillation, the action of R2>R3 is necessary. However, back authoritative the amount of R3 baby compared with R2, the achievement voltage becomes small. The abreast amount is acceptable for R2 and R3. You may accomplish adverse if not aquiver application the resistor with the aforementioned value. The ambit diagram aloft is application the resistor with the amount which is altered to accomplish oscillate surely.	2014-03-09 11:04:04	{"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": 0.8271344900131226, "perplexity": 11638.28587345496}, "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-2014-10/segments/1393999677441/warc/CC-MAIN-20140305060757-00034-ip-10-183-142-35.ec2.internal.warc.gz"}
https://cs.stackexchange.com/tags/oracle-machines/hot	# Tag Info 17 Just take a problem whose Turing degree is above $0'$, which is the degree of The Halting Oracle. In terms of the arithmetical hierarchy you want problems which are above $\Sigma^0_1$. Examples of such problems (where $\phi_n$ is the $n$-th partial computable function and $W_n = \{k \in \mathbb{N} \mid \text{$\phi_n(k)$is defined}\}$ is the $n$-th ... 13 $\mathrm{BQP}^{\mathrm{BQP}} = \mathrm{BQP}$ has been proved in Strengths and Weaknesses of Quantum Computing Bennett et al. (arXiv). According to the complexity zoo, $\mathrm{ZBQP}^{\mathrm{ZBQP}} = \mathrm{ZBQP}$. 13 Sure, you just have to be careful thinking about what it means to have an oracle. The problem comes from an annoying abuse of notation we use in CS: In the statement $P=NP$, $P$ refers to a set of languages. But in the statement $P^A = NP^A$, $P$ refers to a class of Turing Machines (determinstic polytime TMs). You should think of these two $P$s as of ... 13 Why are oracles used in the context you mentioned (where we have an oracle for the halting problem)? Because that allows us to answer questions that are fascinating, questions like "Are there problems that are even harder than the halting problem?". I'm not saying these questions are necessarily useful or important in practice -- but they are fascinating, ... 12 Oracles are a very general formalization of the idea, "If I could solve $X$ efficiently, I could use that to solve $Y$ efficiently." I accept that it sounds a bit silly to go as far as "If I could solve problem $X$ in constant time, I could use that to solve $Y$ efficiently" but, actually, that doesn't make any real difference at the ... 11 Here are some answers to some of the questions, but certainly not all of them: Apparently, according to Wikipedia, we have $P^P=P$, $BPP^{BPP}=BPP$, $PSPACE^{PSPACE}=PSPACE$, $L^L=L$, and $\oplus P^{\oplus P} = \oplus P$. See also What is complexity class $\oplus P^{\oplus P}$, which observes that $\oplus P^{\oplus P} = \oplus P$. Also, if $C^C=C$, then $... 11 No, an oracle is a black box that solves a problem in a single step. The problem that it solves can be any problem, it doesn't need to be the halting problem. What Scott is saying is that there is some black box that a BQP machine with it can do more than what a BPP machine can do with it. However, it doesn't mean that without that black box BQP is more ... 10 There are several applications to oracles. First, there is usage in proving lower bounds (i.e. Turing reductions): if you know that a problem$L$cannot be solved within some complexity (or computability) class$C$, and you show that an oracle to$L'$allows you to solve$L$within$C$, then you can conclude that$L'$is also not in$C$. Second, there is ... 8 Let me try to answer your multifaceted question using an analogy from number theory (or rather, Peano arithmetic). The platonist point of view holds that every question about natural numbers has a YES/NO answer. This is known as "true arithmetic". However, as Gödel showed, some propositions concerning natural numbers can be neither proved nor disproved. The ... 8 It is not true that for$A$being$\sf EXP$-complete${\sf DTIME}^A(n^k) = {\sf EXP}$, but you are right with${\sf P}^A={\sf EXP}$. Here is the reason for this. In order to make use of the oracle you have to transform your problem via a reduction. This reduction is a polynomial reduction. The running time of this particular reduction might need$\omega(n^k)... 8 There's a number of ways to look at this. One is that in proofs, implication is kind of like a function, that takes as input a proof of something, and outputs a proof of something else. We can write functions that operate on values that we don't have. For example, let's consider the halting number $h$, which is not computable. I can write the function $... 8 Please refer Does Cook Levin Theorem relativize?. Also refer to Arora, Implagiazo and Vazirani's paper: Relativizing versus Nonrelativizing Techniques: The Role of local checkability. In the paper by Baker, Gill and Solovay (BGS) on Relativizations of the P =? N P question (SIAM Journal on Computing, 4(4):431–442, December 1975) they give a language$B$... 8 No,$\mathsf{EXP^{EXP}=2EXP}$, a set of languages decidable in$O\left(2^{2^{\mathrm{poly}(n)}}\right)$time. This is just because you can give exponentially long input to an oracle which can solve it. So, the total power is$\exp(\exp(n))\ne \exp(n)$. To see why$\mathsf P$is self-low just take a machine that can run quadratic time and give to it the ... 7 For an oracle$A\in {\sf P}$you have${\sf P}^A={\sf P}$(since you can encode all requests to the oracle as a submodule of the TM). By the same argument you als have that${\sf NP}^A={\sf NP}$. Thus in this case${\sf NP}^A\neq{\sf P}^A$would imply${\sf NP}\neq{\sf P}$. You basically said this already in your question. Note that we cannot query "... 7 The proof of the time hierarchy theorem relativizes. This means that all the steps remain true if all Turing machines are given access to the same oracle$O$(for arbitrary$O$). This implies that the theorem itself remains true if all Turing machines are given access to the oracle$O$. So yes, it is possible to prove that no oracles exist with respect to ... 6 There are many mathematical objects that "do not exist" (afaik, and whatever that means), and which have been the support of mathematical reasonning for centuries (maybe not many centuries). The first example that comes to mind is the real numbers, and more generally non denumerable sets. All you need is that the non-existing object have properties that are ... 6 A complexity class$ C $is called self-low precisely when$ C^C = C $. In general, "lowness" was studied a lot in the 80s and 90s -- google will uncover much for you. 6 Krentel gave two problems complete for$\Delta_2^P$(see Theorem 3.4): Input: Boolean formula$\phi(x_1,\dots,x_n)$. Question: Is$x_n = 1$in the lexicographically largest satisfying assignment of$\phi$? Input: Weighted graph$G$, integer$k$. Question: Is the length of the shortest TSP tour in$G$divisible by$k$? Krentel also states that the only ... 6 An approximation oracle for an optimization problem$X$is an oracle which accepts an instance of$X$and returns an approximate optimum. The parameters$\alpha,\beta$quantify the quality of the approximation. Approximation oracles are a formal way of stating results of the following form: Given a polynomial time$C$-approximation algorithm for$X$, ... 6 The proof that a Turing machine with an oracle for$X$can't solve the halting problem for Turing machines with an oracle for$X$is identical to the proof that an ordinary Turing machine can't solve the halting problem for ordinary Turing machines. 5 It's open whether$\mathsf{BPP} \subseteq \mathsf{P}^{\mathsf{NP}}$. The best we can currently say about$\mathsf{BPP}$is that it is contained in S2P, a class contained in$\Sigma_2 \cap \Pi_2$and$\mathsf{ZPP}^{\mathsf{NP}}$. See references given at Wikipedia. If$\mathsf{BPP} \subseteq \mathsf{P}^{\mathsf{NP}}$, then this fact does not relativize (... 5 Some optimization algorithms are formulated as algorithms for an oracle Turing machine. This is common, among else, in submodular optimization. An algorithm for minimizing or maximizing a submodular function given some constraints typically has oracle access to the submodular function (and sometimes to the list of constraints). This has the advantage of ... 5 Well, nowhere does the answer claim the reduction "implies that the original problem is in NP". So, that explains your confusion. You read something into the answer that isn't actually there. Also, the answer says "Cook reduction". This was a heads-up about the fact that it's not the Karp-style reduction you might be used to. You might like to learn ... 5 The answer in its current form shows only that it belongs to$\Delta_2^P$, or$P^{NP}$. This is a (not necessarily strict) subset of$\Sigma_2^P$, or$NP^{NP}$, which is what the asker had mentioned. The answerer said that they don't know whether it's$\Delta_2$-complete, but doubt it; and it similarly seems unlikely that this in$NP$. (The next answer ... 4 Take$A=NP$, as you requested.$P^{NP}$is not necessarily equal to$NP$. Let me give an example why not. Consider TAUTOLOGY (given a boolean formula$\varphi$, is it true for all possible assignments to the variables?). TAUTOLOGY is known to be co-NP-complete. Therefore, TAUTOLOGY most likely is not in NP, since if TAUTOLOGY were in NP, it would follow ... 4 An oracle is just a theoretical device which will provide the answer to a given class of decision problems in a single step. We say that a decision problem is in$BPP$relative to the oracle if a turing machine (or whatever model of computation you are using) with access to the oracle can answer the decision problem in a polynomial amount of time with ... 4 Every problem$P$can be solved with an oracle machine with oracle access to$P$. In order to get a more meaningful answer, we consider the concept of Turing semi-degree, which is the set of all problems computable with an oracle to$P$, for some problem$P$. The same diagonalization argument used to prove that the halting problem isn't decidable shows that ... 4 This comment lists L (logspace), NC (polylog depth), P, BPP, BQP, and PSPACE as examples of self-low complexity classes. 4 You have the right idea. Suppose you have a SAT oracle and an instance$I$of 3SAT (or whatever SAT-ish class you like) containing$n$variables,$x_1, x_2, \dotsc x_n\$. You could then do this: send I to the oracle if the oracle answers "not satisfiable" quit else j = 1 I_0 = I repeat transform I_{j-1} to I_j by substituting x_j = 1 ... Only top voted, non community-wiki answers of a minimum length are eligible	2021-06-19 01:10:47	{"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": 0.9236318469047546, "perplexity": 700.203735365359}, "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-2021-25/segments/1623487643354.47/warc/CC-MAIN-20210618230338-20210619020338-00226.warc.gz"}
https://cvgmt.sns.it/paper/5331/	# The equation div$u$+$\langle a, u \rangle=f$ created by csato on 12 Nov 2021 [BibTeX] preprint Inserted: 12 nov 2021 Year: 2019 ArXiv: 1901.05783 PDF Abstract: We study the solutions $u$ to the equation $$\begin{cases} \operatorname{div} u + \langle a , u \rangle = f & \textrm{ in } \Omega,\\ u=0 & \textrm{ on } \partial \Omega, \end{cases}$$ where $a$ and $f$ are given. We significantly improve the existence results of Csat\'o and Dacorogna, A Dirichlet problem involving the divergence operator, \textit{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 33 (2016), 829--848, where this equation has been considered for the first time. In particular, we prove the existence of a solution under essentially sharp regularity assumptions on the coefficients. The condition that we require on the vector field $a$ is necessary and sufficient. Finally, our results cover the whole scales of Sobolev and H\"older spaces. Credits \| Cookie policy \| HTML 5 \| CSS 2.1	2022-08-13 12:08:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6745099425315857, "perplexity": 755.3900562993532}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571950.76/warc/CC-MAIN-20220813111851-20220813141851-00357.warc.gz"}
https://brilliant.org/problems/pulse-transmission/	# Pulse transmission A triangular deformation (a pulse) is travelling on a tightened string with the speed $$5~\mbox{m/s}$$. The length of the base of the triangle is $$30~\mbox{cm}$$. The pulse suddenly reaches a point where the wire connects to a different wire made out of a different material. The pulses travel with the speed $$7~\mbox{m/s}$$ in this material. What will be the length of the base of the triangle (length of the pulse) in cm on the other wire? ×	2017-12-14 17:18:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.511093258857727, "perplexity": 344.343742531039}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948545526.42/warc/CC-MAIN-20171214163759-20171214183759-00718.warc.gz"}
http://www.askiitians.com/forums/Analytical-Geometry/24/44541/triangle.htm	Click to Chat 0120-4616500 CART 0 • 0 MY CART (5) Use Coupon: CART20 and get 20% off on all online Study Material ITEM DETAILS MRP DISCOUNT FINAL PRICE Total Price: R There are no items in this cart. Continue Shopping Get instant 20% OFF on Online Material. coupon code: MOB20 \| View Course list Get extra R 350 off USE CODE: SSPD25 if median of triangle are 5 cm 6cm and 7cm then find area of this triangle. 4 years ago Share Ans:Let the median length be m1, m2& m3. Then area of triangle A is:$A = \frac{4}{3}\sqrt{S(S-m_{1})(S-m_{2})(S-m_{3})}$where S is$S= \frac{m_{1}+m_{2}+m_{3}}{2}$$S= \frac{5+6+7}{2} = 9$$A = \frac{4}{3}\sqrt{S(S-m_{1})(S-m_{2})(S-m_{3})}$$A = \frac{4}{3}\sqrt{9(9-5)(9-6)(9-7)}$$A = \frac{4}{3}\sqrt{9.4.3.2}$$A = 8\sqrt{6}$Thanks & RegardsJitender SinghIIT DelhiaskIITians Faculty 2 years ago # Other Related Questions on Analytical Geometry The orthocentre of a triangle formed by the pair of x+y+1=0lines 2(x^2)-xy-(y^y)+x+2y-1=0. Differentiate the eqn. 2(x^2)-xy-(y^y)+x+2y-1=0. once w.r.t x and y separetly and get the relation in x and y eqns. then u have three eqns. for a triangle. apply the orthocentre formulae. Vikas TU 2 months ago Compete the shortest distance between the circle x^2+y^2-10x-14y-151=0 and the point (-7,2) . Hii You have to find the equation of normal passing from the given point . Find out the length from the point given and the centre of circle, you can get the answer Sourabh Singh 2 months ago Show that the equation x^2 + y^2 - 2x - 2ay - 8 = 0 represents for different values of 'a' a sytem of circles passing through two fixed points a, b on the x - axis and find the equation of... From Famile of circles, Apply S1 + KL = 0 Where S1 = x^2 + y^2 - 2x - 2ay - 8 = 0 L = x + 2y + 5 = 0 and then put y = 0. Vikas TU 2 months ago What will the curve xy-3x-2y-10=0 represent? xy-3x-2y-10=0Take y common,y(x – 2) = 3x + 10y = (3x + 10)/(x-2)Point of discontinuity is at x = 2It represents a curve of which relation. in y and x with other than the ellipse, parabola,... SREEKANTH 4 months ago xy-3x-2y-10=0Take y common,y(x – 2) = 3x + 10y = (3x + 10)/(x-2)Point of discontinuity is at x = 2It represents a curve of which relation. in y and x with other than the ellipse, parabola,... Vikas TU 5 months ago what is vector quantity??????????????????????????????????????????????????? Dillep, The quantity which has both magnitude and direction then it is called vector quantity.For example velocity,acceleration......etc. SAI SARDAR 9 months ago Dear Dilip Vectors have magnitude and direction, scalars only have magnitude. The fact that magnitude occurs for both scalars and vectors can lead to some confusion. There are some... Prabhakar ch 9 months ago Dear Dilip Vectors have magnitude and direction, scalars only have magnitude. The fact that magnitude occurs for both scalars and vectors can lead to some confusion. There are some... SAI SANDY 9 months ago View all Questions » • Complete JEE Main/Advanced Course and Test Series • OFFERED PRICE: R 15,000 • View Details Get extra R 3,750 off USE CODE: SSPD25 Get extra R 350 off USE CODE: SSPD25 More Questions On Analytical Geometry	2016-10-22 07:18:00	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38525015115737915, "perplexity": 3025.1449067813655}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718840.18/warc/CC-MAIN-20161020183838-00185-ip-10-171-6-4.ec2.internal.warc.gz"}
http://naturalunits.blogspot.com/2012/12/newtons-law-of-gravity-for-solar-system.html	## Dec 4, 2012 ### Newton's Law of Gravity for Solar System Planets (visualization) According to Newtonian gravity theory $v(r) =\sqrt{ \frac{G M}{r}}$ If the orbit is circular, the speed is simply proportional to $1/\sqrt{r}$. For general cases, however, after some derivation, the average speed, $\bar{v} = \frac{1}{2\pi}\int_0^{2\pi} \mathrm{d}\theta v(\theta) = \sqrt{\frac{GM}{a} } \frac{2 \mathrm{E}(\frac{2\epsilon}{1+\epsilon})}{\pi \sqrt{1-\epsilon}},$ where $\mathrm{E}(z) = \int_0^{\frac{\pi}{2}} (1-z \sin^2\theta)^{1/2} \mathrm{d}\theta$ is the elliptic function. Note that $\mathrm{E}(0) = \frac{\pi}{2}$, restoring the circular motion result. So, the average speed is proportional to $\frac{1}{\sqrt{a}}$ where $a$ is the semi-major axis. Fig. 1: the semi-major axis vs. average orbital speed for solar system planets in linear coordinates Fig. 2: the semi-major axis vs. average orbital speed for solar system planets in logarithmic coordinates The best fit of the slope gives 29.779763 km/s/AU, which is about the earth average orbital speed. Using the data of solar mass and gravitational constant, the average eccentricity is about 0.0195386. This is the absolute value.	2017-11-20 07:27:42	{"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": 0.8016054630279541, "perplexity": 467.8136650505205}, "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-2017-47/segments/1510934805923.26/warc/CC-MAIN-20171120071401-20171120091401-00796.warc.gz"}
https://www.bartleby.com/solution-answer/chapter-17-problem-65e-precalculus-mathematics-for-calculus-6th-edition-6th-edition/9780840068071/f7e28fbf-710b-40c0-8dba-2f208f69ffce	the nonlinear inequality. Express the solution using interval notation and graph the solution set. Precalculus: Mathematics for Calcu... 6th Edition Stewart + 5 others Publisher: Cengage Learning ISBN: 9780840068071 Precalculus: Mathematics for Calcu... 6th Edition Stewart + 5 others Publisher: Cengage Learning ISBN: 9780840068071 Solutions Chapter 1.7, Problem 65E To determine To solve: the nonlinear inequality. Express the solution using interval notation and graph the solution set. Expert Solution x(2,1)(0,1) . Explanation of Solution Given: The given inequality is 1+2x+12x . Concept used: Guidelines for solving nonlinear inequality: 1. Move all terms to one side. 2. Factor the non-zero side of the inequality. 3. Find the value for which each factor is zero. The number will divide the real lines into interval. List the interval determined by these numbers. 4. Make a table or diagram by using test values of the signs of each factor on each interval. In the last row of the table determining the sign of the product of these factors. 5. Determine the solution of the inequality from the last row of the sign table. Calculation: The given inequality can be expressed as 1+2x+12x0{subtract both sides from 2x}x(x+1)+2x2(x+1)x(x+1)0{simplify}x2+x+2x2x2x(x+1)0{simplify}x2+x2x(x+1)0(x1)(x+2)x(x+1)0 . Firstto find the zeros of the expression in the numerator and demniminator, then x1=0x=1x+2=0x=2x+1=0x=1x=0 From the two zeros above, it extracts the following intervals: (,2)(2,1)(1,0)(0,1)(1,) Now, make a table by using test values of the signs of each factor on each interval. (−∞,−2) (−2,−1) (−1,0) (0,1) (1,∞) (3x+2) − − − − + (x+2) − + + + + x − − − + + (x+1) − − + + + quotient + − − − + As it is seen that the less than or equal to 0 in the interval (2,1) and (0,1) . Hence,the solution set is x(2,1)(0,1) . The solution set of the inequality graphed on the number line. The graph of the non-linear inequality 1+2x+12x is: Have a homework question? Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!	2021-09-27 17:09:28	{"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": 0.843515932559967, "perplexity": 1907.0774723773568}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780058456.86/warc/CC-MAIN-20210927151238-20210927181238-00387.warc.gz"}
https://www.spp2026.de/projects/19/	19 Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces A locally compact separable metric space together with a regular Dirichlet form is called a Dirichlet space. There is a strong interplay between geometric properties of the Dirichlet space, spectral features of the generator of the Dirichlet form and stochastic features of the associated Markov process. The project studies this interplay focusing on global properties viz. on properties of the geometry "far out" and corresponding spectral and stochastic features. One approach is centered around the compactification via the Royden boundary, boundary terms and Greens formulae. The other approach is centered around harmonic functions and (generalized) eigenfunctions. Both approaches capture geometry "far out" via specific tools and concepts. The approaches are strongly related and exhibiting the relationship will lead to additional insights. The project will focus on the non-smooth non-local situation of graphs. ## Publications Given two weighted graphs $(X,b_k,m_k)$, $k=1,2$ with $b_1\sim b_2$ and $m_1\sim m_2$, we prove a weighted $L^1$-criterion for the existence and completeness of the wave operators $W_{\pm}(H_{2},H_1, I_{1,2})$, where $H_k$ denotes the natural Laplacian in $\ell^2(X,m_k)$ w.r.t. $(X,b_k,m_k)$ and $I_{1,2}$ the trivial identification of $\ell^2(X,m_1)$ with $\ell^2(X,m_2)$. In particular, this entails a general criterion for the absolutely continuous spectra of $H_1$ and $H_2$ to be equal. Journal Math. Phys. Anal. Geom. Pages 21-28 Link to preprint version We study the Kazdan-Warner equation on canonically compactifiable graphs. These graphs are distinguished as analytic properties of Laplacians on these graphs carry a strong resemblance to Laplacians on open pre-compact manifolds. • 1 ## Team Members Prof. Dr. Matthias Keller Universität Potsdam mkeller(at)math.uni-potsdam.de Prof. Dr. Daniel Lenz	2019-04-22 22:24:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7183489203453064, "perplexity": 724.9094686356016}, "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-2019-18/segments/1555578582736.31/warc/CC-MAIN-20190422215211-20190423001211-00176.warc.gz"}
https://blog.quantinsti.com/epat-webinar-28-june-2018/	### Session Outline If you've been looking to build a career into the quantitative and algorithmic trading domain, there is a high probability that you would have heard about the EPAT programme. But is it something that can help you in achieving your career & learning objectives in this domain? This informative session on EPAT addresses this question while covering various practical aspects of the EPAT programme. In addition to the detailed overview of each of the EPAT module, it also covers how EPAT fills in the existing skill gaps and addresses the learning requirements of the industry & the individuals. You'll get to know all this and much more in this brief but effective informative session on EPAT: • Why has EPAT been working for professionals from 60+ countries? • Learn about the practical aspects of EPAT.	2021-04-11 15:53:51	{"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": 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": 0.19514471292495728, "perplexity": 1792.1833605720285}, "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-2021-17/segments/1618038064520.8/warc/CC-MAIN-20210411144457-20210411174457-00372.warc.gz"}
https://math.stackexchange.com/questions/2882242/there-is-a-sequence-p-n-of-polynomials-such-that-p-n-cos-x-to-fx	# There is a sequence $(P_{n})$ of polynomials such that $P_{n}(\cos x) \to f(x)$ uniformly over $[0,\pi]$. Show that for any function continuous $f:[0,\pi] \to \mathbb{R}$, there is a sequence $(P_{n})$ of polynomials such that $$P_{n}(\cos x) \to f(x)\;\text{uniformly over}\;[0,\pi].$$ The Weierstrass Theorem says that there is a sequence of polynomials $p_{n}$ such that $p_{n} \to f$ uniformly. We cannot have $p_{n} = P_{n}$ if no, $P_{n}(\cos x) \to f(\cos x)$, right? I know that also, there is a sequence $q_{n}$ such that $q_{n} \to \cos$ uniformly. I'm trying to somehow use these two sequences $p_{n},q_{n}$, but I'm stuck. I don't want the solution to the exercise, but I would like a hint. $f\circ \cos^{-1}$ is a continuous function on $[-1,1]$. Approximate this by polynomials. $p_n$ and that will do the trick.	2019-07-23 03:17:48	{"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": 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": 0.9603862762451172, "perplexity": 36.73154326875253}, "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/1563195528687.63/warc/CC-MAIN-20190723022935-20190723044935-00355.warc.gz"}
http://physics.bgu.ac.il/~dcohen/View.php?page=Research+QCC	Highlights ## Diffractive energy spreading and its semiclassical limit (2006)() We consider driven systems where the driving induces jumps in energy space: • particles pulsed by a step potential; • particles in a box with a moving wall; • particles in a ring driven by an electro-motive-force. In all these cases the route towards quantum-classical correspondence is highly non-trivial. Some insight is gained by observing that the dynamics in energy space, where n is the level index, is essentially the same as that of Bloch electrons in a tight binding model, where n is the site index. The mean level spacing is like a constant electric field and the driving induces long range hopping 1/(n-m). In the illustration below the EMF is concentrated at one point along the ring. Whenever a particle cross the EMF region its kinetic energy is boosted. The energy jump is eV. From quantum mechanical point of view this constitutes a non-perturbative effect. It is neither "adiabatic" nor "diabatic" but rather a "semiclassical" transition. In the analagous tight binding model the semicalssical dynamics is regarded as uni-directional Bloch oscillations. [1] A. Stotland and D. Cohen, J. Phys. A 39, 10703 (2006). [arXiv] [pdf]	2018-04-23 00:15:43	{"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": 0.8010458946228027, "perplexity": 1129.5947037477054}, "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-2018-17/segments/1524125945668.34/warc/CC-MAIN-20180422232447-20180423012447-00485.warc.gz"}
http://mathhelpforum.com/calculus/220346-complex-numbers-ellipse-print.html	# Complex Numbers and Ellipse • July 3rd 2013, 07:35 PM lovesmath Complex Numbers and Ellipse Using the fact that \|z1-z2\| is the distance between two points z1 and z2, give a geometric argument that \|z-4i\|+\|z+4i\|=10 represents an ellipse whose foci are (0,4) and (0,-4). Can you help me get started, please? • July 3rd 2013, 07:54 PM HallsofIvy Re: Complex Numbers and Ellipse The [n]defining[/b] property of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is a constant. • July 4th 2013, 10:09 AM lovesmath Re: Complex Numbers and Ellipse Do I need to use the fact that z=x+iy? • July 4th 2013, 12:38 PM HallsofIvy Re: Complex Numbers and Ellipse No! You could but the problem says "give a geometric argument". Just look closely at the problem! What points in the complex plane do "i " and "-i " correspond to? • July 4th 2013, 02:01 PM lovesmath Re: Complex Numbers and Ellipse They correspond to (0,1) and (0,-1), so the distance between those two points is 2. The distance from each of those points to the foci is 3. • July 4th 2013, 02:15 PM Plato Re: Complex Numbers and Ellipse Quote: Originally Posted by lovesmath Using the fact that \|z1-z2\| is the distance between two points z1 and z2, give a geometric argument that \|z-4i\|+\|z+4i\|=10 represents an ellipse whose foci are (0,4) and (0,-4). Can you help me get started, please? Because an ellipse is the set of all points such that the sum of there distances to two fixed points is constant. How does that definition apply to $\|z+4i\|+\|z-4i\|=10~?$ How does it make the equation an ellipse ? What are the two points in question? • July 7th 2013, 02:03 PM lovesmath Re: Complex Numbers and Ellipse The two points in question are (0, 4i) and (0, -4i). The distance between those points is 8, so I don't understand where the 10 comes from. Am I interpreting the question incorrectly? • July 7th 2013, 02:41 PM Plato Re: Complex Numbers and Ellipse Quote: Originally Posted by lovesmath The two points in question are (0, 4i) and (0, -4i). The distance between those points is 8, so I don't understand where the 10 comes from. Am I interpreting the question incorrectly? With all due respect, you real problem is that you have no idea what an ellipse really is. You might study this webpage. For your ellipse the focii are $(0,4)~\&~(0,-4)$,. Thus the major axis is vertical.	2014-03-07 19:57:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/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": 0.8218231201171875, "perplexity": 880.2045551246922}, "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-2014-10/segments/1393999650773/warc/CC-MAIN-20140305060730-00059-ip-10-183-142-35.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/737886/writing-square-root-of-square-free-numbers-as-sum-of-square-roots	# Writing square root of square-free numbers as sum of square roots. Some days ago i came across a question about writing $\sqrt {2001}$ as sum of two other square roots. I managed to prove that this is not possible unless one of them is zero and the other one is $2001$. The proof was as following: $\sqrt{2001}=\sqrt a+\sqrt b$, $\sqrt{2001}-\sqrt a=\sqrt b$ so $2001+a-2\sqrt{2001a}=b$. This shows that $2\sqrt{2001a}$ is an integer so $2001a$ is a perfect square. We also know that $2001=323*29$ which is a square-free number. so $a$ must divide all of $3,23,29$ which means $a\geq2001$ so$\sqrt a\geq\sqrt{2001}$ and $\sqrt{b}\leq 0$ which means $b=0$. With exact method we can prove that $\sqrt{s}=\sqrt a+\sqrt b$ does not have any natural solutions with $s$ being a square-free number. Then I tried to generalize the proof for $3$ or more square roots but i failed. The only thing I always get is $\sqrt {ab}+\sqrt {bc}+\sqrt {ac}$ is an integer which does not help at all. For what numbers can we write the square root of a square-free number as sum of three or more non-zero square roots? I would appreciate any help. • I remember an excercise from a book that asked a proof of that if $p_n$ is the $n$th prime, then the field $\Bbb Q(\sqrt{p_1},\ldots,\sqrt{p_n})$ does not contain $\sqrt{p_{n+1}}$. I think that this statement must be related with your question. Sadly, I couldn't solve the exercise. Apr 3, 2014 at 8:40 • @user2425 Thanks you, but i would prefer an elementary proof like the first case i proved :) – CODE Apr 3, 2014 at 8:51 • I don't think there will be an elementary solution. The general case has already been answered here: math.stackexchange.com/a/437374/43288 Apr 8, 2014 at 14:11 • A related question: Integer solutions to $\sqrt{a} + \sqrt{b} = \sqrt{c}$. Sep 23, 2020 at 2:21 Lemma 1. If $m$ is a positive integer and $\sqrt m$ is rational, then $\sqrt m$ is an integer. Proof. Easy. Lemma 2. If $m,n$ are positive integers and $\sqrt m+\sqrt n$ is rational, then both $\sqrt m$ and $\sqrt n$ are integers. Proof. Say $\sqrt m+\sqrt n=x\in\Bbb Q$. Then $$\sqrt m-\sqrt n=\frac{m-n}{x}$$ is rational and so is $$\sqrt m=\frac{(\sqrt m+\sqrt n)+(\sqrt m-\sqrt n)}{2}\ ,$$ and likewise $\sqrt n\,$. By lemma 1, $\sqrt m$ and $\sqrt n$ are integers. Now suppose that $$\sqrt a+\sqrt b+\sqrt c=\sqrt s\ ,$$ where $a,b,c,s$ are positive integers and $s$ is squarefree. Squaring and rearranging, $$2\bigl(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\bigl)=s-a-b-c\ .$$ Now add to this equation the identity $2\sqrt a\sqrt a=2a$ and factorise to obtain $$2\sqrt{bc}+2\sqrt{as}=s+a-b-c\ .$$ By lemma 2, we see that $\sqrt{as}$ is an integer; since $s$ is squarefree, $a$ must be a square times $s$, say $a=p^2s$. Similarly $b=q^2s$ and $c=r^2s$, so $$p\sqrt s+q\sqrt s+r\sqrt s=\sqrt s\ ,$$ but as $p+q+r>1$, this is impossible. • Thank you! Very nice answer. But can we do the same for 4 or more square roots? – CODE Apr 3, 2014 at 12:37 • No real idea. I suspect it might be significantly more difficult. I guess the first thing would be to try to generalise lemma 2 to more than two square roots. Apr 3, 2014 at 21:23 Note: Squaring only helps when you have 5/6 or fewer terms. Otherwise, you run the risk of introducing too many square roots. As such, you need something more powerful to deal with the general case. Theorem. Let $SF$ be the set of positive integers that are not divisible by the square of any prime. $SF = \{1, 2, 3, 5, 6, 7, 10, 11, 13, \ldots \}$. If $\{a_i \} _{i=1}^n$ are distinct numbers from the set $SF$, and $\{b_i\}_{i=1}^n$ are any integers, then $S = \sum b_i \sqrt{a_i} = 0$ if and only if all $b_i = 0$. Corollary No square free integer can be written as the sum of 3 or more non-zero square roots. Proof. The simplest approach is to use Galois Theory, which might be beyond OP. I'd present an 'elementary' approach which I first saw from Feng Zuming. Recall the idea of conjugates. Consider the linear expression $L(x_1, x_2, \ldots, x_n) = a_1 x_1 + a_2 x_2 + \ldots + a_n x_n$. Consider the conjugate expressions, which have the form $L' (x_1, x_2, \ldots, x_n) = a_1 x_1 \pm a_2 x_2 \pm \ldots \pm a_n x_n$. [There are $1 \times 2 \times \ldots \times 2 = 2^{n-1}$ such expressions.] Let $T$ be a variable and consider the polynomial $$F_{L(x_1, x_2, \ldots , x_n)} (T) = \prod_{L'} \big(T - L'(x_1, x_2, \ldots, x_N) \big).$$ Consider it as a polynomial in $x_i, i\neq 1$. Changing any of the signs of $x_i$ doesn't change $F$, since the set $\{ L' \}$ stays the same. So, $$F_{L(x_1, x_2, \ldots , x_n)} (T) = F_{L(x_1, \pm x_2, \ldots ,\pm x_n)} (T) .$$ As such, the polynomial expansion only contains even powers of $x_i, i\neq 1$. It is clear that it can contain odd or even powers of $x_1$, hence we have $$F_{L(x_1, x_2, \ldots , x_n)} (T) = x_1 P ( x_1^2, x_2 ^2, \ldots, x_n ^2 , T) + Q( x_1^2, x_2 ^2, \ldots, x_n ^2 , T).$$ Since $F$ is a polynomial with integer coefficients, it follows that $P$ and $Q$ are also polynomials with integer coefficients. We will show a slight variant of the original problem, namely that no non-zero integer $M$ can be represented as a nontrivial canonical integer sum of radicals. What this means, is that the square roots have all been simplified, so we're left with square free terms under the root sign. In this case, we are excluding the radical $\sqrt{1}$, which is why we now include the non-zero integer $M$. Hence this problem is equivalent. We will prove this statement by induction. Base case: Clearly, if $b_1 \sqrt{a_1} = M$, then $M^2 = b_1^2 a_1$, which means that $a_1$ must be a square, which is not possible. Suppose that we have an expression of the form $\sum b_i \sqrt{a_i}$ such that $\sum b_i \sqrt{a_i} = M \neq 0$. Then, the polynomial $F_{L(\sqrt{a_1}, \sqrt{a_2}, \ldots , \sqrt{a_n})} (M) = 0$. From the previous discussion, we have $0 = \sqrt{a_1} P(a_1, a_2, \ldots, a_n, M) + Q(a_1, a_2, \ldots, a_n, M)$ Each of these polynomials has integer coefficients, and integer variables, hence when evaluated at an integer, is equal to an integer. By the base case, this shows that $P(a_1, a_2, \ldots, a_n, M) = Q(a_1, a_2, \ldots, a_n, M) =0$. As such, this gives us $0 = - \sqrt{a_1} P(a_1, a_2, \ldots, a_n, M) + Q(a_1, a_2, \ldots, a_n, M)$ Now, consider the expression $G_{L(x_1, x_2, \ldots, x_n)} (T) = \prod \big( T + L' (x_1, x_2, \ldots, x_n) \big)= - x_1 P ( x_1^2, x_2 ^2, \ldots, x_n ^2 , T) + Q( x_1^2, x_2 ^2, \ldots, x_n ^2 , T).$ We know that $\prod(M + L') = 0$, and thus $M = -b_1 \sqrt{a_1} \pm a_2 x_2 \pm \ldots \pm a_n x_n$, for some combination of signs. Adding this to $M = b_1 \sqrt{a_1} + b_2 \sqrt{a_2} + \ldots + b_n \sqrt{a_n}$, we obtain that $2M = (b_2 \pm b_2) \sqrt{a_2} + \ldots + (b_n \pm b_n) \sqrt{a_n}$, which contradicts the induction hypothesis.	2022-05-25 16:39:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9058103561401367, "perplexity": 79.28486927483517}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662588661.65/warc/CC-MAIN-20220525151311-20220525181311-00653.warc.gz"}
http://math.stackexchange.com/questions/289271/unknown-result-in-probability-theory-relating-cdf-of-any-density-to-the-cdf-of-n	# Unknown result in probability theory relating CDF of any density to the CDF of normal distribution There is apparently a result in probability theory saying: If $A(z)$ is any cumulative distribution function, $\alpha(t)$, the corresponding characteristic function and $\Phi(z) = \int_{-\infty}^{z}e^{-\frac{t^{2}}{2}}\mathrm{d}t$ is the cumulative distribution of the normal distribution, then, for any $T > 0$: $$\|A(z) - \Phi(z)\| \leq \int_{-T}^{T}\mathrm{d}t\left\|\dfrac{\alpha(t) - e^{-\frac{t^{2}}{2}}}{t}\right\| + \dfrac{24}{T \pi \sqrt{2 \pi}}$$ Reference: Eq. 4, Page 11 of http://www.glassonion.org/ecc.pdf Could anyone tell me what name this theorem goes by ? I am unable to find any in the above form. Could anyone tell what parameter of $\Phi(z)$ is specific to the CDF function $A(z)$ ? I presume that for a different CDF, say, $A^{\prime}(z)$ satisfying the above result, the corresponding $\Phi(z)$ would be different. - I don't understand the last part of your question. There is only one function $\Phi$ and the inequality presumable holds for any $z$. – Hagen von Eitzen Jan 28 '13 at 21:52 I meant that, in the above case, $\Phi(x)$ is the CDF of a normal distribution with mean 0 and variance 1. I was asking how the mean and variance values depend on the particular CDF function $A(z)$. – Pavithran Iyer Jan 28 '13 at 21:59	2016-06-25 12:37:26	{"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": 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": 0.9028469920158386, "perplexity": 177.97686097444205}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783393146.70/warc/CC-MAIN-20160624154953-00166-ip-10-164-35-72.ec2.internal.warc.gz"}
https://zbmath.org/?q=an:0915.17004	# zbMATH — the first resource for mathematics A criterion for polynomial growth of varieties of Lie superalgebras. (English. Russian original) Zbl 0915.17004 Izv. Math. 62, No. 5, 953-967 (1998); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 5, 103-116 (1998). Suppose that $$V$$ is a variety of Lie superalgebras, i.e. a class of Lie superalgebras that satisfy some set of graded identical relations. Suppose that $$x_1,\dots,x_n$$ are arbitrary (nonhomogeneous) elements in an algebra from $$V$$, one considers the dimension of the spaces of multilinear polynomials in these variables. The supremum of these dimensions is called the codimension growth sequence $$c_n(V)$$. The authors find a criterion for a variety $$V$$ of Lie superalgebras over a field of characteristic zero to have polynomial codimension growth. Namely, $$V$$ has polynomial growth iff the following three conditions hold: 1) $$V$$ has a nilpotent commutator subalgebra, 2) each multilinear polynomial containing at least $$k$$ even and $$k$$ odd variables is an identity for $$V$$, 3) $$V$$ satisfies some additional specific identities. ##### MSC: 17B01 Identities, free Lie (super)algebras 17B30 Solvable, nilpotent (super)algebras 17B65 Infinite-dimensional Lie (super)algebras 16R10 $$T$$-ideals, identities, varieties of associative rings and algebras Full Text:	2022-01-26 10:50:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.5568473935127258, "perplexity": 546.7698180591494}, "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/1642320304947.93/warc/CC-MAIN-20220126101419-20220126131419-00216.warc.gz"}
https://www.lesswrong.com/users/daniel-paleka	Daniel Paleka Sorted by New 164mo 0 # Wiki Contributions I don't know why it sent only the first sentence; I was drafting a comment on this. I wanted to delete it but I don't know how. EDIT: wrote the full comment now. Let me first say I dislike the conflict-theoretic view presented in the "censorship bad" paragraph. On the short list of social media sites I visit daily, moderation creates a genuinely better experience. Automated censorship will become an increasingly important force for good as generative models start becoming more widespread. Secondly, there is a danger of AI safety becoming less robust—or even optimising for deceptive alignment—in models using front-end censorship.[3] This one is interesting, but only in the counterfactual: "if AI ethics technical research focused on actual value alignment of models as opposed to front-end censorship, this would have higher-order positive effects for AI x-safety". But it doesn't directly hurt AI x-safety research right now: we already work under the assumption that that output filtering is not a solution for x-risk. It is clear improved technical research norms on AI non-x-risk safety can have positive effects on AI x-risk. If we could train a language model to robustly align to any set of human-defined values at all, this would be an improvement to the current situation. But, there are other factors to consider. Is "making the model inherently non-racist" a better proxy for alignment than some other technical problems? Could interacting with that community weaken the epistemic norms in AI x-safety? Calling content censorship "AI safety" (or even "bias reduction") severely damages the reputation of actual, existential AI safety advocates. I would need to significantly update my prior if this turns out to be a very important concern. Who are people, whose opinions will be relevant at some point, that understand both what AI non-x-safety and AI x-safety are about, dislike the former, are sympathetic to the latter, but conflate them? Git Re-Basin: Merging Models modulo Permutation Symmetries [Ainsworth et al., 2022] and the cited The Role of Permutation Invariance in Linear Mode Connectivity of Neural Networks [Entezari et al., 2021] seem several years ahead. I cannot independently verify that their claims about SGD are true, but the paper makes sense on the first glance. Opinion: Symmetries in NNs are a mainstream ML research area with lots of papers, and I don't think doing research "from first principles" here will be productive. This also holds for many other alignment projects. However I do think it makes sense as an alignment-positive research direction in general. This is a mistake on my own part that actually changes the impact calculus, as most people looking into AI x-safety on this place will not actually ever see this post. Therefore, the "negative impact" section is retracted.[1] I point to Ben's excellent comment for a correct interpretation of why we still care. I do not know why I was not aware of this "block posts like this" feature, and I wonder if my experience of this forum was significantly more negative as a result of me accidentally clicking "Show Personal Blogposts" at some point. I did not even know that button existed. No other part of my post is retracted. In fact, I'd like to reiterate a wish for the community to karma-enforce [2] the norms of: • the epistemic standard of talking about falsifiable things; • the accepted rhetoric being fundamentally honest and straightforward, and always asking "compared to what?" before making claims; • the aversion to present uncertainties as facts. Thank you for improving my user experience of this site! 1. ^ I am now slightly proud that my original disclaimer precisely said that this was the part I was unsure of the most. 2. ^ As in, I wish to personally be called out on any violations of the described norms. Do you intend for the comments section to be a public forum on the papers you collect? I definitely endorse reading the ROME paper, although the popular-culture claims about what the second part of the paper actually shows seem a bit overblown. They do not seem to claim "changing facts in a generalizable way" (it's likely not robust to synonyms at all)". I am also vary of "editing just one MLP for a given fact" being the right solution, given that the causal tracing shows the fact being stored in several consecutive layers. Refer to a writeup by Thibodeau et al. sometime in the future. That being said, if you are into interpretability, you have to at least skim the paper. It has a whole bunch of very cool stuff in it, from the causal tracing to the testing of whether making Eistein a physician changes the meaning of the word "physics" itself. Just don't overfit on the methods there being exactly the methods that will solve interpretability of reasoning in transformers. I somewhat agree, athough I obviously put a bit less weight on your reason than you do. Maybe I should update my confidence of the importance of what I wrote to medium-high. Let me raise the question of continuously rethinking incentives on LW/AF, for both Ben's reason and my original reason. The upvote/karma system does not seem like it incentivizes high epistemic standards and top-rigor posts, although I would need more datapoints to make a proper judgement. I am very sorry that you feel this way. I think it is completely fine for you, or anyone else, to have internal conflicts about your career or purpose. I hope you find a solution to your troubles in the following months. Moreover, I think you did an useful thing, raising awareness about some important points: • "The amount of funding in 2022 exceeded the total cost of useful funding opportunities in 2022." • "Being used to do everything in Berkeley, on a high budget, is strongly suboptimal in case of sudden funding constraints." • "Why don't we spend less money and donate the rest?" Epistemic status for what follows: medium-high for the factual claims, low for the claims about potential bad optics. It might be that I'm worrying about nothing here. However, I do not think this place should be welcoming of posts displaying bad rhetoric and epistemic practices. Posts like this can hurt hurt the optics of the research done in the LW/AF extended universe. What does a prospective AI x-safety researcher think when they get referred to this site and see this post above several alignment research posts? EDIT: The above paragraph was off. See Ben's excellent reply for a better explanation of why anyone should care. I think this place should be careful about maintaining: • the epistemic standard of talking about falsifiable things; • the accepted rhetoric being fundamentally honest and straightforward, and always asking "compared to what?" before making claims; • the aversion to present uncertainties as facts. For some examples: My hotel room had the nightly price written on the inside of the door: $500. Shortly afterwards, I found out that the EA-adjacent community had bought the entire hotel complex. I tried for 15 minutes to find a good faith reading of this, but I could not. Most people would read this as "the hotel room costs$500 and the EA-adjacent community bought the hotel complex in which that hotel is a part of", while being written in a way that only insinuates and does not commit to meaning exactly that. Insinuating bad optics facts while maintaining plausible deniability, without checking the facts, is a horrible practice, usually employed by politicians and journalists. The poster does not deliberately lie, but this is not enough when making a "very bad optics" statement that sounds like this one. At any point, they could have asked for the actual price of the hotel room, or about the condition of the actual hotel that might be bought. I have never felt so obliged, so unpressured. If I produce nothing, before Christmas, then nothing bad will happen. Future funds will be denied, but no other punishment will ensue. This is true. But it is not much different from working a normal software job. The worst thing that can happen is getting fired after not delivering for several months. Some people survive years coasting until there is a layoff round. An important counterfactual for a lot of people reading this is a PhD degree. There is no punishment for failing to produce good research, except getting dropping out of the program after a few years. After a while I work out why: every penny I’ve pinched, every luxury I’ve denied myself, every financial sacrifice, is completely irrelevant in the face of the magnitude of this wealth. I expect I could have easily asked for an extra 20%, and received it. This might be true. Again, I think it would be useful to ask: what is the counterfactual? All of this is applicable for anyone that starts working for Google or Facebook, if they were poor beforehand. This feeling (regretting saving and not spending money) is incredibly common in all people that have good careers. One way to get rid of it and save several lives in the meantime is to donate the surplus to the Maximum Impact Fund. I would suggest going through the post with a cold head and removing parts which are not up to the standards. Again, I am very sorry that you feel like this. On the other hand, the current community believes that getting AI x-safety right is the most important research question of all time. Most people would not publish something just for their career advancement, if it meant sucking oxygen from more promising research directions. This might be a mitigating factor for my comment above. I am curious about what happened research fields which had "change/save the world' vibes. Was environmental science immune to similar issues? because LW/AF do not have established standards of rigor like ML, they end up operating more like a less-functional social science field, where (I've heard) trends, personality, and celebrity play an outsized role in determining which research is valorized by the field. In addition, the AI x-safety field is now rapidly expanding. There is a huge amount of status to be collected by publishing quickly and claiming large contributions. In the absence of rigor and metrics, the incentives are towards: - setting new research directions, and inventing new cool terminology; - using mathematics in a way that impresses, but is too low-level to yield a useful claim; - and vice versa, relying too much on complex philosophical insights without empirical work - getting approval from alignment research insiders.	2022-10-01 10:51:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34947505593299866, "perplexity": 2089.6625017872134}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335609.53/warc/CC-MAIN-20221001101652-20221001131652-00344.warc.gz"}
https://www.zbmath.org/?q=an%3A0572.73059	## On the convergence of the energy, stress tensors, and eigenvalues in homogenization problems of elasticity.(English)Zbl 0572.73059 In the framework of homogenization problems the authors study the convergence of the energy integrals, stress tensors and eigenvalue for elastic problems. The domain with smooth boundary belongs to a class of perforated domains. The medium is supposed nonhomogeneous and porous elastic with a periodic structure of period $$\epsilon$$ which tends to zero. Starting from the estimates of the solutions in the norm $$L^ 2$$ and, using correctors, in $$H^ 1$$, the convergence of energy integrals is valuated. Moreover, using correctors, an estimate is furnished for the difference between the stress tensors of the problems with $$\epsilon >0$$ and $$\epsilon =0$$. Some inequality is also obtained for the frequencies of free vibrations. Reviewer: M.Codegone ### MSC: 74H45 Vibrations in dynamical problems in solid mechanics 74E05 Inhomogeneity in solid mechanics 35J25 Boundary value problems for second-order elliptic equations Full Text: ### References: [1] De Giorgi, Boll. Un. Mat. Ital., (4) 8 pp 391– (1973) [2] Zhikov, Uspehi Mat. Nauk. 34 pp 63– (1979) [3] ; , On homogenization of system of elasticity with almost periodic coefficients. Vestnik. Mosc. Univ. ser. 1, mat. mech., 1982, no. 6, p. 62–70. [4] ; ; , Homogenization of eigenvalues and eigenfunctions of the boundary value problem of elasticity in a perforated domain. Vestn. Mosc. Univ., ser. 1, mat., mech., 1983, no. 4, p. 53–63. · Zbl 0567.73019 [5] ; ; , Homogenization of eigenvalues of the boundary value problem of elasticity with rapidly oscillating periodic coefficients. Sibirsk. Matem. Zh. 1983, no. 5. [6] Shamaev, Uspehi. Mat. Nauk. 37 pp 243– (1982) [7] Oleinik, Dokl. Akad. Nauk SSSR 266 pp 18– (1982) [8] Oleinik, Matem. Sbornik 120 pp 22– (1983) [9] ; ; , Asymptotic analysis for periodic structures, North Holland Publ. Co., 1978. [10] Non-homogeneous media and vibration theory, Lect. Notes in Physics, Springer Verlag, 1980, 127. · Zbl 0432.70002 [11] Oleinik, Uspehi Mat. Nauk 37 pp 195– (1982) [12] Regular convergence of operators and approximate solution of equations. VINITI, Itogi Nauki i Techniki, ser. ”Math. analysis” v. 16, 1979. 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.	2022-07-06 08:04:39	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.6401659846305847, "perplexity": 1831.904640450608}, "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-2022-27/segments/1656104668059.88/warc/CC-MAIN-20220706060502-20220706090502-00775.warc.gz"}
https://brilliant.org/problems/new-year-gift-1/	# New year gift 1 Algebra Level 4 Suppose that $$s_1,s_2,s_3,\ldots$$ is a strictly increasing sequence of positive integers such that the subsequences$$\large s_{s_1}, s_{s_2}, s_{s_3}, \ldots$$ and $$\large s_{s_1+1}, s_{s_2+1}, s_{s_3+1}, \ldots$$ are both arithmetic progressions. Then $$s_1,s_2,s_3,\ldots$$ follows a/an: ×	2017-12-12 04:47:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.37094131112098694, "perplexity": 843.0596558624344}, "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-2017-51/segments/1512948515165.6/warc/CC-MAIN-20171212041010-20171212061010-00609.warc.gz"}
https://qetlab.com/IsAbsPPT	# IsAbsPPT Other toolboxes required IsAbsPPT Determines whether or not a density matrix is absolutely PPT none AbsPPTConstraintsInSeparableBall Ball of separability IsAbsPPT is a function that determines whether or not a density matrix $\rho$ is "absolutely PPT" (that is, whether or not $U\rho U^\dagger$ has positive partial transpose for all unitary matrices $U$). The conditions that determine whether or not a state is absolutely PPT were derived in [1]. This function returns 1 if $\rho$ is absolutely PPT, 0 if it is not absolutely PPT, and -1 if it was unable to determine whether or not $\rho$ is absolutely PPT within a reasonable amount of time. ## Syntax • IAPPT = IsAbsPPT(RHO) • IAPPT = IsAbsPPT(RHO,DIM) ## Argument descriptions • RHO: A bipartite density matrix (or any bipartite positive semidefinite operator). • DIM (optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the two subsystems that X acts on. ## Examples The maximally-mixed state is the simplest example of an absolutely PPT state: >> d = 5; >> rho = eye(d^2); >> IsAbsPPT(rho) ans = 1 ## Notes • This function always gives an answer of either 0 or 1 if at least one of the local dimensions is 6 or less. If both local dimensions are 7 or higher, than sometimes an answer of -1 is returned, indicating that the script was unable to determine whether or not RHO is absolutely PPT within a reasonable amount of time (but these situations are still relatively rare). • Absolutely PPT states are sometimes said to be "PPT from spectrum".	2023-02-09 00:33:36	{"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": 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": 0.7131219506263733, "perplexity": 951.4350892000884}, "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/1674764500983.76/warc/CC-MAIN-20230208222635-20230209012635-00459.warc.gz"}
https://socratic.org/questions/5a1cc9b87c014927bf605787#535195	# Question #05787 Jan 14, 2018 $x + 2 = \frac{7}{8} \times x - 15$ #### Explanation: Probably the question is : What would be the equation for : Two more than a certain number is 15 less than the product of 7/8 and the number? Let the number be $x$, then Two more than the number will be $x + 2$ The product of 7/8 and the number will be $\frac{7}{8} \times x$ And 15 less than the product of 7/8 and the number will be : $\frac{7}{8} \times x - 15$ Therefore according to given statement, the equation will be: $x + 2 = \frac{7}{8} \times x - 15$ Solving it: $\implies x - \frac{7}{8} x = - 15 - 2 = - 17$ $\implies x \times \frac{8}{8} - \frac{7}{8} x = - 17$ $\implies \frac{1}{8} x = - 17$ $\implies x = - 17 \times 8$ $\implies x = - 136$	2022-08-09 13:33:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 11, "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": 0.6961610913276672, "perplexity": 747.9190359067478}, "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/1659882570977.50/warc/CC-MAIN-20220809124724-20220809154724-00006.warc.gz"}
https://doc.cgal.org/latest/Apollonius_graph_2/group__PkgApolloniusGraph2.html	CGAL 4.12 - 2D Apollonius Graphs (Delaunay Graphs of Disks) 2D Apollonius Graphs (Delaunay Graphs of Disks) Reference Menelaos Karavelas and Mariette Yvinec Algorithms for computing the Apollonius graph in two dimensions. The Apollonius graph is the dual of the Apollonius diagram, also known as the additively weighted Voronoi diagram. The latter can be thought of as the Voronoi diagram of a set of disks under the Euclidean metric, and it is a generalization of the standard Voronoi diagram for points. The algorithms provided are dynamic. Introduced in: CGAL 3.0 Depends on: 2D Triangulation Data Structure BibTeX: cgal:ky-ag2-18a Windows Demo: 2D Apollonius Graph Common Demo Dlls: dlls An Apollonius graph is the dual of the Apollonius diagram, also known as the additively weighted Voronoi diagram. It is essentially the Voronoi diagram of a set of disks, where the distance of a point of the plane from a disk is defined as the Euclidean distance of the point and the center of the circle, minus the radius of the disk. CGAL provides the class CGAL::Apollonius_graph_2<Gt,Agds> for computing the 2D Apollonius graph. The two template parameters must be models of the ApolloniusGraphTraits_2 and ApolloniusGraphDataStructure_2 concepts. The first concept is related to the geometric objects and predicates associated with Apollonius graphs, whereas the second concept refers to the data structure used to represent the Apollonius graph. The classes Apollonius_graph_traits_2<K,Method_tag> and Triangulation_data_structure_2<Vb,Fb> are models of the aforementioned concepts. ## Concepts • ApolloniusSite_2 • ApolloniusGraphDataStructure_2 • ApolloniusGraphVertexBase_2 • ApolloniusGraphTraits_2 • ApolloniusGraphHierarchyVertexBase_2 ## Classes • CGAL::Apollonius_graph_2<Gt,Agds> • CGAL::Apollonius_site_2<K> • CGAL::Apollonius_graph_vertex_base_2<Gt,StoreHidden> • CGAL::Apollonius_graph_traits_2<K,Method_tag> • CGAL::Apollonius_graph_filtered_traits_2<CK,CM,EK,EM,FK,FM> • CGAL::Apollonius_graph_hierarchy_2<Gt,Agds> • CGAL::Apollonius_graph_hierarchy_vertex_base_2<Agvb> Concepts ## Classes class CGAL::Apollonius_graph_2< Gt, Agds > The class Apollonius_graph_2 represents the Apollonius graph. More... class CGAL::Apollonius_graph_filtered_traits_2< CK, CM, EK, EM, FK, FM > The class Apollonius_graph_filtered_traits_2 provides a model for the ApolloniusGraphTraits_2 concept. More... class CGAL::Apollonius_graph_hierarchy_2< Gt, Agds > We provide an alternative to the class Apollonius_graph_2<Gt,Agds> for the dynamic construction of the Apollonius graph. More... class CGAL::Apollonius_graph_hierarchy_vertex_base_2< Agvb > The class Apollonius_graph_hierarchy_vertex_base_2 provides a model for the ApolloniusGraphHierarchyVertexBase_2 concept, which is the vertex base required by the Apollonius_graph_hierarchy_2<Gt,Agds> class. More... class CGAL::Apollonius_graph_traits_2< K, Method_tag > The class Apollonius_graph_traits_2 provides a model for the ApolloniusGraphTraits_2 concept. More... class CGAL::Apollonius_graph_vertex_base_2< Gt, StoreHidden > The class Apollonius_graph_vertex_base_2 provides a model for the ApolloniusGraphVertexBase_2 concept which is the vertex base required by the ApolloniusGraphDataStructure_2 concept. More... class CGAL::Apollonius_site_2< K > The class Apollonius_site_2 is a model for the concept ApolloniusSite_2. More...	2018-08-18 18:09:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.37373360991477966, "perplexity": 5405.429871255259}, "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-2018-34/segments/1534221213693.23/warc/CC-MAIN-20180818173743-20180818193743-00271.warc.gz"}
https://tex.stackexchange.com/questions/315720/use-endcomment-instead-of-begin-endcomment	In my document, the comment environment works fine, but \comment and \endcomment stop processing of what comes next in the document: \documentclass{article} \usepackage{verbatim} \begin{document} X \begin{comment} 1 \end{comment} Y \comment 2 \endcomment Z \end{document} Why is that? Is that similar to End \verbatim command? • 'Yes': this is verbatim-like – Joseph Wright Jun 20 '16 at 15:22 • Never use the \foo...\endfoo commands in the body of a document. Ever. Is that clear? ;-) You immediately get into troubles if you do, not only for this kind of verbatim-like environments. – egreg Jun 20 '16 at 15:29 The comment environment is defined by the verbatim package as a variant of verbatim: instead of printing each line, it simply throws it away. The package defines its verbatim-like environments by doing some steps. First all special characters (well, almost all) become non special; then TeX is instructed to absorb one line at a time, checking whether it contains the string \end (with a non-special backslash). If it does, a further check is done, to see whether it is followed by the string {foo}, where foo is the name the current environment. If this string is not found, the line is treated like the ones not containing \end. Otherwise, the line is thrown away and the verbatim-like environment is finished up, executing the macro \endfoo and closing the group started by \begin{foo}. Thus, when seeing \endcomment, the line is recognized to contain \end, but no {comment} string follows, so TeX continues its work processing line by line until finding \end{document}, which does satisfy the requirements, because the current environment is indeed document. As a general rule, never use \foo...\endfoo in the document body. You'll get into troubles. Just for an easy example, try \documentclass{article} \begin{document} \quote Some text that should wrap around, let's type something long enough. Hope this will suffice. \endquote \quote Some text that should wrap around, let's type something long enough. Hope this will suffice. \endquote \end{document} Here's the (perhaps surprising) output: Using \foo and \endfoo when defining environments is OK and, in some cases, mandatory. But use it with care.	2021-06-21 01:25:31	{"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": 0.9374402761459351, "perplexity": 2946.609696997003}, "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/1623488259200.84/warc/CC-MAIN-20210620235118-20210621025118-00215.warc.gz"}
http://sbwt.ciiz.pw/affine-transformation-c++.html	# Affine Transformation C++ The multiplication of 2 affine transformations and is defined as the affine transformation which, when applied on any vecor , results in the same vector that one would obtain by first transforming it with and then with. The value of the input at those coordinates is determined by spline interpolation of the requested order. , the midpoint of a line segment remains the midpoint after transformation). I am trying to implement affine transformation on two images. Affine Transformation¶ In affine transformation, all parallel lines in the original image will still be parallel in the output image. Affine Transformation. The Transformation Matrix. Now that you understand the basics of drawing shapes like triangles and rectangles, let's take another step and try to move (translate), rotate, and scale the triangle and display the results on the screen. The affine transformation has an important property that two successive affine transformations combine also into an affine transformation. A 2D point transformation requires 9 multiplies and 6 adds But since affine transformations have always the form: The number of operations can be reduced to 4 multiplies and 4 adds ab c x ax by cz defydx eyfz g hi zg xhy iz ++ =++ ++ 00 1 1 1 ab c x ax by c. Eden, "B-Spline Signal Processing: Part I--Theory," IEEE Transactions on Signal Processing, vol. It also contains face drawing function that make it easy to plot the 3d figure with respect to a specific question. Also, sets of parallel lines remain parallel after an affine transformation. A C++ library for Affine transformation. Download Anaconda. It can be obtained from the previous description by applying affine transformations to keep the polygon bounded as it degenerates. particular transformations that are not supported by ITransform2D and also for performing numerous transformations in one go. My question is, what did people find useful in this article, and I mean it sincerely. Pluto transforms C programs from source to source for coarse-grained parallelism and data locality simultaneously. The matrix operation is applied to each location (x, y) that is then transformed to (x', y') of the new array. An affine transformation can differentially scale the data, skew it, rotate it, and translate it. I am implementing some affine transformations and I am aware that there are several algorithms for efficient matrix multiplication, like Strassen. But are there some algorithms that are especially efficient for matrices that small?. That is, the shape is defined up to an affine transformation in space. So, this class library implements affine transformations on images such as translation, rotation, scaling, schear. gdTransformAffineCopy: Applies an affine transformation to a region and copy the result in a destination to the given position. Lowe, International Journal of Computer Vision, 60, 2 (2004), pp. This example illustrates how to use the maketform and imtransform functions to perform a 2-D spatial transformation of an image. Affine Partitioning Based Algorithms. , DoG, Harris-Affine, Harris-Laplace) and corresponding feature descriptors (SIFT, raw patches). In this question, f and g are both affine transformations. The name affine differential geometry follows from Klein's Erlangen program. Property 1 An affine transformation of the plane is defined uniquely by three pairs of points. An affine transformation is any transformation that preserves collinearity (i. , all points lying on a line initially still lie on a line after transformation) and ratios of distances (e. The chapter reviews some properties of affine mappings through theorems and discusses the representation of any affine transformation as a product of affine transformations of the simplest types. affine invariant subspaces of c(c) 233 2. Affine Transformation. An affine transform is a special case of a perspective transform. Thus, rectified text region is required for most text recognition algorithm. For each of the configuration, calculate the Sum of Absolute Differences (or other scoring method), but only for a…. org, freedictionary. Just an update, I changed the CvInvoke. Affine (distance) ratio from 3 parallel lines. Moreover, the generalized de Casteljau approach is computationally more efficient than trivariate de Casteljau, because it is an affine transformation plus a univariate interpolation in space vs. 1 #include "rotate 44 // our affine transformation. There are two important particular cases of such transformations: A nonproportional scaling transformation centered at the origin has the form (x,y,z) (ax,by,cz), where a,b,c 0 are the. The image of a line under any affine transformation is a line. Transformation. It successfully performs affine transformations or more general non-affine transformations such as tiling on the polyhedron, and then converts the transformed polyhedron into equivalent, but optimized (depending on targeted optimization goal), loop nests through polyhedra scanning. C++ : Will an (affine transformationno) better pow() function improve accuracy? By random_thinker , August 20, 2005 in General and Gameplay Programming This topic is 5131 days old which is more than the 365 day threshold we allow for new replies. These n+1-dimensional transformation matrices are called, depending on their application, affine transformation matrices, projective transformation matrices, or more generally non-linear transformation matrices. Notice that in order to check this, we needed only the matrix C, and not the entire affine transformation. The goal of the localisation network is to spit out the parameters of the affine transformation that’ll be applied to the input feature map. cvEstimateAffineTransformation function to wrap the opencv C function defined in the opencv_video module. origin is moved, axes do not rotate diagram u = x - a v = y - b. The usual way to represent an Affine Transform is by using a 2 \times 3 matrix. As an example we will convert world coordinates to pixel (screen or image) coordinates. An unsupervised algorithm for learning lie group transformations. Affine transformation is a linear mapping method that preserves points, straight lines, and planes. Affine transformations as matrices. Here is it , My First article for codeproject, I hope you enjoy it. With input feature map U, with width W, height H and C channels, outputs are θ, the parameters of transformation Tθ. Mathematically, it is represented as e(x) = (ax + b) mod m. Good parametrisations of affine transformations are essential to interpolation, deformation, and analysis of shape, motion, and animation. It does not necessarily preserve angles or lengths, but does have the. The normalisation stage, where the homogeneous vectors are scaled so that their third component is one, and is set to one, is carried out to make the equations linear. which is should not get otherwise if the transformation is right. Affine transformation is a linear mapping method that preserves points, straight lines, and planes. The inverse of a transformation L, denoted L−1, maps images of L back to the original points. Affine transformations are composed of Affine transformations are composed of elementary ones. An unsupervised algorithm for learning lie group transformations. in an output image) by applying a linear combination of translation, rotation, scaling and/or shearing (i. And the second one is, if I take the transformation of any scaled up version of a vector -- so let me just multiply vector a times some scalar or some real number c. The ModelView matrix combined the model and view transformations into one. Affine transformation software Author: Philippe Thévenaz This C routine is based on the following two papers: M. Pitchaiah, Philemon Daniel, Praveen Abstract—Cryptography is the study of mathematical techniques related to aspects of information security such as confidentiality, data integrity, entity authentication and data origin authentication. Transformation matrixes for affine transformations are as follows: 9 DOF transformation matrix which includes scale parameters Sx, Sy and Sz looks as follows \begin{bmatrix}. Javascript isomorphic 2D affine transformations written in ES6 syntax. Control points are used to define the mapping. Write a 3-by-3 matrix representing this transformation. Translation. Making these things is very easy with grid package. With beginners, trying to implement an affine transformation in a programming language (C/C++) is really a challenge. See Also EGS_AffineTransform. whereas affine transformations have the form € xnew=ax+by+e ynew=cx+dy+f € ⇔ (xnew,ynew)=(x,y)∗ ac bd +(e,f). Introduction to Transformations n Introduce 3D affine transformation: n Position (translation) n Size (scaling) n Orientation (rotation) n Shapes (shear) n Previously developed 2D (x,y) n Now, extend to 3D or (x,y,z) case n Extend transform matrices to 3D n Enable transformation of points by multiplication. Anaconda Cloud. ch Abstract A well-known modular software for analysis and performance has been redesigned in Java for distributed components and. A spatial transformation is a mapping function that establishes a spatial correspondence between all points in an image and its. The new affine. Define an affine transformation on the CMF’s: For the case of monochromatic SRD, let )w1 (λ), w2 (λ , w3 (λ) be three auxiliary functions of the wavelength and define the affine tri-stimulus values )(Xa ,Ya ,Za through Eqn. Mark these points and their images on the same diagram, making it clear which points map to which. YANO, A class of. For this instruction, an affine transformation is defined by A * x + b where "A" is an 8 by 8 bit matrix, and "x" and "b" are 8-bit vectors. Notice that in order to check this, we needed only the matrix C, and not the entire affine transformation. This leads to the following differences in operations properties:. Implementation of Affine Cipher The Affine cipher is a type of monoalphabetic substitution cipher, wherein each letter in an alphabet is mapped to its numeric equivalent, encrypted using a simple mathematical function, and converted back to a letter. How to use this projective transformation with CSS. If matrix tf is is a 3x2 matrix, an affine transformation will be performed. Creation You can create an affine2d object using the following methods:. ‘Our interest is in the space of affine equivalence classes of equal-area polygons. This does shearing , scaling , translating and scaling. In geometry, an affine transformation, affine map or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. The technical definition of an affine transformation is one that preserves parallel lines, which basically means that you can write them as matrix transformations, or that a rectangle will become a parallelogram under an affine transformation (see fig 10. Note: 11 12 1 1 11 1 12 2 1 21 22 2 2 21 1 22 2 2. Affine transformations are composed of Affine transformations are composed of elementary ones. $\endgroup$ – godaygo Jul 11 at 10:26. the transformation φ we used is Affine Transformation (AT), defined by: φ(s ) = Asi + b (6) where = = 2 1 22 12 21 11, b b a a a a A b are six parameters of AT. Affine Transformations. 2D Cartesian coordinate transformations are generally used to assign map coordinates (x,y) to an uncorrected image or scanned map. An affine transformation does not necessarily preserve angles between. Obviously the images represent only partially the same thing (some background is removed and other is added) but after a roto-translation (an affine transformation caused by the camera movement). , & Goel, A. I am an entrepreneur who loves Computer Vision and Machine Learning. AFFINE—Affine transformation requires a minimum of three transformation links. Georgieva, CSI/CUNY 5 Using Transformations cont’d • A designer may want to view an object from different vantage points. Gangopadhyay XLRI C. By affine calibration is meant that besides of projective calibration, some affine information should somehow introduced. There are some common transformations such as:. Ask Question Asked 6 months ago. It does not necessarily preserve angles or lengths, but does have the. Affine Transformation zAffine Transforms needed in OQM since imaging is done on basis of the phase difference between pixels. Suppose an affine transformation T maps P to P', Q to Q', R to R' Suppose some other collineation S maps P to P', Q to Q', R to R' Then T-1 S fixes P, Q, and R and must be the identity. To the point: Unfortunately CF doesn't support this kind of operation on Images in comaprison with. Transformations can be done in two different ways. 2D projective transformations (homographies) Christiano Gava christiano. After beeing multiplied by the ProjectionMatrix, homogeneous coordinates are divided by their own W component. you can see that, in essence, an Affine Transformation represents a relation between two images. Affine Transformations. Imaging Namespace / AffineMatrix Class. Model matrix. (c) Use the expressions that you found for f(x) and g(x) in parts (a) and (b) to calculate f(g(x)), and hence find the affine transformation f g in the same form as you found g in part (a). An affine symmetric space is a connected affinely connected manifold M such that to each point peM there is an involutive (i. Studholme U. c) Find the affine transformation g o f (in the same form as you found g and f in parts a) and b)) d) Hence, or otherwise, find the images of the points (0,0), (4,0), (4,1) and (0,1) under g o f. It also contains face drawing function that make it easy to plot the 3d figure with respect to a specific question. hpp // // Copyright 2005-2007 Adobe Systems Incorporated // // Distributed under the Boost Software License, Version 1. Moving, Rotating, and Scaling. Although not user-friendly, not real. , all points lying on a line initially still lie on a line after transformation) and ratios of distances (e. The algorithm compares an input image to its database of preprocessed images and determines if the input matches any image in the database. When the contents of children change, their parents are automatically invalidated. This is a left side matrix multiplication. Applications: whitening transformation: Suppose X is a column vector zero-centered data. Theorem Affine transformations map affine subspaces to affine subspaces. Affine transformation In geometry, an affine transformation is a transformation which preserves straight lines (all points lying on a line initially still lie on a line after transformation) and ratios of distances between points lying on a straight line. Mathematically, it is represented as e(x) = (ax + b) mod m. We will learn how to apply those transformations to entire render targets, brushes and geometries. Apply an affine transformation. It would not be appropriate for example, for air photos taken in mountainous terrain. Affine Transformation. png picture (with transparent background) placed on a form (using a TImage object). To the point: Unfortunately CF doesn't support this kind of operation on Images in comaprison with. The usual way to represent an Affine Transformation is by using a \(2 \times 3. Affine Transformations •Line preserving •Characteristic of many physically important transformations - Rigid body transformations: translation, rotation - Non-rigid: Scaling, shear •Importance in graphics is that we need only transform vertices (points) of line segments and polygons, then system draws between the transformed points. Transformations of Variables Basic Theory The Problem As usual, we start with a random experiment with probability measure ℙ on an underlying sample space. affine: [adjective] of, relating to, or being a transformation (such as a translation, a rotation, or a uniform stretching) that carries straight lines into straight lines and parallel lines into parallel lines but may alter distance between points and angles between lines. A 2D transformation matrix is an array of numbers with three rows and three columns for performing algebraic operations on a set of homogeneous coordinate points (regular points, rational points, or vectors) that define a 2D graphic. This program is able to load one PCD or PLY file; apply a matrix transformation on it and display the original and transformed point cloud. This means that, each pixel is localized by two coordinates, in the rectangular domain of the image. Geospatial software of all varieties use an affine transform (sometimes refered to as "geotransform") to go from raster rows/columns to the x/y of the coordinate reference system. So this article will show you guys some simple examples that apply affine transformations. To the point: Unfortunately CF doesn't support this kind of operation on Images in comaprison with. Download Anaconda. Affine Transformations Every linear transformation is equivalent to a change in frames Every affine transformation preserves lines However, an affine transformation has only 12 degrees of freedombecause 4 of the elements in the matrix are fixed and are a subset of all possible 4 x 4 linear transformations. • They can be inferred by giving the correspondence of three 2-D points between the input and output images. • Affine transformations map triangles onto triangles. Translation of axes. King ([email protected] For the medium faculty senate there exist collineations that are not affine transformations. 3D Transformations Yong Cao Virginia TechVirginia Tech 5. If we do a translation, T, by by^we convert the problem to reection about a line passing through the origin; the translation matrix is T= 2 4 1 0 0 0 1 b 0 0 1 3 5:. Manually annotated macro‐structures on both pathology and MRI were used to assist registration using a relaxed local affine transformation approximation. Show that an affine transformation is rigid its homogeneous part is an orthogonal matrix. Parameters. The affine transformation Imagine you have a ball lying at (1,0) in your coordinate system. Include translations, rotations, scales, and/or skewing parameters. Fast logarithm and exponential for 3D transformations ( rotational and shearing transformations ) Euclidean parametrisation map for 3D affine transformation (see [1]) Fast polar decomposition ( without SVD. Excised prostate specimens underwent quarter mount step-section pathologic processing, digitization, annotation, and assembly into a PWM. A transformation changes the positions of points in the plane. Transformation matrixes for affine transformations are as follows: 9 DOF transformation matrix which includes scale parameters Sx, Sy and Sz looks as follows \begin{bmatrix}. An affine transformation can differentially scale the data, skew it, rotate it, and translate it. Where I is the identity matrix and N is the unit vector for the surface normal of the plane. The image of a line under any affine transformation is a line. A perspective transformation is not affine, and as such, can't be represented entirely by a matrix. Transformations of Variables Basic Theory The Problem As usual, we start with a random experiment with probability measure ℙ on an underlying sample space. On infinitesimal affine and isometric transformations preserving respective vector fields. C A; simply represents an arbitrary a ne transformation, having 12 degrees of freedom. Flash uses matrices to define affine transformations. This can be computed from the "nice" format transformation information. Mathematically, it is represented as e(x) = (ax + b) mod m. Affine transformation) with that of the transformations () was first proved (for ${\rm char}\; k = 0$) by H. PDF \| This paper deals with surface normal estimation from calibrated stereo images. This means points on a line will remain in a line after an affine transformation is applied to the coordinate space in which that line exists. Affine Transformation zAffine Transforms needed in OQM since imaging is done on basis of the phase difference between pixels. The usual way to represent an Affine Transform is by using a matrix. glTranslate only allows you to change b. Given the affine map C == aP +b(mod N), where a E (ZINZ), b E (ZINZ). Usage with GIS data packages. This produces a transformation whose effect is that of A followed by B. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. These transformations and coordinate systems will be discussed below in more detail. Theorem Affine transformations map affine subspaces to affine subspaces. A single multi-dimensional affine function can represent a long and complex sequence of simpler transformations. (c) Extra discussion on the plane-to-plane projectivity. the result will be the equivalent of doing first the transformation m1 and then m2. not involving gamma correction or whatnot), this is actually pretty simple. Decryption is a slightly different formula, d(x) = a-1 (x - b) mod m. Affine transformations. With beginners, trying to implement an affine transformation in a programming language (C/C++) is really a challenge. Creation You can create an affine2d object using the following methods:. If v is such a vector, then we say that X(v) is the affine transformation of the vector. Include translations, rotations, scales, and/or skewing parameters. Copying and pasting the three blocks of PostScript into a text file with a. A 4x4 matrix can represent all affine transformations (including translation, rotation around origin, reflection, glides, scale from origin contraction and expansion, shear, dilation, spiral similarities). CCS CONCEPTS. [5] (d) Use your answer to part (c) to determine any points (x, y) that are left unchanged by the transformation f g, or to show that there are. So take the image below as my input example: After successfully detecting the area that corresponds to the p…. Manually annotated macro-structures on both pathology and MRI were used to assist registration using a relaxed local affine transformation approximation. It successfully performs affine transformations or more general non-affine transformations such as tiling on the polyhedron, and then converts the transformed polyhedron into equivalent, but optimized (depending on targeted optimization goal), loop nests through polyhedra scanning. The ModelView matrix combined the model and view transformations into one. We will apply a rotation and a translation to a loaded point cloud and display then result. In all this, it is only necessary to keep track 00185 * of the shear angles and translations of points during the shears. Solving for T requires a minimum of 3 pairing points (that aren't degenerate!). With input feature map U, with width W, height H and C channels, outputs are θ, the parameters of transformation Tθ. This repository uses dlib's real-time pose estimation with OpenCV's affine transformation to try to make the eyes and bottom lip appear in the same location on each image. Automatic face recognition usually normalizes the face images as the preprocessing step and then proceeds with the recognition. It is capable of the following operations: Declare Vectors, matrices, quaternions. THis program does some simple graphics transformation in 2D. The affine transformation function is. Sometimes a similarity transformation doesn't do the trick. 9/10/2016 4 Affine properties: composition Affine transforms are closed under composition. affine invariant subspaces of c(c) 233 2. Affine Transformation Shear, Skew, & Rotate Add an Affine transform (sometimes called a Free Transform) to any graphic, then use the interactive on-screen handles to apply a distortion or scaling transform. In the case of affine transformation the scaling can be non-uniform, that is different in each direction. Geometric image transformation, such as translation, scaling, or rotation A transformation that involves a linear transformation followed by a translation Function expressed in the form of the equation of a line or plane see above (Also call 1st order) Relevante Übersetzungen affine transformations plural form of affine transformation. Affine transformations and their inverse When you're programming games or other 3d applications using OpenGL or DirectX, it is often required to use affine transformation matrices to describe transformations in space (4x4 matrices and the like). particular transformations that are not supported by ITransform2D and also for performing numerous transformations in one go. For an affine transformation there are 6 transformation parameters, so you need at least 3 control points (each control point implies 4 coordinates: Xsource, Ysource, Xtarget, Ytarget), but more control points are recommended to have redundancy and thus be able to apply Least. Usage with GIS data packages. Gangopadhyay XLRI C. In the AIR package, the 2D affine model is parameterized in terms of six parameters defined below. In a translation, you shift an image in coordinate space by adding a specified value to the x- and y. The performance of fractal image coding mainly depends on the affine transformations. What happens if I multiply TxP? Let's do that right over here. One such measure is the Arrow-. Find an afne transformation to reect two-dimensional points about this line. The methods are essential in handling digitized locational data and are applicable widely in other graphical applications such as calibrating data sets for plotting, and in. That is, the shape is defined up to an affine transformation in space. convert Matrix4f to Affine transformation in Eigen To rename all the files in a folder; Errors the errors that I get while compiling GTSA adding / Linking GTSAM with ROS; linking external c++ library to ROS January 2014 (4) 2013 (61) December 2013 (10) November 2013 (8) October 2013 (7). Refer to the transformations. Jung ; the case of arbitrary ground field was proved by W. This page documents progress on automating the computation of the transformation matrix by least-squares (Bruce Rindahl) via SQL. Any number of points are said to be collinear when they lie on one line. Next, we will cover some interesting applications and concepts like Face Detection, Image Recognition, Object Detection and Facial Landmark Detection. Define an affine transformation on the CMF’s: For the case of monochromatic SRD, let )w1 (λ), w2 (λ , w3 (λ) be three auxiliary functions of the wavelength and define the affine tri-stimulus values )(Xa ,Ya ,Za through Eqn. Two-Dimensional Affine Transformations Affine transformations of the plane in two dimensions include pure translations, scaling in a given direction, rotation, and shear. For images gray level transformations these take the form g(m,n) = af(m,n) + b (2. Generate a unique affine transformation for each ops. Usage with GIS data packages. Note that the horizontal and vertical grids are perpendicular to each other. The class of bi-affine functions is not closed under composition, but the composition of a bi-affine and an affine function is bi-affine. Depending on the input imagery and output coordinate systems, a nonsymmetric transformation may be REQUIRED to properly fit the points. //get the affine transformation. In addition, to strengthen the S-Box against algebraic attacks, the affine transformation was added. affine transformations keep parallel lines parallel are four different types (primitives): handout - Affine transformation primitives 1. Transformations of Variables Basic Theory The Problem As usual, we start with a random experiment with probability measure ℙ on an underlying sample space. However, since expected utility functions are not uniquely defined (are defined only up to affine transformations), a measure that stays constant with respect to these transformations is needed. In fact, to avoid sampling artifacts, the mapping is done in the reverse order, from destination to the source. Many transformations have been proposed in the past including unimodular transformations (interchange, skew and reversal), fusion, fission, reindexing, scaling, and statement reordering. "A transformation that maps lines to lines (but does not necessarily preserve parallelism) is a projective transformation. For example, because paper maps expand and contract more along the paper grain than across the grain in response to changes in humidity, the scale of a paper map is likely to be slightly greater along one axis than the other. In this paper, we proposed a text detection method which can provide accurate text region. An affine2d object stores information about a 2-D affine geometric transformation and enables forward and inverse transformations. $\endgroup$ – godaygo Jul 11 at 10:26. Then A n C°°(C) is dense in A. An algebraic normalization. This gives us a new view of the intrinsic matrix: a sequence of 2D affine transformations. Note that the matrix form of an affine transformation is a 4-by-4 matrix with the fourth row 0, 0, 0 and 1. To scale the object's texture, set the Scale property. Firstly, and most commonly within the geometry model, the AffineTransformation2D object can be used in the ITransform2D::Transform method to transform an existing Geometry. It is assumed that the reader knows what a matrices are and how to multiply them. That is, the shape is defined up to an affine transformation in space. 2 A ne transformations In geometry, an a ne transformation is a function that maps an object from an a ne space to an other and which preserve structures. Tool to decrypt/encrypt with Affine automatically. Linear transformations A ne transformations Transformations in 3D Graphics 2009/2010, period 1 Lecture 5 Linear and a ne transformations Graphics, 1st period 2009/2010 Lecture 5: linear and a ne transformations. org, freedictionary. Find an afne transformation to reect two-dimensional points about this line. Still to be implemented are geometric primitives and affine transformation of images. For affine transformations, the first two elements of this line are zeros. The usual way to represent an Affine Transform is by using a matrix. First i find the matching pairs in both of the images. , all points lying on a line initially still lie on a line after transformation) and ratios of distances (e. Carlson Center for Imaging Science Rochester Institute of Technology [email protected] Sometimes a similarity transformation doesn't do the trick. 2D and 3D Transformations Doug Bowman Adapted from notes by Yong Cao Virginia Tech. If X and F are compact connected metric abelian groups, T=a + A an affine transformation of X and S = b + Ban affine transformation of Y, what are necessary and sufficient conditions. Fast logarithm and exponential for 3D transformations ( rotational and shearing transformations ) Euclidean parametrisation map for 3D affine transformation (see [1]) Fast polar decomposition ( without SVD. 31: Using the transformation matrix to shear text 104 Figure 10. The affine transformation Imagine you have a ball lying at (1,0) in your coordinate system. There are various ways of achieving 180° or even 360° view, with their distinct pros and cons. In general, an affine transformation is a composition of rotations. Zapraszam, zapraszam kolejny niezrównoważony pseudomatematyczny bełkot. The transformations you can do with a 2D matrix are called affine transformations. Geospatial software of all varieties use an affine transform (sometimes refered to as "geotransform") to go from raster rows/columns to the x/y of the coordinate reference system. , or be the result of operators like vector_angle_to_rigid. Attributes affines list of AffineTransform objects. Full 2D affine transform. When a transformation takes place on a 2D plane, it is called 2D transformation. If A, B and C are collinear, so are their images under any affine map. Cartesian is a type of affine coordinate space, but we can transform it to other affine spaces as we prefer. If and are affine spaces, then every affine transformation : → is of the form ↦ +, where is a linear transformation on the space , is a vector in , and is a vector in. A 2D point transformation requires 9 multiplies and 6 adds But since affine transformations have always the form: The number of operations can be reduced to 4 multiplies and 4 adds ab c x ax by cz defydx eyfz g hi zg xhy iz ++ =++ ++ 00 1 1 1 ab c x ax by c defydx eyf + + =++. برای پردازش‌های بعدی، صرفا تبدیل‌های پایه‌ای عکس شامل چرخش و تبدیل مقیاسی که خطوط موازی را حفظ می‌کند انجام خواهد شد (به آن «تبدیل آفین» (Affine transformation) گفته می‌شود). They preserve straight lines but necessarily not angles or lengths. The transformation between images is modeled as locally affine but globally smooth, and explicitly accounts for local and global variations in image intensities. txt for the full license. In the AIR package, the 2D affine model is parameterized in terms of six parameters defined below. The algorithm compares an input image to its database of preprocessed images and determines if the input matches any image in the database. Combined Rotation and Translation using 4x4 matrix. Therefore, the scale invariant detectors fail in the case of significant affine transformations. Consequently, the distorted text region can be rectified according to the affine parameters. Affine Transformations take place in three steps (TRS) in. Real-Time Tool for Affine Transformations of Two Dimensional IFS Fractal 151 2. And there is even more general formula that covers affine transformations as a corner case but introduces projection. (Method: Since is affine, one can write (i) , where is the homogeneous part of and is the translation; then also (ii). ) Since afﬁne transformations form a group, group theory shows that there exist matrix representations for the group – that is, there is a group of matrices that follows the same rules as the group, with the matrices being. Any plane projective transformation can be expressed by an invertible 3×3 matrix in homogeneous coordinates; conversely, any invertible 3×3 matrix defines a projective transformation of the plane. This paper also introduces a secure and efficient symmetric cryptosystem based on affine transformation. A Retrospective: A Data Locality Optimizing Algorithm M. mapping required in enciphering and deciphering and number of affine transformations. De Bruijn-like digraphs 3 Affine TCP digraphs 4. Similarity transformations preserve the angles of the original object, but not necessarily the size. Transformation Matrix. However, it considers only rotations, translations, and uniform scale changes in finding the mapping. (c) Extra discussion on the plane-to-plane projectivity. どこで出てくるか 3. We can apply affine transformations to an image by describing the transformation with a 3x3 matrix and applying the transformation to each pixel location in the original image to get that pixel's location in the target image. Generic affine transformations are represented by the Transform class which internally is a (Dim+1)^2 matrix. If we do a translation, T, by by^we convert the problem to reection about a line passing through the origin; the translation matrix is T= 2 4 1 0 0 0 1 b 0 0 1 3 5:. This code registers 2-D images. A linear transformation on a vector space can be represented (in a particular basis) as a matrix; an affine transformation can be represented (in a particular coordinate system) as a matrix together with a translation vector. Using matrix multiplication, find the image of the point (3, 4). Copying and pasting the three blocks of PostScript into a text file with a. From the above, We can use an Affine Transformation to express: Rotations (linear transformation) Translations (vector addition) Scale operations (linear transformation) you can see that, in essence, an Affine Transformation represents a relation between two images. Using matrix multiplication, find a point whose image is (13, –4). In linear algebra, a frame of reference is like a vector basis [2], and the transformation to another frame of reference is a change of basis. compose(B) = T B x T A. There are various ways of achieving 180° or even 360° view, with their distinct pros and cons. Affine Transformation in Image Processing: Explained with C++ Transformations are used to change the geometry of the contents within the image.	2020-04-03 06:41:47	{"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": 0.7506680488586426, "perplexity": 1006.9113200639567}, "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-2020-16/segments/1585370510352.43/warc/CC-MAIN-20200403061648-20200403091648-00046.warc.gz"}
https://plainmath.net/college-statistics/32392-to-find-the-95-confidence-interval-for-the-comparison	Caelan 2021-10-22 To find: The $95\mathrm{%}$ confidence interval for the comparison. odgovoreh Calculation: The study is conducted to see that whether a sample of children of different age groups consumed an adequate amount of calcium or not. In the study, samples are considered from two independent populations (different age groups). First, the proportion of children aged 5 to 10 year who met the calcium requirement is defined by the formula: ${\stackrel{^}{p}}_{1}=\frac{\text{Count}}{\text{Sample size}}$ Substitute the values in the above formula: ${\stackrel{^}{p}}_{1}=\frac{\text{Count}}{\text{Sample size}}$ $=\frac{861}{1055}$ $=0.8161$ Now, the proportion of children aged 11 to 13 years who met the calcium requirement is defined by the formula: ${\stackrel{^}{p}}_{2}=\frac{\text{Count}}{\text{Sample size}}$ Substitute the values in the above formula: ${\stackrel{^}{p}}_{2}=\frac{\text{Count}}{\text{Sample size}}$ $=\frac{417}{974}$ $=0.4281$ The $95\mathrm{%}$ confidence interval is defined by the formula: $CI=\left({p}_{1}-{p}_{2}\right)±{z}_{\frac{\alpha }{2}}×PSK\sqrt{\frac{{p}_{1}\left(1-{p}_{1}\right)}{{n}_{1}}+\frac{{p}_{2}\left(1-{p}_{2}\right)}{{n}_{2}}}$ Substitute the values in the above formula: $CI=\left({p}_{1}-{p}_{2}\right)±{z}_{\frac{\alpha }{2}}×PSK\sqrt{\frac{{p}_{1}\left(1-{p}_{1}\right)}{{n}_{1}}+\frac{{p}_{2}\left(1-{p}_{2}\right)}{{n}_{2}}}$ $=\left(0.8161-0.4281\right)±1.96×\sqrt{0.8161\left(1-0.8161\right)}\left\{1055\right\}+\frac{0.4281\left(1-0.4281\right)}{974}$ $=0.388±1.96×0.01983$ $=0.388±0.03886$ $=\left(0.3492,0.4268\right)$ Hence, the required confidence interval is (0.3492, 0.4268). Do you have a similar question?	2023-03-27 05:08:31	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 16, "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": 0.7547377347946167, "perplexity": 944.4108356884636}, "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/1679296946637.95/warc/CC-MAIN-20230327025922-20230327055922-00105.warc.gz"}
http://weblib.cern.ch/collection/ALICE%20Preprints?ln=ka	# ALICE Preprints უკანასკნელი დამატებები: 2020-07-01 04:17 Heavy-flavour correlations and jets with ALICE at the LHC / Oliveira Da Silva, Antonio Carlos This contribution summarises the results on heavy-flavour correlations and jets measured with ALICE detector. [...] arXiv:2006.12606. - 7 p. Fulltext 2020-07-01 04:17 Design and Implementation of Detector Control System for Muon Forward Tracker at ALICE / Yamakawa, K. ; Augustinus, A. ; Batigne, G. ; Chochula, P. ; Oya, M. ; Panebianco, S. ; Pinazza, O. ; Shigaki, K. ; Tieulent, R. ; Yamaguchi, Y. ALICE is the experiment at the CERN LHC devoted to study heavy-ion collisions. [...] arXiv:2006.07224. - 15 p. Fulltext 2020-07-01 04:16 GPU-based reconstruction and data compression at ALICE during LHC Run 3 / Rohr, David In LHC Run 3, ALICE will increase the data taking rate significantly to 50 kHz continuous read out of minimum bias Pb-Pb collisions. [...] arXiv:2006.04158. - 7 p. Fulltext 2020-05-27 13:15 Elliptic and triangular flow of (anti)deuterons in Pb-Pb collisions at $\sqrt{s_{NN}}$= 5.02 TeV / ALICE Collaboration, CERN /ALICE The measurements of the (anti)deuterons elliptic flow ($v_2$) and the first measurements of triangular flow ($v_3$) in Pb-Pb collisions at a center-of-mass energy per nucleon-nucleon collisions $\sqrt{s_{NN}}$ = 5.02 TeV are presented. A mass ordering at low transverse momentum ($p_{\rm T}$) is observed when comparing these measurements with those of other identified hadrons, as expected from relativistic hydrodynamics. [...] CERN-EP-2020-099.- Geneva : CERN, 2020 Draft (restricted): PDF; 2020-05-27 10:06 Constraining the Chiral Magnetic Effect with charge-dependent azimuthal correlations in Pb-Pb collisions at $\sqrt{\it{s}_{\mathrm{NN}}}$ = 2.76 and 5.02 TeV / ALICE Collaboration, CERN /ALICE Systematic studies of charge-dependent two- and three-particle correlations in Pb--Pb collisions at $\sqrt{\it{s}_\mathrm{{NN}}} =$ 2.76~and 5.02~TeV used to probe the Chiral Magnetic Effect (CME) are presented. These measurements are performed for charged particles in the pseudorapidity ($\eta$) and transverse momentum ($p_{\rm{T}}$) ranges $\left\|\eta \right\| < 0.8$ and $0.2 < p_{\mathrm{T}} < 5$~GeV/$c$. [...] CERN-EP-2020-098.- Geneva : CERN, 2020 Draft (restricted): PDF; 2020-05-26 02:36 Measurement of isolated photon-hadron correlations in 5 TeV pp and pPb data / ALICE Collaboration, CERN /ALICE This paper presents isolated photon-hadron correlations using pp and p-Pb data collected by the ALICE detector at the LHC. For photons with \|$\eta$\| < 0.67 and 12 < $p_{\rm{T}}$ < 40 GeV/$c$, the associated yield of charged particles in the range \|$\eta$\| < 0.80 and 0.5 < $p_{\rm{T}}$ < 10 GeV/$c$ is presented. [...] CERN-EP-2020-097.- Geneva : CERN, 2020 - 19. Draft (restricted): PDF; 2020-05-21 19:12 J/$\psi$ elliptic and triangular flow in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV / ALICE Collaboration, CERN The inclusive J/$\psi$ elliptic ($v_{2}$) and triangular ($v_{3}$) flow coefficients are measured at forward rapidity (2.5 < y < 4) and the $v_{2}$ at midrapidity (\|y\| < 0.9) in PbPb collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV using the ALICE detector at the LHC. The entire Pb-Pb data sample collected during Run 2 is employed, amounting to integrated luminosities of about 750 $\mu\rm{b}^{-1}$ at forward rapidity and 93 $\mu\rm{b}^{-1}$ at midrapidity. [...] CERN-EP-2020-094.- Geneva : CERN, 2020 - 22. Draft (restricted): PDF; 2020-05-21 18:55 Soft-dielectron excess in proton-proton collisions at $\sqrt{s}$ = 13 TeV / ALICE Collaboration, CERN A measurement of dielectron production in proton-proton (pp) collisions at $\sqrt{s}$ = 13 TeV, recorded with the ALICE detector at the CERN LHC, is presented in this Letter. The data set was recorded with a reduced magnetic solenoid field. [...] CERN-EP-2020-095.- Geneva : CERN, 2020 - 13. Draft (restricted): PDF; 2020-05-20 19:58 First measurement of quarkonium polarization in nuclear collisions at the LHC / ALICE Collaboration, CERN The polarization of inclusive J/$\psi$ and $\Upsilon(1{\rm S})$ produced in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}=5.02$ TeV at the LHC is measured with the ALICE detector. The study is carried out by reconstructing the quarkonium through its decay to muon pairs in the rapidity region \$2.5 2020-05-20 11:15 A new laboratory to study hadron-hadron interactions / ALICE Collaboration, CERN One of the big challenges for nuclear physics today is to understand, starting from first principles, the effective interaction between hadrons with different quark content. First successes have been achieved utilizing techniques to solve the dynamics of quarks and gluons on discrete space-time lattices. [...] CERN-EP-2020-091.- Geneva : CERN, 2020 Draft (restricted): PDF;	2020-07-06 00:31:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9486756324768066, "perplexity": 6185.283472561273}, "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-2020-29/segments/1593655889877.72/warc/CC-MAIN-20200705215728-20200706005728-00269.warc.gz"}
https://ftp.aimsciences.org/article/doi/10.3934/jcd.2020015	# American Institute of Mathematical Sciences December 2020, 7(2): 369-399. doi: 10.3934/jcd.2020015 ## A tale of two vortices: How numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia * Corresponding author: uqcblach@uq.edu.au Received October 2019 Published July 2020 Fund Project: This work has been partially supported by an Australian Research Council Discovery Early Career Researcher Award (DE160100147) and by an Australian Government Research Training Program Stipend Scholarship (CB) Coherent structures are spatially varying regions which disperse minimally over time and organise motion in non-autonomous systems. This work develops and implements algorithms providing multilayered descriptions of time-dependent systems which are not only useful for locating coherent structures, but also for detecting time windows within which these structures undergo fundamental structural changes, such as merging and splitting events. These algorithms rely on singular value decompositions associated to Ulam type discretisations of transfer operators induced by dynamical systems, and build on recent developments in multiplicative ergodic theory. Furthermore, they allow us to investigate various connections between the evolution of relevant singular value decompositions and dynamical features of the system. The approach is tested on models of periodically and quasi-periodically driven systems, as well as on a geophysical dataset corresponding to the splitting of the Southern Polar Vortex. Citation: Chantelle Blachut, Cecilia González-Tokman. A tale of two vortices: How numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems. Journal of Computational Dynamics, 2020, 7 (2) : 369-399. doi: 10.3934/jcd.2020015 ##### References: show all references ##### References: Figures 1a and 1b show almost invariant structures as described by the (evolved) subdominant eigenvector of an Ulam matrix approximation to the transfer operator in the periodically driven double gyre flow, with parameters as in [40]. Figures 1c and 1d show finite-time coherent structures as described by the (evolved) subdominant initial time singular vector of a composition of $10$ Ulam matrices describing the evolution of the transitory double gyre flow introduced by [31]. See [17] for a thorough discussion of both models Evolution in non-autonomous dynamical systems: driving system (above the arrow), particle evolution (2nd row), transfer operators (3rd row) and Ulam's method (bottom row) Selected vector field instances for the periodically forced double well potential An illustration of the behaviour of $\alpha(t)$ and $\tilde{\alpha}(t)$ over $5$ periods Tracking modes over rolling windows for the periodically forced double well potential Tracking modes for time windows of length $n = 50$, evolved using Algorithm 4 Crossing introduced by shifting from $n = 54$ to $n = 51$ for the periodically forced double well potential Equivariance mismatch for the periodically forced double well potential when $n = 50$ Leading $6$ of $\mathcal{N} = 6$ modes for the periodically forced double well potential when $n = 50$ Leading $4$ of $\mathcal{N} = 4$ modes for the periodically forced double well potential when $n = 100$ Mean equivariance mismatch, as per Algorithm 5, for the leading $4$ of $\mathcal{N}$ modes using the two pairing methods given by Algorithms 2 ($\bar{\varsigma}_{S}$) and 3 ($\bar{\varsigma}_{U}$) for $n = 50$ Leading $4$ from a total $\mathcal{N} = 5$ tracked modes for $n = 50$ using Algorithms 3 (top) and 5 (bottom) Leading $4$ from a total $\mathcal{N} = 7$ tracked modes for $n = 50$ using Algorithms 3 (top) and 5 (bottom) Consecutive windows corresponding to reasonable equivariance for $S_{U}^{(4)}$ of Figure 12 Initial time singular vectors corresponding to rolling windows initialised at the various $t_{0}$ indicated by colour coded bars and column headings. These are paired according to the paths illustrated in Figure 12 Evolved $u^{(50)}_{75,4}$ (top) and $u^{(50)}_{274,4}$ (bottom) of mode $S_{U}^{(4)}$ in Figure 15, evolved as per Algorithm 4 Southern hemisphere wind speed (easterly and northerly) on the $850$ K isentropic surface Mean equivariance mismatch, as per Algorithm 5, for the leading $3$ of $\mathcal{N}$ modes using the two pairing methods given in Algorithms 2 and 3 with $n = 56$ and $t_0 \in[0000 \: 1 \: \text{August}, 1800 \: 30 \: \text{September}]$. Here the Ulam matrices, describing transitions for the area south of $30^{\circ}$S, are of dimension $m \times m$ for $m = 2^{14}$ Leading $3$ of $\mathcal{N} = 3$ tracked paths of singular values of rolling windows paired using Algorithm 2 for $n = 56$ Leading singular vectors, for various $t_{0}$, of matrix compositions associated with Figure 19b where time windows are of length $n = 56$. The area illustrated is south of $50^{\circ}$S and the time given in the label is the relevant $t_{0}$ for that window Evolved leading mode associated with Figure 19a for a time window centred on the peak at $1800$ on $23$ Sep. This is illustrated on the area south of $15^{\circ}$S Evolved subdominant mode associated with Figure 19b for a time window centred on the peak at $0600$ on $24$ Sep. This is illustrated on the area south of $15^{\circ}$S Evolved leading singular vectors for time windows centred at $0000$ on $24$ Sep. for $m = 12,800$ initially seeded bins whose centres are south of $20^{\circ}$S. This is illustrated on the full southern hemisphere Evolved subdominant mode normalised as in [19] for time windows centred at $0000$ on $24$ Sep. for $m = 12,800$ initially seeded bins whose centres are south of $20^{\circ}$S. This is illustrated on the full southern hemisphere [1] Yangrong Li, Shuang Yang, Qiangheng Zhang. 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Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020381 Impact Factor: ## Metrics • HTML views (219) • Cited by (0) • on AIMS	2020-11-28 14:11:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.474622905254364, "perplexity": 3526.482759249047}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141195656.78/warc/CC-MAIN-20201128125557-20201128155557-00465.warc.gz"}
https://mitpress.mit.edu/books/final-over-final-condition	Hardcover \| Out of Print \| 464 pp. \| 6 x 9 in \| 1 graph \| October 2017 \| ISBN: 9780262036696 Paperback \| $40.00 X \| £32.95 \| 464 pp. \| 6 x 9 in \| 1 graph \| October 2017 \| ISBN: 9780262534161 eBook \|$28.00 X \| October 2017 \| ISBN: 9780262342001 Mouseover for Online Attention Data ## The Final-Over-Final Condition A Syntactic Universal Foreword by David Pesetsky ## Overview This book presents evidence for a universal word order constraint, the Final-over-Final Condition (FOFC), and discusses the theoretical implications of this phenomenon. FOFC is a syntactic condition that disallows structures where a head-initial phrase is contained in a head-final phrase in the same extended projection/domain. The authors argue that FOFC is a linguistic universal, not just a strong tendency, and not a constraint on processing. They discuss the effects of the universal in various domains, including the noun phrase, the adjective phrase, the verb phrase, and the clause. The book draws on data from a wide range of languages, including Hindi, Turkish, Basque, Finnish, Afrikaans, German, Hungarian, French, English, Italian, Romanian, Arabic, Hebrew, Mandarin, Pontic Greek, Bagirmi, Dholuo, and Thai. FOFC, the authors argue, is important because it is the only known example of a word order asymmetry pertaining to the order of heads. As such, it has significant repercussions for theories connecting the narrow syntax to linear order. ## About the Authors Michelle Sheehan is Reader in Linguistics in the Department of English and Media at Anglia Ruskin University. Theresa Biberauer is Principal Research Associate in the Department of Theoretical and Applied Linguistics at the University of Cambridge and Professor in the Department of General Linguistics at Stellenbosch University, South Africa. Ian Roberts is Professor of Linguistics and Professorial Fellow at Downing College at the University of Cambridge. Anders Holmberg is Professor of Theoretical Linguistics at Newcastle University and Director of Research at the University of Cambridge. ## Endorsements “I consider the discovery and development of the ‘Final-Over-Final Condition’ to be the most exciting advance in the study of word order in recent years—and therefore in the broader study of crosslinguistic variation. This volume is fascinating in the range of languages and constructions it draws on, the range of linguistic considerations it brings to bear on the issue, and—not least—in the different complementary perspectives that each of the authors brings to the topic. It will repay careful consideration on every level.” Mark Baker, Distinguished Professor, Linguistics and Cognitive Science, Rutgers University; author of The Syntax of Agreement and Concord “This book reminds us how much fun it is to do syntax. Such a combination of detailed analysis with serious architectural assessment should open the kind of discussion that makes our field young all over again.” Juan Uriagereka, Professor, University of Maryland; author of Spell-Out and the Minimalist Program and Syntactic Anchors	2018-02-19 05:57:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.287507027387619, "perplexity": 3486.9995650115}, "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-2018-09/segments/1518891812405.3/warc/CC-MAIN-20180219052241-20180219072241-00489.warc.gz"}
https://physics.stackexchange.com/questions/604519/are-there-black-body-like-thermal-emissions-of-gravitational-waves	Are there black body-like thermal emissions of gravitational waves? According to the answer to this previous question: Yes, all of the fields in quantum electrodynamics are excited in blackbody radiation, not just the electromagnetic field. But, (as I understand) there is currently no experimental evidence that gravitational waves are described by a quantum field theory. Is there scientific consensus that there are thermal (black body) gravitational waves? If so, is thermal gravitational radiation negligible? I imagine it would be impossible/difficult to directly detect, but does it factor into any cosmological theories? Clarification: I say "black body radiation," but here I'm more interested in whether there are emissions of gravitational waves related to temperature (as opposed to macroscopic motion), and not whether the system would be able to reach equilibrium or meet a strict definition of thermal/black body radiation. • does it factor into any cosmological theories? Quick googling produced the following paper: Cosmological decoherence from thermal gravitons, (note, “thermal gravitons” there are from de Sitter horizon). Jan 1 at 19:23 • A good comment by @ProfRob makes me think that I should have requested clarification before posting an answer. I interpreted the statement "all of the fields in quantum electrodynamics are excited in blackbody radiation, not just the electromagnetic field" as the focal point of the question, so that the question is really "Does the 'all' in this statement include gravitational radiation?" Is that the right way to read the question? Or are you really asking the more general question "Can gravitational radiation ever have a thermal spectrum, and is such radiation ever significant for cosmology?" Jan 2 at 17:30 In this answer, I'm interpreting the question like this: Does a blackbody excite thermal gravitational radiation, like it excites thermal electromagnetic radiation? And can this ever be significant in the real world? From both theory and observation, we know that gravitational radiation can carry energy away from a system,$$^\dagger$$ such as a system of two compact objects orbiting each other, but those gravitational waves never reached equilibrium with the system that produced them, so this is not a blackbody. $$^\dagger$$ Defining the "energy" of gravitational disturbances in general relativity can be problematic, but I won't try to address that here. This answer is long enough already. Reaching equilibrium takes time. The weaker the interaction, the longer it takes. And if the radiation escapes too quickly after it is produced, then equilibrium may never be reached. According to ref 1, that's exactly what happens in the case of gravitational radiation. Gravitational radiation is hard to contain in a bounded region even temporarily (this post addresses a related issue), and gravity is very weak, so gravitational radiation tends to escape long before reaching equilibrium with the rest of the system. The same paper proposes a range of conditions under which gravitational radiation could theoretically reach equilibrium, but the paper does not show numeric estimates, and I don't know if the proposed conditions correspond to anything realistic. Hawking radiation from black holes is special. I'll say more about that below, but first, here are a few miscellaneous comments: • Cosmology: According to ref 2, the current temprature of the gravitational wave background (resulting from gravitational radiation that may have been in thermal equilibrium with other entities in the very early universe) is expected to be much lower than the already-low temperature of the cosmic microwave background. Regarding the not-so-early universe: as far as I know, gravitational waves being produced today can't reach equilibrium on cosmological scales (even electromagnetic radiation isn't doing that, and it interacts much more strongly), but I don't know enough about cosmology to say anything enlightened. • If equilibrium is reached, then the resulting properties of blackbody radiation can be calculated without worrying about the how strong or weak the interactions are. The Boltzmann distribution, where the probability of a state of energy $$E$$ being occupied is $$e^{-E/kT}$$, can be used for gravitational radiation just like it's used for electromagnetic radiation, using a quantum version of linearized general relativity to define the "energy" of a graviton. • Gravitons in quantum physics: Whether or not a full theory of quantum gravity would have gravitons is a question that has been debated in the literature, but we can say this: string theory, the most-studied theory of quantum gravity by far, does have gravitons at least in its perturbative expansion(s). Gravitons can also be included in quantum field theory in a perturbative expansion (ref 3). The resulting quantum field theory theory is not renormalizable, but that's okay as long as we treat it perturbatively as a low-energy effective theory with a high-energy cutoff to hide the deeper physics that quantum field theory (probably) doesn't know about. • Gravitationally bound systems: The thermodynamics of systems that are held together by gravity is interesting, because such systems have negative heat capacity: their temperature increases when they lose energy, and putting more energy into them makes them colder. This is true whether or not gravitational radiation plays any role. It's true even in Newton's model of gravity, which doesn't have gravitational waves. But the negative heat capacity does have an interesting consequence in one type of system where (quantum) gravitational radiation does play a role: black holes. That brings us to the subject of Hawking radiation... Hawking radiation is a quantum effect that all black holes are expected to exhibit. Hawking radiation has the characteristics of blackbody radiation. Hawking's original derivation of Hawking radiation did not use a full theory of quantum gravity, and the process that produces the radiation in Hawking's original approach is different than an ordinary blackbody. A full theory of quantum gravity undoubtedly has something interesting to say about the actual process, which is probably something like thermalization except that it must involve spacetime geometry in a novel way. Significant recent progress has been made in understanding how this probably works (ref 4), but I'm only barely beginning to study that subject, so I won't try to say anything else about it here. Regardless of exactly how it is produced, the temperature of Hawking radiation is predicted to be exceedingly low for black holes of stellar mass or more. (Remember: gravitationally-bound systems have negative heat capacity, so larger black holes are colder.) As a result, the radiation is expected to be dominated by massless entities — photons and gravitons — even though it can contain anything in principle (ref 5). Even neutrinos might be too massive to make a significant contribution. The quantitative details are reviewed in ref 6, which says that the power emitted as gravitons is expected to be roughly ten times less than the power emitted as photons, according to the text above equation (1) in ref 6. 1. Padmanabhanan and Singh, "A note on the thermodynamics of gravitational radiation" (https://arxiv.org/abs/gr-qc/0305030) 2. Press and Thorne (1972), "Gravitational-wave astronomy", Annual Rev. Astron Astrophys 10:355-374 (https://www.annualreviews.org/doi/pdf/10.1146/annurev.aa.10.090172.002003) 3. Donoghue, "Introduction to the Effective Field Theory Description of Gravity" (https://arxiv.org/abs/gr-qc/9512024) 4. Raju, "Lessons from the Information Paradox", (https://arxiv.org/abs/2012.05770) 5. Harlow and Ooguri, "Symmetries in Quantum Field Theory and Quantum Gravity" (https://arxiv.org/abs/1810.05338) 6. Don Page, "Time Dependence of Hawking Radiation Entropy" (https://arxiv.org/abs/1301.4995) • I agree that thermal equilibrium is required for a blackbody, but thermal radiation requires no such equilibrium. Thermal radiation is simply chacterised by a temperature. Cooling objects can still emit thermal radiation. Jan 2 at 9:16 • @ProfRob Right, but "cooling object" implies an object with a practically well-defined temperature, which is all I meant by equilibrium: different parts of the system have reached a state that is "steady" enough so that all of its parts (including the gravitational waves) share a practically well-defined temperature, even if that temperature is changing on a more gradual timescale. Jan 2 at 15:03 • An object with a multi-temperature or continuous range of temperatures can still emit thermal radiation - e.g. the solar corona. The solar corona isn't at a single temperature and isn't in equilibrium. There is a clear distinction. Or to put it another way - the definition of "equilibrium" you are using is not strict enough to imply blackbody radiation. Jan 2 at 15:07 • @ProfRob That's a great point. Would it be fair to say that the gravitational radiation from a globular cluster of closely-space neutron stars is "thermal"? (Or even a globular cluster of regular stars... I just picked closely-spaced neutron stars to try to make the radiation a little more noticeable.) I'm asking because I haven't done the calculation to see if the radiation in that case actually has a spectrum characterized by a single temperature. Jan 2 at 16:03 • @ProfRob, This is what I meant. I added a clarification to the my question, but I was wondering if there were "thermal radiation" rather than something that meets the strictest definition of a black body. Jan 4 at 12:57 I'm adding this rather than extending the discussion below Chiral Anomaly's answer. I think that answer is correct, but I do think some clarification is required. The requirements for black body radiation (which is not a radiation mechanism in itself) are that the radiation emitted by the body is "thermal", which means that the emission spectrum could be characterised by a temperature; that the object absorbs all radiation incident upon it; and that the object described as a blackbody is in equilibrium at a single temperature. The second of these conditions is very unlikely to apply anywhere in the present-day universe. Matter is almost transparent to gravitational waves which is why they are so difficult to detect. However, conditions are considerably different in the early universe. There are indeed predictions that the relic gravitational waves arising from the epoch of inflation will have a thermal spectrum and could have been characterised by a temperature $$T > 10^{28}$$ eV (!) in the pre-inflationary universe and subsequently has a blackbody spectrum of frequencies (e.g. Bhattacharaya et al. 2006; Zhao et al. 2009; Wang et al. 2017). This radiation will have decoupled from the rest of the universe after inflation and will have cooled to an extremely low temperature today ($$\sim 10^{-26}$$) K, but with wavelengths that may have been sufficiently large to imprint subtle signatures on the cosmic microwave background. But is there any way in which the gravitational wave emission process could be described as "thermal" or be assigned a "temperature" in the present day? I don't think you can ascribe a gravitational wave temperature to single macroscopic objects, or even to binary systems because the microscopic components of these systems are behaving in concert. It would be akin to trying to assign a temperature to a single atom or molecule. Chiral Anomaly suggests (in comments) a cluster of neutron stars. Clusters can be assigned a "temperature" that is essentially proportional to the rms speed of the component stars, in much the same way that molecules in a gas have a temperature. There could be some merit in this. Undoubtedly the cluster as a whole would emit low frequency gravitational wave radiation (with wavelengths that could be larger than the cluster) that had a spectrum that could be connected with this rms speed.	2021-12-01 10:02:41	{"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": 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": 6, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6774184107780457, "perplexity": 463.514125249657}, "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/1637964359976.94/warc/CC-MAIN-20211201083001-20211201113001-00128.warc.gz"}
https://community.atlassian.com/t5/Jira-questions/Escalation-Service-not-rendering-newline-in-setComment/qaq-p/179402	cancel Showing results for Did you mean: See all See all ##### Groups Explore all groups # Escalation Service not rendering newline in setComment Any idea why my newline code (2 forward slashes \\) below is not being rendered properly in the comments? They are displaying as is. issueInputParameters.setComment('Resource Admin, \\ \\ We have noticed that this issue has not been updated for a period of 7 days. \\ We will assume that this issue is resolved and close the ticket. If you still need help, just reply comment on the ticket and we will reopen it. You can also request a delay (freeze) now if you need more time to respond. \\ If you have managed to resolve this issue locally, please take a moment to share your solution for future reference. \\ \\ Kind Regards, \\ Support Team') Not Sure if you already found the answer to this question, try using "\\\\" for line break. issueInputParameters.setComment('Resource Admin, \\\\ We have noticed that this issue has not been updated for a period of 7 days.....') Community showcase Published Nov 27, 2018 in Portfolio for Jira ### Introducing a new planning experience in Portfolio for Jira (Server/DC) In the past, Portfolio for Jira required a high degree of detail–foresight that was unrealistic for many businesses to have–in order to produce a reliable long-term roadmap. We're tur... 2,650 views 18 21 ### Atlassian User Groups Connect with like-minded Atlassian users at free events near you! Connect with like-minded Atlassian users at free events near you! ##### Find my local user group Unfortunately there are no AUG chapters near you at the moment.	2018-12-14 05:43:37	{"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": 0.9975159764289856, "perplexity": 3915.181979951582}, "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-2018-51/segments/1544376825363.58/warc/CC-MAIN-20181214044833-20181214070333-00262.warc.gz"}
https://support.bioconductor.org/p/83349/	Search Question: Unable to load/install rtracklayer (v1.33.2) on OS X (possibly SSL-related?) 0 2.1 years ago by Peter Hickey370 Johns Hopkins University, Baltimore, USA Peter Hickey370 wrote: I could do with some help figuring out why I can't load/properly install rtracklayer (v1.33.2) on my laptop running OS X. The error message suggests to me some issue with SSL, but I don't really understand what's going wrong. Please let me know if I can provide additional information to help solve this (I had to trim the output in order to get post under 15000 characters) Help is much appreciated, Pete Just realised I have multiple versions of openssl installed (unsure if helpful) # Version installed by conda (and default on my system) peters-mbp-2:~ Peter$which openssl /Users/Peter/anaconda/bin/openssl peters-mbp-2:~ Peter$ openssl version OpenSSL 1.0.2g 1 Mar 2016 # Version install by homebrew peters-mbp-2:~ Peter$/usr/local/Cellar/openssl/1.0.2d_1/bin/openssl version OpenSSL 1.0.2d 9 Jul 2015 R details > library(BiocInstaller) Bioconductor version 3.4 (BiocInstaller 1.23.4), ?biocLite for help > biocLite('rtracklayer') BioC_mirror: https://bioconductor.org Using Bioconductor 3.4 (BiocInstaller 1.23.4), R 3.3.0 (2016-05-03). Installing package(s) ‘rtracklayer’ trying URL 'https://bioconductor.org/packages/3.4/bioc/bin/macosx/mavericks/contrib/3.3/rtracklayer_1.33.2.tgz' Content type 'application/x-gzip' length 1908273 bytes (1.8 MB) ================================================== downloaded 1.8 MB The downloaded binary packages are in /var/folders/f1/6pjy5xbn0_9_7xwq6l7fj2yc0000gn/T//RtmpzfApQX/downloaded_packages > suppressPackageStartupMessages(library(rtracklayer)) Error in dyn.load(file, DLLpath = DLLpath, ...) : unable to load shared object '/Library/Frameworks/R.framework/Versions/3.3/Resources/library/rtracklayer/libs/rtracklayer.so': dlopen(/Library/Frameworks/R.framework/Versions/3.3/Resources/library/rtracklayer/libs/rtracklayer.so, 6): Symbol not found: _BIO_new_ssl_connect Referenced from: /Library/Frameworks/R.framework/Versions/3.3/Resources/library/rtracklayer/libs/rtracklayer.so Expected in: flat namespace in /Library/Frameworks/R.framework/Versions/3.3/Resources/library/rtracklayer/libs/rtracklayer.so Error: package or namespace load failed for ‘rtracklayer’ Trying to install from source ultimately leads to the same error (and looks like openssl isn't being picked up by configure?): > biocLite("rtracklayer", type = "source") BioC_mirror: https://bioconductor.org Using Bioconductor 3.4 (BiocInstaller 1.23.4), R 3.3.0 (2016-05-03). Installing package(s) ‘rtracklayer’ trying URL 'https://bioconductor.org/packages/3.4/bioc/src/contrib/rtracklayer_1.33.2.tar.gz' Content type 'application/x-gzip' length 1378796 bytes (1.3 MB) ================================================== downloaded 1.3 MB * installing source package ‘rtracklayer’ ... checking for pkg-config... /usr/local/bin/pkg-config checking pkg-config is at least version 0.9.0... yes checking for OPENSSL... no configure: creating ./config.status config.status: creating src/Makevars libs clang -I/Library/Frameworks/R.framework/Resources/include -DNDEBUG -I/usr/local/opt/openssl/include -DUSE_SSL -D_FILE_OFFSET_BITS=64 -I/usr/local/include -I/usr/local/include/freetype2 -I/opt/X11/include -I"/Library/Frameworks/R.framework/Versions/3.3/Resources/library/S4Vectors/include" -I"/Library/Frameworks/R.framework/Versions/3.3/Resources/library/IRanges/include" -I"/Library/Frameworks/R.framework/Versions/3.3/Resources/library/XVector/include" -fPIC -Wall -mtune=core2 -g -O2 -c S4Vectors_stubs.c -o S4Vectors_stubs.o clang -I/Library/Frameworks/R.framework/Resources/include -DNDEBUG -I/usr/local/opt/openssl/include -DUSE_SSL -D_FILE_OFFSET_BITS=64 -I/usr/local/include -I/usr/local/include/freetype2 -I/opt/X11/include -I"/Library/Frameworks/R.framework/Versions/3.3/Resources/library/S4Vectors/include" -I"/Library/Frameworks/R.framework/Versions/3.3/Resources/library/IRanges/include" -I"/Library/Frameworks/R.framework/Versions/3.3/Resources/library/XVector/include" -fPIC -Wall -mtune=core2 -g -O2 -c IRanges_stubs.c -o IRanges_stubs.o # Clip a bunch of output in order for post to have <= 15000 characters installing to /Library/Frameworks/R.framework/Versions/3.3/Resources/library/rtracklayer/libs R data demo inst preparing package for lazy loading Creating a generic function for ‘offset’ from package ‘stats’ in package ‘rtracklayer’ Creating a generic function from function ‘uri’ in package ‘rtracklayer’ help * installing help indices building package indices installing vignettes ** testing if installed package can be loaded Error in dyn.load(file, DLLpath = DLLpath, ...) : unable to load shared object '/Library/Frameworks/R.framework/Versions/3.3/Resources/library/rtracklayer/libs/rtracklayer.so': dlopen(/Library/Frameworks/R.framework/Versions/3.3/Resources/library/rtracklayer/libs/rtracklayer.so, 6): Symbol not found: _BIO_new_ssl_connect Referenced from: /Library/Frameworks/R.framework/Versions/3.3/Resources/library/rtracklayer/libs/rtracklayer.so Expected in: flat namespace in /Library/Frameworks/R.framework/Versions/3.3/Resources/library/rtracklayer/libs/rtracklayer.so Error: loading failed Execution halted ERROR: loading failed * removing ‘/Library/Frameworks/R.framework/Versions/3.3/Resources/library/rtracklayer’ * restoring previous ‘/Library/Frameworks/R.framework/Versions/3.3/Resources/library/rtracklayer’ The downloaded source packages are in ‘/private/var/folders/f1/6pjy5xbn0_9_7xwq6l7fj2yc0000gn/T/RtmpzfApQX/downloaded_packages’ Warning message: In install.packages(pkgs = doing, lib = lib, ...) : installation of package ‘rtracklayer’ had non-zero exit status Here's my session info: > sessionInfo() R version 3.3.0 (2016-05-03) Platform: x86_64-apple-darwin13.4.0 (64-bit) Running under: OS X 10.11.4 (El Capitan) locale: [1] en_AU.UTF-8/en_AU.UTF-8/en_AU.UTF-8/C/en_AU.UTF-8/en_AU.UTF-8 attached base packages: [1] stats4 parallel stats graphics grDevices utils datasets [8] methods base other attached packages: [1] GenomicRanges_1.25.0 GenomeInfoDb_1.9.1 IRanges_2.7.1 [4] S4Vectors_0.11.2 BiocGenerics_0.19.0 BiocInstaller_1.23.4 [7] repete_0.0.0.9004 devtools_1.11.1 loaded via a namespace (and not attached): [1] Rcpp_0.12.5 XVector_0.13.0 [3] magrittr_1.5 GenomicAlignments_1.9.0 [5] zlibbioc_1.19.0 BiocParallel_1.7.2 [7] munsell_0.4.3 colorspace_1.2-6 [9] stringr_1.0.0 plyr_1.8.3 [11] tools_3.3.0 SummarizedExperiment_1.3.2 [13] Biobase_2.33.0 withr_1.0.1 [15] digest_0.6.9 pryr_0.1.2 [17] bitops_1.0-6 codetools_0.2-14 [19] RCurl_1.95-4.8 memoise_1.0.0 [21] stringi_1.1.1 Rsamtools_1.25.0 [23] Biostrings_2.41.1 scales_0.4.0 [25] XML_3.98-1.4 ADD COMMENTlink modified 2.1 years ago by Michael Lawrence10.0k • written 2.1 years ago by Peter Hickey370 1 2.1 years ago by Michael Lawrence10.0k United States Michael Lawrence10.0k wrote: When building from source on the Mac, it looks specifically for the homebrew installation, even if the pkg-config query fails. I guess the conda openssl is coming first on the search path, so you end up with a binary incompatibility. I think you can fix this by setting the DYLD_LIBRARY_PATH environment variable to point to the directory containing the homebrew-installed library. I guess you will want to do this inside .Renviron so that it always works. One general concern is that if the distributed Mac binary depends on having openssl available at run time, many will not be able to load rtracklayer. We might want to consider not building with openssl support on the Mac, or perhaps not distributing a binary at all. Another idea would be to automatically install openssl via homebrew, although I'm not sure it's worth it given how SSL support for bigwig files is a minor feature of the package. ADD COMMENTlink written 2.1 years ago by Michael Lawrence10.0k Thanks, Michael, but I'm still having problems. I tried the following: Add DYLD_LIBRARY_PATH="/usr/local/Cellar/openssl/1.0.2d_1/lib" to .Renviron and confirmed this set in a new R session. Re-ran biocLite("rtracklayer", type = "source") and hit same error. ADD REPLYlink modified 2.1 years ago • written 2.1 years ago by Peter Hickey370 Do a R CMD ldd path/to/your/rtracklayer.so, trying with and without the DYLD_LIBRARY_PATH. Might also try LD_LIBRARY_PATH. ADD REPLYlink written 2.1 years ago by Michael Lawrence10.0k$ R CMD ldd /Library/Frameworks/R.framework/Versions/3.3/Resources/library/rtracklayer/libs/rtracklayer.so 1 Sorry you need otool -L instead of ldd on the Mac. Actually, I tried that myself and found that the libraries were not linking at runtime, and I fixed it just now in rtracklayer 1.33.4. Not sure why it wasn't breaking for me. Thanks Michael! Would it make sense to backport this change to release? The bug does not exist in release, since the homebrew support is only in devel (due to the potential for this sort of bug).	2018-06-23 15:49:18	{"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": 0.3246341645717621, "perplexity": 4626.39182947877}, "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-2018-26/segments/1529267865098.25/warc/CC-MAIN-20180623152108-20180623172108-00316.warc.gz"}
https://www.nature.com/articles/s41535-019-0162-3?error=cookies_not_supported&code=84c639e4-9fb1-467b-b059-55d59e7acac2	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. # Lattice frustration in spin-orbit Mott insulator Sr3Ir2O7 at high pressure ## Abstract The intertwined charge, spin, orbital, and lattice degrees of freedom could endow 5d compounds with exotic properties. Current interest is focused on electromagnetic interactions in these materials, whereas the important role of lattice geometry remains to be fully recognized. For this sake, we investigate pressure-induced phase transitions in the spin-orbit Mott insulator Sr3Ir2O7 with Raman, electrical resistance, and x-ray diffraction measurements. We reveal an interesting magnetic transition coinciding with a structural transition at 14.4 GPa, but without a concurrent insulator-metal transition. The conventional correlation between magnetic and Mott insulating states is thereby absent. The observed softening of the one-magnon mode can be explained by a reduced tetragonal distortion, while the actual magnetic transition is associated with tilting of the IrO6 octahedra. This work highlights the critical role of lattice frustration in determining the high-pressure phases of Sr3Ir2O7. The ability to control electromagnetic properties via manipulating the crystal structure with pressure promises a new way to explore new quantum states in spin-orbit Mott insulators. ## Introduction Exotic ground states in 5d quantum materials, such as spin liquids,1,2 Weyl semimetals,3 topological insulators,4,5 and superconductors,6,7 are commonly believed to be driven by the competition between electron interaction U, spin-orbit coupling (SOC) ξ, and hopping amplitude t. In particular, for Ir5+ atoms in octahedral crystal field, the 5d orbital degeneracy is lifted by the strong on-site SOC, and the ground state is formed by effective S = 1/2 pseudospins.8 The lattice degree of freedom and its coupling to electron orbital angular momenta therefore have long been thought to play a trivial role.9 However, recently it has been recognized that pseudospin-lattice coupling may have strong impact on the low-energy physics of 5d materials.10 In the single-layer perovskite Sr2IrO4, canted S= 1/2 pseudospins in the antiferromagnetic phase are shown to rigidly lock to IrO6 octahedra due to strong SOC, and the pseudospin orientations rotate together with the octahedra under applied electric current.11 Moreover, pseudospin-lattice coupling can induce a tetragonal-to-orthorhombic structural transition and explain the in-plane magnon gaps of Sr2IrO4.12 Jahn–Teller effect also can explain some high energy features of different iridates in resonant inelastic x-ray scattering (RIXS),13 and the avoidance of metallization of Sr2IrO4 under pressure.14 These findings suggest that subtle structural changes may influence critically the low-energy Hamiltonian. While determining the exact role of lattice variable in 5d materials remains a challenge, applying pressure opens up an avenue for such research, since it could possibly decouple entangled degrees of freedom during phase transitions.15 In this work, we apply pressure to the double-layered perovskite Sr3Ir2O7, which is the middle member of the Ruddlesden-Popper series Srn+1IrnO3n+1 (n = 1, 2, ∞). This material provides an interesting playground to study phase transitions, as it is considered in the weak Mott limit16 with a relatively small charge gap and tiny magnetic moment.17 Sr3Ir2O7 has been extensively studied recently,8,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31 and its crystal structure is commonly reported to stabilize in I4/mmm symmetry.18 However, when the rotation angle α (describing in-plane IrO6 rotation, ~11–12°) and the tilt angle β (describing the deviation of IrO6 c-axis from the z-direction, ~less than 1°) are taken into account, the symmetry is reduced to Bbca,17 or Bbcb,19 or even C2/c.20 Furthermore, the IrO6 octahedron itself has a slight tetragonal distortion. Therefore, Sr3Ir2O7 serves as an ideal candidate for studying how lattice frustrations affect the electromagnetic properties and structural stability at high pressure. ## Results and discussion Figure 1 shows the temperature-dependent Raman spectra of Sr3Ir2O7 at ambient pressure. Six Raman modes are identified, respectively, at 146, 181, 269, 392, 592, and 1360 cm−1 based on the room-temperature spectrum. The frequencies are close to those reported in the literature.21 As temperature decreases, a new Raman peak appears at 780 ± 8 cm−1 (96.7 ± 1.4 meV) at 270 K. We assign this new peak to the one-magnon mode, as its peak intensity decreases significantly with increasing temperature, and its energy is also consistent with the zone-center magnon excitation (~90 meV) reported in RIXS studies of Sr3Ir2O7.22,23 The one-magnon peak exhibits an asymmetric Fano-like line shape, which could result from interaction with a continuum of excitations. Above TN ~285 K,17 the magnon peak disappears. The Raman spectra also reveal a two-magnon mode at 1360 ± 2.3 cm−1 (168.62 ± 0.3 meV) [Fig. 1a], which is different from the previously reported energy ~1500 cm−1 (~185 meV) measured by 632.8 nm laser.21 The two-magnon peak increases slowly with decreasing temperature, and its intensity remains appreciable even at room temperature, suggesting a short-range spin-correlation character. In contrast, the previously reported two-magnon mode was strongly suppressed with temperature and vanished at TN = 285 K.21 Given that the two-magnon mode in our study also could be excited by 488 nm laser, different excitation source is probably not the reason for the observed discrepancy. Rather, the discrepancy could originate from subtle variation in the compositions of different samples. Apart from the assigned magnon modes at 780 and 1360 cm−1, the other five Raman peaks are assigned to phonon modes, which could be indexed according to the I4/mmm space group of tetragonal symmetry.18 The fourteen Raman active modes expected from group theory include ΓRaman = 5A1g + 2B1g + 1B2g + 6Eg.24 However, only one B2g mode at 392 cm−1 and four A1g phonon modes are observable in our Raman experiments, whereas the rest of the modes are absent probably due to their weak Raman scattering cross sections. As previously reported, the A1g mode at 146 cm−1 corresponds to the stretching of Sr atoms against the IrO6 octahedra.24 The A1g mode at 181 cm−1 involves displacements of Sr atoms along the c-axis with antiphase motion of adjacent layers and in-plane rotations of O atoms.21 The A1g mode at 269 cm−1 is attributed to the bending of the Ir–O–Ir bond21,24 due to IrO6 rotation, and the B2g mode at 392 cm−1 is associated with some out-of-plane atomic displacement of in-plane oxygens.21 Finally, the A1g mode at 592 cm−1 originates from the vibration of apical oxygen atoms in IrO6 octahedra.32 These modes are illustrated in Fig. 1a. When cooled down to 10 K, these Raman modes shift to higher frequencies as shown in Fig. 1b. Figure 1b shows the pressure-dependent Raman spectra of Sr3Ir2O7 at 10 K. First, we note that two weak phonon modes emerge close to the A1g mode (187 cm−1) and B2g mode (403 cm−1) at 1.9 GPa. These two emergent modes, indicated by arrow and asterisk, respectively, in Fig. 1b, have been discovered previously24 and are visible below TN owing to magnetic interaction. Second, the one-magnon peak softens continuously and broadens with increasing pressure; the peak eventually disappears around 14.4 GPa, evincing a magnetic transition. In addition, two new modes appear below 152 cm−1 and the above A1g mode weakens at around 14.4 GPa. Similarly, the A1g mode at 288 cm−1 (the bending of the Ir–O–Ir bond) and 595 cm−1 (the vibration of apical oxygens in IrO6 octahedra) harden with pressure prior to almost vanishing around 14.4 GPa, which indicates increased IrO6 rotation and tilt angles at high pressure. These changes in phonon modes suggest a structural transition concurring with the magnetic transition around 14.4 GPa. We also performed room temperature and high-pressure Raman measurements to provide further evidence of the structural transition at 23.2 GPa. The results are given in the Supplementary Material in Fig. S2. The pressure-induced structural phase transition is also confirmed by X-ray diffraction (XRD) at room temperature. The XRD measurements are performed on single crystals, and the results confirm that the sample is stable in an I4/mmm phase up to 21.2 GPa at room temperature, while the new phase was fitted with space group C2 (Fig. 2b). The integrated XRD patterns of Sr3Ir2O7 up to 33.2 GPa are presented in the Supplementary Material in Fig. S3a. At 33.2 GPa, the X-ray diffraction pattern fitting using the symmetry of I4/mmm starts to fail (see Fig. S3b in the Supplementary Material), suggesting the structures of the new phase should adopt a lower symmetry, i.e., C2. Detailed XRD analysis of the Sr3Ir2O7 crystal performed to exclude possible admixture of Sr2IrO4 is also shown in the Supplementary Material in Fig. S4. Our discovery of magnetic and structural transitions at 14.4 GPa now could explain the mysterious origin of a second-order phase transition reported by Zhao et al.25 In their work, the second-order transition derived from the P-V data was attributed to an insulator-metal like transition at ~13 GPa,26 but they also suspected that magnetic transition might be a possible cause.25 To investigate if there is a concurring insulator-metal transition (similar to that observed by in pump-probe experiment at ambient conditions27), we also perform electrical transport measurement on a Sr3Ir2O7 single crystal. The results are plotted in Fig. 3a. The electrical resistance within the a-b plane follows an activation law Ra–b (T) = exp (Δ/2kBT), where Δ is the charge gap and kB is the Boltzmann’s constant.26,33 We obtain the value of Δ at each pressure point from linear fitting of lnR (T) vs. 1/T. The gap energy (black square) as a function of pressure is plotted in Fig. 3b. From this analysis, Sr3Ir2O7 remains an insulator even when the structural and magnetic transitions concur at 14.4 GPa. It is expected to metalize at 55.6 GPa, which is close to the critical pressure reported previously.26,28,29 Pressure thus decouples the insulator-metal transition from the magnetic and structural transitions in Sr3Ir2O7, while these transitions remain coupled in pump-probe measurements.27 To further address how lattice frustration affects the magnetic order, we use spin-wave theory to study the magnon excitations. In particular, we adopt the magnetic exchange Hamiltonian from the reference,22 which describes well the spin-wave dispersion of Sr3Ir2O7 in RIXS measurements. The pressure evolution of magnon dispersion could originate from a change in three microscopic parameters: the ratio of Hund’s coupling to the on-site Coulomb interaction η (=JH/U), the IrO6 rotation angle α, and the effective tetragonal distortion θ that parametrizes the tetragonal splitting of t2g levels. It is important to understand which parameter dominates the softening of single-magnon energy under pressure. First, pressure could enhance η via screening the Hubbard U while leaving JH nearly unchanged. Since we do not observe any metallization at 14.4 Gpa, the impact of pressure on η should be small. Second, it has been shown that pressure can increase the rotation angle α by a few degrees.28 However, our numerical calculations indicate that such a small change in α only lowers the magnon frequency by a few percent (see Fig. 4), which is not enough to account for the experimentally observed magnon softening. On the other hand, decreasing θ can significantly reduce the zone-center magnon energy and lift the degeneracy of the magnon branches (see Fig. S1 in the Supplementary Material). The evidence for reduced tetragonal distortion at high pressure is indeed found in the RIXS experiment by Ding et al.,28 in which the peak width of spin-orbiton excitation at ~0.5 eV reduces from 0.56 eV at 0.98 GPa to 0.48 eV at 12.4 GPa. Such a reduction strongly suggests a reduced tetragonal distortion, as in principle the peak width should have increased under pressure due to the broadening of Jeff = 1/2 and Jeff = 3/2 bands. We thus conclude that one major effect of pressure on the material is to reduce the anisotropy arising from tetragonal distortion of IrO6 octahedra (see Fig. 4). Although the model could explain the magnon softening, it is unable to illustrate the disappearance of magnon. To explain this effect, the IrO6 tilting angle β should be considered. It already has been reported that the actual symmetry of Sr3Ir2O7 should be C2/c with a tilting angle β less than 1 degree.20 Such a small tilting usually has been regarded as having a trivial effect. In our high-pressure study, however, the tilting appears to be important. In fact, the observed blue shift and disappearance of the 152 cm−1 and 595 cm−1 Raman modes suggest that pressure enhances both α and β, which could induce buckling or disordering of IrO6 octahedra along the c-axis and results in a local symmetry breaking (structural transition). The antiferromagnetic order (that gives rise to the magnon dispersion) in Sr3Ir2O7 stems from the interlayer exchange coupling Jc between Ir atoms mediated through apical oxygens.30 A strong IrO6 tilting could suppress Jc and result in the disappearance of magnon excitation. In principle, full suppression of the magnon peak also could arise from other effects, such as staggered distortions where the intralayer interactions change differently in two neighboring layers. Spin-phonon coupling mediated by single-ion anisotropy also can induce a small in the spin gap, although the phonon renormalization after magnetic transition in Sr3Ir2O7 appears weaker compared with that in other 5d compounds, such as Cd2Os2O7.34,35,36,37,38 A future comprehensive theoretical investigation of these effects is greatly needed. Finally, we summarize our findings in a phase diagram shown in Fig. 5. After the magnetic transition at 14.4 GPa and 10 K, Sr3Ir2O7 is still an insulator. It remains challenging to characterize the magnetic structure and the underlying mechanism of phase transition. Based on similar phenomena observed in doping experiments, the insulating phase above 14.4 GPa could be paramagnetic, spin frustrated,31 or even a totally new quantum state that has never been reported. Sr3Ir2O7 at ambient conditions is located approximately at the center of the phase diagram plotted as functions of U/t and ξ/t.9 When pressure increases, both U/t and ξ/t could decrease, and the system could eventually reach the metallic region. We have reported previously an insulator-metal transition in Sr3Ir2O7 around 55–59.5 GPa,28 which is consistent with other studies.26,29 According to Fig. 5, the pathway for such a pressure evolution could possibly pass though the axion insulator phase3 before it reaches the metallic regime. Therefore, it is not impossible that the insulating phase discovered here might be an axion state. We speculate that applying pressure could be a promising way to search for various predicted phases in 5d spin-orbit materials.9 In summary, we have revealed an interesting magnetic transition that concurs with a structural transition around 14.4 GPa by Raman measurements on Sr3Ir2O7. We attribute the origin of these transitions to pressure-enhanced rotation and tilt angles of IrO6 octahedra. The magnetic transition is shown to decouple from insulator-metal transition. The absence of usual correlation between magnetic and insulating phases in Sr3Ir2O7 is similar to that in Sr2IrO4.39,40 Our high-pressure study together with previous discoveries manifests the critical role of lattice frustration in determining the ground state properties of Sr3Ir2O7, and maybe generically of 5d materials. Our work calls for more theoretical studies to unravel the interplay of intertwined degrees of freedom in spin-orbit systems and the exact mechanism of their phase transitions. ## Methods ### Sample synthesis Sr3Ir2O7 single crystal was grown from flux method. High-purity SrCO3, IrO2 and SrCl2·6H2O powders were mixed together and placed in platinum crucible. The molar ratios of the source materials was 2:1:20. The crucible was heated to 1573 K and dwelt for 10 h and then slowly cooled down to room temperature. After that the single crystals were separated from the flux by washing with deionized water. The obtained single crystal had typical dimensions 0.8 × 0.8 × 0.3 mm3. The experimental XRD pattern of Sr3Ir2O7 and calculated pattern based on the standard ICSD (075587) date are shown in the Supplementary Material in Fig. S4a. This result indicated that our sample is tetragonal phase (space group I4/mmm) at ambient condition and has good quality. ### High-pressure Raman measurements Raman spectra were collected on a single-crystal Sr3Ir2O7 at beamline 22-IR-1 of the National Synchrotron Light Source II, Brookhaven National laboratory. The sample was loaded inside a symmetric-type diamond anvil cell (DAC) and the DAC was placed in a microscopy cryostat system (Cryo Industries of America, Inc.). A pair of ultralow fluorescence type II diamonds with culet size ~300 μm were used. A Spectra-Physics Excelsior solid-state laser with a wavelength of 532 nm was used in the Raman measurement. Potassium bromide (KBr) was used as the pressure transmitting medium, and the pressure inside the cell was determined by the shift of the ruby fluorescence line. The laser power was less than 1 mW. In our measurements, 300 grooves/mm grating and ~1–3 μm beam spot were applied. ### High-pressure transport measurements Electrical resistance was measured with a standard four-probe-electrode circuit on a single-crystal Sr3Ir2O7. A T301 stainless steel gasket with cubic boron nitride/epoxy mixture powder inserts was used. Si oil was used as a pressure medium and the pressure was determined using ruby fluorescence technique. Four thin gold probes were attached to the samples with silver glue to measure the resistance. ### High-pressure synchrotron diffraction measurements The in situ high-pressure XRD measurements were carried out on a single-crystal Sr3Ir2O7 at beamline 16-BM-D of the Advanced Photon Source (APS), Argonne National Laboratory. A symmetric type DAC with culet size of 300 μm was used for the measurement. Neon was used as a pressure medium and the pressure was determined using ruby fluorescence technique. The incident monochromatic x-ray beam energy was set to 30 keV (λ = 0.4133 Ǻ). Diffraction patterns were recorded on a MAR345 image plate. We aligned the sample to the vertical-rotation axis and collected diffraction patterns in a step-scan method (1.5 s/step) with 1.0° step over the range from −20° to 20° up to 33.2 GPa, similar to the “rotation method” used in conventional single crystal crystallography. ### Model Hamiltonian and spin-wave theory We describe the pressure evolution of magnon excitation using the spin model reported in the ref. 22 Below we review the interaction terms in the model, which includes both intralayer (Hab) and interlayer (Hc) Hamiltonians: $$H_{ab} =\sum \limits_{ < i,j > } \left[ JS_{i}{\cdot}S_{j} + \Gamma S_{i}^{z}S_{j}^{z} + D(S_{i}^{x}S_{j}^{y} - S_{i}^{y}S_{j}^{x}) \right] + \sum \limits_{ < < i,j > > } J_{2}S_{i}{\cdot}S_{j} + \sum \limits_{ < < < i,j > > > } J_{3}S_{i}{\cdot}S_{j}$$ $$H_c = \sum \limits_i \left[ {J_cS_i{\cdot}S_{i + z} + \Gamma _cS_i^{z}S_{i + z}^{z} + D_c(S_i^{x}S_{i + z}^y - S_i^{y}S_{i + z}^x)} \right] + \sum \limits_{ < i,j > } J_{2c}S_i{\cdot}S_{j + z}$$ where <i,j>, <<i,j>>, and <<< i,j >>> denote the first, second, and third nearest neighbors within the a-b plane. J, J2, and J3 represent the isotropic coupling constants. The anisotropic coupling term Γ stems from Hund’s exchange interaction and staggered rotations of octahedra. The latter also results in a Dzyaloshinsky-Moriya (DM) interaction,22 characterized by the constant D. For the nearest-interlayer interactions, Jc, Γc, and Dc were adopted for the similar coupling constants along the c-axis, while J2c stands for the next-nearest-neighbor interlayer coupling. All these isotropic and anisotropic exchange coupling constants (except for the long-range interactions J2, J3, and J2c) can be expressed in terms of the three microscopic parameters: the IrO6 rotation angle α, the effective tetragonal distortion θ, and the ratio of Hund’s coupling to onsite Coulomb interaction η = (JH/U). Here, we use the parameters in the reference for ambient pressure. We also assume J2c = 0.25 Jc and keep J2 = 11.9 meV, J3 = 14.6 meV, and η as constants.22 Within linear spin-wave theory (applicable in the antiferromagnetic phase), the single-magnon excitation energy in the Brillouin zone center (i.e., that observed by Raman measurements) is expressed by $$\omega = 1/2\sqrt {\left( {4\Gamma + \Gamma _c} \right)\left( {8J + 2J_c + 4\Gamma + \Gamma _c} \right) - \left( {4D + D_c} \right)^2}$$. Our calculations reveal that the major influence of pressure is to reduce the anisotropic coupling Γ, e.g., from 4.4 meV at ambient pressure to 2.0 meV at around 20 GPa. Also, both Jc and Dc are enhanced, while other coupling constants are only slightly affected. We notice that tilting of the IrO6 octahedra along the c-axis can reduce the interlayer coupling Jc and thus soften the magnon energy, although overall Jc is enhanced due to a pressure-enhanced bandwidth. ## Data availability The data that support the findings of this study are available from the corresponding author upon reasonable request. ## References 1. 1. Wang, F. & Senthil, T. Twisted Hubbard model for Sr2IrO4: magnetism and possible high temperature superconductivity. Phys. Rev. Lett. 106, 136402 (2011). 2. 2. Kitagawa, K. et al. A spin-orbital-entangled quantum liquid on a honeycomb lattice. Nature 554, 341–345 (2018). 3. 3. Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011). 4. 4. Price, C. C. & Perkins, N. B. Critical properties of the Kitaev-Heisenberg model. Phys. Rev. Lett. 109, 187201 (2012). 5. 5. 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Lett. 114, 147205 (2015). 37. 37. Shinaoka, H., Miyake, T. & Ishibashi, S. Noncollinear magnetism and spin-orbit coupling in 5d pyrochlore oxide Cd2Os2O7. Phys. Rev. Lett. 108, 247204 (2012). 38. 38. Bogdanov, N. A. et al. Magnetic state of pyrochlore Cd2Os2O7 emerging from strong competition of ligand distortions and longer-range crystalline anisotropy. Phys. Rev. Lett. 110, 127206 (2013). 39. 39. Ge, M. et al. Lattice-driven magnetoresistivity and metal-insulator transition in single-layered iridates. Phys. Rev. B 84, 100402 (2011). 40. 40. Haskel, D. et al. Pressure tuning of the spin-orbit coupled ground state in Sr2IrO4. Phys. Rev. Lett. 109, 027204 (2012). ## Acknowledgements This work is supported by National Key R&D Program of China No. 2018YFA0305703. The x-ray diffraction measurements were performed at sectors 16 BM-D of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science user facility operated by Argonne National Laboratory (ANL) supported by the U.S. DOE Award No. DE-AC02-06CH11357. The Raman experiments were performed at beamline 22-IR-1 of the National Synchrotron Light Source II (NSLS-II), Brookhaven National Laboratory, supported by NSF (Cooperative Agreement EAR 1606856, COMPRES) and DOE/NNSA (DE-NA-0002006, CDAC). NSLS-II is supported by the DOE/BES (DE-SC0012704). The electric transport measurements were performed at the Center for High-Pressure Science and Technology Advanced Research. The authors thank S. Tkachev for help with the gas loading at the Advanced Photon Source. Y.D. and H.-k. M. acknowledges the support from DOE-BES under Award No. DE-FG02-99ER45775 and NSFC Grant No. U1530402. This work is also supported by National Key R&D Program of China No. 2016YFA0300604, 2017YFA0302901, and the National Natural Science Foundation of China No. 11774399, 11474330, 91750111, and Science Challenge Project, No. TZ2016001, and the Fundamental Research Funds for the Central Universities, China, No. GK201801009. ## Author information Authors ### Contributions Y.D. and J.Z. designed the project. J.Z., H.D., and Z.L. performed the Raman measurements. J.Z., S.Y., and H.D. carried out the XRD and transport measurements. C.Y., D.Y., and Y.S. synthesized the single crystals. J.C. performed the theoretical calculations. J.Z., Y.D., and J.C. analyzed the data. Y.D. supervised the project. All the authors helped with the project and read and commented on the manuscript. ### Corresponding authors Correspondence to Jianbo Zhang or Jun Chang or Yang Ding. ## 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 Zhang, J., Yan, D., Yesudhas, S. et al. Lattice frustration in spin-orbit Mott insulator Sr3Ir2O7 at high pressure. npj Quantum Mater. 4, 23 (2019). https://doi.org/10.1038/s41535-019-0162-3 • Accepted: • Published: • ### Exciton condensation in bilayer spin-orbit insulator • Hidemaro Suwa • , Shang-Shun Zhang • & Cristian D. Batista Physical Review Research (2021) • ### The phase multiformity and domain structure of Sr3Ir2O7 • Huaixiang Wang • , Xudong Shen • , Weipeng Wang • , Yifan Ding • , Yu Ji • , Junkai Yang • , Xi Shen • , Yuan Yao • , Youwen Long • & Richeng Yu Journal of Physics and Chemistry of Solids (2021) • ### Physical Properties of Molecules and Condensed Materials Governed by Onsite Repulsion, Spin-Orbit Coupling and Polarizability of Their Constituent Atoms • Paul A. Maggard • , Xiyue Cheng • , Shuiquan Deng • & Myung-Hwan Whangbo Molecules (2020) • ### Anisotropic lattice compression and pressure-induced electronic phase transitions in Sr2IrO4 • K. Samanta • , R. Tartaglia • , U. F. Kaneko • , N. M. Souza-Neto Physical Review B (2020) • ### Possible Quantum Paramagnetism in Compressed Sr2IrO4 • , G. Fabbris • , J. H. Kim • , L. S. I. Veiga • , J. R. L. Mardegan • , C. A. Escanhoela • , S. Chikara • , V. Struzhkin • , T. Senthil • , B. J. Kim • , G. Cao • & J.-W. Kim Physical Review Letters (2020)	2021-04-12 02:00:44	{"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": 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": 0.6474462747573853, "perplexity": 3821.699732136032}, "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/1618038065903.7/warc/CC-MAIN-20210411233715-20210412023715-00601.warc.gz"}
http://mathhelpforum.com/latex-help/42396-test.html	Test Hi, you'll have to learn how to calculate square roots with continued fractions. Since formatting continued fractions is a nightmare of parentheses, I'll refer you to Continued Fraction -- from Wolfram MathWorld and ask you to read equation (3), which gives the basic form of the continued fractions we'll need and the shortcut form for this (4) so I don't scratch my eyeballs out trying to format things. Also see (11) for the finite version of this. Another bit of slightly confusing notation, [x] will mean the greatest integer less than x. This shouldn't be confused with our continued fraction notation, since that will always have more than one term. Assume D is not a perfect square. To find the continued fraction expression of \sqrt{D}, we first set a_{0}=[\sqrt{D}]. This is a very crude approximation to \sqrt{D}. At this point we have \sqrt{D}=a_{0}+(\sqrt{D}-a_{0})=a_{0}+\frac{1}{\frac{1}{\sqrt{D}-a_{0}}} We apply the same procedure to \frac{1}{\sqrt{D}-a_{0}}=\frac{\sqrt{D}+a_{0}}{D-a_{0}^{2}} and get a_{1}=[\frac{\sqrt{D}+a_{0}}{D-a_{0}^{2}}] Now we have \sqrt{D}=a_{0}+\frac{1}{a_{1}+(\frac{\sqrt{D}+a_{0 }}{D-a_{0}^{2}}-a_{1})} Now a_{2}=[\frac{1}{\frac{\sqrt{D}+a_{0}}{D-a_{0}^{2}}-a_{1}}]. Continue to get the rest of the a's. Eventually you'll get something that repeats like \sqrt{D}=[a_{0}, a_{1}, \ldots, a_{k}, a_{0}+\sqrt{D}] Heres where you stop. If k is odd find integers x and y where x/y=[a_{0}, a_{1}, \ldots, a_{k}]. These are your minimal solutions to Pell's equation. If k is even, you do something similar, I'm not positive exactly what, sorry. I'll hopefully come back tomorrow with an answer. An example: D=14 a_{0}=[\sqrt{14}]=3 so \sqrt{14}=3+\frac{1}{\frac{1}{\sqrt{14}-3}} a_{1}=[\frac{1}{\sqrt{14}-3}]=[\frac{\sqrt{14}+3}{5}]=1 So \sqrt{14}=3+\frac{1}{1+(\frac{\sqrt{14}+3}{5}-1)}=3+\frac{1}{1+\frac{1}{\frac{1}{\frac{\sqrt{14} +3}{5}-1}}} a_{2}=[\frac{1}{\frac{\sqrt{14}+3}{5}-1}]=[\frac{5}{\sqrt{14}-2}]=[\frac{\sqrt{14}+2}{2}]=2 So \sqrt{14}=3+\frac{1}{1+\frac{1}{2+(\frac{\sqrt{14} +2}{2}-2)}} ok I'm stopping here. Go a couple more steps and you'll get \sqrt{14}=[3,1,2,1,3+\sqrt{14}], so we've started to repeat. Now find [3,1,2,1]=3+\frac{1}{1+\frac{1}{2+\frac{1}{1}}}=15/4, so x=15 and y=4 are the minimal solutions in this case.	2014-09-22 21:18:19	{"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": 0.9146742820739746, "perplexity": 940.1126138047382}, "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-2014-41/segments/1410657137190.70/warc/CC-MAIN-20140914011217-00000-ip-10-234-18-248.ec2.internal.warc.gz"}
https://www.csauthors.net/vinay-jethava/	Vinay Jethava According to our database1, Vinay Jethava authored at least 15 papers between 2009 and 2019. Collaborative distances: Book In proceedings Article PhD thesis Other Bibliography 2019 On weighted uncertainty sampling in active learning. CoRR, 2019 2017 GANs for LIFE: Generative Adversarial Networks for Likelihood Free Inference. CoRR, 2017 2015 Finding Dense Subgraphs in Relational Graphs. Proceedings of the Machine Learning and Knowledge Discovery in Databases, 2015 2014 Integrative Analysis of Dynamic Networks. PhD thesis, 2014 Global graph kernels using geometric embeddings. Proceedings of the 31th International Conference on Machine Learning, 2014 2013 Lovász ϑ function, SVMs and finding dense subgraphs. J. Mach. Learn. Res., 2013 Lovasz ϑ, SVMs and applications. Proceedings of the 2013 IEEE Information Theory Workshop, 2013 DLOREAN: Dynamic Location-Aware Reconstruction of Multiway Networks. Proceedings of the 13th IEEE International Conference on Data Mining Workshops, 2013 Entity disambiguation in anonymized graphs using graph kernels. Proceedings of the 22nd ACM International Conference on Information and Knowledge Management, 2013 2012 "The Lovasz $\theta$ function, SVMs and finding large dense subgraphs". Proceedings of the Advances in Neural Information Processing Systems 25: 26th Annual Conference on Neural Information Processing Systems 2012. Proceedings of a meeting held December 3-6, 2012 Intent-aware temporal query modeling for keyword suggestion. Proceedings of the 5th Ph.D. Workshop on Information and Knowledge Management, 2012 2011 NETGEM: Network Embedded Temporal GEnerative Model for gene expression data. BMC Bioinform., 2011 Scalable multi-dimensional user intent identification using tree structured distributions. Proceedings of the Proceeding of the 34th International ACM SIGIR Conference on Research and Development in Information Retrieval, 2011 2009 Randomized Algorithms for Large scale SVMs CoRR, 2009 Extension of Path Probability Method to Approximate Inference over Time CoRR, 2009	2020-07-07 03:31:45	{"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": 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": 0.3941652476787567, "perplexity": 9759.951752966685}, "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/1593655891640.22/warc/CC-MAIN-20200707013816-20200707043816-00099.warc.gz"}
http://gallery2020.vupinteractive.com/blue-colour-brgrwxk/690caf-symbolab-definite-integral	30 Dec ### symbolab definite integral $\mathrm {If}\:f\left (x\right)=-f\left (-x\right)\Rightarrow\int_ {-a}^af\left (x\right)dx=0$. First, we must identify a section within the integral with a new variable (let's call it. When evaluating definite integrals for practice, you may use your calculator to inspect the answers. - [Voiceover] So we wanna evaluate the definite integral from negative one to negative two of 16 minus x to the third over x to the third dx. Symbolab Integrals Cheat Sheet. If one or both integration bounds a and b are not numeric, int assumes that a <= b unless you explicitly specify otherwise. ... Symbolab. Pre-Álgebra. Each part of the symbol makes sense. The definite integral f(x) from, say, x=a to x= b, is defined as the signed area between f(x) and the x-axis from the point x = a to the point x = b. You can learn how to calculate definite integrals by using our free definite integral calculator. This means . Conic Sections. Functions. Type in any integral to get the solution, free steps and graph. Free definite integral calculator - solve definite integrals with all the steps. We can read the integral sign as a summation, so that we get "add up an infinite number of infinitely skinny rectangles, from x=1 to x=2, with height x^2 times width dx." Definite Integral Calculator Symbolab Solutions My Notebook Practice Blog English New! Example: A definite integral of the function f (x) on the interval [a; b] is the limit of integral sums when the diameter of the partitioning tends to zero if it exists independently of the partition and choice of points inside the elementary segments.. Type in any integral to get the solution, free steps and grap Because integral psychotherapy is a wide philosophy, anyone may opt to practice iteven without formal mental wellness training. Loading... Definite Integral Calculator. Advanced Math Solutions – Integral Calculator, common functions. Thanks to all of you who support me on Patreon. Partial fractions decomposition is the opposite of adding fractions, we are trying to break a rational expression... High School Math Solutions – Polynomial Long Division Calculator. this website uses cookies to ensure you get the best experience. i know for a fact that the answer should be 9/2 because i solved for the horizontal strips. Definite integral could be represented as the signed area in the XY-plane bounded by the function graph as shown on the image below. 2. Make your first steps in evaluating definite integrals, armed with the Fundamental theorem of calculus. Help De‫צ‬nite Integral Calculator Solve de‫ﺡ‬nite integrals step by step Enter a topic Algebra Matrices & Vectors Functions & Graphing Like full pad » x 2 x d dx (☐)' x log √☐ √☐ ≤ ≥ ∂ ∂x ∫ ∫ lim ∑ ∞ ( f g) x When you use IgnoreAnalyticConstraints, int applies these rules: Find more Mathematics widgets in Wolfram\|Alpha. You da real mvps! The calculator will approximate the integral using the trapezoidal rule, with steps shown. Detailed step by step solutions to your Improper integrals problems online with our math solver and calculator. setting up the definite integral. Your private math tutor, solves any math problem with steps! Advanced Math Solutions – Integral Calculator, integration by parts, Part II. Log InorSign Up. 3. I'm trying to evaluate the following definite integral: $\int_{0}^{1} \frac{x}{\sqrt{x+1}} dx$ Well I put the integral on symbolab to know its value ($4/3$) However when I tried to calculate the To create your new password, just click the link in the email we sent you. Submit Assignment Start Over Back. Definite and Improper Integral Calculator. ... Related Symbolab blog posts. Definite Integrals Calculator. Matrices & Vectors * Matrix Add/Subtract com, the most comprehensive source for safe, trusted, and spyware-free Symbolab - Math solver. Example: Proper and improper integrals. Submit Assignment Start Over Back. Input a function, the integration variable and our math software will give you the value of the integral covering the selected interval (between the lower limit and the upper limit). Advanced Math Solutions – Integral Calculator, the complete guide. Definite Integrals Rules. Free calculus calculator - calculate limits, integrals, derivatives and series step-by-step This website uses cookies to ensure you get the best experience. Enter your function in line 2 below... 1. f x = xsinx. Message received. Definite Integrals Calculator. Example: A definite integral of the function f (x) on the interval [a; b] is the limit of integral sums when the diameter of the partitioning tends to zero if it exists independently of the partition and choice of points inside the elementary segments.. the task is to set up the definite integral. ∫ 1 2 x 2 d x. change password email address. Free definite integral calculator - solve definite integrals with all the steps. You can also check your answers! This website and its content is subject to our Terms and Conditions. Integral dx Use latex commands: * is multiplication oo is $\infty$ pi is $\pi$ x^2 is x 2 sqrt(x) is $\sqrt{x}$ sqrt[3](x) is $\sqrt[3]{x}$ (a+b)/(c+d) is $\frac{a+b}{c+d}$ Powered by Sympy. setting up the definite integral. = limx → b − ( F ( x)) − limx → a + ( F ( x)) Odd function. Interactive graphs/plots help visualize and better understand the functions. Definite Integral Calculator Added Aug 1, 2010 by evanwegley in Mathematics This widget calculates the definite integral of a single-variable function given certain limits of integration. Summary. Line Equations Functions Arithmetic & Comp. The calculus integrals of function f(x) represents the area under the curve from x = a to x = b. U-substitution in definite integrals is just like substitution in indefinite integrals except that, since the variable is changed, the limits of integration must be changed as well. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. i have been studying this problem. … It is used to transform the integral of a product of functions into an integral that is easier to compute. The integral calculator allows you to solve any integral problems such as indefinite, definite and multiple integrals with all the steps. This website uses cookies to ensure you get the best experience. Show More Show Less. ... Related Symbolab blog posts. By … View integral x^2(2x^3+3)^3dx - Indefinite Integral Calculator - Symbolab from MATH 122 at Oakland University. For more about how to use the Integral Calculator, go to "Help" or take a look at the examples. the solution shown in the picture is from symbolab. Keywords Learn how to evaluate the integral of a function. Our mission is to provide a free, world-class education to anyone, anywhere. =ln() ∫ \| =√. \int x\left (x^2-3\right)dx ∫ x(x2 −3)dx by applying integration by substitution method (also called U-Substitution). Both types of integrals are tied together by the fundamental theorem of calculus. ∫ is the Integral Symbol and 2x is the function we want to integrate. Definite Integral Calculator. Type in any integral to get the solution, free steps and graph This website uses cookies to ensure you get the best experience. Solve definite integrals with us! By using this website, you agree to our Cookie Policy. The Integral Calculator supports definite and indefinite integrals (antiderivatives) as well as integrating functions with many variables. Improper integrals Calculator online with solution and steps. Type in any integral to get the solution, free steps and graph This website uses cookies to ensure you get the best experience. Example: Proper and improper integrals. Adjust the lower and upper bound of the integral by … You can check your own solution or get rid of unnecessary labour-intensive calculations and to confide in a high-tech automated machine when solving the definite integral with us. All online services are accessible even for unregistered users and absolutely free of charge. $\int_a^bf\left (x\right)dx=F\left (b\right)-F\left (a\right)$. Solved exercises of Improper integrals. This calculator is convenient to use and accessible from any device, and the results of calculations of integrals and solution steps can be easily copied to the clipboard. Definite integrals calculator. This states that if is continuous on and is its continuous indefinite integral, then . An absolutely free online step-by-step definite and indefinite integrals solver. I use for a starter or plenary or occasionally a homework. This website uses cookies to ensure you get the best experience. Advanced Math Solutions – Integral Calculator, trigonometric substitution. ∫ x ( x 2 − 3) d x. Common Integrals: ∫−1 =ln() ∫ �� . The definite integral has both the start value & end value. Advanced Math Solutions – Integral Calculator, advanced trigonometric functions. Derivatives Derivative Applications Limits Integrals Integral Applications Riemann Sum Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series. Advanced Math Solutions – Integral Calculator, integration by … The dx shows the direction alon the x-axis & dy shows the direction along the y-axis. 3. If you have a table of values, see trapezoidal rule calculator for a table. :) https://www.patreon.com/patrickjmt !! Definite Integrals . ... Related Symbolab blog posts. Summary. In general, you can skip the multiplication sign, so 5 x is equivalent to 5 ⋅ x. Pre-Álgebra. The definite integral of a non-negative function is always greater than or equal to zero: $${\large\int\limits_a^b\normalsize} {f\left( x \right)dx} \ge 0$$ if $$f\left( x \right) \ge 0 \text{ in }\left[ {a,b} \right].$$ The definite integral of a non-positive function is always less than or equal to zero: The calculator will evaluate the definite (i.e. ∫sin() =−cos() ∫cos() =sin() Trigonometric Integrals: Definite Integral Calculator - Symbolab (2 days ago) Free definite integral calculator - solve definite integrals with all the steps. In this integral equation, dx is the differential of Variable x. Posted by 4 days ago. By using this website, you agree to our Cookie Policy. Orden (jerarquía) de operaciones Factores y números primos Fracciones Aritmética Decimales Exponentes y radicales Módulo Aritmética con notación científica. Show Instructions. - [Voiceover] So we wanna evaluate the definite integral from negative one to negative two of 16 minus x to the third over x to the third dx. Definite Integral Calculator. Polynomial long division is very similar to numerical long division where you first divide the large part of the... partial\:fractions\:\int_{0}^{1} \frac{32}{x^{2}-64}dx, substitution\:\int\frac{e^{x}}{e^{x}+e^{-x}}dx,\:u=e^{x}. … We can solve the integral. To create your new password, just click the link in the email we sent you. Symbolab: equation search and math solver - solves algebra, trigonometry and calculus problems step by step This website uses cookies to ensure you get the best experience. Free definite integral calculator - solve definite integrals with all the steps. type in any integral to get the solution, free steps and graph. Definite Integrals. In general, you can skip parentheses, but be very careful: e^3x is e 3 x, and e^ (3x) is e 3 x. The definite integral is denoted by a f(x) d(x). Advanced Math Solutions – Integral Calculator, integration by parts Integration by parts is essentially the reverse of the product rule. Free definite integral calculator - solve definite integrals with all the steps. Definite Integrals . Message received. The definite integral of the function $$f\left( x \right)$$ over the interval $$\left[ {a,b} \right]$$ is defined as the limit of the integral sum (Riemann sums) as the maximum length … Functions 3D Plotter is an application to drawing functions of several variables and surface in the space R3 and to calculate indefinite integrals or definite integrals. Advanced Math Solutions – Integral Calculator, the complete guide. 2 ∫ b a f x dx. ... Symbolab. Related Symbolab blog posts Advanced Math Solutions – Ordinary Differential Equations Calculator, Bernoulli ODE Last post, we learned about separable differential equations. By using this website, you agree to our Cookie Policy. u = sin x. u=\sin {x} u = sinx to find limits of integration in terms of. There is also the issue that the symbols make more sense in the definite integral. Log InorSign Up. Thanks for the feedback. Definite Integral Calculator computes definite integral of a function over an interval using numerical integration. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Thanks for the feedback. A free graphing calculator - graph function, examine intersection points, find maximum and minimum and much more This calculus video tutorial explains how to evaluate definite integrals using u-substitution. ∫ = . Given the condition mentioned above, consider the function F\displaystyle{F}F(upper-case "F") defined as: (Note in the integral we have an upper limit of x\displaystyle{x}x, and we are integrating with respect to variable t\displaystyle{t}t.) The first Fundamental Theorem states that: Proof 2. In the previous post we covered integrals involving powers of sine and cosine, we now continue with integrals involving... partial\:fractions\:\int_{0}^{1} \frac{32}{x^{2}-64}dx, substitution\:\int\frac{e^{x}}{e^{x}+e^{-x}}dx,\:u=e^{x}. Related Symbolab blog posts Advanced Math Solutions - Integral Calculator, integration by parts, Part II In the previous post we covered integration by parts ; Free definite integral calculator - solve definite integrals with all the steps. Indefinite Integral Calculator - Symbolab Solutions My Notebook Practice Blog English New! Matrices & … Definite integrals calculator. Tes Global Ltd is registered in England (Company No 02017289) with its registered office at 26 Red Lion Square London WC1R 4HQ. Free antiderivative calculator - solve integrals with all the steps. The definite integral of from to , denoted , is defined to be the signed area between and the axis, from to . 2 ∫ b a f x dx. Homework later than 1 class period won't be accepted. Input a function, the integration variable and our math software will give you the value of the integral covering the selected interval (between the lower limit and the upper limit). Now at first this might seem daunting, I have this rational expression, I have xs in the numerators and xs in the denominators, but we just have to remember, we just have to do some algebraic manipulation, and this is going to seem a lot more attractable. Advanced Math Solutions – Integral Calculator, integration by parts, Part II. In the previous posts we covered substitution, but standard substitution is not always enough. 1. ∫abf ( x) dx = F ( b) − F ( a) $=\lim_ {x\to b-}\left (F\left (x\right)\right)-\lim_ {x\to a+}\left (F\left (x\right)\right)$. Type in any integral to get the solution, free steps and graph - [Instructor] What we're gonna do in this video is introduce ourselves to the notion of a definite integral and with indefinite integrals and derivatives this is really one of the pillars of calculus and as we'll see, they're all related and we'll see that more and more in future videos and we'll also get a better appreciation for even where the notation of a definite integral comes from. Definite Integral Calculator. Related Symbolab blog posts Advanced Math Solutions – Limits Calculator, Infinite limits In the previous post we covered substitution, where the limit is simply the function value at the point. Orden (jerarquía) de operaciones Factores y números primos Fracciones Aritmética Decimales Exponentes y radicales Módulo Aritmética con notación científica. Symbolab – Math solver Pro. ... Related Symbolab blog posts. Show More Show Less. Khan Academy is a 501(c)(3) nonprofit organization. Free definite integral calculator - solve definite integrals with all the steps. Definite Integral Calculator Added Aug 1, 2010 by evanwegley in Mathematics This widget calculates the definite integral of a single-variable function given certain limits of integration. u. u u, instead of. u. u u ), which when substituted makes the integral easier. Free definite integral calculator - solve definite integrals with all the steps. It highlights that the Integration's variable is x. ... * Integrals (definite, indefinite, multiple) * Derivatives * Partial derivatives * Series * ODE * Laplace Transform * Inverse Laplace Transform. This website uses cookies to ensure you get the best experience. Now at first this might seem daunting, I have this rational expression, I have xs in the numerators and xs in the denominators, but we just have to remember, we just have to do some algebraic manipulation, and this is going to seem a lot more attractable. Type in any integral to get the solution, free steps and graph. Get the free "Triple Integral Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. with bounds) integral, including improper, with steps shown. The usual stuff, solve the problems to discover the punchline to the joke. Type in any integral to get the solution, steps and graph This website uses cookies to ensure you get the best experience. $1 per month helps!! History! Algorithms. If you don’t change the limits of integration, then you’ll need to back-substitute for the original variable at the en High School Math Solutions – Partial Fractions Calculator. Definite Integrals. partial fractions \int_{0}^{1} \frac{32}{x^{2}-64}dx, Please try again using a different payment method. i need help. Advanced Math Solutions – Integral Calculator, integration by parts, Part II. In general, you can skip the multiplication sign, so 5x is equivalent to 5x. Show Instructions. Close. The Integral Calculator supports definite and indefinite integrals (antiderivatives) as well as integrating functions with many variables. Free Series Integral Test Calculator - Check convergence of series using the integral test step-by-step This website uses cookies to ensure you get the best experience. Also, be careful when you write fractions: 1/x^2 ln (x) is 1 x 2 ln ⁡ ( … First of all I would like to start off by asking why do they have different change of variable formulas for definite integrals than indefinite...why cant we just integrate using U substitution as we normally do in indefinite integral and then sub the original U value back and use that integrand for definite integral?. It is important to note that both the definite and indefinite integrals are interlinked by … Integrals involving... Advanced Math Solutions – Integral Calculator, advanced trigonometric functions, Part II. Please try again using a different payment method. Funcions 3D plotter calculates the analytic and numerical integral and too calculates partial derivatives with respect to x and y for 2 variabled functions. Definite Integral Boundaries. For definite integrals, int restricts the integration variable var to the specified integration interval. Enter your function in line 2 below... 1. f x = xsinx. Type in any integral to get the solution, free steps and graph. x. x x. U-substitution in definite integrals is just like substitution in indefinite integrals except that, since the variable is changed, the limits of integration must be changed as well. Free Series Comparison Test Calculator - Check convergence of series using the comparison test step-by-step send reset link. Of charge n't be accepted product rule f ( x ) ) Odd function in line 2 below... f! With steps shown ) ( 3 ) d x any Math problem with steps shown the along!... advanced Math Solutions – integral Calculator supports definite and indefinite integrals solver Blog, Wordpress, Blogger, iGoogle! With a new variable ( let 's call it i solved for the horizontal strips ).! Let 's call it calculates partial derivatives with respect to x = b Terms and Conditions by function... Integrals solver problems online with our Math solver to use the integral the... Integral x^2 ( 2x^3+3 ) ^3dx - indefinite integral, including improper, with steps shown the task to. Content is subject to our Cookie Policy of from to, denoted, is defined be. More Symbolab – Math solver improper integrals problems online with our Math solver functions into an integral that is to. '' or take a look at the examples a new variable ( let call... Anyone, anywhere both types of integrals are tied together by the function graph as shown on the below... ( antiderivatives ) as well as integrating functions with many variables and absolutely free of charge function, intersection... Calculate definite integrals with all the steps, see trapezoidal rule, with steps shown must identify a section the. Symbolab ( 2 days ago ) free definite integral Calculator, the complete guide Calculator a! A section within the integral Symbol and 2x is the differential of variable x, examine intersection points find. Wo n't be symbolab definite integral more about how to use the integral of from to, denoted is. Dx ∫ x ( x ) ) − limx → a + ( f ( x ) Odd... Free, world-class education to anyone, anywhere it highlights that the 's! Function over an interval using numerical integration, go to help '' or take a look at the.! Dx=F\Left ( b\right ) -F\left ( a\right )$ in line 2 below... 1. f x b. C ) ( 3 ) d ( x ) ) Odd function wellness.. 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Opt to Practice iteven without formal mental wellness training by substitution method ( also called )!	2021-04-21 11:31:56	{"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": 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": 0.8556149005889893, "perplexity": 1328.6747538342684}, "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/1618039536858.83/warc/CC-MAIN-20210421100029-20210421130029-00351.warc.gz"}
https://proofwiki.org/wiki/Closed_Set_of_Uncountable_Finite_Complement_Topology_is_not_G-Delta	# Closed Set of Uncountable Finite Complement Topology is not G-Delta ## Theorem Let $T = \struct {S, \tau}$ be a finite complement topology on an uncountable set $S$. Let $V \in \tau$ be a closed set of $T$. Then $V$ is not a $G_\delta$ set. ## Proof Let $V$ be a closed set of $T$. Aiming for a contradiction, suppose $V$ is $G_\delta$ set. $S \setminus V$ is an $F_\sigma$ set. By definition of closed set, $S \setminus V$ is an open set of $T$. $S \setminus V$ is not an $F_\sigma$ set. It follows by Proof by Contradiction that $V$ is not a $G_\delta$ set. $\blacksquare$	2021-06-20 15:45:42	{"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": 0.944719672203064, "perplexity": 129.42376928896687}, "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/1623488249738.50/warc/CC-MAIN-20210620144819-20210620174819-00199.warc.gz"}
http://golem.ph.utexas.edu/category/2012/05/integrating_against_the_euler.html	## May 17, 2012 ### Integrating Against the Euler Characteristic #### Posted by Tom Leinster The Euler characteristic of topological spaces behaves something like a measure. For example, under suitable hypotheses, $\chi(X \cup Y) = \chi(X) + \chi(Y) - \chi(X \cap Y).$ One of the main things you can do with a measure is integrate with respect to it — or ‘against’ it, as they say. So: what happens if you try to integrate against the Euler characteristic? I don’t completely understand the answer myself, but I’ll explain as well as I can. Along the way, we’ll see: • how this train of thought helps us to define Euler characteristic • how it also leads to the notion of curvature. ### Simple functions on the line Let’s begin in one dimension. Our aim is to define the integral $\int f \; d \chi$ for suitable functions $f\colon \mathbf{R} \to \mathbf{R}$. Whatever we think Euler characteristic is, the Euler characteristic of a compact, nonempty interval $A$ should be $1$. So, writing $I_A$ for the indicator function (or characteristic function) of $A$, we should have $\int I_A \;d \chi = 1.$ Since integration is supposed to be linear, this tells us how we must integrate any finite linear combination of indicator functions of compact nonempy intervals. I’ll call these simple functions on $\mathbf{R}$. So, for a simple function $f = \sum_{r = 1}^k c_r I_{A_r},$ where each $A_r$ is a compact nonempty interval, we should have $\int f \;d \chi = \sum_{r = 1}^k c_r.$ Do I hear you sigh? If you’ve seen this kind of thing before, you’ll recognize the standard problem: the definition isn’t obviously consistent, since $f$ can be expressed as a combination of indicator functions in multiple ways, and maybe these give multiple different values for the integral. In that case, you’ll also know that the standard solution, involving common refinements, is pretty tedious work. Happily, we can avoid it. Here’s how. For a simple function $f$, put $J(f) = \sum_{x \in \mathbf{R}} (f(x) - f(x-))$ where $f(x-)$ means $lim_{\varepsilon \to 0+} f(x - \varepsilon)$. This quantity $J(f)$ is well-defined, linear in $f$, and takes value $1$ when $f$ is the indicator function of a compact nonempty interval. So, $J(f) = \sum c_r$ whenever $f = \sum c_r I_{A_r}$. We therefore put $\int f \;d \chi = J(f),$ and the consistency problem evaporates. Let’s have some examples. What, for instance, is the integral against the Euler characteristic of this function? The solid and empty circles indicate that $f(a) = f(b) = 0$. So $f = 3I_{(a, b)} = 3I_{[a, b]} - 3I_{[a, a]} - 3I_{[b, b]}$ and $\int f \;d\chi = 3 - 3 - 3 = -3.$ Since $f = 3I_{(a, b)}$, this tells us that we’re treating the Euler characteristic of the open interval $(a, b)$ as $-1$. That might strike you as wrong if you’re used to Euler characteristic being invariant under homotopy equivalence. But as James Propp pointed out long ago, there’s a tension between the requirement that Euler characteristic is homotopy-invariant and the requirement that it behaves like a finitely additive measure. You can’t have both at once. (See also John Baez’s excellent talk on the mysteries of counting.) Here we’re not worrying about homotopy invariance; we’re using what Propp would call ‘Euler measure’. It’s not hard to see that $\int g \;d\chi = 7$, whatever the unlabelled values on the axes might happen to be. I chose the letter $J$ to stand for ‘jump’: $J(f) = \int f \;d\chi$ is the total vertical jump occurring at jump discontinuities from the left. One more example: what is the integral against the Euler characteristic of the following function? Here $h = 3 I_{[a, d]} + 5 I_{[a, b]} + I_{[c, d]},$ so $\int h \;d\chi = 3 + 5 + 1 = 9$ (regardless of the values of $a$, $b$, $c$ and $d$). Alternatively, you can calculate $\int h \;d\chi$ via the formula for $J(f)$, giving $8 + (4 - 3) = 9$ again. ### More general functions on the line Classical measure theory also involves things called ‘simple functions’ (with a different but analogous meaning). There, defining integration for simple functions is just a prelude to defining integration for a larger class of functions. Can we do something similar here, extending our integral to a larger class of functions? We can. Indeed, the formula $\sum_{x \in \mathbf{R}} (f(x) - f(x-))$ for $\int f \;d\chi$ immediately makes sense for more than just the simple functions. But before we charge ahead and generalize, let’s correct the ugly asymmetry you see here. In everything so far, we could equally well have used the formula $\sum_{x \in \mathbf{R}} (f(x) - f(x+)).$ It makes no difference: this is still equal to $\int f \;d\chi$ for simple functions $f$. Of course, it’s just as asymmetric. However, taking the average of the two formulas, we also have $\int f \;d\chi = \frac{1}{2} \sum_{x \in \mathbf{R}} \bigl[-f(x-) + 2f(x) - f(x+)\bigr].$ This is now symmetric, and therefore more likely to give us a useful definition for more general functions $f$. (Maybe you can see a hint of how curvature is going to enter the story: this looks like the expression $f''(x) = \displaystyle\lim_{\varepsilon\to 0} \frac{f(x-\varepsilon) - 2f(x) + f(x+\varepsilon)}{\varepsilon^2}$ for a second derivative, and second derivatives have something to do with curvature.) So: let $f\colon \mathbf{R} \to \mathbf{R}$ be a function such that the limits $f(x-)$ and $f(x+)$ exist for all $x \in \mathbf{R}$, and are equal to $f(x)$ for all but finitely many $x \in \mathbf{R}$. In other words, $f$ is continuous except for a finite number of jump discontinuities. The integral against the Euler characteristic of such a function $f$ is defined by the formula above: $\int f \;d\chi = \frac{1}{2} \sum_{x \in \mathbf{R}} \bigl[ -f(x-) + 2f(x) - f(x+) \bigr].$ Be warned: this has some properties that you might not expect of an integral. For instance, the integral of any continuous function is $0$, the integral of a function that is everywhere strictly positive can be strictly negative, and changing the value of a function at a single point can change the value of the integral. On the other hand, this integral has some interesting properties too, as we’ll see when we get to higher dimensions. ### Simple functions in higher dimensions Let’s now consider functions on $\mathbf{R}^n$, for $n \geq 1$. The role of intervals will be played by convex sets. For brevity, I’ll use ‘convex’ to mean ‘compact, nonempty and convex’. A function $f\colon \mathbf{R}^n \to \mathbf{R}$ is simple if it can be expressed as a finite linear combination of indicator functions of convex sets. Again, we want to ‘define’ $\int f \;d\chi = \sum_{r = 1}^k c_r$ whenever $f = \sum_{r = 1}^k c_r I_{A_r}$ for some convex sets $A_r$, and again we might groan at the prospect of having to do those tedious consistency checks. But once more, the jump functional $J$ comes to the rescue. I’ll explain in the case $n = 2$; the strategy for higher dimensions should be clear. Let $f\colon \mathbf{R}^2 \to \mathbf{R}$ be a simple function. For each $x \in \mathbf{R}$, the function $f(x, -)\colon \mathbf{R} \to \mathbf{R}$ is simple, and so we can define a function $F\colon \mathbf{R} \to \mathbf{R}$ by $F(x) = J(f(x, -))$. This function $F$, too, is simple, so we get a real number $J(F)$. We define $J(f)$ to be this number: $J(f) = J(F)$. Thus, we have defined $J(f)$ for every simple function $f$ on $\mathbf{R}^2$. It is linear in $f$, with $J(1_A) = 1$ whenever $A$ is convex. So just as in the one-dimensional case, $J(f) = \sum c_r$ when $f = \sum c_r I_{A_r}$. This solves the consistency problem, and $\int f \;d\chi = J(f)$. I learned this from Chapter 5 of Klain and Rota’s Introduction to Geometric Probability (source of so many wonderful things). It is remarkably little effort, and is even based on a very standard technique: reducing a multivariable integral to a sequence of single-variable integrals. But it has an immediate nontrivial consequence: the definition of Euler characteristic for a large class of subsets of $\mathbf{R}^n$. Indeed, call a subset $S$ of $\mathbf{R}^n$ polyconvex if it can be expressed as a finite union of convex sets. For example, any picture on a black and white television is polyconvex (assuming that each pixel is convex). And quite simply, we define $\chi(S) = \int I_S \;d\chi.$ You can prove, as Klain and Rota do, that this coincides with the usual definition. ### More general functions in higher dimensions, and curvature The rough idea now is that given a function $f\colon \mathbf{R}^n \to \mathbf{R}$ whose discontinuities are no worse than those of a simple function, we should be able to define $\int f \;d\chi$ by repeating verbatim the definition for simple functions. In the one-dimensional case, we first had to deal with the pesky problem that the formula $J(f) = \sum_{x \in \mathbf{R}} (f(x) - f(x-))$ isn’t symmetric, so probably wouldn’t generalize well. To fix that, we considered the formula $\sum (f(x) - f(x+))$ obtained by reversing the orientation of the line, and then we averaged over the two orientations to get something symmetric. In higher dimensions, the asymmetry problem can no longer be brushed aside with this algebraic flick of the wrist. In fact, some interesting geometry comes in here. This asymmetry issue, which looked like a nuisance distracting us from the main business, turns out to be exactly the reason why integration against the Euler characteristic is closely related to curvature. Again I’ll stick to $n = 2$, leaving higher dimensions to your imagination. In defining $J(f)$ for simple functions $f\colon \mathbf{R}^2 \to \mathbf{R}$, we used the standard coordinate system on $\mathbf{R}^2$. When $f$ is simple, the choice of basis does not affect the value of $J(f)$ (which is always equal to $\sum c_r$, if $f = \sum c_r I_{A_r}$). But for more general functions $f$, it certainly does make a difference. What we should do is consider all ordered orthonormal bases of $\mathbf{R}^2$, calculate $J(f)$ with respect to each basis, and define $\int f \;d\chi$ to be the average. (You can see that this generalizes what we did for $\mathbf{R}^1$: there we took the average of two things, and there are two orthonormal bases of $\mathbf{R}^1$.) I don’t want to make this post any longer by explaining exactly what, for instance, ‘average’ means. Nor will I say much about which functions $f$ this will be a reasonable definition for. Instead, I’ll focus on the geometric interpretation of $\int f \;d\chi$. Let’s think about a function $f\colon \mathbf{R}^2 \to \mathbf{R}$ of the form $g \cdot I_A$, where $g$ is continuous and $A$ is convex. Thus, $f$ is supported on $A$ and continuous everywhere except perhaps on the boundary of $A$, where it might jump in value as it crosses the boundary. I’ve just been reading something that talks about functions `suffering a jump discontinuity’. The suffering of $f$ is limited to $\partial A$. What is $\int f \;d\chi$? How can we understand it? Well, $\int f \;d\chi$ is the average over all orthonormal bases of the quantity “$J(f)$ with respect to that basis”. So, take an orthonormal basis — a coordinate system — and let’s consider $J(f)$. Recall that the definition of $J(f)$ was slicewise. For each $x \in \mathbf{R}$, we take the function $f(x, -)\colon \mathbf{R} \to \mathbf{R}$ and put $F(x) = J(f(x, -))$. In the picture, the value of $x$ shown is in the vertical shadow of $A$, so $F(x)$ is equal to $f(x, y)$. Now since $f$ is continuous, $F$ is too, except that $F$ has a jump discontinuity at each end of the shadow. So, $J(F) = f(x_0, y_0)$. Finally, by definition, $J(f) = J(F)$, and so $J(f) = f(x_0, y_0)$. So in the end, $J(f)$ is something really trivial: $J(f)$ is the value of $f$ at the leftmost point of $A$ where ‘leftmost’ refers to the basis concerned. (I’m assuming for simplicity that the boundary of $A$ is smooth and contains no line segments. This isn’t crucial.) But this isn’t the same as $\int f \;d\chi$. To get that, we have to average over all orthonormal bases of $\mathbf{R}^2$. That is, we slowly rotate our coordinate axes through $360^\circ$, at each moment recording the value of $f$ at the ‘leftmost’ point of $A$ (with respect to the current axes), then taking the mean. As we rotate, that leftmost point works itself around the whole boundary of $A$, never backtracking. And here’s the important thing: It spends more time at some boundary points than others. To see why, consider a convex set like this: For almost all choices of axes, the leftmost point of $A$ will be in one of the red parts of the boundary. Only rarely will it be elsewhere. So as we rotate the axes, the leftmost point with respect to the axes moves quickly over parts of the boundary with low curvature, and lingers where the curvature is high. We’d therefore imagine that $\int f \;d\chi$ would be the integral of $f$ over the boundary of $A$ with respect to some kind of curvature measure on $\partial A$. This turns out to be true. In fact, $\int f \;d\chi = \int_{\partial A} f d\phi$ where $\phi$ is the angle that the tangent makes with some fixed, arbitrarily chosen reference line: This is good, but there’s another way to put it too. When we integrate along a curve, we usually do it with respect to the arclength measure, typically written as $d s$. And the rate of change of the angle $\phi$ per unit arclength is nothing but the classical curvature $\kappa$. That is, $\kappa = d\phi/d s$. Putting this together with the last equation, we get $\int f \;d\chi = \int_{\partial A} f\cdot\kappa \; d s.$ So, the integral against the Euler characteristic of a function of this type is naturally expressed in terms of curvature. You can go further down this track. If you know about intrinsic volumes, you can ask and answer the question: what does it mean to integrate against an intrinsic volume? You’ll see that each intrinsic volume corresponds to a different curvature measure; in $n$ dimensions, we get $n$ different curvature measures on the boundary of a convex set. What I’d like to know is how much of this story is well-known. Curvature measures are extremely well-studied, and connections between curvature and Euler characteristic go back to the Gauss–Bonnet theorem at least. On the other hand, I’ve never heard anyone talking explicitly about integration against the Euler characteristic. Does anyone know where this stuff is written up? Posted at May 17, 2012 4:29 AM UTC TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2528 ### Re: Integrating Against the Euler Characteristic Well, what it makes me think of isn’t terribly new (I don’t suppose), except that the later formulas look like inventing a change-of-variables from the geometric measure theorem $V_{k} (C) = c_{n,k} \int_{W : AGr_k} \chi (C \cap W) d\mu_W$ writing the $k$-homogeneous invariant volume of polyconvex (or very smooth, or …) object $C$, as an integral over an affine Grassmannian. It’s a different change-of-variables that leads to those formulas involving Second Fundamental Forms and all that. Intriguing! I will have to re-read soon. Posted by: Jesse C. McKeown on May 17, 2012 5:48 AM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic Thanks for your comment! I had in mind to use that formula to integrate against the higher intrinsic volumes $V_k$. In the case $k = n$, integration against $V_k$ is just integration against Lebesgue measure. In the case $k = n$ we have $V_0 = \chi$, so this is exactly what we’ve been talking about. In the general case, one can use the formula you cite to reduce integration against $V_k$ to integration against $V_0$. For example, suppose we have a function $f\colon \mathbf{R}^2 \to \mathbf{R}$ of the type discussed at the end of the post: continuous except for some jumps across the boundary of a convex set $A$. What is its integral against $V_1$? In this case $V_1$ is just perimeter (to within a scale factor), so we are ‘integrating against perimeter’. What we do is take a line $W$, not necessarily through the origin, and consider $\int (f\mid_W) \;d\chi.$ Then we take the average over all lines $W$: $\int f \;d V_1 = c_{2,1} \int_{AGr_1} \Bigl( \int (f\mid_W) \;d\chi \Bigr) d\mu_1(W)$ where I’m following your notation by using $AGr_1$ for the space of all lines in the plane, $\mu_1$ for the canonical measure on $AGr_1$, and $c_{2, 1}$ for a positive constant (whose value will depend on conventions). In the end, this is nothing but the integral of $f$ around $\partial A$, taken with respect to arclength measure. So this is nothing new. But it does point the way to how to do it in higher dimensions. Posted by: Tom Leinster on May 17, 2012 9:45 AM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic … of course, what I mean by “change of variables” I think I ought to have called “changing order of integration”, or “integration by parts”… the renovators accross the alley from me had just woken me up dropping things into their dumpster, and then I remembered. Posted by: Jesse C. McKeown on May 17, 2012 12:51 PM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic This is very interesting! I observe that while you mention the Gauss-Bonnet theorem, one noticeable difference between that and the situation you’re describing here is that the Gauss-Bonnet theorem (and its higher-dimensional generalization) is about intrinsic curvature (the Gaussian curvature of a surface depends only on its Riemannian structure), whereas what you have here is an extrinsic curvature (the curvature of a curve depends on its embedding into the plane). Posted by: Mike Shulman on May 17, 2012 9:04 AM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic That’s a very good point. It makes me wonder about different uses of the word “intrinsic”. Euler characteristic, for example, is one of the “intrinsic volumes”, but here “intrinsic” means that embedding the convex set into an $\mathbf{R}^n$ of higher-than-necessary dimension doesn’t change the value of the invariant. For any $n$, there are the $n + 1$ intrinsic volumes $V_0, \ldots, V_n$, and the fact that it’s safe to call them $V_k$ rather than (say) $V^{(n)}_k$ is a reflection of the wise normalization chosen: if $A \subseteq \mathbf{R}^n$ and $i\colon \mathbf{R}^n \hookrightarrow \mathbf{R}^N$ is an isometry, then it’s always the case that $V^{(n)}_k(A) = V^{(N)}_k(i A).$ That’s the reason for the word “intrinsic”. But it doesn’t seem to be quite the same usage as the more classical one that you mention. Simon and I have this conjecture that for any convex subset $A$ of Euclidean space, the magnitude of $A$ is given by $\|A\| = \sum_{k = 0}^\infty \frac{1}{k!\omega_k} V_k(A)$ where $\omega_k$ is the volume of the unit $k$-ball. If this is true then $\|t A\| = \sum_{k = 0}^\infty \frac{1}{k!\omega_k} V_k(A) t^k$ whenever $t \gt 0$, and we can therefore recover all the intrinsic volumes from the function $t \mapsto \|t A\|$. Since the magnitude of a space $X$ depends purely on the structure of $X$ as an abstract metric space, this would imply that the intrinsic volumes are intrinsic in a stronger sense too. Posted by: Tom Leinster on May 17, 2012 9:59 AM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic Come to think of it, I guess it’s true that if two subsets of $\mathbf{R}^n$ are isometric then there’s a self-isometry of $\mathbf{R}^n$ carrying one to the other. (I haven’t thought this through completely.) If so, the intrinsic volumes are metric invariants for this more direct reason. Still, it would be useful to have a formulation of the intrinsic volumes that (i) doesn’t even appear to depend on a choice of embedding into $\mathbf{R}^n$, and (ii) makes sense for non-Euclidean spaces. That’s what the conjecture I mentioned would provide. Posted by: Tom Leinster on May 17, 2012 11:01 AM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic Robert Ghrist ( http://www.math.upenn.edu/~ghrist/research.html ) together with Yuriy Baryashnikov are working out most of this story, and its implications for signals, radar work, and various other applications. They count target foot prints in sensor networks with Euler integration, and use Euler integral transforms for things like target recognition in radar. (I would sign the comment, but MacGPG is failing on me) Posted by: Mikael Vejdemo-Johansson on May 17, 2012 9:17 AM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic That’s very interesting. Thanks! From the first paper of Ghrist’s that I clicked on, I immediately see the relevance. Posted by: Tom Leinster on May 17, 2012 10:02 AM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic Incidentally, I see that Robert Ghrist was pointing us towards integration against Euler characteristic in this related conversation last year. I didn’t pick up on that at the time. Posted by: Tom Leinster on May 17, 2012 10:35 PM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic This is called “Euler Calculus” http://en.wikipedia.org/wiki/Euler_calculus and have a lots of applicatons in tropical geometry, toric varieties etc. Posted by: Nikolai Mnev on May 17, 2012 9:53 AM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic Thanks! I’m genuinely glad this has been worked out: I want to use it rather than develop it for its own sake. Posted by: Tom Leinster on May 17, 2012 10:05 AM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic I’ve had an hour or two to check this out now, and I have a question. Am I right in understanding that the Euler calculus is principally about integer-valued functions? Section 24 of this — Justin Curry, Robert Ghrist, Michael Robinson, Euler calculus with applications to signals and sensing, www.math.upenn.edu/~ghrist/preprints/eulertome.pdf — says so in black and white. They write at the start of that section: The Euler calculus, being integer-valued, has a delimited purview. But maybe that’s only the opinion of these authors, and others think differently — which is why I’m asking. They do then go on to look at how Euler calculus could work for $\mathbf{R}$-valued functions, but it’s more tentative. (Indeed, that part of the paper is called “Toward a $\mathbf{R}$-valued Euler calculus”.) The application I’m interested in very much involves non-integer-valued functions. I’m getting the impression now that this is closer to the fringes of what’s been done. Posted by: Tom Leinster on May 17, 2012 10:31 PM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic A minor point: being integer-valued is a distraction. You just need the function to be continuous wrt discrete topology in the range. Posted by: ymb on May 18, 2012 4:18 AM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic Thanks, but I think I must be misunderstanding. A continuous function from a connected space (such as $\mathbf{R}^n$) into a space with the discrete topology is, of course, constant. So you must mean something that I’m not getting. Posted by: Tom Leinster on May 20, 2012 12:04 AM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic Integrating against a universal Euler characteristic, taking values in the Grothendieck group of a suitable collection of subsets, is precisely the idea behind motivic integration. I’m not sure about curvature… Posted by: Moshe on May 17, 2012 3:06 PM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic Ah, good. Yes. Seeing as I never seem to find the time to sit down and learn about motivic integration and measure properly, maybe this is a good moment to find out whether what I’ve absorbed by osmosis is accurate. I’ll sketch what I think is the basic set-up, in the hope that some kind expert points out my mistakes. We begin by distinguishing certain subsets of $\mathbf{R}^n$ as “nice”. I think the official word for this is probably “tame”, or possibly “definable” or “constructible”, but as I’m not sure, I’ll just go with “nice” for now. For what I’m about to say, I need the class of nice subsets to be closed under finite unions and intersections. (My memory is that in the long run, the right thing to do is consider $n$ as varying, and to take a class of subsets of $\mathbf{R}^n$ for each $n$. These classes should interact well; for example, the product of a nice subset of $\mathbf{R}^n$ with a nice subset of $\mathbf{R}^m$ should be a nice subset of $\mathbf{R}^{n+m}$. With appropriate axioms, such a sequence of classes of subsets is called an o-minimal structure, I think.) We want to be able to measure nice sets, but we try to do so in as free a way as possible. (That’s “free” in the sense of “free group”.) So, we don’t require that the measure of a nice set is a real number; we simply ask that every nice set does have a measure, and that measures can be added. We also ask, naturally, that measures satisfy some measure-like axioms. I think it’s just finite additivity we want. In practice, this means the following (I believe). We form the free commutative monoid on the set of nice subsets of $\mathbf{R}^n$, then quotient out by the relations $[A \cup B] + [A \cap B] = [A] + [B], [\emptyset] = 0$ where $[X]$ denotes the equivalence class of $X$ in this quotient. Let’s call this commutative monoid $\mathbf{M}$. The motivic measure of a nice set $A$ is just $[A] \in \mathbf{M}$. Or is that called the Euler characteristic of $A$? I don’t know. Actually, Moshe mentioned a Grothendieck group, so I guess all occurrences of “commutative monoid” above should be “abelian group”. I don’t see immediately why we need subtraction, but subtraction has been known to come in handy. For motivic integration, I guess we begin by doing something basic: integrating finite $\mathbf{Z}$-linear combinations of indicator functions of nice sets. If $f = \sum_{r=1}^k c_r I_{A_r},$ where each $c_r$ is an integer and each $A_r$ is a nice subset of $\mathbf{R}^n$, then we must have $\int f = \sum_{r = 1}^k c_r [A_r] \in \mathbf{M}.$ (I guess the notation $\int f$ isn’t quite right; there should be something after the $f$ to show what kind of integration we’re doing. Is it “$d\chi$”?) It might be that the nice sets all have Euler characteristic in some traditional numerical sense — let’s say with values in $\mathbf{Z}$, for sake of argument. If this Euler characteristic $\chi\colon \{nice sets\} \to \mathbf{Z}$ satisfies the inclusion-exclusion principle (that is, finite additivity), then it induces a homomorphism $\mathbf{M} \to \mathbf{Z}$, which I’ll also call $\chi$. Then $\chi\Bigl(\int f\Bigr) = \sum_{r=1}^k c_r \chi(A_r),$ which is more like the integral formulas in my post. No one reading this should assume any of it’s correct! I hope someone who knows this stuff will tell me what I’ve done wrong or what crucial things I’ve left out. Posted by: Tom Leinster on May 17, 2012 11:08 PM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic Yes, this is essentially the idea. Usually you would like the measure to be invariant under appropriate (“measure preserving”) bijections, for example translations, so you first pass to isomorphism classes under such bijections. You are right about monoid vs. group: it’s better to have a theory with monoids, it’s just easier to have it with groups. But you lose information: for example, in the real line, $[[0,\infty)]=[[0,1)]+[[1,\infty)]$, and $[[0,\infty)]=[[1,\infty)]$ (by translation) so if you have cancellation, $[[0,1)]$ must by $0$. More remarks: • It is possible to integrate more general functions, essentially functions whose graph is a nice set. The values lie in a localisation of the Grothendieck semi-ring, though. • Actual motivic integration takes place in valued fields, and the measure then turns out to have essentially two “components”: one in the value group, where the theory looks essentially as you described, and the other in the Grothendieck group of algebraic varieties, which is the same kind of construction, but for nice sets in the residue field. Posted by: Moshe on May 18, 2012 3:06 PM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic It seems like you’ve rediscovered a fair bit of the Euler calculus for yourself! First off, you need to work with Euler characteristic with compact supports in order to get the inclusion-exclusion principle. The integer-valued theory works very nicely for the following reason: integer-valued functions can be thought of as the image of a constructible sheaf in the Grothendieck group. Euler integration is simply the derived pushforward to a point (with compact supports), other standard operations on sheaves manifest themselves as operations on constructible functions in the euler calculus. I’ve often wondered how the to get the real-valued theory as a de-categorification. For a more concise treatment of the real-valued theory, see Baryshnikov and Ghrist’s paper: http://www.math.upenn.edu/~ghrist/preprints/definable.pdf I believe it is the first appearance in the literature of such a theory. Although folklore has it that other people developed it on the side and got discouraged by the non-linearity of the integral. However, like you observed in your ellipse example, the integral does have interesting index theory – it “sees” critical points. I’d highly recommend Brocker and Kuppe’s paper “Integral Geometry over Tame Sets” for more on this Morse-theoretic perspective, as well as proofs of Gauss-Bonnet using this language. Also, Matthew Wright’s thesis with Rob, dealt with these Hadwiger measures that you are referring to http://arxiv.org/abs/1203.6120 -Justin Posted by: Justin Curry on May 17, 2012 11:38 PM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic Superb; thanks. I was reading your big paper on the train, saw the reference to Matthew Wright’s thesis, thought to myself “I’d like to see that”, and wondered how I was going to get hold of it. (Apparently I’m an old-fashioned enough person that being on a train means no internet.) So I’m glad to have the link to the arXiv paper — and 13 pages is probably better than a whole thesis, to be honest… The non-linearity of that integral is definitely a shocker. I don’t know what to think about that. Posted by: Tom Leinster on May 17, 2012 11:47 PM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic One more thing. For more on the motivic perspective, I’d like to recommend the following paper: Gusein-Zade, “Integration with respect to the Euler characteristic and its applications,”; Russ. Math. Surv., 65:3,2010, 399-432. Posted by: Justin Curry on May 17, 2012 11:47 PM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic 1) i found euler integration to arise from the independent works of macpherson and kashiwara on constructible sheaves in the 1970s, as popularized by schapira and viro later. 2) the sheaf-theoretic definition (the euler integral of a constructible function h over a space X is the euler characteristic of X with coefficients in a complex of sheaves F_h associated to h) is very appealing, and avoids combinatorics of jumps, etc. 3) euler integration extends to, e.g., continuous functions, with a nice relationship to stratified morse theory; the price one pays is that the euler measure splits into two poincare-dual measures, and the integral operator is nonlinear. 4) the connection to curvature can be understood through the microlocal fourier transform from sheaf theory. brocker and kuppe worked out (though, alas, without using the language of euler integrals) the appropriate gauss-bonnet theorem. 5) matthew wright’s ph.d. thesis from 2011 is on extending euler integration to the other hadwiger/intrinsic volumes. Posted by: Robert Ghrist on May 18, 2012 4:43 AM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic Thanks! What about distributions? I’d really like a way to differentiate the intrinsic volumes. I guess the nonlinearity of your integral puts a very different complexion on the concept of distribution. Posted by: Tom Leinster on May 20, 2012 12:10 AM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic Back in the 80’s when I worked for SUNY Buffalo, I was introduced to this theme by Schanuel. He published in Schanuel, Stephen H.(1-SUNYB) What is the length of a potato? An introduction to geometric measure theory. Categories in continuum physics (Buffalo, N.Y., 1982), 118–126, Lecture Notes in Math., 1174, Springer, Berlin, 1986. Posted by: Charles Frohman on May 19, 2012 3:02 PM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic I first came across this lovely paper years ago, and have been citing it enthusiastically since, but I hadn’t realized how close it came to the subject of this post until I looked at it again just now. So, thanks. It really is a great piece of work. Posted by: Tom Leinster on May 20, 2012 12:07 AM \| Permalink \| Reply to this ### Re: Integrating Against the Euler Characteristic Corresponding to this notion of integration is a numerical version that can be used to compute Euler integrals for functions sampled into a grid. It’s curious because like in the examples you draw above, you need to have a notion of whether or not a point is “attached” to the left or right side of a discontinuity. So instead of an ordinary rectangular grid of samples (with each sample thought of as living on a little square) you must keep track of the value of your function on the edges between squares and at the vertices where edges meet. I wrote some code for this a couple of years back, all based on one of Ghrist’s paper. It’s a pretty surprising way to count (simply connected) blobs. Posted by: Dan Piponi on May 22, 2012 5:38 PM \| Permalink \| Reply to this Post a New Comment	2014-07-26 01:03:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 244, "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": 0.9332126379013062, "perplexity": 414.03302031680187}, "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/1405997894931.59/warc/CC-MAIN-20140722025814-00244-ip-10-33-131-23.ec2.internal.warc.gz"}
https://socratic.org/questions/what-is-a-measurement-of-the-amount-of-solute-that-is-dissolved-in-a-given-quant	# What is a measurement of the amount of solute that is dissolved in a given quantity of solvent usually expressed as mol/L? The molarity (symbol $M$) of a solution is also called its (aptly named) "molar concentration", the number of moles of solute in one liter of the solution: $M = \text{mol solute"/"L soln}$	2021-09-24 18:20:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 2, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8121020793914795, "perplexity": 997.6365854608475}, "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/1631780057564.48/warc/CC-MAIN-20210924171348-20210924201348-00628.warc.gz"}
http://mail.calcidrata.pt/o288zi/what-is-the-product-of-a-number-e0624f	the product of a 3-digit number and a 1-digit number will be either a 3-digit number or 4-digit number.Does Emma's statement make sense? The sum of the digit is 12 . . Rules for finding the product of a fraction and a whole number. the number uses only two different digits .find the number, Jason has 4 tiles. = a We are going to use a prime factor tree to do this.. Start by writing out the first few prime numbers, as they are going to help you. 4 x 71, A number rounds off 4000 the digit in the hundred places is twice the digit in the tens place. The sum of this 4-digit number and the original 5-digit number is 52713. The digit in the ones place is 4. {\displaystyle a_{i}} {\displaystyle \prod _{i=1}^{n}} ∏ This notation (or way of writing) is in some ways similar to the Sigma notation of summation. For example, the expression $${\displaystyle \textstyle \prod _{i=1}^{6}i^{2}}$$is another way of writing $${\displaystyle 1\cdot 4\cdot 9\cdot 16\cdot 25\cdot 36}$$. The numbers are 2,3,6, and 8. a. Which of the following expressions is equivalent to x-y? The Péclet number (Pe) is a class of dimensionless numbers relevant in the study of transport phenomena in a continuum. ∏ the number of factors are odd hence in that case required number of ways in which we can write perfect square number as a product of its two factors are (n – 1)/2 if we do not include the square root of the number and Unlike summation, the sums of two terms cannot be separated into different sums. a Place Value of a Number This selection will help you to find what the place value is of a particular digit in a number. so your choice for the described condition is we define A product is the answer that you get when you multiply numbers together. 1 Given any whole number take the sum of the digits, and the product of the digits, and multiply these together to get a new whole number; for example, starting with $6712$, the sum of the digits is $6+7+1+2=16$, and the product of the digits is $6\times 7\times 1\times 2=84$. a Clues: digit in tens place is the greatest, A warranty identification number for a certain product consists of a letter of the alphabet followed by a seven-digit number. The product of a number and its multiplicative inverse is … In a three digit number, the hundreds digit is twice the units digit. . Math (help) A coin is tossed, and a standard number cube is rolled. As another example, the product of 6 and 4 is 24, because 6 times 4 is 24. Product of a Sum and a Difference What happens when you multiply the sum of two quantities by their difference? In mathematics, a product is a number or a quantity obtained by multiplying two or more numbers together. {\displaystyle \prod } & Calculus. In a three digit number, the hundreds digit is twice the units digit. A warranty identification number for a certain product consists of a letter of the alphabet followed by a seven-digit number. General Math. The sum of the digits in the 3 digit number is 11. ⋯ n As another example, the product of 6 and 4 is 24, because 6 times 4 is 24. For example: 4 × 7 = 28 Here, the number 28 is called the product of 4 and 7. In this article, we will discussed about definition of factors of number, formulas for finding number of factors, sum of factors, product of factors, even number of factors, odd number of factors, perfect square factors and perfect cube factors for any number. If the digits are reversed, the new number is 396 less than the original number. For every positive 2-digit number, x, with tens digit t and units digit u , let y be the 2 digit number formed by reversing the digits of x. is 9(u-t) c. 9t-u d.0. This page was last changed on 16 August 2020, at 17:48. (Hint: Let x = tens-place digit, 1. Your problem states that the PRODUCT of 3 and some number x is AT MOST 21. Find the number. := The product of 1, 2 and 5 is 10.The product of 1, 2 and 5 is 10.The product of 1, 2 and 5 is 10.The product of 1, 2 and 5 is 10. i i So, if you multiply 3 and x (that's what PRODUCT means), the result is at most 21. Derek is studying the change in the price of two products, A and B, over time. Place Value of a Number. i My 3rd grade son get this math problem. Illustrated definition of Product: The answer when two or more values are multiplied together.	2023-03-25 17:32:48	{"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": 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": 0.6780405640602112, "perplexity": 240.71563255651986}, "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/1679296945368.6/warc/CC-MAIN-20230325161021-20230325191021-00445.warc.gz"}
https://turbomachinery.asmedigitalcollection.asme.org/article.aspx?articleid=1468209	0 Research Papers Aerodynamic Performance of Suction-Side Gill Region Film Cooling [+] Author and Article Information Justin Chappell Department of Mechanical Engineering, University of Utah, 50 S. Central Campus Drive, MEB 2110, Salt Lake City, UT 84112-9208 Phil Ligrani1 University of Oxford, 17 Foundry House, Walton Well Road, Oxford OX2 6AQ, Englandp_ligrani@msn.com Sri Sreekanth Pratt and Whitney Canada, 1801 Courtney Park Drive East, Mississauga, ON L5A 3S8, Canadasri.sreekanth@pwc.ca Terry Lucas Edward Vlasic 1 Corresponding author. J. Turbomach 132(3), 031020 (Apr 07, 2010) (11 pages) doi:10.1115/1.3151603 History: Received February 12, 2009; Revised February 26, 2009; Published April 07, 2010; Online April 07, 2010 Abstract The performance of suction-side gill region film cooling is investigated using the University of Utah transonic wind tunnel and a simulated turbine vane in a two-dimensional cascade. The effects of film cooling hole orientation, shape, and number of rows, and their resulting effects on the aerodynamic losses, are considered for four different hole configurations: round axial (RA), shaped axial (SA), round radial (RR), and round compound (RC). The mainstream Reynolds number based on axial chord is 500,000, exit Mach number is 0.35, and the tests are conducted using the first row of holes, or both rows of holes at blowing ratios of 0.6 and 1.2. Carbon dioxide is used as the injectant to achieve density ratios of 1.77–1.99 similar to values present in operating gas turbine engines. Presented are the local distributions of total pressure loss coefficient, local normalized exit Mach number, and local normalized exit kinetic energy. Integrated aerodynamic losses $(IAL)$ increase anywhere from 4% to 45% compared with a smooth blade with no film injection. The performance of each hole type depends on the airfoil configuration, film cooling configuration, mainstream flow Mach number, number of rows of holes, density ratio, and blowing ratio, but the general trend is an increase in $IAL$ as either the blowing ratio or the number of rows of holes increase. In general, the largest total pressure loss coefficient $Cp$ magnitudes and the largest $IAL$ are generally present at any particular wake location for the RR or SA configurations, regardless of the film cooling blowing ratio and number of holes. The SA holes also generally produce the highest local peak $Cp$ magnitudes. $IAL$ magnitudes are generally lowest with the RA hole configuration. A one-dimensional mixing loss correlation for normalized $IAL$ values is also presented, which matches most of the both rows data for RA, SA, RR, and RC hole configurations. The equation also provides good representation of the RA, RC, and RR first row data sets. <> Figures Figure 12 Aerodynamic loss profiles for the RA holes: (a) pressure loss coefficients, (b) normalized Mach numbers, and (c) normalized kinetic energies Figure 19 Normalized integrated aerodynamic loss values for different hole configurations for film injection from the first row of holes only, including comparisons with the data of Jackson (21) Figure 20 Comparison of area-averaged loss coefficients with values from Ref. 28 Figure 1 University of Utah TWT Figure 3 Film cooling hole configurations: (a) RA, (b) RR, (c) SA, and (d) RC Figure 2 Schematic of the test section Figure 4 Film cooling hole locations Figure 5 Vane Mach number distribution Figure 6 Vane film cooling hole discharge coefficients for first row of holes only Figure 7 Vane film cooling hole discharge coefficients for both rows of holes. Symbols and lines are defined in Fig. 6. Figure 8 Local pressure loss coefficient profiles for injection from the first row of holes only with a blowing ratio of 0.6 Figure 9 Local pressure loss coefficient profiles for injection from the first row of holes only with a blowing ratio of 1.2 Figure 10 Local pressure loss coefficient profiles for injection from both rows of holes with a blowing ratio of 0.6 Figure 11 Local pressure loss coefficient profiles for injection from both rows of holes with a blowing ratio of 1.2 Figure 13 Aerodynamic loss profiles for the RR holes: (a) pressure loss coefficients, (b) normalized Mach numbers, and (c) normalized kinetic energies Figure 14 Aerodynamic loss profiles for the RC holes: (a) pressure loss coefficients, (b) normalized Mach numbers, and (c) normalized kinetic energies Figure 15 Aerodynamic loss profiles for the SA holes: (a) pressure loss coefficients, (b) normalized Mach numbers, and (c) normalized kinetic energies Figure 16 Dimensional integrated aerodynamic loss values for different hole configurations, number of rows of holes, and blowing ratios Figure 17 Normalized integrated aerodynamic loss values for different hole configurations for film injection from the first row of holes only Figure 18 Normalized integrated aerodynamic loss values for different hole configurations for film injection from both rows of holes. Symbols are defined in Fig. 1. Errata Some tools below are only available to our subscribers or users with an online account. Related Content Customize your page view by dragging and repositioning the boxes below. Related Journal Articles Related Proceedings Articles Related eBook Content Topic Collections	2019-03-19 01:34:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "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": 0.2647689878940582, "perplexity": 7015.1080753380975}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912201882.11/warc/CC-MAIN-20190319012213-20190319034213-00460.warc.gz"}
https://stats.stackexchange.com/questions/76789/distribution-of-logically-constrained-parameters-in-monte-carlo-simulation	# Distribution of logically constrained parameters in Monte Carlo simulation Papers like Briggs et al. 2002 say that logical constraints on inputs such as probability parameters exclude the the Normal distribution from consideration due to its unboundedness. In this example, values below 0 and above 1. They briefly describe the Bayesian derivation from the proportion data (binomial) to the beta, for use as the sampling distribution for probabilities in the probabilistic sensitivity analysis via Monte Carlo simulation. The beta can take on a normal-like shape, but it can also have very non-normal shapes. 1. Can this Bayesian derivation be reconciled with the frequentist central limit theorem that says with a sufficiently large sample size the sampling distribution of the sampling mean will be approximately normal? For resource use (ie. number of visits to the doctor) Briggs et al 2002 use a Gamma distribution, as shaped in Table 4, which is certainly not Normal. Normality is what would be expected for sufficiently large samples in the frequentist interpretation. 2. In a Monte Carlo simulation, if a Normal distribution was used to draw realized values for probabilities, there would be some iterations where values <0 or >1 are observed. However, for symmetric distributions this should cancel out asymptotically. The distribution is still centered around the mean which is within its logical bounds, and Monte Carlo considers aggregate results not results of a single iteration. So I'm wondering why we care about the logical constraints of parameters such as probabilities, costs, etc. within the context of a Monte Carlo simulation? For your first point: Not sure what you mean by reconciled, as I think Bayesians do not challenge the CLT. The use of the Gamma appears fine...note that the Gamma $\rightarrow$ Normal as the mean approaches $\infty$. For a Bayesian analysis, your prior should exclude parameter values that cannot happen, hence they are correct than an unbounded normal would not be applicable. What the CLT says is that the normal distribution becomes a better and better approximation to the sample mean distribution, hence also to the sum. The Beta, Gamma and Normal form an asymptotic triad, with the Gamma being the limit of the Beta as you increase the right tail to infinity, holding the mean and variance constant. The Normal is the limit of the Gamma as you allow the mean to go to infinity holding the variance constant (hence the centered Gamma gets a longer and longer left tail too). So I don't see an issue here. Item 2: Depending on your problem, the tail values can make a large impact on your analysis. Also, if you used the normal to generate probabilities, how would you actaully use a value of -4 or 50? Those are not proabiliites and I don't know how you would use them to make the simulatoin "balance out". Also, having these illicit tails in your simulation will jack up your variance without increasing yoru accuracy, which is not helpful. If you think that the normal distribution, as a shape, holds well, I would use a truncated Normal. If it is a good approximation, the tail probability you are truncating should be very small. • With regard to your first paragraph: what do you mean by "as the mean approaches infinity?" – kirk Nov 18 '13 at 15:07 • And again, a value of -4 is definitely not logical, but my question is why does that matter in the context of a Monte Carlo simulation? It is the aggregate of the iterations not the individual draws that matter. That illogical draw will not be used in isolation – kirk Nov 18 '13 at 15:13 • If you select $k$ and $\alpha$ for the Gamma such that the mean approaches infinity with constant standard deviation, it will approach a normal distribution. – user31668 Nov 18 '13 at 20:07 • As for monte carlo, it doesn't consider anything. you need to tell it what to calculate. What statistics are you calculating from the Monte Carlo simulation? – user31668 Nov 18 '13 at 20:08 • (cont'd): Also note that they are using the beta in a sensitivity analysis. The sensitivity depends not only on the mean of the input, but the model's response to changes in that input. In general, the function of the mean of a random quantity does not equal to mean of a function of a random quanity: $f(E[X])\neq E[f(X)]$, so while simply drawing proportions and averaging them will be insensitive to outliers, the model's response to such outliers may not be linear and symmetric. Also, its good form to not allow your model to take on unrealistic values, inviting more criticism – user31668 Nov 19 '13 at 14:24	2022-01-28 12:28:15	{"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": 0.8418764472007751, "perplexity": 454.1588638796821}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305494.6/warc/CC-MAIN-20220128104113-20220128134113-00181.warc.gz"}
http://en.wikipedia.org/wiki/Hammer_retroazimuthal_projection	# Hammer retroazimuthal projection The frontside hemisphere of the Hammer retroazimuthal projection. 15° graticule; center point at 45°N, 0°E. The backside hemisphere of the Hammer retroazimuthal projection. 15° graticule; center point at 45°N, 0°E. The Hammer retroazimuthal projection is a modified azimuthal proposed by Ernst Hermann Heinrich Hammer in 1910. As a retroazimuthal projection, azimuths (directions) are correct from any point to the designated center point.[1] In whole-world presentation, the back and front hemispheres overlap, making the projection a surjective function. Given a radius R for the projecting globe, the projection is defined as: $x = R K \cos \phi_1 \sin (\lambda-\lambda_0)$ $y = -R K [\sin \phi_1 \cos \phi - \cos \phi_1 \sin \phi \cos (\lambda-\lambda_0)]$ where $K = z/\sin z$ and $\cos z = \sin \phi_1 \sin \phi + \cos \phi_1 \cos \phi \cos (\lambda - \lambda_0)$ The latitude and longitude of the point to be plotted are φ and λ respectively, and the center point to which all azimuths are to be correct is given as φ1 and λ0. ## References 1. ^ Snyder, John P. (1993). Flattening the Earth: Two Thousand Years of Map Projections. Chicago: University of Chicago Press. pp. 228–229. ISBN 0-226-76747-7. Retrieved 2011-11-14.	2013-12-09 15:53:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 4, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/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": 0.9137917757034302, "perplexity": 7510.54386117804}, "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-2013-48/segments/1386163986869/warc/CC-MAIN-20131204133306-00001-ip-10-33-133-15.ec2.internal.warc.gz"}
http://techiemathteacher.com/2017/06/08/2017-mmc-national-finals/	• Uncategorized # 2017 MMC National Finals Grade 6 Individual Competition 15-second Questions 1. Four mangoes and 3 oranges cos Php 62.00. Four mangoes and 5 oranges cos Php 88.00. How much does one orange cost? [Sol]By subtracting 5 oranges and 4 mangoes to 3 oranges and 4 mangoes, the result will be 5 – 3 = 2 oranges. Thus, $2 \, \mathrm{oranges} = 88 -62 = \, \mathrm{Php}\, 26$. Therefore, 1 orange is $26/2 = \boxed{ \mathrm{Php} \, 13.00}$ 2. Forty three is 12.5% of what number? [Sol]Notice that $12.5\% = \dfrac{1}{8}$. Thus, $43 = \dfrac{1}{8} \times n$. The number, therefore, is $8 \times 43 = \boxed{344}$. 3. What is $(4 + 4 - 4 \times 4 \div 4 )^4$ ?[Sol] 4. If $12.8 \times 3.4 = 43.52$, what is $0.128 \times 0.34$?[Sol] 5. How many positive divisors does the product $(2^3)(3^4)(5^6)$ have? [Sol]$(3 + 1) \times (4 + 1) \times (6 + 1) = \boxed{140}$ 6. What is the sum of all prime numbers from 1 to 20? [Sol]$2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = \boxed{77}$ 7. If January 1 is a Wednesday, what day is February 1? [Sol]There are $31$ days between January 1 and February 1. Thus, 31 divided by 7 gives remainder 3. Wednesday + 3 days is a $\boxed{\mathrm{Saturday}}$. 8. The average of 11 positive consecutive odd integers is 37. What is the smallest number? [Sol]Since the difference of 2 odd numbers is 2, then the middle number should be subtracted by $5 \times 2 = 10$. Since 11 is odd, then the middle number is its average. Therefore the smallest number is $37 - 10 = \boxed{27}$. 9. What is $33 \frac{1}{3} \%$ of $60 + 55 \frac{5}{9} \%$ of $45 + 37 \frac{1}{2} \%$ of $64$ ? [Sol]Converting the percentages into decimals, we get $60 \left( \dfrac{1}{3} \right) + 45 \left( \dfrac{5}{9} \right) + 64 \left( \dfrac{3}{8} \right) = 20 + 25 + 24 = \boxed{69}$ 10. What is the product of $\left(1 - \dfrac{1}{2} \right)\left(1 - \dfrac{1}{3} \right)\left(1 - \dfrac{1}{4} \right) \cdots \left(1 - \dfrac{1}{10} \right)$ ?[Sol] Simplifying the values inside the parentheses and by cancellation, $\dfrac{1}{2} \cdot \dfrac{2}{3} \cdot \dfrac{3}{4} \dots \cdot \dfrac{9}{10} = \boxed{\dfrac{1}{10}}$	2017-12-13 18:46:24	{"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": 22, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/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": 0.759214460849762, "perplexity": 1657.5000641898487}, "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/1512948530668.28/warc/CC-MAIN-20171213182224-20171213202224-00454.warc.gz"}
https://zbmath.org/?q=an:0987.14009	## Birational quadratic transformations of the three dimensional complex projective space. (Transformations birationnelles quadratiques de l’espace projectif complexe à trois dimensions.)(French)Zbl 0987.14009 The paper concerns the classical subject of the classification of all birational morphisms (i.e. Cremona transformations) of $$\mathbb P^n(\mathbb C)$$ in particular case $$n=3$$ and degree of morphisms 2. The main result is a finite list of birational morphisms of $$\mathbb P^3(\mathbb C)$$ of degree 2 (a geometric description of them is also given) such that any other birational morphism $$\phi :\mathbb P^3_x(\mathbb C) \to \mathbb P^3_y(\mathbb C)$$ of degree 2 is equal to one in this list up to linear changes of variables in $$\mathbb P^3_x(\mathbb C)$$ and $$\mathbb P^3_y(\mathbb C)$$. Besides, the authors divide the whole class of birational morphisms of degree 2 in three natural subclasses (non-disjoint) which are locally closed subvarieties in an appropriate Grassmannian. ### MSC: 1.4e+08 Birational automorphisms, Cremona group and generalizations ### Keywords: birational morphism; Cremona group Full Text: ### References: [1] Algebraische Transformationen und Korrespondenzen, 2.2.B, (1932), Teubner · JFM 59.1291.01 [2] Le superficie razionali, (1939), Zanichelli, Bologna · Zbl 0021.05306 [3] Sulle transformazioni razionali nello spazio, Annali di Mat. ser. II, V, 131-162, (18711873) · JFM 04.0418.02 [4] On varieties of minimal degree, Algebraic Geometry, Bowdoin 1985, 46, 3-13, (1987), Amer. Math. Soc. · Zbl 0646.14036 [5] Classification of degree 2 polynomial automorphisms of $${\Bbb C}^3,$$ Publ. Mat., 42, 195-210, (1998) · Zbl 0923.58006 [6] Algebraic Geometry, (1992), Springer Verlag · Zbl 0779.14001 [7] Algebraic Geometry, (1979), Springer Verlag · Zbl 0367.14001 [8] Cremona transformation in Plane and Space, (1927), University Press, Cambridge · JFM 53.0595.01 [9] Sur le multidegré des transformations de Cremona, C.R. Acad. Sci. Paris, Série I, 330, 297-300, (2000) · Zbl 1011.14003 [10] Introduction to Algebraic Geometry, (1949), Claredon Press, Oxford · Zbl 0041.27903 [11] Selected Topics in Algebraic Geometry, (1970), Chelsea Pub. Company, Washington · Zbl 0213.47101 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.	2022-08-15 16:10:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.6391263008117676, "perplexity": 2733.7651903516216}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572192.79/warc/CC-MAIN-20220815145459-20220815175459-00299.warc.gz"}
https://mathhelpboards.com/threads/greatest-common-divisor-of-two-polynomials.4186/	# Greatest common divisor of two polynomials #### Peter ##### Well-known member MHB Site Helper I am working on Exercise 8 of Dummit and Foote Section 9.2 Exercise 8 ==================================================================================== Determine the greatest common divisor of [TEX] a(x) = x^3 - 2 [/TEX] and [TEX] b(x) = x + 1 [/TEX] in [TEX] \mathbb{Q} [x] [/TEX] and write it as a linear combination (in [TEX] \mathbb{Q} [x] [/TEX] ) of a(x) and b(x). ===================================================================================== In working on this I applied the Division Algorithm to a(x) and b(x) resulting in [TEX] x^3 - 2 = (x^2 - x + 1) (x+ 1) + (-3) [/TEX] then [TEX] (x + 1) = (1/3 x + 1/3) + 0 [/TEX] Last non-zero remainder is -3 Therefore, gcd is -3 BUT! This does not seem to be correct because -3 does not divide either a(x) and b(x) Peter #### Opalg ##### MHB Oldtimer Staff member I am working on Exercise 8 of Dummit and Foote Section 9.2 Exercise 8 ================================================== Determine the greatest common divisor of [TEX] a(x) = x^3 - 2 [/TEX] and [TEX] b(x) = x + 1 [/TEX] in [TEX] \mathbb{Q} [x] [/TEX] and write it as a linear combination (in [TEX] \mathbb{Q} [x] [/TEX] ) of a(x) and b(x). ================================================== In working on this I applied the Division Algorithm to a(x) and b(x) resulting in [TEX] x^3 - 2 = (x^2 - x + 1) (x+ 1) + (-3) [/TEX] then [TEX] (x + 1) = (1/3 x + 1/3) + 0 [/TEX] Last non-zero remainder is -3 Therefore, gcd is -3 BUT! This does not seem to be correct because -3 does not divide either a(x) and b(x) $-3$ is a unit in $\mathbb{Q} [x]$, so is equivalent to $1$. You have shown that $$-\tfrac13(x^3-2) + \tfrac13(x^2-x+1)(x+1) = 1.$$ Thus $p(x)a(x)+q(x)b(x) = 1$, where $p(x) = -\frac13$ and $q(x) = \frac13(x^2-x+1)$. The polynomials $p(x)$ and $q(x)$ are both in $\mathbb{Q} [x]$.	2020-11-30 04:25:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.6298971176147461, "perplexity": 1248.090358543103}, "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-2020-50/segments/1606141205147.57/warc/CC-MAIN-20201130035203-20201130065203-00343.warc.gz"}
https://combine.se/blog/page/10/	Blog - Page 10 of 10 - Combine \| Combine # Blog When building business-critical applications for an enterprise environment, it is common to first gather requirements from domain experts using a business analyst. The business analyst then formulates a set of requirements which are given to an architect. The architect, in turn, creates some design documents which the development team translates into code. One problem here is that the distance between the ones who implement the solution and the domain experts is considerable. Information is filtered through many persons. An advantage is that we end up with documentation describing the product. In agile development, the domain experts talk directly to the development team. Changes are fast, and the development pace can be high, but the process is likely to produce some waste as well. The development team might not be proficient in communicating with the domain expert. The business analyst in the previous example is an expert in interviewing domain experts and writing down requirements. The agile process might also produce less documentation which makes it harder for developers entering the project late to understand the big picture. In domain driven development domain experts, the development team and other stakeholders strive to build a shared mental model of the business process. Having a programming language with a strong type system also helps to model the domain directly in code meaning that if the requirements change the code will not compile anymore (see “Domain Modeling Made Functional“). The claimed advantage of aligning the software model with the business domain is a faster time to market, more business value, less waste and easier maintenance and evolution. Recommended guidelines for working with a domain-driven design is to focus on what is called business events and workflows instead of data structures. This way the business requirements are captured in a way that hidden requirements are not lost as easily while the developer is not superimposing his or her technical solutions on the design too early. The problem domain needs to be partitioned into smaller subdomains such that the subdomains are not too large. The subdomains should match domains in the organization, not necessarily the actual company hierarchy, but instead real domains. Each subdomain has to be modeled in the solution in such a way that the solution does not share any persistent data with other subdomains. If a subdomain needs information from another subdomain, it has to ask for it instead of just accessing directly in the database. A so-called ubiquitous language needs to be developed. The language is shared among all the participants of the project and in the code as well. There might be local variations in the meaning of words in different domains, and that is okay as long as those differences are understood, otherwise unnecessary conflicts and misunderstandings could arise. The shared mental model of the domain allows other stakeholders to understand what is going on as well since business processes are described on a high level while the code is documenting the requirements directly and dictates how data structures should be designed instead of the other way around. ## Introduction Functional Principal Component Analysis (FPCA) is a generalization of PCA where entire functions act as samples ($$X \in L^2(\mathcal{T})$$ over time a interval $$\mathcal{T}$$) instead of scalar values ($$X \in \mathbb{R}^p$$). The FPCA can be used to find the dominant modes of a set of functions. One of the central ideas is to redefine the scalar product from $$\beta^T x = \left \langle \beta, x \right \rangle = \sum_j \beta_j x_j$$ into a functional equivalent $$\left\langle \beta, x \right\rangle = \int_{\mathcal{T}} \beta(s) x(s) ds$$. ## Temperature in Gothenburg Using data from SMHI, we are going to look at variations of temperature over the year in Gothenburg. The data spans from 1961 to today and all measurements have been averaged per month and grouped by year. To be able to do an FPCA we need to remove the mean from the data. The first principal component of the data which explains 94% of the total variation is unsurprisingly the variation over seasons followed by the second and third principal components at 1.8% and 0.95% respectively. Looking at the scores for the two first components gives us an idea which years differ the most from each other, i.e., the points which are farthest away from each other. Horizontally, the years 1989 and 2010 seem to be different for the first principal component. Apparently, the winter of 1989 was much warmer than 2010. The years 1971 and 2004 are very close to each other which suggests that they should be very similar, and they are. The second principal component represents a mode where the late winter differs from the autumn/early winter between the years. The year 2006 had a cold early year and a warm late year while 2002 was warm to start with and cold at the end. ## Conclusion The FPCA is a powerful tool when analyzing variations in functional data. It applies to multidimensional functional data as well. Functional data analysis, in general, is a powerful tool which also can be used to categorization where different clusters of, e.g., motion trajectories needs to be found. #### Introduction Scheduling constrained resources over time is a tough problem. In fact, the problem is NP-hard. One part of the problem is to find a feasible solution where all constraints are satisfied simultaneously. Another part is to also find a solution which also satisfies some measure of optimality. In practice, it is often sufficient to solve the problem partially such that the solution is better than the first feasible solution which has been found. Solving combinatorial scheduling problems can be done using mixed-integer linear programming (MIP) or some heuristic approaches such as the Tabu search. #### The Flow Approach There are several ways to formulate the scheduling problem. The flow formulation presented by Christian Artigues in 2003 is one interesting approach. Assume for simplicity a process with two tasks: The nodes 1 and 2 are representing the tasks. “S” is the start task and no edges can go back here. “E” is the end task and only incoming edges are allowed. The directed graph shows all possible edges for this configuration per resource type. The graphs tell us how resources can be transported between different tasks. Hence, “S” can give resources to task 1 and task 2 while also sending resources which are not needed directly to “E.” Tasks can handle resources in three different ways: 1. A task can consume a resource such that it disappears. 2. A task can produce a new resource making it available for someone else to interact with. 3. A task can pass a resource through for others to use (e.g., a person or a tool). Assume that we have the following resources available in “S” to start with: • One operator. • Two tools. • Three raw materials. Task 1 requires the following to be able to produce one product A. • One operator. • One tool. • One raw material. Task 2 requires the following to produce one product B. • One operator. • Two tools. • One product like the one produced by task 1. • One raw material. This problem can easily be solved by hand yielding: The solution is an acyclic directed graph where resources flow from “S” to “E” fulfilling the constraints given by each node. Based on the solution task 2 must wait for task 1 to finish. Also, note that the edge between 2 and 1 has been removed since it is not needed. Solving more complex resource constrained scheduling problems increases the available number of combinations rapidly making the problem harder and harder to solve as can be seen for the full graphs for 3, 6 and 10 tasks: #### Scheduling In some cases, it is possible to change the order between tasks or even execute them in parallel without violating any constraints. The graph tells us whether any task must follow any other task(s), either directly or through a chain of events. What is left is to take the duration of each task into consideration to produce the final time schedule. #### Conclusion The resource-constrained scheduling problem is relatively simple to solve if there is either excess of tasks or resources compared to the other. The number of combinations is much reduced then. When the distribution of resources is close to the number of tasks involved the problem gets much harder to solve. Playing around with the measure of optimality can yield many different results depending on the formulation. For example, some tasks could be prioritized to be executed before others and tasks could be distributed over a time interval to maximize robustness for deviations and so forth. The problem can be solved using mixed-integer linear programming (MIP) for which there are several mature solvers available on the market. #### Introduction Ordinary linear differential equations can be solved as trajectories given some initial conditions. But what if your initial conditions are given as distributions of probability? It turns out that the problem is relatively simple to solve. #### Transformation of Random Variables If we have a random system described as $$\dot{X}(t) = f(X(t),t) \qquad X(t_0) = X_0$$ we can write this as $$X(t) = h(X_0,t)$$ which is an algebraic transformation of a set of random variables into another representing a one-to-one mapping. Its inverse transform is written as $$X_0 = h^{-1}(X,t)$$ and the joint density function $$f(x,t)$$ of $$X(t)$$ is given by $$f(x,t) = f_0 \left[ x_0 = h^{-1}(x,t) \right] \left\| J \right\|$$ where $$J$$ is the Jacobian $$J = \left\| \frac{\partial x^T_0}{\partial x} \right\|$$. #### Solving Linear Systems For a system of differential equations written as $$\dot{x}(t) = A x(t) + B u(t)$$ a transfer matrix can be defined $$\Phi(t,t_0) = e^{A(t-t_0)}$$ which can be used to write the solution as $$x(t) = \Phi(t,t_0) x(0) + \int_{t_0}^{t} {\Phi(t,s) B u(t) ds}$$. The inverse formulation of this solution is $$x(0) = \Phi^{-1}(t,t_0) x(t) – \Phi^{-1}(t,t_0) \int_{t_0}^{t} {\Phi(t,s) B u(t) ds}$$. #### Projectile Trajectory Example Based on the formulations above we can now move on to a concrete example where a projectile is sent away in a vacuum. The differential equations to describe the motion are $$\left\{ \begin{array}{rcl} \dot{p}_{x_1}(t) & = & p_{x_2}(t) \\ \dot{p}_{x_2}(t) & = & 0 \\ \dot{p}_{y_1}(t) & = & p_{y_2}(t) \\ \dot{p}_{y_2}(t) & = & -g \end{array} \right.$$ where $$p_{x_1}$$ and $$p_{y_1}$$ are cartesian coordinates of the projectile in a two dimensional space while $$p_{x_2}$$ is the horizontal velocity and $$p_{y_2}$$ is the vertical velocity. We only have gravity as an external force, $$-g$$, and no wind resistance which means that the horizontal velocity will not change. The matrix representation of this system becomes $$A = \left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right)$$ with $$B^T = \left( \begin{array}{cccc} 0 & 0 & 0 & 1 \end{array} \right)$$. The transfer matrix is (matrix exponential, not element-wise exponential) $$\Phi(t,t_0) = e^{A(t-t_0)} = \left( \begin{array}{cccc} 1 & 0 & t-t_0 & 0 \\ 0 & 1 & 0 & t-t_0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$$ Calculating the solution of the differential equation gives $$x(t) = \Phi(t,0) x(0) + \int_0^t {\Phi(t,s) B u(t) ds}$$ where $$u(t) = -g$$ and $$x^T(0) = \left( \begin{array}{cccc} 0 & 0 & v_x & v_y \end{array} \right)$$. The parameters $$v_x$$ and $$v_y$$ are initial velocities of the projectile. The solution becomes $$x(t) = \left( \begin{array}{c} v_x t \\ v_y t – \frac{g t^2}{2} \\ v_x \\ v_y – g t \end{array} \right)$$ and the time when the projectile hits the ground is given by $$p_y(t) = v_y t – \frac{g t^2}{2} = 0 \qquad t > 0$$ as $$t_{y=0} = 2 \frac{v_y}{g}$$. A visualization of the trajectory given $$v_x = 1$$ and $$v_y = 2$$ with gravity $$g = 9.81$$ shows an example of the motion of the projectile: Now, if assume that the initial state x(0) can be described by a joint Gaussian distribution we can use the formula shown earlier to say that $$f(x,t) = f_0\left[x(0)=h^{-1}(x,t)\right] \left\|J\right\| = \frac{1}{\sqrt{\left\|2 \pi \Sigma \right\|}} e^{-\frac{1}{2}(x(0)-\mu)^T \Sigma^{-1} (x(0)-\mu)}$$, where $$\left\| J \right\| = \left\| \Phi^{-1}(t) \right\|$$, $$\mu^T = \left( \begin{array}{cccc} 0 & 0 & v_x & v_y \end{array} \right)$$ and $$\Sigma = \left( \begin{array}{cccc} 0.00001 & 0 & 0 & 0 \\ 0 & 0.00001 & 0 & 0 \\ 0 & 0 & 0.01 & 0 \\ 0 & 0 & 0 & 0.01 \end{array} \right)$$ which means that we have high confidence in the firing position but less in the initial velocity. We are only interested in where the projectile lands and we can marginalize the velocities to get: $$f\left(p_{x_1},p_{y_1},t\right) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,t) dp_{x_2} dp_{y_2}$$ which when plotted gives Since we have used the landing time for the deterministic trajectory, we get a spread across the y-axis as well (the ground is located at $$p_y = 0$$). We could marginalize the y-direction as well to end up with: This shows the horizontal distribution of the projectile at the time when the deterministic trajectory of the projectile is expected to hit the ground. #### Conclusion Given a set of ordinary differential equations, it is possible to derive the uncertainty of the states given a probability distribution in the initial conditions. There are two other important cases to look into as well: stochastic input signals and random parameters. For more complex models it is possible to solve the Bayesian problem numerically using, for example, MCMC (Markov-Chain Monte-Carlo). It is a computationally expensive method which gives the solution as a set of points in the parameter space which are distributed according to the likelihood of the parameters given the data at hand. #### Regression Example For this example, we are working with a linear model of the form $$f(x)=a+bx + \epsilon$$, where $$\varepsilon \sim N\left(0,\sigma^2\right)$$ (normal distributed noise). First, we need to generate some random data starting choosing 50 samples where $$a=1$$, $$b=2$$, $$\sigma^2=1$$: One simple way to solve the MCMC-problem is the Metropolis-Hastings method. It is based on evaluating changes in the posterior likelihood function one parameter at a time doing a random walk trying to stay in a region with high probability all the time. If the likelihood is multi-modal, it is, of course, possible to get stuck in one mode. The resulting estimated likelihood given 100,000 samples for the linear regression is shown below where the red dot represents the highest likelihood, and the blue dot is the real parameters. The contours show a smoothed kernel estimate of the density of the distribution. Note that there is a slight covariance between parameters a and b which means that if you change one of the parameters the other has to change as well. It turns out that the maximum likelihood of the MCMC-estimate and the Least-Squares method gives the same result, which is expected since maximum likelihood and least-squares are equal in the presence of Gaussian noise. This example has just been a simple demonstration of how to find a good fit for model parameters given some data measurement. Based on the likelihood plots above we obtain some understanding of the sensitivity of changes in the parameters and if they are likely to be correlated to obtain maximum likelihood. In the presence of non-gaussian noise and high dimensional complex models, MCMC might be your only way to obtain a solution at all at the cost of long durations of computation. Combine Control Systems visited this year’s UAS Forum conference in Linköping. UAS is the abbreviation for Unmanned Aerial System which has been an interest at the company since the first master’s thesis regarding the subject started in 2012. For two days the UAS Forum with lectures, presentations and fair was held in connection with the international workshop on research, education and development on unmanned aerial systems (RED-UAS) which has been held every second year since 2011. This substantiates the fact that Linköping is Sweden’s capital regarding aviation. During Combine Control Systems visit at the conference the major topics were “Introduction of UAS in an organization”, “How do we share the airspace”, “The need for artificial intelligence in UAS” as well as presentations of companies in the region such as CybAero and UMS Skeldar. It was very interesting to hear from both known and unknown actors in the industry and understanding their thoughts about using UAS in their businesses. Thanks to all participants and organizers and a special thanks to Jan Holmbom who moderated the event. If you are interested in talking more about UAS or other flight systems, please contact us at Combine Control Systems. Some of our previous projects related to UAS can be found on the webpage, such as cluster behavior for UAS. This time, we will consider the development of an arbitrary mechatronic system, a system consisting of both hardware and software. Into the hardware, we count all physical components that range all the way from processors and integrated circuits through actuators and sensor to engines and ventilation circuits. While with the term software, we refer to the embedded code uploaded on processors and integrated circuits. Back in the days, which is not that many years ago, the graph in the figure above could have been a good schematic view of a development process. Here, we have time along the x-axis, a start point to the left and a delivery point to the right. At this period, it was more or less necessary to have a sequential process, where the actual hardware had to be available before the development of the software could be started. In this graph, it can be noticed that, • The knowledge about the system, the green curve, is increasing with time. The knowledge is obtained by testing different solutions and the more tests that can be performed, the more knowledge the developers will obtain about the system. • The possibility to make changes, the red curve, is instead decreasing with time. The closer one gets to the point of delivery the more limited is the possibility to make changes. The short period left to the delivery makes it hard to get large changes in place and even small changes can rock the foundation of the rigid structure that the system has become close to the delivery point. • The yellow marked area between the two curves and x-axis, within the considered developing time, is a measure of the effective work that can be performed during the process. Clearly, the goal should be to maximize the efficient work during the process. The more useful work, the more tests can be performed and more bugs and faults can be found. Fewer bugs and errors result in a better quality of the system and a better final product. One important variable that we have disregarded in this graph is the cost. We do all know that there always exist alignments within companies whose main responsibility is to keep the costs as low and the income as high as possible. One efficient way to obtain this is to reduce the development time, to shorten the time-to-market, which is exactly what is visualized in the figure below. Directly, two significant consequences can be noticed. First, the amount of productive work is more limited. Secondly, the sequential procedure, where the software development starts first after the hardware is in place, does not longer fit within the time for development. The introduction of Model-Based Design, MBD, has made it possible to separate the software into different components. Some of these are in direct contact with the hardware and are interacting with it. While others, like the controller software components C-SWC, which have the purpose to control the behavior of some physical quantities, have several layers of software between themself and the hardware. To test the C-SWC, it might not be necessary to have the actual device in an early stage of the project. Instead, virtual models describing the dynamics of the physical quantities, and how they are perturbed by other quantities, can be the object to test the C-SWC code against, a so-called plant model. The entry of plant models and separable software components made it possible to start the development of the software earlier and test it on desktop computers instead of on physical test benches. The effect of the virtual testing is visualized by the graph in the figure above. Before virtual testing was introduced, one had to wait to have an available test bench before testing the software, described by the blue curve in the graph. With the introduction of MBD, “only” plant models were needed to verify part of the software for bugs and faults in a virtual environment. The bugs and errors are found at an earlier stage of the process, which is what the red curve is telling us. Still, there is a need for physical testing, but the amount has now been reduced. The look of the schematic view of the development process changes with the introduction of virtual testing, see the figure above. The curve for obtaining knowledge about the system has a steeper behaviour at the beginning of the process. This corresponds to the possibility to perform virtual tests at an earlier stage. Faster obtained knowledge increases the effective work performed during the process, and the question now is, how can one get even more knowledge at an early stage? One way is to increase the number of tests that are performed and a virtual environment is an ideal location for performing tests in large numbers, see the figure below. A physical test bench is usually designed for a specific type of test, if one wants to do something outside its specification one has to rearrange the setup or build a new test bench. This can be both expensive and cost a lot of time. With virtual testing, a new test bench can just be some lines in a script away, which makes it simple to switch between test configurations and set up automated processes. A growing field within virtual testing is Model-Based Testing, MBT, where software algorithms are used to design the test cases, run the test procedures and analyze the result. These algorithms can automatically produce a substantial number of test cases and do even feedback information from the results back to the process in order to create new and better test cases. An example is the TestWeaver algorithm that is described to play chess with the system under test. Testing a system under test (SUT) is like playing chess against the SUT and trying to drive it into a state where it violates its specification. Most, if not all, applications presented so far have been introduced to benefit software development. Plant models and separable software gave the developers access to the virtual test benches. Will it also be possible to use virtual hardware models to actually improve the development of the physical hardware, as well as the software? If the virtual hardware can be in place early in the process, it would be possible to test combinations of different components in virtual test benches and obtain early knowledge for both hardware and software developers. This will, of course, require much more detailed models of the hardware that exist today, including non-trivial behaviours and limitations that could be triggered from the virtual test environment. Hardware models of fine granularity will benefit the development of both the hardware and the software. With a structure of common virtual test benches, into which both hardware and software teams are delivering models, it will be possible to test the robustness of the systems in new ways. For example, how the software will react to signals coming from hardware components that are old and not functioning perfectly anymore? Or, how the hardware components should be designed in order to hold for the large forces which can appear with rapid actions from the control algorithms? To be able to test this kind of scenarios at an early stage will not only generate knowledge within the hardware and software teams themself but also put the teams closer together to make it possible for them to find solutions together. Model-Based Testing and virtual hardware are both two examples of concepts that will increase the knowledge about the system at an early stage and decrease the need for expensive physical test environments. ### Progressive Self-Exploring Design of Experiments In a classical design of experiments (DoE) you usually choose a set of points according to some rule and perform experiments to be able to, for example, create a response surface. But when the properties of the process you are trying to describe is difficult to understand and can be destroyed if wrong parameters are applied we have to try something different. ## Introduction In a classical design of experiments (DoE) you usually choose a set of points according to some rule and perform experiments to be able to, for example, create a response surface. But when the properties of the process you are trying to describe is difficult to understand and can be destroyed if wrong parameters are applied we have to try something different. One solution could be to build a predictive model each time a new sample has been taken and decide where to take the next sample given information taken from the updated model. I am going to show you how Gaussian Processes (see the introduction) can be used to collect samples efficiently. In short, the algorithm teaches itself how the process works by asking the correct questions based on what is known, slowly expanding its knowledge safely. ## Ingredients The properties of the Gaussian Process relies on the chosen kernel. In this example, the squared exponential is used which for $$e^{-x^2}$$ looks like: This kernel is used to control the curvature of the estimated function. The formula for estimating the conditional distribution of the Gaussian Process gives us an expression the covariance: $$\text{cov}(\mathbf{f}_)=K(X_,X_)-K(X_,X)\left[K(X,X)+\sigma_n^2I\right]^{-1} K(X,X_*)$$ What is nice about this formula is that it is not dependent on any measurements. Given a kernel and a set of hyperparameters you only need to decide where you want to measure to understand what uncertainty you should expect when predicting the function. This fact makes it possible to design a space-filling experiment design for a given assumption of the properties of the model. Now recall when we have some measurements we can generate a model such as: The gray area shows one standard deviation. When the standard deviation is small, we can make good predictions about the function while higher standard deviation indicates that we lack information. Given the four samples, we should be tempted to measure where the standard deviation is high. Just looking at the standard deviation as a function clarifies this thought: If we have defined a limited domain on the horizontal axis, it should be straightforward to choose the point with the highest standard deviation. This is ok as long as the process cannot be destroyed for a set of parameters. Assume that we do not know exactly for which parameters we reach safety limits, then we need to expand slowly from the measurements we are aware are safe. One way of doing this is to use the squared exponential kernel to include an allowed action radius. Drawing some kernels around the measurements looks like: And if we take the maximum of these four functions we get: Notice that the maximum is small between the two points on the left while the kernels are smeared together on the right since they are closer together. This function can be used to describe how safe it is to measure at a given set of parameters. We can now combine the kernels with the standard deviation by taking the product ending up with: Now we are encouraged to measure in the vicinity of each data point, but not too close and not too far away. Since the standard deviation is lower when points are closer to each other exploration is often prioritized before refining. ## Simulation We are going to try to generate a model of the function $$f(x) = (x-0.5)^2 + (x+0.5)^2 + \sin(1.1 \pi x)$$ on the interval $$x \in [-2,2]$$ constrained by $$f(x) < 4$$ as seen here: We need to have some knowledge about the process to be able to give the process one or several safe points to start from. We are going to start with $$x_0 = 0$$ and the goal is to obtain a sequence of $$p_i = \left(x_i, f(x_i)\right)$$ for which we can predict the function with good precision. To find a new candidate we need to have a set of candidates to choose from. The set of candidates are generated using a space-filling random algorithm, in our case the Sobol sequence. Here is a sequence of 21 samples taken using the method described above. Notice how the algorithm is cautious to start with and then starts expanding to the right and left, occasionally going back to refine the model instead of exploring. It also does not violate the condition $$f(x) \leq 4$$. ## Final Discussion Progressive sampling is useful when the process you want to describe is nonlinear and when you need to avoid breaking any constraints. The method scales well to many dimensions and can be automated in actual physical testing environments. We can also handle noisy measurements which would result in slower propagation since the uncertainty of predictions would be larger. We could add additional constraints which are tailored to the problem at hand, for example scaling the width of the kernel depending on the estimated magnitude of the gradient for each measurement or adding other functions which control how samples are chosen.	2022-01-28 09:25:15	{"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": 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": 0.522671103477478, "perplexity": 428.07609947020626}, "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-2022-05/segments/1642320305423.58/warc/CC-MAIN-20220128074016-20220128104016-00703.warc.gz"}
https://www.aimsciences.org/article/doi/10.3934/jimo.2015.11.185	# American Institute of Mathematical Sciences January 2015, 11(1): 185-198. doi: 10.3934/jimo.2015.11.185 ## A $2.28$-competitive algorithm for online scheduling on identical machines 1 Department of Automation, Xiamen University, 422 South Siming Road, Xiamen, 361005, China, China, China Received March 2013 Revised January 2014 Published May 2014 Online scheduling on identical machines is investigated in the setting where jobs arrive over time. The goal is to minimize the total completion time. A waiting strategy based online algorithm is designed and is proved to be $2.28$-competitive. The result improves the current best online algorithm from the worse-case prospective. Citation: Jiping Tao, Ronghuan Huang, Tundong Liu. A $2.28$-competitive algorithm for online scheduling on identical machines. Journal of Industrial & Management Optimization, 2015, 11 (1) : 185-198. doi: 10.3934/jimo.2015.11.185 ##### References: [1] E. J. Anderson and C. N. Potts, Online scheduling of a single machine to minimize total weighted completion time, Mathematics of Operations Research, 29 (2004), 686-697. doi: 10.1287/moor.1040.0092. Google Scholar [2] M. C. Chou, M. Queyranne and D. Simchi-Levi, The asymptotic performance ratio of an on-line algorithm for uniform parallel machine scheduling with release dates, Mathematical Programming, 106 (2006), 137-157. doi: 10.1007/s10107-005-0588-1. Google Scholar [3] J. R. Correa and M. R. Wagner, LP-based online scheduling: From single to parallel machines, Mathematical Programming, 119 (2009), 109-136. doi: 10.1007/s10107-007-0204-7. Google Scholar [4] A. Fiat and G. J. Woeginger, Competitive analysis of algorithms, Lecture Notes in Computer Science, 1442 (1998), 1-12. doi: 10.1007/BFb0029562. Google Scholar [5] M. X. 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Potts, Online scheduling of a single machine to minimize total weighted completion time, Mathematics of Operations Research, 29 (2004), 686-697. doi: 10.1287/moor.1040.0092. Google Scholar [2] M. C. Chou, M. Queyranne and D. Simchi-Levi, The asymptotic performance ratio of an on-line algorithm for uniform parallel machine scheduling with release dates, Mathematical Programming, 106 (2006), 137-157. doi: 10.1007/s10107-005-0588-1. Google Scholar [3] J. R. Correa and M. R. Wagner, LP-based online scheduling: From single to parallel machines, Mathematical Programming, 119 (2009), 109-136. doi: 10.1007/s10107-007-0204-7. Google Scholar [4] A. Fiat and G. J. Woeginger, Competitive analysis of algorithms, Lecture Notes in Computer Science, 1442 (1998), 1-12. doi: 10.1007/BFb0029562. Google Scholar [5] M. X. Goemans, Improved approximation algorithms for scheduling with release dates, in Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms, New Orleans, (1997), 591-598. Google Scholar [6] M. X. Goemans, M. Queyranne, A. S. Schulz, M. Skutella and Y. Wang, Single machine scheduling with release dates, SIAM Journal on Discrete Mathematics, 15 (2002), 165-192. doi: 10.1137/S089548019936223X. Google Scholar [7] L. A. Hall, A. S. Schulz, D. B. Shmoys and J. Wein, Scheduling to minimize average completion time: Off-line and on-line approximation algorithms, Mathematics of Operations Research, 22 (1997), 513-544. doi: 10.1287/moor.22.3.513. Google Scholar [8] J. A. Hoogeveen and A. P. A. Vestjens, Optimal on-line algorithms for single-machine scheduling, Lecture Notes in Computer Science, 1084 (1996), 404-414. doi: 10.1007/3-540-61310-2_30. Google Scholar [9] P. H. Liu and X. W. Lu, On-line scheduling of parallel machines to minimize total completion times, Computers & Operations Research, 36 (2009), 2647-2652. doi: 10.1016/j.cor.2008.11.008. Google Scholar [10] X. Lu, R. A. Sitters and L. Stougie, A class of on-line scheduling algorithms to minimize total completion time, Operations Research Letters, 31 (2003), 232-236. doi: 10.1016/S0167-6377(03)00016-6. Google Scholar [11] N. Megow and A. S. Schulz, On-line scheduling to minimize average completion time revisited, Operations Research Letters, 32 (2004), 485-490. doi: 10.1016/j.orl.2003.11.008. Google Scholar [12] C. Phillips, C. Stein and J. Wein, Minimizing average completion time in the presence of release dates, Mathematical Programming, 82 (1998), 199-213. doi: 10.1007/BF01585872. Google Scholar [13] M. Pinedo, Scheduling: Theory, Algorithms, and Systems, 4nd edition, Springer-Verlag, New York, 2012. doi: 10.1007/978-1-4614-2361-4. Google Scholar [14] R. Sitters, Efficient algorithms for average completion time scheduling, Lecture Notes in Computer Science, 6080 (2010), 411-423. doi: 10.1007/978-3-642-13036-6_31. Google Scholar [15] J. P. Tao, H. Jiang and T. D. Liu, A 2.5-competitive Online Algorithm for $P_m\|r_j\|\sum w_jC_j$, in the 24th Chinese Control and Decision Conference (CCDC), Taiyuan, China, IEEE, (2012), 3184-3188. Google Scholar [16] J. P. Tao, Z. J. Chao and Y. G. Xi, A semi-online algorithm and its competitive analysis for a single machine scheduling problem with bounded processing times, Journal of Industrial and Management Optimization, 6 (2010), 269-282. doi: 10.3934/jimo.2010.6.269. Google Scholar [17] J. P. Tao, Z. J. Chao, Y. G. Xi and Y. Tao, An optimal semi-online algorithm for a single machine scheduling problem with bounded processing time, Information Processing Letters, 110 (2010), 325-330. doi: 10.1016/j.ipl.2010.02.013. Google Scholar [1] Ran Ma, Jiping Tao. An improved 2.11-competitive algorithm for online scheduling on parallel machines to minimize total weighted completion time. Journal of Industrial & Management Optimization, 2018, 14 (2) : 497-510. doi: 10.3934/jimo.2017057 [2] Jiping Tao, Zhijun Chao, Yugeng Xi. A semi-online algorithm and its competitive analysis for a single machine scheduling problem with bounded processing times. Journal of Industrial & Management Optimization, 2010, 6 (2) : 269-282. doi: 10.3934/jimo.2010.6.269 [3] P. Liu, Xiwen Lu. Online scheduling of two uniform machines to minimize total completion times. Journal of Industrial & Management Optimization, 2009, 5 (1) : 95-102. doi: 10.3934/jimo.2009.5.95 [4] Ling Lin, Dong He, Zhiyi Tan. Bounds on delay start LPT algorithm for scheduling on two identical machines in the $l_p$ norm. Journal of Industrial & Management Optimization, 2008, 4 (4) : 817-826. doi: 10.3934/jimo.2008.4.817 [5] Ning Zhang, Qiang Wu. Online learning for supervised dimension reduction. 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Journal of Industrial & Management Optimization, 2017, 13 (2) : 977-993. doi: 10.3934/jimo.2016057 [10] Tugba Sarac, Aydin Sipahioglu, Emine Akyol Ozer. A two-stage solution approach for plastic injection machines scheduling problem. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1289-1314. doi: 10.3934/jimo.2020022 [11] Xianzhao Zhang, Dachuan Xu, Donglei Du, Cuixia Miao. Approximate algorithms for unrelated machine scheduling to minimize makespan. Journal of Industrial & Management Optimization, 2016, 12 (2) : 771-779. doi: 10.3934/jimo.2016.12.771 [12] Alireza Goli, Taha Keshavarz. Just-in-time scheduling in identical parallel machine sequence-dependent group scheduling problem. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021124 [13] Roberto C. Alamino, Nestor Caticha. Bayesian online algorithms for learning in discrete hidden Markov models. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 1-10. doi: 10.3934/dcdsb.2008.9.1 [14] Donglei Du, Xiaoyue Jiang, Guochuan Zhang. Optimal preemptive online scheduling to minimize lp norm on two processors. Journal of Industrial & Management Optimization, 2005, 1 (3) : 345-351. doi: 10.3934/jimo.2005.1.345 [15] Zhenhuan Yang, Yiming Ying, Qilong Min. Online optimization for residential PV-ESS energy system scheduling. Mathematical Foundations of Computing, 2019, 2 (1) : 55-71. doi: 10.3934/mfc.2019005 [16] Matthew M. Dunlop, Andrew M. Stuart. The Bayesian formulation of EIT: Analysis and algorithms. Inverse Problems & Imaging, 2016, 10 (4) : 1007-1036. doi: 10.3934/ipi.2016030 [17] Hua-Ping Wu, Min Huang, W. H. Ip, Qun-Lin Fan. Algorithms for single-machine scheduling problem with deterioration depending on a novel model. Journal of Industrial & Management Optimization, 2017, 13 (2) : 681-695. doi: 10.3934/jimo.2016040 [18] Ethel Mokotoff. 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Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021144 2020 Impact Factor: 1.801	2022-01-27 13:47:09	{"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": 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": 0.7793856859207153, "perplexity": 7340.271882545968}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305266.34/warc/CC-MAIN-20220127133107-20220127163107-00142.warc.gz"}
https://examcraze.in/ask/average-of-ten-positive-numbers-is-overline-x-if-each-number/	# Average of ten positive numbers is overline x If each number… Are you looking for correct answer of Average of ten positive numbers is overline x If each number…? Here we have shared detailed answer with explanations. ### Average of ten positive numbers is $$\overline x$$. If each number is increased by 10%, then $$\overline x$$ – 1. A. Remains unchanged 2. B. May decrease 3. C. May increase 4. D. Is increased by 10% eqalign{ & Rightarrow frac{{{x_1} + {x_2} + ..... + {x_{10}}}}{{10}} = overline x cr & Rightarrow {x_1} + {x_2} + ..... + {x_{10}} = 10overline x cr & Rightarrow frac{{110}}{{100}}{x_1} + frac{{110}}{{100}}{x_2} + ..... + frac{{110}}{{100}}{x_{10}} = frac{{110}}{{100}} times 10overline x cr & Rightarrow frac{{frac{{110}}{{100}}{x_1} + frac{{110}}{{100}}{x_2} + ..... + frac{{110}}{{100}}{x_{10}}}}{{10}} = frac{{11}}{{10}}overline x cr}	2022-09-29 10:49:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.8176878094673157, "perplexity": 9104.509046265455}, "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/1664030335350.36/warc/CC-MAIN-20220929100506-20220929130506-00050.warc.gz"}
http://www.physicsforums.com/showthread.php?t=173218&page=2	# Saver or spender by wolram Tags: saver, spender PF Gold P: 3,673 Is your philosophy to put money away for retirement or spend it and enjoy it while you can, i seem to take a 50/50 approach, just can not resit buying but know i should save. Sci Advisor P: 5,095 My philosophy has definitely changed with age. When I was just starting out, I spent, but I'd like to think it was for good things, like working on my first house, etc...Still, there was a lot of spending to do. I think now that I am established, I don't have the need to spend a lot other than pay the bills. Every once in a while the mood hits when a cool toy comes along, but that's not very often. Now a days, it's definitely save time. I try to sock away as much as possible into ye olde 401k. HW Helper PF Gold P: 2,328 My approach right now is just to get the debt paid off first and then start saving. I'm not a big spender really. Not a big saver either. I do plan on saving and investing once I'm done school though. P: 2,163 ## Saver or spender Neither, poor me. P: 113 Spender. My life philosophy is that I'm only here for a good time -- not a long time. HW Helper PF Gold P: 2,328 Quote by Beeza Spender. My life philosophy is that I'm only here for a good time -- not a long time. Becareful though. If debt gets out of hand, the debt will control your life and ruin it, so that good time won't exist and a couple years because you have to make up for that last 2 years. PF Gold P: 8,961 I try to save, particularly now that I'm getting old, but whenever there's a surplus something uncontrollable happens to make it go away. For instance, my entire paycheque that I got last Friday was gone by Monday night due to a funeral in another town along with our rent coming due. Then there are vehicle repairs and whatnot that can't be avoided. PF Gold P: 3,673 Quote by jimmysnyder Neither, poor me. Money will come your way some time P: 113 Quote by JasonRox Becareful though. If debt gets out of hand, the debt will control your life and ruin it, so that good time won't exist and a couple years because you have to make up for that last 2 years. I don't have any debt except for the inevitable student loans, which I take sparingly. I just don't save much of my money, but do live within my means. P: 2,163 Quote by wolram Money will come your way some time Not if my wife and kids have any say. Emeritus Sci Advisor PF Gold P: 12,257 My natural inclination has always been to save. Whenever someone gave me money as a present when I was a kid, I'd want to go straight to the bank to put it in my savings account. My one grandmother used to try to insist I was supposed to spend some of it on something fun, but that never stopped me from squirreling it all away. As I've gotten older, I've learned to spend some. I still lean toward saving...I have my retirement account that gets a fixed amount every paycheck (and every few years, I adjust that amount up a bit as I earn more and have a little extra to set aside), and then I have other accounts for goal-oriented saving toward things that are more like investments (i.e., toward a downpayment for my next house), and some for rainy day funds for emergencies, but I also don't want to be such a penny-pincher that I never do anything enjoyable with my life, so I do spend on things like vacations, nice furniture to make my home a pleasant place to live, the ocassional splurge on a gadget or clothes or dinner out. I just don't do it often, and always have in mind a portion of the budget that is available for little splurges that won't cut into my long-term savings so I don't have to worry about going into debt should some emergency come up (i.e., hospitalization, major home repairs, etc.) P: 4,780 I live at home, so I don’t really have any expenses and it allows me to save a Ton of money from work. My parents pay for school, but I try to help out by buying my own books (which end up being around $600 bucks each semester, ouch!). That’s the reason why I’m flying and bought a new bike. I saved that money from working throughout college. There are things I don’t mind paying top dollar for though (bikes, motorcycles, airplanes, RC airplanes) because I learn how they work, how to operate them, get to use them all the time. But there are some things I don’t like buying and complain about. I don’t care for jewelry (other than buying a class ring, as an accomplishment). I dont wear a watch unless I have to dress up and use the one my grandfather gave me (a simple black band swiss watch). I don’t buy CD's that often. I have about as many movies or DVDs as fingers. ( I don’t download them off the net, if I want a song Ill buy it from I-tunes for$0.99). I don’t supped-up my car and put rims on it, or other obnoxious stuff people do around here. I don’t believe in putting money in a bank and letting it sit there. It’s useless. The interest is peanuts. Hopefully if I get into grad school they will pay me $30k a year, so when I graduate I will come out with around$50k. Either I can use that as a down payment on a house or I can buy a used Cessna or something. But I don’t wanna buy something like a new car with it. Honestly, buying a house with it would be the smartest move. My school is only about $4,500 a semester, so its not that bad considering that you can put that on your taxes. I am glad I dont go to a private school that would have cost me almost$20-30k a year. Not worth it IMO. Im not married, dont have kids, dont have cost of living, and I dont give people expensive gifts or any gifts (and I dont want gifts in return, just give me a card at most). I think I spend my money in a smart way. PF Gold P: 3,673 I think i feel guilty for spending a large amount on my (hobby) i could have put it to better uses, when i reckon up how much i have spent the figures seem to scream out IDIOT, but then i think i have done my spending for now and can start saving again. P: 4,780 Hobbies are expensive. If you spent it on motorbikes, money well spent. You got to take apart your bike, fix it, learn about it, and put it back together. I think hobbies are the best way to blow your money. Many people don’t have a hobby. I find those people are the ones obsessed with having a nice car or clothes, because they have nothing else to do. PF Gold P: 3,673 You guessed :smile P: 1,401 My philosophy is to make so much money I cant help but save easier to say than implement.. P: 2,891 right now I have a net total of.. 300 dollars. Canadian. Hey, I'm a student. I tend to save a lot during the summer, and then lose it all somehow during the semester. Related Discussions General Discussion 1 Biology 0	2014-04-24 21:46:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.1889301985502243, "perplexity": 1897.4051470629774}, "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-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00426-ip-10-147-4-33.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/calculating-the-magnetic-field-from-the-hall-voltage.387228/	# Calculating the magnetic field from the Hall Voltage ## Homework Statement You have built a sensor to detect the strength of unknown magnetic fields. You use a rectangular sample of copper that is 14.2 cm wide and 0.5 cm thick. You apply a current of 2.4 A to the copper. You know that there is a magnetic field perpendicular to the current because you measure a Hall Voltage of 0.1μV. What was the magnitude of the magnetic field that you detected? Assume that one electron per atom is available for conduction. (2) Copper has a density of 8.93 g/cm3 and a molar mass of 63.55 g/mol. (3) Remember that 1 mol of any substance contains 6.02 x1023 atoms (Avogadro’s number). ## Homework Equations q vd B = q EH VH = EH d = vd B d n = $$\frac{\rho N_{A}}{M}$$ B = $$\frac{E_{H}}{v_{d}}$$ EH = $$\frac{V_{H}}{d}$$ ## The Attempt at a Solution B = $$\frac{E_{H}}{v_{d}}$$ EH = $$\frac{V_{H}}{d}$$ Therefore B = $$\frac{V_{H}}{v_{d} d}$$ ... If vd = $$\frac{I}{n q A}$$ ... Then B = $$\frac{n q A V_{H}}{I d}$$ If n = $$\frac{\rho N_{A}}{M}$$ Then B = $$\frac{\rho N_{A} q A V_{H}}{M I d}$$ I worked out the following numbers (I don't know whether my error lies here or not)... $$\rho$$ = 8.93 x 10-9 kgm-3 NA = 6.02 x 1023 atoms q = 1.602 x 10-19 C A = 14.2 x 10-2 x 0.5 x 10-2 = 7.1 x 10-4m2 VH = 0.1 x 10-6 v M = 63.55 x 10-3 kgmol-1 I = 2.4 A d = 0.5 x 10-2 m I have been stuck on this question for several hours now, and can't see where I'm going wrong. The answer I'm getting is 8.018 x 10-11T, when the answer expected is between 2 and 4 T apparently. Any help would be much appreciated :) $$\rho$$ = 8.93 x 10-9 kgm-3	2021-02-24 18:30:49	{"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": 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": 0.48182472586631775, "perplexity": 1211.6859869009825}, "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/1614178347293.1/warc/CC-MAIN-20210224165708-20210224195708-00222.warc.gz"}
https://www.gradesaver.com/textbooks/engineering/other-engineering/materials-science-and-engineering-an-introduction/chapter-3-the-structure-of-crystalline-solids-questions-and-problems-page-97/3-5	## Materials Science and Engineering: An Introduction $APF={\text{volume of atomic unit cell}}/{\text{total unit cell volume}}$ $V_{s}=\frac{4}{3}\pi r^3$ As we know BCC has a total of 2 atoms: $V_{s}=2\times\frac{4}{3}\pi r^3=\frac{8}{3}\pi r^3$ Thus: $V_{c}=a^3={\frac{4}{\sqrt{3}}r}^3$ $V_{c}=\frac{64}{3\sqrt{3}}r^3$ APF =$\frac{\frac{8}{3}\pi r^3}{\frac{64}{3\sqrt{3}}r^3}=0.68$	2019-11-22 05:42:08	{"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": 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": 0.3314146399497986, "perplexity": 1759.1524272298773}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496671239.99/warc/CC-MAIN-20191122042047-20191122070047-00286.warc.gz"}
https://everything.explained.today/Arithmetical_hierarchy/	# Arithmetical hierarchy explained Arithmetical hierarchy should not be confused with Levy hierarchy. In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy is important in recursion theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic. The Tarski–Kuratowski algorithm provides an easy way to get an upper bound on the classifications assigned to a formula and the set it defines. The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy to classify additional formulas and sets. ## The arithmetical hierarchy of formulas The arithmetical hierarchy assigns classifications to the formulas in the language of first-order arithmetic. The classifications are denoted 0 \Sigma n and 0 \Pi n for natural numbers n (including 0). The Greek letters here are lightface symbols, which indicates that the formulas do not contain set parameters. If a formula \phi is logically equivalent to a formula without quantifiers, then \phi is assigned the classifications 0 \Sigma 0 and 0 \Pi 0 . Since any formula with bounded quantifiers can be replaced by a formula without quantifiers (for example, \existsx<2,\phi(x) is equivalent to \phi(0)\vee\phi(1) ), we can also allow \phi to have bounded quantifiers. The classifications 0 \Sigma n and 0 \Pi n are defined inductively for every natural number n using the following rules: • If \phi is logically equivalent to a formula of the form \existsm1\existsm2 … \existsmk\psi , where \psi is 0 \Pi n , then \phi is assigned the classification 0 \Sigma n+1 . • If \phi is logically equivalent to a formula of the form \forallm1\forallm2 … \forallmk\psi , where \psi is 0 \Sigma n , then \phi is assigned the classification 0 \Pi n+1 . A 0 \Sigma n formula is equivalent to a formula that begins with some existential quantifiers and alternates n-1 times between series of existential and universal quantifiers; while a 0 \Pi n formula is equivalent to a formula that begins with some universal quantifiers and alternates analogously. Because every first-order formula has a prenex normal form, every formula is assigned at least one classification. Because redundant quantifiers can be added to any formula, once a formula is assigned the classification 0 \Sigma n or 0 \Pi n it will be assigned the classifications 0 \Sigma r and 0 \Pi r for every r > n. The only relevant classification assigned to a formula is thus the one with the least n; all the other classifications can be determined from it. ## The arithmetical hierarchy of sets of natural numbers A set X of natural numbers is defined by a formula φ in the language of Peano arithmetic (the first-order language with symbols "0" for zero, "S" for the successor function, "+" for addition, "×" for multiplication, and "=" for equality), if the elements of X are exactly the numbers that satisfy φ. That is, for all natural numbers n, n\inX\LeftrightarrowN\models\varphi(\underlinen), where \underlinen is the numeral in the language of arithmetic corresponding to n . A set is definable in first-order arithmetic if it is defined by some formula in the language of Peano arithmetic. Each set X of natural numbers that is definable in first-order arithmetic is assigned classifications of the form 0 \Sigma n , 0 \Pi n , and 0 \Delta n , where n is a natural number, as follows. If X is definable by a 0 \Sigma n formula then X is assigned the classification 0 \Sigma n . If X is definable by a 0 \Pi n formula then X is assigned the classification 0 \Pi n . If X is both 0 \Sigma n and 0 \Pi n then X 0 \Delta n . Note that it rarely makes sense to speak of 0 \Delta n formulas; the first quantifier of a formula is either existential or universal. So a 0 \Delta n set is not defined by a 0 \Delta n formula; rather, there are both 0 \Sigma n and 0 \Pi n formulas that define the set. For example, the set of odd natural numbers n is definable by either \forallk(n2 x k) or \existsk(n=2 x k+1) . A parallel definition is used to define the arithmetical hierarchy on finite Cartesian powers of the set of natural numbers. Instead of formulas with one free variable, formulas with k free number variables are used to define the arithmetical hierarchy on sets of k-tuples of natural numbers. These are in fact related by the use of a pairing function. ## Relativized arithmetical hierarchies Just as we can define what it means for a set X to be recursive relative to another set Y by allowing the computation defining X to consult Y as an oracle we can extend this notion to the whole arithmetic hierarchy and define what it means for X to be 0 \Sigma n , 0 \Delta n or 0 \Pi n in Y, denoted respectively 0,Y \Sigma n 0,Y \Delta n and 0,Y \Pi n . To do so, fix a set of integers Y and add a predicate for membership in Y to the language of Peano arithmetic. We then say that X is in 0,Y \Sigma n if it is defined by a 0 \Sigma n formula in this expanded language. In other words, X is 0,Y \Sigma n if it is defined by a 0 \Sigma n formula allowed to ask questions about membership in Y. Alternatively one can view the 0,Y \Sigma n sets as those sets that can be built starting with sets recursive in Y and alternately taking unions and intersections of these sets up to n times. For example, let Y be a set of integers. Let X be the set of numbers divisible by an element of Y. Then X is defined by the formula \phi(n)=\existsm\existst(Y(m)\landm x t=n) so X is in 0,Y \Sigma 1 (actually it is in 0,Y \Delta 0 as well since we could bound both quantifiers by n). ## Arithmetic reducibility and degrees Arithmetical reducibility is an intermediate notion between Turing reducibility and hyperarithmetic reducibility. A set is arithmetical (also arithmetic and arithmetically definable) if it is defined by some formula in the language of Peano arithmetic. Equivalently X is arithmetical if X is 0 \Sigma n or 0 \Pi n for some integer n. A set X is arithmetical in a set Y, denoted X\leqAY , if X is definable as some formula in the language of Peano arithmetic extended by a predicate for membership in Y. Equivalently, X is arithmetical in Y if X is in 0,Y \Sigma n or 0,Y \Pi n for some integer n. A synonym for X\leqAY is: X is arithmetically reducible to Y. The relation X\leqAY is reflexive and transitive, and thus the relation \equivA defined by the rule X\equivAY\LeftrightarrowX\leqAY\landY\leqAX is an equivalence relation. The equivalence classes of this relation are called the arithmetic degrees; they are partially ordered under \leqA . ## The arithmetical hierarchy of subsets of Cantor and Baire space The Cantor space, denoted 2\omega , is the set of all infinite sequences of 0s and 1s; the Baire space, denoted \omega\omega or l{N} , is the set of all infinite sequences of natural numbers. Note that elements of the Cantor space can be identified with sets of integers and elements of the Baire space with functions from integers to integers. The ordinary axiomatization of second-order arithmetic uses a set-based language in which the set quantifiers can naturally be viewed as quantifying over Cantor space. A subset of Cantor space is assigned the classification 0 \Sigma n if it is definable by a 0 \Sigma n formula. The set is assigned the classification 0 \Pi n if it is definable by a 0 \Pi n formula. If the set is both 0 \Sigma n and 0 \Pi n then it is given the additional classification 0 \Delta n . For example, let O\subset2\omega be the set of all infinite binary strings which aren't all 0 (or equivalently the set of all non-empty sets of integers). As O=\{X\in2\omega\|\existsn(X(n)=1)\} we see that O is defined by a 0 \Sigma 1 formula and hence is a 0 \Sigma 1 set. Note that while both the elements of the Cantor space (regarded as sets of integers) and subsets of the Cantor space are classified in arithmetic hierarchies, these are not the same hierarchy. In fact the relationship between the two hierarchies is interesting and non-trivial. For instance the 0 \Pi n elements of the Cantor space are not (in general) the same as the elements X of the Cantor space so that \{X\} is a 0 \Pi n subset of the Cantor space. However, many interesting results relate the two hierarchies. There are two ways that a subset of Baire space can be classified in the arithmetical hierarchy. • A subset of Baire space has a corresponding subset of Cantor space under the map that takes each function from \omega to \omega to the characteristic function of its graph. A subset of Baire space is given the classification 1 \Sigma n , 1 \Pi n , or 1 \Delta n if and only if the corresponding subset of Cantor space has the same classification. • An equivalent definition of the analytical hierarchy on Baire space is given by defining the analytical hierarchy of formulas using a functional version of second-order arithmetic; then the analytical hierarchy on subsets of Cantor space can be defined from the hierarchy on Baire space. This alternate definition gives exactly the same classifications as the first definition. A parallel definition is used to define the arithmetical hierarchy on finite Cartesian powers of Baire space or Cantor space, using formulas with several free variables. The arithmetical hierarchy can be defined on any effective Polish space; the definition is particularly simple for Cantor space and Baire space because they fit with the language of ordinary second-order arithmetic. Note that we can also define the arithmetic hierarchy of subsets of the Cantor and Baire spaces relative to some set of integers. In fact boldface 0 \Sigma n is just the union of 0,Y \Sigma n for all sets of integers Y. Note that the boldface hierarchy is just the standard hierarchy of Borel sets. ## Extensions and variations It is possible to define the arithmetical hierarchy of formulas using a language extended with a function symbol for each primitive recursive function. This variation slightly changes the classification of 0 \Sigma 0 , since using primitive recursive functions in first-order Peano arithmetic requires, in general, an unbounded existential quantifier, and thus some sets that are in 0 \Sigma 0 by this definition are in 0 \Sigma 1 0 \Sigma 1 and thus all higher classes in the hierarchy remain unaffected. A more semantic variation of the hierarchy can be defined on all finitary relations on the natural numbers; the following definition is used. Every computable relation is defined to be 0 \Sigma 0 . The classifications 0 \Sigma n and 0 \Pi n are defined inductively with the following rules. • If the relation R(n1,\ldots,nl,m1,\ldots,mk) is 0 \Sigma n then the relation S(n1,\ldots,nl)=\forallm1 … \forallmkR(n1,\ldots,nl,m1,\ldots,mk) is defined to be 0 \Pi n+1 • If the relation R(n1,\ldots,nl,m1,\ldots,mk) is 0 \Pi n then the relation S(n1,\ldots,nl)=\existsm1 … \existsmkR(n1,\ldots,nl,m1,\ldots,mk) is defined to be 0 \Sigma n+1 This variation slightly changes the classification of some sets. In particular, 0 \Sigma 0 , as a class of sets (definable by the relations in the class), is identical to 0 \Delta 1 as the latter was formerly defined. It can be extended to cover finitary relations on the natural numbers, Baire space, and Cantor space. ## Meaning of the notation The following meanings can be attached to the notation for the arithmetical hierarchy on formulas. The subscript n in the symbols 0 \Sigma n and 0 \Pi n indicates the number of alternations of blocks of universal and existential number quantifiers that are used in a formula. Moreover, the outermost block is existential in 0 \Sigma n formulas and universal in 0 \Pi n formulas. The superscript 0 in the symbols 0 \Sigma n , 0 \Pi n , and 0 \Delta n indicates the type of the objects being quantified over. Type 0 objects are natural numbers, and objects of type i+1 are functions that map the set of objects of type i to the natural numbers. Quantification over higher type objects, such as functions from natural numbers to natural numbers, is described by a superscript greater than 0, as in the analytical hierarchy. The superscript 0 indicates quantifiers over numbers, the superscript 1 would indicate quantification over functions from numbers to numbers (type 1 objects), the superscript 2 would correspond to quantification over functions that take a type 1 object and return a number, and so on. ## Examples • The 0 \Sigma 1 sets of numbers are those definable by a formula of the form \existsn1\existsnk\psi(n1,\ldots,nk,m) where \psi has only bounded quantifiers. These are exactly the recursively enumerable sets. • The set of natural numbers that are indices for Turing machines that compute total functions is 0 \Pi 2 . Intuitively, an index e falls into this set if and only if for every m "there is an s such that the Turing machine with index e halts on input m after s steps”. A complete proof would show that the property displayed in quotes in the previous sentence is definable in the language of Peano arithmetic by a 0 \Sigma 1 formula. • Every 0 \Sigma 1 subset of Baire space or Cantor space is an open set in the usual topology on the space. Moreover, for any such set there is a computable enumeration of Gödel numbers of basic open sets whose union is the original set. For this reason, 0 \Sigma 1 sets are sometimes called effectively open. Similarly, every 0 \Pi 1 set is closed and the 0 \Pi 1 sets are sometimes called effectively closed. • Every arithmetical subset of Cantor space or Baire space is a Borel set. The lightface Borel hierarchy extends the arithmetical hierarchy to include additional Borel sets. For example, every 0 \Pi 2 subset of Cantor or Baire space is a G\delta set (that is, a set which equals the intersection of countably many open sets). Moreover, each of these open sets is 0 \Sigma 1 and the list of Gödel numbers of these open sets has a computable enumeration. If \phi(X,n,m) is a 0 \Sigma 0 formula with a free set variable X and free number variables n,m then the 0 \Pi 2 set \{X\mid\foralln\existsm\phi(X,n,m)\} is the intersection of the 0 \Sigma 1 sets of the form \{X\mid\existsm\phi(X,n,m)\} as n ranges over the set of natural numbers. • The 0 \Sigma 0 formulas can be checked by going over all cases one by one, which is possible because all their quantifiers are bounded. The time for this is polynomial in their arguments (e.g. polynomial in n for \varphi(n) ); thus their corresponding decision problems are included in E (as n is exponential in its number of bits). This no longer holds under alternative definitions of 0 \Sigma 0 , which allow the use of primitive recursive functions, as now the quantifiers may be bound by any primitive recursive function of the arguments. • The 0 \Sigma 0 formulas under an alternative definition, that allows the use of primitive recursive functions with bounded quantifiers, correspond to sets of integers of the form \{n:f(n)=0\} for a primitive recursive function f. This is because allowing bounded quantifier adds nothing to the definition: for a primitive recursive f, \forallk<n:f(k)=0 is the same as f(0)+f(1)+...f(n)=0 , and \existsk<n:f(k)=0 is the same as f(0)f(1)...f(n)=0 ; with course-of-values recursion each of these can be defined by a single primitive recursion function. ## Properties The following properties hold for the arithmetical hierarchy of sets of natural numbers and the arithmetical hierarchy of subsets of Cantor or Baire space. • The collections 0 \Pi n and 0 \Sigma n are closed under finite unions and finite intersections of their respective elements. • A set is 0 \Sigma n if and only if its complement is 0 \Pi n . A set is 0 \Delta n if and only if the set is both 0 \Sigma n and 0 \Pi n , in which case its complement will also be 0 \Delta n . • The inclusions 0 \Pi n \subsetneq 0 \Pi n+1 and 0 \Sigma n \subsetneq 0 \Sigma n+1 hold for all n . Thus the hierarchy does not collapse. This is a direct consequence of Post's theorem. • The inclusions 0 \Delta n \subsetneq 0 \Pi n , 0 \Delta n \subsetneq 0 \Sigma n and 0 \Sigma n \cup 0 \Pi n \subsetneq 0 \Delta n+1 hold for n\geq1 . • For example, for a universal Turing machine T, the set of pairs (n,m) such that T halts on n but not on m, is in 0 \Delta 2 (being computable with an oracle to the halting problem) but not in 0 \Sigma 1 \cup 0 \Pi 1 , . 0 \Sigma 0 = 0 \Pi 0 = 0 \Delta 0 = 0 \Sigma 0 \cup 0 \Pi 0 \subset 0 \Delta 1 . The inclusion is strict by the definition given in this article, but an identity with 0 \Delta 1 holds under one of the variations of the definition given above. ## Relation to Turing machines ### Computable sets If S is a Turing computable set, then both S and its complement are recursively enumerable (if T is a Turing machine giving 1 for inputs in S and 0 otherwise, we may build a Turing machine halting only on the former, and another halting only on the latter). By Post's theorem, both S and its complement are in 0 \Sigma 1 . This means that S is both in 0 \Sigma 1 and in 0 \Pi 1 , and hence it is in 0 \Delta 1 . Similarly, for every set S in 0 \Delta 1 , both S and its complement are in 0 \Sigma 1 and are therefore (by Post's theorem) recursively enumerable by some Turing machines T1 and T2, respectively. For every number n, exactly one of these halts. We may therefore construct a Turing machine T that alternates between T1 and T2, halting and returning 1 when the former halts or halting and returning 0 when the latter halts. Thus T halts on every n and returns whether it is in S, So S is computable. ### Summary of main results The Turing computable sets of natural numbers are exactly the sets at level 0 \Delta 1 of the arithmetical hierarchy. The recursively enumerable sets are exactly the sets at level 0 \Sigma 1 . No oracle machine is capable of solving its own halting problem (a variation of Turing's proof applies). The halting problem for a 0,Y \Delta n oracle in fact sits in 0,Y \Sigma n+1 . Post's theorem establishes a close connection between the arithmetical hierarchy of sets of natural numbers and the Turing degrees. In particular, it establishes the following facts for all n ≥ 1: • The set \emptyset(n) (the nth Turing jump of the empty set) is many-one complete in 0 \Sigma n . • The set N\setminus\emptyset(n) is many-one complete in 0 \Pi n . • The set \emptyset(n-1) is Turing complete in 0 \Delta n . The polynomial hierarchy is a "feasible resource-bounded" version of the arithmetical hierarchy in which polynomial length bounds are placed on the numbers involved (or, equivalently, polynomial time bounds are placed on the Turing machines involved). It gives a finer classification of some sets of natural numbers that are at level 0 \Delta 1 of the arithmetical hierarchy.	2023-03-24 21:25:12	{"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": 0.9894222617149353, "perplexity": 682.9403794933232}, "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-20230324211121-20230325001121-00435.warc.gz"}
https://www.gamedev.net/forums/topic/629022-distance-over-time/	Public Group # Distance over Time This topic is 2175 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic. ## Recommended Posts i am currently trying to model a projectile in Physx. I would like a force to be applied to this projectile every ten seconds. I have never used time functions before. I currently have access to time using QueryPerformanceFrequency(&freq); QueryPerformanceCounter(&previousTime); and with this i get the elapsed time. but i am not sure how to create a ten second counter. something that will just return tru when the elapsed time reaches ten seconds. I know this is simple but for some reason my mind is stumped. i just need some one to potentially explain how this works. thanks Edited by greenzone ##### Share on other sites // simplified pseudocode class Timer { public: Timer(int duration) { this.duration = duration; this.start = GetCurrentTime(); // replace GetCurrentTime() with whatever the right function is for your target API } void restart() { this.start = GetCurrentTime(); } bool finished() { return GetCurrentTime() >= this.start + this.duration; } private: int start, duration; }; Of course, you're free to ask questions, but I tried to make it uber simple and hopefully self explanatory. ##### Share on other sites i am currently trying to model a projectile in Physx. I would like a force to be applied to this projectile every ten seconds. I have never used time functions before. I currently have access to time using QueryPerformanceFrequency(&freq); QueryPerformanceCounter(&previousTime); and with this i get the elapsed time. but i am not sure how to create a ten second counter. something that will just return tru when the elapsed time reaches ten seconds. I know this is simple but for some reason my mind is stumped. i just need some one to potentially explain how this works. thanks If you are trying to model a projectile, why are you applying a force every 10 seconds? By the definition of a projectile, the only force acting on the projectile should be gravity. but gravity is a constant force... it is not applied every 10 seconds. If a force is being applied to the body other than gravity, than it is not a projectile and you are trying to model something different (imo). Even if I assume you mean some sort of thrust (not a projectile) that is being applied to the "projectile", what sorts of thrust would occur every 10 seconds? I imagine that it is possible that you want some sort of "homing" projectile which must constantly correct its angle to locate the target. In that context perhaps a 10 second interval for course correction would make sense. If that is the case then this link: http://answers.unity3d.com/questions/48836/determining-the-torque-needed-to-rotate-an-object.html may help I only bring this up because from the statement of your problem it is possible that you may need to reconsider the approach. Look at the PhysX documentation and look to see if you can find something about applying forces and impulses. I think this may help you. ##### Share on other sites shadowisadog, your right i am setting up a homing missile and i need to correct the tangent velocity angle for the uniform circular motion to work correctly. I only wrote every seconds because i didn't have a precise time in my head lol. realistically it would be tenth or hundredth of a second depending on how fast the missile is moving. thanks for the link as well. 1. 1 2. 2 Rutin 23 3. 3 JoeJ 20 4. 4 5. 5 • 22 • 40 • 23 • 13 • 13 • ### Forum Statistics • Total Topics 631733 • Total Posts 3001927 ×	2018-07-17 23:46:58	{"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": 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": 0.3094382882118225, "perplexity": 894.4556646971104}, "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-2018-30/segments/1531676589932.22/warc/CC-MAIN-20180717222930-20180718002930-00096.warc.gz"}
https://academic.oup.com/cercor/article/11/10/946/280028/Spectral-and-Temporal-Processing-in-Human-Auditory	## Abstract We used positron emission tomography to examine the response of human auditory cortex to spectral and temporal variation. Volunteers listened to sequences derived from a standard stimulus, consisting of two pure tones separated by one octave alternating with a random duty cycle. In one series of five scans, spectral information (tone spacing) remained constant while speed of alternation was doubled at each level. In another five scans, speed was kept constant while the number of tones sampled within the octave was doubled at each level, resulting in increasingly fine frequency differences. Results indicated that (i) the core auditory cortex in both hemispheres responded to temporal variation, while the anterior superior temporal areas bilaterally responded to the spectral variation; and (ii) responses to the temporal features were weighted towards the left, while responses to the spectral features were weighted towards the right. These findings confirm the specialization of the left-hemisphere auditory cortex for rapid temporal processing, and indicate that core areas are especially involved in these processes. The results also indicate a complementary hemispheric specialization in right-hemisphere belt cortical areas for spectral processing. The data provide a unifying framework to explain hemispheric asymmetries in processing speech and tonal patterns. We propose that differences exist in the temporal and spectral resolution of corresponding fields in the two hemispheres, and that they may be related to anatomical hemispheric asymmetries in myelination and spacing of cortical columns. ## Inroduction A salient aspect of hemispheric specialization in the human brain pertains to the processing of auditory speech signals. Much evidence exists indicating that speech sounds are preferentially processed within left auditory cortical areas. Classic observations of aphasic patients have pointed to the importance of left posterior temporal cortex in speech comprehension, and more recent evidence from neuroimaging also indicates recruitment of left temporal and frontal cortical regions for the analysis of speech information [see Zatorre and Binder for review (Zatorre and Binder, 2000)]. However, the basis for this specialization remains unclear. One relevant set of findings that is often overlooked in examining the specialization of speech relates to evidence for specialization of pitch processing. Lesions to right but not left primary auditory cortical areas, for example, specifically impair the perception of missing fundamental pitch (Zatorre, 1988), and direction of pitch change (Johnsrude et al., 2000); more generally, damage to the superior temporal cortex on the right also affects a variety of other tonal or spectral processing tasks (Milner, 1962; Zatorre, 1985; Sidtis and Volpe, 1988; Divenyi and Robinson, 1989; Robin et al., 1990; Zatorre and Samson, 1991; Zatorre and Halpern, 1993). Also, considerable functional imaging evidence is consistent with a greater role for right auditory cortices in tonal pitch perception (Zatorre et al., 1994; Binder et al., 1997; Griffiths et al., 1999; Halpern and Zatorre, 1999; Hugdahl et al., 1999; Perry et al., 1999). The complementary specializations for speech and tonal materials revealed by this literature suggest a hypothesis to explain the functional hemispheric differences. Whereas the analysis of speech requires good temporal resolution (Tallal et al., 1993) to process rapidly changing energy peaks (formants) that are characteristic of many speech consonants, it can be argued that tonal processes instead require good frequency resolution. The hypothesis we wish to test is that human auditory cortex is functionally segregated, such that different fields are selectively sensitive to temporal or spectral acoustic features, and that differences exist in the temporal and spectral resolving power between corresponding cortical fields in the two hemispheres. Considerable evidence already exists that cortical specialization for speech may be related to rapid auditory processing. For example, deficits in temporal processing in the tens of milliseconds range have been demonstrated in a variety of language disorders (Efron, 1963; Tallal and Newcombe, 1978; Phillips and Farmer, 1990; Fitch et al., 1997). Recent brain imaging data also indicate better temporal resolution for the left auditory cortex (Belin et al., 1998). It has also been shown that behaviorally, speech recognition can be accomplished with primarily temporal cues (Shannon et al., 1995). What is not yet known is whether distinct cortical fields are primarily responsive to temporal as opposed to spectral sound features, and whether their response properties differ in the two hemispheres. The present investigation was carried out to examine this question directly using functional imaging. We constructed nonverbal stimuli that varied independently and systematically along spectral and temporal dimensions. We predicted that increasing the rate of temporal change would preferentially recruit left auditory cortical areas, while increasing the number of spectral elements would engage right auditory cortical regions more strongly. ## Materials and Methods ### Subjects Twelve healthy, right-handed volunteers (half of each sex) with normal hearing participated in the experiment after giving informed consent. ### Stimuli The stimuli consisted of pure-tone patterns in which the frequency and duration of the individual elements were varied systematically. At each frequency transition, tones were ramped on and off in counterphase using a cosine function with a 10 ms duration, thus maintaining a nearly constant total amplitude envelope and avoiding transients (Fig. 1a). Two stimulus parameters — rate of frequency modulation and spectral distribution of elements within the pattern — were varied independently in the two conditions of the experiment, while keeping the frequency range and amplitude constant at all times. The parametric changes in the two dimensions were applied starting with the same standard stimulus condition, consisting of two tones separated by one octave with a relatively slow rate of alternation (see Fig. 1b). This standard condition was then subjected to five levels of temporal variation and five levels of spectral variation, for a total of 11 scans. Throughout a given scan the parameters were held constant, so that variation in the desired parameter occurred only across scans. In the temporal condition, the frequencies used were fixed at 500 and 1000 Hz, thus keeping a constant separation of one octave. The rate of alternation of these two frequencies was varied from relatively slower to relatively faster across each of five different scans, doubling the speed with each successive condition. In order to avoid spectral changes that would emerge if a constant rate of alternation were used, the stimuli were generated using a randomly determined duty cycle, sampled from a distribution with the following characteristics: if the shortest duration used for a given condition is t, then the range of durations sampled varied from t to nt, where n is an integer, and the probability of occurrence of any given duration is ½n. Thus, the duration with the highest probability of occurrence (½1 = 0.50) will be t, the shortest one for a given condition; the next longer tone would have a duration of 2t, and would occur with a probability of ½2 = 0.25; the duration 3t would occur with a probability 0.125, and so forth. These stimulus sequences are illustrated in Figure 1b (left side). In each of the five scanning conditions, the value of the parameter t varies from the slowest, at 667 ms (the standard stimulus), to the next faster with t = 333 ms, and so forth to the fastest condition, where t = 21 ms. The latter value corresponds to the fastest range of temporal change most relevant for speech analysis (Phillips and Farmer, 1990; Tallal et al., 1993). The figure also illustrates Fourier spectra of these sequences, showing that there is essentially no change in spectral distribution across conditions, apart from some minimal spread of energy for the fastest (t = 21) condition. The spectral peaks corresponding to the two tones remain fixed, and what differences do exist are on the order of 20 dB below these peaks. Thus, the random duty cycle manipulation successfully avoids major artifacts in the spectral domain arising from the change in rate of presentation. ### Procedure Stimuli were presented binaurally over insert earphones at an intensity of between 73 and 78 dB SPL. The order of presentation of all conditions was counterbalanced across subjects according to a Latin square design. Subjects were familiarized with the appropriate stimulus sequence prior to each scan; they were then instructed to listen carefully to the continuous stimulus sequence with their eyes closed but not to perform any explicit task. Stimulation was started several seconds before scanning began, and continued uninterrupted for the 60 s of scanning. In addition, a silent baseline condition, in which no stimulation was presented, was interspersed with the others. ### Positron Emission Tomography (PET) Scanning PET scans were obtained with a Siemens Exact HR+ tomograph operating in three-dimensional acquisition mode. The distribution of cerebral blood flow (CBF) was measured during each 60 s scan using the H2O15 water bolus method (Raichle et al., 1983). Magnetic resonance imaging (MRI) scans (160 1-mm-thick slices) were also obtained for each subject with a 1.5 T Phillips ACS system to provide anatomical detail. CBF images were reconstructed using a 14 mm Hanning filter, normalized for differences in global CBF and co-registered with the individual MRI data (Evans et al., 1992). Each matched MRI/PET data set was then linearly resampled into the standardized Talairach and Tournoux stereotaxic coordinate system (Talairach and Tournoux, 1988) via an automated feature-matching algorithm (Collins et al., 1994). ### Statistical Analysis of CBF Changes Regression maps (Paus et al., 1996) were calculated to assess the significance of the relationship between each input parameter (spectral and temporal variation) and CBF (i.e. their linear regression). The data set for this analysis consisted of normalized CBF values obtained in each subject during each of the six conditions (standard plus five levels of parametric variation), yielding a total of 72 image volumes. The effect of the variation in spectral or temporal parameter was assessed by means of analysis of covariance, with subjects as a main effect and the parameter number as a covariate. The following model was fitted: $\mathit{E}(\mathit{y}_{\mathit{ij}})\ {=}\ \mathit{a}_{\mathit{i}}\ {+}\ {\beta}_{P}\mathit{s}_{\mathit{ij}}$ where yij is the normalized CBF of subject i on scan j, and sij is the parameter value at scan j. The subject effect (ai) is removed and the parameter of interest is the slope βP of the effect of the parametric change on CBF. In addition to the covariation analysis, a categorical comparison was also performed, contrasting the spectral conditions to the temporal conditions directly. For this analysis the CBF values corresponding to the five conditions in which spectral variation occurred were averaged, and this volume was compared with the average of the five conditions in which temporal variation occurred. The resulting volume was searched for regions of significant CBF change. Differences reflect areas whose CBF response is greater to stimuli varying in one parameter than the other, and vice versa. The significance of focal CBF changes for both types of analysis was assessed by a method based on three-dimensional Gaussian random-field theory (Worsley et al., 1992), which corrects for the multiple comparisons involved in searching across a volume. Because we wished to test hypotheses concerning the temporal neocortex, we restricted the search volume a priori, which allows the use of a lower threshold value and hence greater sensitivity within this region. Values equal to or exceeding a criterion of t = 3.2 were considered significant (P < 0.01, two-tailed), yielding a false-positive rate of 0.52 in 60 resolution elements (each of which has dimensions 14 × 14 × 14 mm), if the volume of temporal cortex gray matter is roughly 170 cm3. ## Results PET data were analyzed using a two-step approach. First, the image volumes were searched for CBF changes in the superior temporal region that covaried significantly as a function of the two input parameters (temporal and spectral) in order to identify the location of significant CBF change in temporal cortices. The response within these regions was extracted by computing the CBF value relative to the silent baseline condition in a spherical region of interest (radius 5 mm) centered on the coordinates from the covariation analysis. These CBF values were then used to compute slopes as a function of each parameter, in order to test for the predicted hemispheric differences. Secondly, as an additional way to examine hemispheric asymmetries, the direct comparison of spectral to temporal conditions was analyzed via subtraction of these conditions from one another, and region of interest analyses were also conducted on the resulting CBF values. ### Covariation Analyses The analysis of covariation for the temporal parameter revealed only two regions of significant CBF covariation (Fig. 2; Table 1); these foci were located in the left and right Heschl's gyri (HG), as determined from inspection of the average anatomical MRIs, and by comparison to anatomical probabilistic maps (Penhune et al., 1996). Analysis of covariation for the spectral parameter revealed three regions located within the superior temporal gyri (Fig. 2; Table 1): two roughly symmetrically located in a region of the STG anterior to HG in each hemisphere, and a third in a region in the upper bank of the right superior temporal sulcus (STS), posterior to HG, which was not matched by a symmetrical area on the left (Fig. 2). ## Discussion ### Core versus Belt Areas The finding that primary auditory regions (core areas) in both hemispheres preferentially respond to acoustic temporal features, whereas more anterior STG regions (belt or parabelt areas) respond preferentially to spectral features fits with recent views about the hierarchical arrangement of primate auditory cortex (Kaas et al., 1999). One consistent finding is that pure tone responses are generally best observed in core regions, whereas lateral belt areas have more complex response properties and are often sensitive to stimuli with wider bandwidths (Rauschecker et al., 1995; Rauschecker, 1998). Since belt areas receive projections from the core areas, and hence integrate inputs from more narrowly tuned core units, one might expect the spectral set of stimuli to recruit belt over core areas (Rauschecker et al., 1997) because they involve complex spectral changes over time, and this was indeed what we observed. However, spectral integration is usually understood in terms of multiple frequencies present simultaneously. In the present study, since only a single frequency was presented at any one time, the effect we observed most likely reflects the integration of different frequencies over time, and may therefore be tied to interactions between units tuned to different frequency bands (Brosch et al., 1999). The observation that a region in the upper bank of the right STS also responds to increasing spectral complexity fits with recent fMRI findings that STS areas are sensitive to the acoustic shape of human vocal sounds (Belin et al., 2000), since both situations would require sensitivity to subtle changes in spectral energy distribution. More generally, the finding that an STS region responds to these sorts of stimuli is in agreement with the proposal that these regions form part of a ventral stream specialized for object-feature processing, in which spectral information would play an important role (Rauschecker, 1998). The high CBF response to the temporally varying stimuli in the core area (HG) in both hemispheres presumably reflects the high temporal sensitivity of neurons in this region, but because the amplitude envelope was constant (Fig. 1a), this sensitivity must be specifically related to the temporal rate of frequency change, independently of amplitude modulation. ### Hemispheric Differences The findings relating to hemispheric differences support the hypothesis that a relative tradeoff may exist between temporal and spectral resolution of the two hemispheres. In a linear system, temporal and spectral resolution are inversely related; although the auditory nervous system is a nonlinear, distributed system, and hence would be unlikely to show a direct reciprocal relation, the differential hemispheric responses observed in the present study may nevertheless reflect the auditory nervous system's adaptation to the presence of spectral and temporal features in the acoustic environment. That is, maximizing the processing of rapidly changing information (high temporal resolution) might set an upper limit on frequency resolution; this might explain why the changes in CBF observed within the left anterior STG area were greater to the temporal than to the spectral parameter. Conversely, higher frequency resolution might entail a relative decrement in temporal processing, hence the inverse pattern seen in the right anterior STG and STS regions. Neurophysiological recordings in macaque monkeys have shown that auditory cortical neurons are highly sensitive to both spectral and temporal features of sounds (Phillips, 1993; Steinschneider et al., 1995; deCharms et al., 1998). Data from several species indicate that core regions often have higher frequency resolution than do belt areas, which may not fit with the current data. However, Eggermont noted that in the AI and AII fields of the cat there was an inverse relation between bandwidth and temporal resolution (Eggermont, 1998), consistent with our proposal. Existing single-unit data are insufficiently certain about the response properties of cortical fields, particularly outside the core, to allow prediction of their responses to stimuli such as those used here; moreover, the relation between blood flow measures and single-unit responses is complex. Further, homologies between monkey and human auditory cortex have not been worked out, and it is not yet known whether auditory cortical architecture in humans and other primates differs qualitatively. Despite these uncertainties, the neuronal functional properties derived from animal studies suggest that the model proposed above is at least plausible, in terms of the type of acoustic information to which neurons are sensitive. Another parallel with the animal literature is provided by evidence that hemispheric differences also exist in several species. For example, Fitch et al. found rate-dependent left-hemisphere lateralization effects for rapidly changing tone sequences in the rat (Fitch et al., 1993). Wetzel et al. report right-hemisphere lateralization for processing frequency modulation in the Mongolian gerbil (Wetzel et al., 1998). These findings suggest that the laterality effects discussed in the present paper may not be limited to the human nervous system; if so, then perhaps they should best be characterized as precursors to the development of speech and tonal processing, rather than consequences thereof. The differential temporal and spectral processing capacities in the auditory areas of the two hemispheres, and the hypothesis that these arise from differences in spectral/temporal resolution, help to explain a variety of previous results. The present findings confirm and extend the idea that left auditory regions may be specialized for rapid temporal processing (Tallal et al., 1993), but additionally they provide a unifying framework to account for the complementary specialization of the right auditory cortex. The hypothesis is consistent with prior studies which have shown that damage to left auditory cortical areas often results in deficits in temporal processing that are manifested as speech disorders (Efron, 1963; Phillips and Farmer, 1990; Tallal et al., 1993). The hypothesis is also supported by functional imaging data that indicate left auditory cortex advantages for processing rapidly changing formant transitions in speech or pseudospeech syllables (Fiez et al., 1995; Belin et al., 1998). Additional evidence specifically indicating that left auditory cortical units have higher temporal resolution comes from recent electrophysiological recordings (Liégeois-Chauvel et al., 1999) showing that responses within left HG encode the voice-onset time of a consonant, whereas the right HG did not show sensitivity to this temporal parameter. Even more striking, Liégeois-Chauvel et al. (2001) showed that intracortically recorded auditory evoked potentials were more sharply tuned to frequency in the right auditory cortex than in the left, consistent with the pattern observed in the present study. The hypothesis proposed in this paper also helps to clarify findings of impaired pitch and spectral processing after lesions of right auditory cortical areas, as well as brain imaging evidence in a wide variety of tonal processing tasks. The findings of Johnsrude et al. (Johnsrude et al., 2000) are particularly relevant in this respect: they found that lesions encroaching onto the right (but not left) HG resulted in a specific deficit for judging direction of pitch change. Notably, this impairment did not preclude most patients from completing the task, but raised their threshold by a factor of four. Such a finding can be explained if spectral resolution underlies hemispheric differences: damage to the right auditory cortex would disrupt the finer-grained mechanism but would leave intact the left auditory cortical areas which presumably have coarser resolution, hence leading to increased thresholds but not a complete inability to discriminate. The findings of Robin et al. (Robin et al., 1990) are also relevant, as they too found that damage to association cortices in the right hemisphere resulted in spectral but not temporal processing deficits, while the converse was observed after left-hemisphere damage. Conversely, aspects of tonal processing that depend on small temporal differences, such as discrimination of the temporal microstructure of familiar tunes, are more affected by damage to the left than to the right temporal lobe (Samson et al., 2001). ### Anatomical Considerations If the hypothesis suggested here is valid, the question arises as to how it is implemented in the auditory cortex. A variety of findings of anatomical and structural asymmetries in human auditory cortices are relevant to this question. In-vivo volumetric measures of HG from MRI scanning have shown a greater volume of white matter underlying the left HG as compared with the right, which could be a consequence of a greater number or denser myelination of fibers into the primary auditory cortex (Penhune et al., 1996). The latter interpretation is supported by post-mortem tissue analysis of the posterior temporal lobe (Anderson et al., 1999), which confirmed a greater volumer of white-matter tissue on the left than on the right, and showed that this was due to greater myelin sheath thickness on the left. In addition, Seldon (Seldon, 1981) reported that cortical columns in the left auditory cortex were more widely spaced than those on the right, and Galuske et al. (Galuske et al., 2000) found wider spacing of intrinsic connections on the left. In addition, Hutsler and Gazzaniga (Hutsler and Gazzaniga, 1996) reported larger left than right layer IV pyramidal cells in the human auditory cortex. The foregoing anatomical differences are consistent with the model proposed here in that greater myelination on the left would allow for faster conduction, thereby leading to greater sensitivity to rapid acoustic changes. At the same time, one may speculate that the wider spacing of cortical columns and greater intrinsic connections on the left would allow integration over larger tonotopically organized areas, thereby leading to poorer spectral resolution. The converse arguments apply to the right auditory cortices, since the structural features would appear to favor higher frequency resolution but slower transmission. Thus, these microstructural anatomical differences between the two hemispheres might provide the neural substrate for the functional differences in spectral/temporal resolution observed in the present study (Klingberg et al., 2000). The present model thus suggests that relatively subtle quantitative differences in neural response properties, present early in the cortical processing stream, may lead to qualitatively distinct functional roles for higher-order processes. The organization of intra-hemispheric local circuits (Ringo et al., 1994) might be sensitive to these initial processing advantages and could lead to more general functional specialization within the hemisphere. The left hemisphere's predominant role in many complex linguistic functions may thus be tied to a slight initial advantage in decoding speech sounds. We speculate that the important role of the right hemisphere in many, though not necessarily all, aspects of musical perception [cf. Peretz et al. (Peretz et al., 1994)], might then conceivably have arisen as a consequence of this specialization. ## Notes We thank B. Pike and A.C. Evans for access to the McConnell Brain Imaging Centre, and its personnel for their assistance; P. Ahad and M. Bouffard for technical expertise; and W. Serniclaes for helpful discussion. Supported by grants MT11541 and GR13972 from the Canadian Institute for Health Research and an award from the McDonnell Pew Cognitive Neuroscience Program to the first author, and by fellowships from France-Télécom and INSERM-FRSQ to the second author. Table 1 Covariation analysis Parameter Region x y z t Abbreviations: HG: Heschl's gyrus; STG: superior temporal gyrus; STS: superior temporal sulcus. Temporal left HG –47 –18 6.14 right HG 50 –11 5.61 Spectral left STG anterior –51 –13 3.24 right STG anterior 51 –2 3.50 right STS 62 –37 3.30 Parameter Region x y z t Abbreviations: HG: Heschl's gyrus; STG: superior temporal gyrus; STS: superior temporal sulcus. Temporal left HG –47 –18 6.14 right HG 50 –11 5.61 Spectral left STG anterior –51 –13 3.24 right STG anterior 51 –2 3.50 right STS 62 –37 3.30 Table 2 Subtraction analysis Comparison Region x y z t Abbreviations: HG: Heschl's gyrus; STG: superior temporal gyrus. Temporal–spectral left HG –47 –18 5.08 right HG 40 –23 3.43 Spectral–temporal left STG anterior –44 14 3.80 right STG anterior 50 10 –6 4.58 Comparison Region x y z t Abbreviations: HG: Heschl's gyrus; STG: superior temporal gyrus. Temporal–spectral left HG –47 –18 5.08 right HG 40 –23 3.43 Spectral–temporal left STG anterior –44 14 3.80 right STG anterior 50 10 –6 4.58 Figure 1. (a) Detail of transition between frequencies used in all stimulus sequences. The top graph shows how the amplitude profile of each tone was ramped in cosine counterphase; the bottom graph shows the resulting waveform. Note the smooth change of frequency, devoid of discontinuities or any variation in amplitude envelope at the transition point. (b) Schematic representation of stimuli used. Each pair of panels shows an excerpt of a stimulus sequence represented on the left as a spectrogram (frequency as a function of time) and on the right as a Fourier spectrum (amplitude as a function of frequency; integrated over a 10 s time window). The top pair of panels represents the standard stimulus: two tones with a frequency separation f of 1200 cents (one octave), with a fastest temporal change of t = 667 ms. The three sets of panels along the left of the figure illustrate three levels of change on the temporal parameter, such that t = 333, 83 or 21 ms, the latter corresponding to the fastest value used, while f is held constant. The spectrogram shows how the stimuli change progressively faster, while the Fourier spectra show two fixed spectral peaks corresponding to the two frequencies, with only a minimal spread of energy as the rate of alternation is increased. The three sets of panels along the right side of the figure illustrate three levels of change on the spectral parameter, with the minimum frequency difference f decreasing to 600, 150 and 37.5 cents. The value of t is held constant at 667 ms. Note that the Fourier spectra show increasing numbers of more finely spaced frequency components as the spectral parameter changes. Figure 1. (a) Detail of transition between frequencies used in all stimulus sequences. The top graph shows how the amplitude profile of each tone was ramped in cosine counterphase; the bottom graph shows the resulting waveform. Note the smooth change of frequency, devoid of discontinuities or any variation in amplitude envelope at the transition point. (b) Schematic representation of stimuli used. Each pair of panels shows an excerpt of a stimulus sequence represented on the left as a spectrogram (frequency as a function of time) and on the right as a Fourier spectrum (amplitude as a function of frequency; integrated over a 10 s time window). The top pair of panels represents the standard stimulus: two tones with a frequency separation f of 1200 cents (one octave), with a fastest temporal change of t = 667 ms. The three sets of panels along the left of the figure illustrate three levels of change on the temporal parameter, such that t = 333, 83 or 21 ms, the latter corresponding to the fastest value used, while f is held constant. The spectrogram shows how the stimuli change progressively faster, while the Fourier spectra show two fixed spectral peaks corresponding to the two frequencies, with only a minimal spread of energy as the rate of alternation is increased. The three sets of panels along the right side of the figure illustrate three levels of change on the spectral parameter, with the minimum frequency difference f decreasing to 600, 150 and 37.5 cents. The value of t is held constant at 667 ms. Note that the Fourier spectra show increasing numbers of more finely spaced frequency components as the spectral parameter changes. Figure 2. (Top panel) Three-dimensional rendering of the CBF data from the covariation analyses onto a magnetic resonance image of a representative individual subject's brain, viewed from the right. The level of the section (z = 3mm) is indicated in the inset. The green areas correspond to the regions showing significant covariation of CBF with increasing rate of temporal change, while the red areas correspond to regions whose CBF increased as a function of change in the spectral parameter (precise stereotaxic coordinates are given in Table 1). H indicates the stem of Heschl's gyrus in each hemisphere; STS indicates the superior temporal sulcus. Note that the temporal covariation sites are located within Heschlapos;s gyri in the two hemispheres, while the spectral covariations are located anterior to the sites covarying with the temporal stimulus parameter. An additional posterior site of spectral covariation is located within the STS in the right hemisphere only. (Top panel) Three-dimensional rendering of the CBF data from the covariation analyses onto a magnetic resonance image of a representative individual subject's brain, viewed from the right. The level of the section (z = 3mm) is indicated in the inset. The green areas correspond to the regions showing significant covariation of CBF with increasing rate of temporal change, while the red areas correspond to regions whose CBF increased as a function of change in the spectral parameter (precise stereotaxic coordinates are given in Table 1). H indicates the stem of Heschl's gyrus in each hemisphere; STS indicates the superior temporal sulcus. Note that the temporal covariation sites are located within Heschlapos;s gyri in the two hemispheres, while the spectral covariations are located anterior to the sites covarying with the temporal stimulus parameter. An additional posterior site of spectral covariation is located within the STS in the right hemisphere only. Figure 3. Slopes of CBF changes in three selected cortical regions as a function of spectral and temporal input parameters. Symbols indicate average CBF with one standard error of the mean; lines are drawn through the least-squares linear solution corresponding to these points. Figure 3. Slopes of CBF changes in three selected cortical regions as a function of spectral and temporal input parameters. 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J Neurosci 14 : 1908 –1919.	2017-02-20 08:40:06	{"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": 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": 0.7114928960800171, "perplexity": 3743.033384059838}, "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-2017-09/segments/1487501170434.7/warc/CC-MAIN-20170219104610-00330-ip-10-171-10-108.ec2.internal.warc.gz"}
https://dr.lib.iastate.edu/handle/20.500.12876/90247	## Communication-Efficient Network-Distributed Optimization with Differential-Coded Compressors 2019-01-01 ##### Authors Zhang, Xin Zhu, Zhengyuan Liu, Jia Zhu, Zhengyuan Bentley, Elizabeth ##### Organizational Units Computer Science Organizational Unit Statistics Organizational Unit ##### Department Computer ScienceStatistics ##### Abstract Network-distributed optimization has attracted significant attention in recent years due to its ever-increasing applications. However, the classic decentralized gradient descent (DGD) algorithm is communication-inefficient for large-scale and high-dimensional network-distributed optimization problems. To address this challenge, many compressed DGD-based algorithms have been proposed. However, most of the existing works have high complexity and assume compressors with bounded noise power. To overcome these limitations, in this paper, we propose a new differential-coded compressed DGD (DC-DGD) algorithm. The key features of DC-DGD include: i) DC-DGD works with general SNR-constrained compressors, relaxing the bounded noise power assumption; ii) The differential-coded design entails the same convergence rate as the original DGD algorithm; and iii) DC-DGD has the same low-complexity structure as the original DGD due to a {\em self-noise-reduction effect}. Moreover, the above features inspire us to develop a hybrid compression scheme that offers a systematic mechanism to minimize the communication cost. Finally, we conduct extensive experiments to verify the efficacy of the proposed DC-DGD and hybrid compressor. ##### Comments This is a pre-print of the proceeding Zhang, Xin, Jia Liu, Zhengyuan Zhu, and Elizabeth S. Bentley. "Communication-Efficient Network-Distributed Optimization with Differential-Coded Compressors." arXiv preprint arXiv:1912.03208 (2019).	2022-07-03 18:54:34	{"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": 0.8109397888183594, "perplexity": 3803.784374482109}, "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-2022-27/segments/1656104248623.69/warc/CC-MAIN-20220703164826-20220703194826-00284.warc.gz"}
http://link-springer-com-443.webvpn.fjmu.edu.cn/chapter/10.1007/978-94-009-4706-1_9	A Classification of Nonlinear Systems Based on the Invariant Subdistribution Algorithm • Maria Domenia Di Benedetto Chapter Part of the Mathematics and Its Applications book series (MAIA, volume 29) Abstract Consider a nonlinear system of the form $$\dot{x} = f(x) + \sum\limits_{{i = 1}}^m {{g_i}(x){u_i}} y = h(x)$$ (1.1) with state x ∈ X Ì ℝn, input u ∈ ℝm and output y ∈ ℝP; f and g1,...,gm are analytic vector fields on X and h is an analytic function. References 1. [1] A.J. Krener: (Adf,g),(adf,g) and Locally (adf,g) Invariant and Controllability Distributions. SIAM J. Control and Optimization, 23, 523–549, (1985). 2. [2] A. Isidori, A.J. Krener, C. Gori-Giorgi, S. Monaco: Nonlinear Decoupling via Feedback: A Differential-Geometric Approach IEEE Trans. Automat. Contr., Vol. AC-26, 331–345, (1981). 3. [3] H. Nijmeijer, J.M. Schumacher: Zeros at Infinity for Affine Nonlinear Control Systems. IEEE Trans. Automat. Contr., Vol. AC-30, 566–573, (1985). 4. [4] M.D. Di Benedetto, A. Isidori: The Matching of Nonlinear Models via Dynamic State Feedback. Proc. 23rd CDC, Las Vegas, (Dec. 1984), to appear on SIAM J. of Contr. and Optimization.Google Scholar 5. [5] H. Nijmeijer: Right-invertibility for nonlinear control systems: a geometric approach. Memo. 484, Dept. Appl. Math., Twente University of Technology, (1984).Google Scholar 6. [6] A. Isidori, A. Ruberti: On the Synthesis of Linear Input-Output Responses for Nonlinear Systems. Sys. Control Lett. 4, 17–22, (1984). 7. [7] A. Isidori: Nonlinear Control Systems: an Introduction. Lect. Notes in Control and Information Science, Vol. 72, Springer-Verlag, (1985). 8. [8] A. Isidori: Control of Nonlinear Systems via Dynamic State Feedback. Algebraic and Geometric Methods in Nonlinear Control Theory, M. Fliess and M. Hazewinkel eds., Reidel, Dordrecht, (1986). This volume.Google Scholar	2020-07-07 11:25:11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5209840536117554, "perplexity": 12912.703301014806}, "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/1593655892516.24/warc/CC-MAIN-20200707111607-20200707141607-00326.warc.gz"}
https://www.tutorialspoint.com/how-to-find-a-document-by-the-non-existence-of-a-field-in-mongodb	# How to find a document by the non-existence of a field in MongoDB? MongoDBDatabaseBig Data Analytics To find a document by the non-existence of a field in MongoDB, the syntax is as follows − db.yourCollectionName.find({ "yourFieldName" : { "$exists" : false } }).pretty(); To understand the above syntax, let us create a collection with the document. The query to create a collection with a document is as follows − > db.findDocumentNonExistenceFieldDemo.insertOne({"StudentName":"John","StudentAge":25}); { "acknowledged" : true, "insertedId" : ObjectId("5c8a5c629064dcd4a68b70e8") } > db.findDocumentNonExistenceFieldDemo.insertOne({"StudentName":"David","StudentAge":26,"StudentMathMarks":78}); { "acknowledged" : true, "insertedId" : ObjectId("5c8a5c809064dcd4a68b70e9") } Display all documents from a collection with the help of find() method. The query is as follows − > db.findDocumentNonExistenceFieldDemo.find().pretty(); The following is the output − { "_id" : ObjectId("5c8a5c629064dcd4a68b70e8"), "StudentName" : "John", "StudentAge" : 25 } { "_id" : ObjectId("5c8a5c809064dcd4a68b70e9"), "StudentName" : "David", "StudentAge" : 26, "StudentMathMarks" : 78 } Here is the query to find a document by the non-existence of a field − > db.findDocumentNonExistenceFieldDemo.find({ "StudentMathMarks" : { "$exists" : false } }).pretty(); The following is the output − { "_id" : ObjectId("5c8a5c629064dcd4a68b70e8"), "StudentName" : "John", "StudentAge" : 25 } Published on 29-Mar-2019 10:05:14	2021-12-06 20:40:15	{"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": 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": 0.5956736207008362, "perplexity": 8052.385750504573}, "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/1637964363312.79/warc/CC-MAIN-20211206194128-20211206224128-00431.warc.gz"}
https://www.physicsforums.com/threads/what-is-the-energy-stored-in-the-capacitor.113309/	# What is the energy stored in the capacitor? 1. Mar 6, 2006 ### donjt81 20 J is placed across a 15uF capacitor. What is the energy stored in the capacitor? any ideas on how to do this? 2. Mar 6, 2006 ### Andrew Mason Check the question. I think it should be 20 volts not 20 J. (If 20 J is added to the capacitor, the energy stored in the capacitor is 20 J. because there is no energy is lost due to resistance). The energy is $\int_0^q Vdq$ where V = potential (energy/charge) between the plates of the capacitor. It is 0 Volts initially but increases to 20 V. when fully charged. Use the relationship between V and Q in a capacitor to express dq in terms of dV and then integrate with respect to V over the range 0 to 20 V. AM	2017-01-23 11:18:01	{"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": 0.724164605140686, "perplexity": 662.8926944109476}, "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-2017-04/segments/1484560282631.80/warc/CC-MAIN-20170116095122-00187-ip-10-171-10-70.ec2.internal.warc.gz"}
https://snowex-2021.hackweek.io/tutorials/gpr/gpr.html	# GPR¶ Lead Developer: Tate Meehan Co-developers: Dan McGrath & Ryan Webb Overview In this tutorial we will request the snow pit 1S1 location and density, ground-penetrating radar (GPR) two-way travel-times (TWT) and geolocation data, and Magnaprobe snow depths and locations to make a quick comparison of the Magnaprobe snow depth measurements and the GPR estimated snow depths. We will calculate the average density from the pit and visualize a set of GPR travel-times around Pit 1S1. Given the average dry snow density of 1S1, we will then use an empirically derived expression from Kovacs et. al (1995) to estimate the radar wave speed. The wave speed allows us to convert the radar two-way travel-time to snow depth. Lastly we will use a few summary statistics to compare the GPR and Magnaprobe snow depths, and we will discuss the various potential sources of error that arise naturally, systematically, and/or algorithmically. Slides ## Retrieve density, GPR, and Magnaprobe data from Pit 1S1¶ Goal: Compare the Magnaprobe snow depth to the GPR estimated snow depth from SnowEx 2020 Grand Mesa IOP Pit 1S1 Approach: 1. Retrieve the pit location from the Layer Data table 2. Build a circle of 50 m radius around the pit location 3. Request the pit data to get density layers and calculate the average 4. Request all the GPR data within a 50 m distance of our pit 5. Plot GPR TWT 6. Convert TWT to depth using snow density 7. Request the Magnaprobe depths around Pit 1S1 8. Interpolate GPR depths to the locations of the Magnaprobe depths 9. Compare statistics of GPR and Magnaprobe depths ## Process¶ ### Step 1: Get the pit/site coordinates¶ We must first import the necessary libraries for operating with the SnowEx SQL database. We then import the Point Data (e.g. GPR) and Layer Data (e.g. snow pit) and GeoPandas, PostGIS, and Python functionality. We also establish the Pit Site ID (1S1) and the buffer radius around the pit (50 m). ## Import our DB access function from snowexsql.db import get_db # Import the two tables we need GPR ---> PointData, Density (Pits) --> LayerData from snowexsql.data import PointData, LayerData from snowexsql.conversions import query_to_geopandas # Import to make use of the postgis functions on the db that are not necessarily in python from sqlalchemy import func, Float # Import datetime module to filter by a date import datetime # use numpy to calculate the average of the density results import numpy as np # Import matplotlib import matplotlib.pyplot as plt %config InlineBackend.figure_format='retina' # PIT Site Identifier site_id = '1S1' # Distance around the pit to collect data in meters buffer_dist = 50 # Connect to the database we made. db_name = 'snow:hackweek@db.snowexdata.org/snowex' engine, session = get_db(db_name) # Grab our pit geometry (position) object by provided site id from the site details table, Since there is multiple layers and dates we limit the request to 1 q = session.query(LayerData.geom).filter(LayerData.site_id == site_id).limit(1) site = q.all() ### Step 2: Build a buffered circle around our pit¶ # Cast the geometry point into text to be used by Postgis function ST_Buffer point = session.query(site[0].geom.ST_AsText()).all() print(point) # Create a polygon buffered by our distance centered on the pit using postgis ST_Buffer q = session.query(func.ST_Buffer(point[0][0], buffer_dist)) buffered_pit = q.all()[0][0] [('POINT(741920 4322845)',)] ### Step 3: Grab Density Profiles¶ We query all Layer Data, cast these values to a float, and then compute the average. Then the query is filtered to only the data type ‘density’. The output rho_avg_all is the average density of all snow pits, we also filter the query again by site_id to extract the average density of pit 1S1. These density values are then printed to the screen for comparison. # Request the average (avg) of Layer data casted as a float. We have to cast to a float in the layer table because all main values are stored as a string to # ...accommodate the hand hardness. qry = session.query(func.avg(LayerData.value.cast(Float))) # Filter our query only to density qry = qry.filter(LayerData.type=='density') # Request the data rho_avg_all = qry.all() # Request the Average Density of Just 1S1 rho_avg_1s1 = qry.filter(LayerData.site_id == site_id).limit(1) # This is a gotcha. The data in layer data only is stored as a string to accommodate the hand hardness values print(f"Average density of all pits is {rho_avg_all[0][0]:0.0f} kg/m3") print(f"Average density of pit 1S1 is {rho_avg_1s1[0][0]:0.0f} kg/m3") # Cast Densities to float rho_avg_all = float(rho_avg_all[0][0]) rho_avg_1s1 = float(rho_avg_1s1[0][0]) Average density of all pits is 266 kg/m3 Average density of pit 1S1 is 245 kg/m3 ### Step 4: Request all GPR TWT measured inside the buffer¶ In this step, we first print all of the instruments and data types contained in PointData. Doing so let’s us know that in order to query the GPR two-way travel-times we use the identifier ‘two_way_travel’. We then apply a filter for the date January 29, 2020, and refine this query further with the filter for TWT only within our buffered region. Using geopandas, the query is cast into a dataframe. By default the queried PointData type is given the name ‘value’. To be more explicit we rename the variable as ‘twt’ within the dataframe. # Collect all Point Data where the instrument string contains the GPR in its name #qry = session.query(PointData).filter(PointData.instrument.contains('GPR')) # Print all types of PointData in the query tmp = session.query(PointData.instrument).distinct().all() print(tmp) # Print all types of PointData in the query tmp = session.query(PointData.type).distinct().all() print(tmp) qry = session.query(PointData).filter(PointData.type == 'two_way_travel') # Additionally Filter by a date qry = qry.filter(PointData.date==datetime.date(2020, 1, 29)) # Grab all the point data in the buffer using the POSTGIS ST_Within, anytime using the postgis functions we typically have to convert to text qry = qry.filter(func.ST_Within(PointData.geom.ST_AsText(), buffered_pit.ST_AsText())) # Use our handy dandy function to execute the query and make it a geopandas dataframe dfGPR = query_to_geopandas(qry, engine) # rename 'value' in dataframe as 'twt' dfGPR.rename(columns={'value': 'twt'},inplace=True ) [('mesa',), ('magnaprobe',), ('camera',), ('pulse EKKO Pro multi-polarization 1 GHz GPR',), ('pit ruler',)] [('depth',), ('swe',), ('two_way_travel',)] ### Step 6: Convert TWT to Depth Using Snow Density¶ We will relate the dry snow density to the electromagnetic wave speed using the Kovacs et. al (1995) formula (1)$\epsilon_{\mathrm{r}}^{\prime}=(1+0.845 \rho)^{2} \quad .$ Equation (1) calculates the dielectric constant $$\epsilon_{\mathrm{r}}^{\prime}$$, provided the snow density $$\rho$$. We must then relate the dielectric constant to the electromagnetic wave speed $$(v)$$ (2)$v = \frac{c}{\sqrt{\epsilon_{\mathrm{r}}^{\prime}}} \quad .$ In equation (2) $$c$$ is the universal constant $$0.3~m/ns$$. We then calculate the depth of the snow (3)$z = \frac{vt}{2} \quad ,$ using the two-way travel-time $$(t)$$ and the electromagnetic velocity. We add the GPR estimated snow depths to the dataframe, and print the head of the dataframe to confirm this addition. # Average Snow Density # all pits rho = rho_avg_all # 1s1 rho = rho_avg_1s1 # convert density to specific gravity rho = rho/1000 # Calculate Dielectric Permittivity of Snow epsilon = (1+0.845rho)2 c = 0.3 # m/ns v = c/np.sqrt(epsilon) # m/ns t = dfGPR.twt z = vt/2100 # Add the GPR depths to the datafram dfGPR['depth'] = z site_name date time_created time_updated id site_id doi date_accessed instrument type ... northing easting elevation utm_zone geom time version_number equipment twt depth 0 Grand Mesa 2020-01-29 2021-06-10 14:54:37.077837+00:00 None 882788 None https://doi.org/10.5067/Q2LFK0QSVGS2 2021-05-30 pulse EKKO Pro multi-polarization 1 GHz GPR two_way_travel ... 4322837.051 741940.717 None 12 POINT (741940.717 4322837.051) 16:53:13.197000-06:00 None None 6.3 78.288242 1 Grand Mesa 2020-01-29 2021-06-10 14:54:37.080197+00:00 None 882789 None https://doi.org/10.5067/Q2LFK0QSVGS2 2021-05-30 pulse EKKO Pro multi-polarization 1 GHz GPR two_way_travel ... 4322837.043 741940.673 None 12 POINT (741940.673 4322837.043) 16:53:13.230000-06:00 None None 6.3 78.288242 2 Grand Mesa 2020-01-29 2021-06-10 14:54:37.017366+00:00 None 882759 None https://doi.org/10.5067/Q2LFK0QSVGS2 2021-05-30 pulse EKKO Pro multi-polarization 1 GHz GPR two_way_travel ... 4322837.268 741941.510 None 12 POINT (741941.510 4322837.268) 16:53:12.233000-06:00 None None 6.0 74.560231 3 Grand Mesa 2020-01-29 2021-06-10 14:54:36.998583+00:00 None 882750 None https://doi.org/10.5067/Q2LFK0QSVGS2 2021-05-30 pulse EKKO Pro multi-polarization 1 GHz GPR two_way_travel ... 4322837.344 741941.641 None 12 POINT (741941.641 4322837.344) 16:53:11.933000-06:00 None None 6.0 74.560231 4 Grand Mesa 2020-01-29 2021-06-10 14:54:37.000674+00:00 None 882751 None https://doi.org/10.5067/Q2LFK0QSVGS2 2021-05-30 pulse EKKO Pro multi-polarization 1 GHz GPR two_way_travel ... 4322837.338 741941.627 None 12 POINT (741941.627 4322837.338) 16:53:11.967000-06:00 None None 6.0 74.560231 5 rows × 24 columns ### Step 7: Get Depth Probes¶ We can recall the PointData types from above in Step 4, and we choose ‘depth’ as the PointData type filter. Again to ensure we are only considering data that was acquired on the same day as the GPR, we filter the depth data by the date January 29, 2020. We further refine this search to the instrument type ‘magnaprobe’ and of those data only query the points within our buffer. Lastly, we send this query to a new dataframe using the geopandas functionality, and rename the ‘value’ column as ‘depth’. # Filter by the dataset type depth qry = session.query(PointData).filter(PointData.type == 'depth') # Additionally Filter by a date qry = qry.filter(PointData.date==datetime.date(2020, 1, 29)) # Additionally Filter by instrument qry = qry.filter(PointData.instrument=='magnaprobe') # Grab all the point data in the buffer qry = qry.filter(func.ST_Within(PointData.geom.ST_AsText(), buffered_pit.ST_AsText())) # Execute the query # Use our handy dandy function to execute the query and make it a geopandas dataframe dfProbe = query_to_geopandas(qry, engine) # rename Probed Depths to dataframe dfProbe.rename(columns={'value': 'depth'},inplace=True ) site_name date time_created time_updated id site_id doi date_accessed instrument type ... longitude northing easting elevation utm_zone geom time version_number equipment depth 0 Grand Mesa 2020-01-29 2021-06-10 13:06:49.779246+00:00 None 80320 None https://doi.org/10.5067/9IA978JIACAR 2021-05-30 magnaprobe depth ... -108.20546 4322840.31 741932.13 3036.2 None POINT (741932.130 4322840.310) 07:14:00-06:00 1 CRREL_C 94.0 1 Grand Mesa 2020-01-29 2021-06-10 13:06:49.703341+00:00 None 80286 None https://doi.org/10.5067/9IA978JIACAR 2021-05-30 magnaprobe depth ... -108.20554 4322850.09 741924.90 3036.1 None POINT (741924.900 4322850.090) 07:10:00-06:00 1 CRREL_C 96.0 2 Grand Mesa 2020-01-29 2021-06-10 13:06:49.705901+00:00 None 80287 None https://doi.org/10.5067/9IA978JIACAR 2021-05-30 magnaprobe depth ... -108.20554 4322853.04 741924.66 3035.2 None POINT (741924.660 4322853.040) 07:10:00-06:00 1 CRREL_C 93.0 3 Grand Mesa 2020-01-29 2021-06-10 13:06:49.708092+00:00 None 80288 None https://doi.org/10.5067/9IA978JIACAR 2021-05-30 magnaprobe depth ... -108.20553 4322853.45 741925.81 3035.5 None POINT (741925.810 4322853.450) 07:10:00-06:00 1 CRREL_C 87.0 4 Grand Mesa 2020-01-29 2021-06-10 13:06:49.710224+00:00 None 80289 None https://doi.org/10.5067/9IA978JIACAR 2021-05-30 magnaprobe depth ... -108.20554 4322854.54 741925.05 3036.1 None POINT (741925.050 4322854.540) 07:11:00-06:00 1 CRREL_C 80.0 5 rows × 23 columns ### Step 8: Average GPR Depths to Compare with Probed Depths¶ In this step we will compare the GPR estimated depths to the Magnaprobe depths. In order to accomplish this, we must interpolate the GPR locations to the locations of the probe. We will use inverse distance weighting as our interpolation method. Example Code taken from https://stackoverflow.com/questions/3104781/inverse-distance-weighted-idw-interpolation-with-python Inverse distance weighting is a weighted average interpolant. The weights are computed as the inverse of the distance between the GPR locations $$(x,y)$$ and the depth probe locations (the interpolated locations $$(x_i,y_i)$$) (4)$\begin{split} d = {\sqrt{(x-x_i)^{2}+(y-y_i)^2}} \quad , \\ w = \frac{1}{d} \quad . \end{split}$ Equation (4) is then normalized (5)$w = \frac{w}{\sum{w}} \quad ,$ to sum to one. For the $$i^{th}$$ location these weights are multiplied by the GPR depths (6)$z_i = w_iz \quad ,$ to compute a weighted average. In the following code implementation of the inverse distance weighting algorithm, the subtract.outer method of the universal functions (ufunc) within numpy is called which computes the distances in equation (4) by looping through the interpolation points. The dot product (inner product) is then used to multiply the weights with the GPR depths. This algorithm relies on the use of a for loop within the ufunc.outer call, yet it seems quite efficient! A notable caveat of this algorithm, and a source of error, is that the interpolation considers all points in the domain, rather than a localized interpolation. An interpolation scheme that employs a search radius of three meters, rather than 50 meters (as established by the buffer distance in step one), would be preferable. We then assign the GPR estimated depths to the probe dataframe, and compute the error between the probed depths and the GPR depths. The head of the depth probe dataframe is printed to show that we have added the interpolated GPR depths and the error. # Inverse Distance Weighting Interpolation def simple_idw(x, y, z, xi, yi): dist = distance_matrix(x,y, xi,yi) # In IDW, weights are 1 / distance weights = 1.0 / dist # Make weights sum to one weights /= weights.sum(axis=0) # Multiply the weights for each interpolated point by all observed Z-values zi = np.dot(weights.T, z) return zi def distance_matrix(x0, y0, x1, y1): obs = np.vstack((x0, y0)).T interp = np.vstack((x1, y1)).T # Make a distance matrix between pairwise observations # Note: from <http://stackoverflow.com/questions/1871536> # (Yay for ufuncs!) d0 = np.subtract.outer(obs[:,0], interp[:,0]) d1 = np.subtract.outer(obs[:,1], interp[:,1]) return np.hypot(d0, d1) # Estimate the GPR Depths at the Probe Locations z = simple_idw(dfGPR.easting, dfGPR.northing, dfGPR.depth, dfProbe.easting, dfProbe.northing) # Assign the GPR depths to the Probe dataframe dfProbe['depthGPR'] = z # Calculate the Error err = dfProbe.depth-dfProbe.depthGPR # Assign the Error to the Probe Dataframe dfProbe['error'] = err site_name date time_created time_updated id site_id doi date_accessed instrument type ... easting elevation utm_zone geom time version_number equipment depth depthGPR error 0 Grand Mesa 2020-01-29 2021-06-10 13:06:49.779246+00:00 None 80320 None https://doi.org/10.5067/9IA978JIACAR 2021-05-30 magnaprobe depth ... 741932.13 3036.2 None POINT (741932.130 4322840.310) 07:14:00-06:00 1 CRREL_C 94.0 82.689476 11.310524 1 Grand Mesa 2020-01-29 2021-06-10 13:06:49.703341+00:00 None 80286 None https://doi.org/10.5067/9IA978JIACAR 2021-05-30 magnaprobe depth ... 741924.90 3036.1 None POINT (741924.900 4322850.090) 07:10:00-06:00 1 CRREL_C 96.0 86.767049 9.232951 2 Grand Mesa 2020-01-29 2021-06-10 13:06:49.705901+00:00 None 80287 None https://doi.org/10.5067/9IA978JIACAR 2021-05-30 magnaprobe depth ... 741924.66 3035.2 None POINT (741924.660 4322853.040) 07:10:00-06:00 1 CRREL_C 93.0 87.286674 5.713326 3 Grand Mesa 2020-01-29 2021-06-10 13:06:49.708092+00:00 None 80288 None https://doi.org/10.5067/9IA978JIACAR 2021-05-30 magnaprobe depth ... 741925.81 3035.5 None POINT (741925.810 4322853.450) 07:10:00-06:00 1 CRREL_C 87.0 87.169223 -0.169223 4 Grand Mesa 2020-01-29 2021-06-10 13:06:49.710224+00:00 None 80289 None https://doi.org/10.5067/9IA978JIACAR 2021-05-30 magnaprobe depth ... 741925.05 3036.1 None POINT (741925.050 4322854.540) 07:11:00-06:00 1 CRREL_C 80.0 87.454136 -7.454136 5 rows × 25 columns ### Step 9. Plot the Depths, Correlation, and Errors¶ In this final step, we will compare the GPR depths and probed depths. We compute the Pearson correlation (7)$r=\frac{\sum\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{\sqrt{\sum\left(x_{i}-\bar{x}\right)^{2} \sum\left(y_{i}-\bar{y}\right)^{2}}} \quad ,$ where $$x$$ represents the probed depths and $$y$$ represents the GPR depths. We calculate the bias of the GPR estimated depths (8)$\mathrm{ME}=\frac{\sum_{i=1}^{N} \left( x_{i}-y_{i} \right) }{N} \quad ,$ as the mean error ($$\mathrm{ME}$$). Our example at pit 1S1 is relatively unbiased with a $$\mathrm{ME}=1.3~cm$$. This indicates that the sources of error are uncorrelated and that systematic errors are small. The root-mean-square error (9)$\mathrm{RMSE}=\sqrt{\frac{\sum_{i=1}^{N}\left(x_{i}-y_{i}\right)^{2}}{N}}$ is a measurement of the standard deviation of the errors, if we assume that the errors are normally distributed and independent. In this example the $$\mathrm{RMSE}=11~cm$$, which is approximately $$1/2$$ of the L-band GPR wavelength. Potential Sources of Error 1. Incorrect density used in depth conversion 2. Depth probe entering the soil or air-gap beneath vegitation 3. Geolocation errors 4. Sample “footprint” size mismatch 5. Interpolation Data biases can be caused by using the incorrect density value. A lower density value will bias the GPR estimated depths to larger values, whereas, a higher density value will bias the GPR depths to lower values. It has also been shown that the point of the probe can enter the soil, which biases the observed depths positively (McGrath et al., 2019). Similarly, vegitation beneath the snow cover can be a source of error. It is possible that the GPR signal is reflected from the air gap beneath snow that is not grounded. In this case the depth probe may contact the ground, though the GPR travel-time does not, leading to a positive bias. Co-location of the GPR and depth probe presents a third possible source of error. Inaccuracy of GPS measurements or probes not coincident with the GPR transect are likely sources of geolocation error. A fourth possible source of error in the comparison of these depths is the disagreement between the size of the GPR “footprint” (known as the fresnel zone radius), which is on the order of one meter, and the area of the probe tip which is about one centimeter. Because these instruments do not sample the same place on the ground, localized variability in the ground topography can lead to errors between the measured and estimated depths. As mentioned in the previous section, the choice of interpolation scheme will affect the accuracy of the co-located depths. It is important to understand the pros and cons of various interpolation algorithms and to document the choice of algorithm used and it’s parameters. In the code ection we, first, display these summary statistics. Then we view this information graphically. The first plot is the scatter of the observed depths versus the estimated depths with the regression line. The second plot is a histogram of the errors (observed - estimated). The histogram shows a slight positive bias, which indicates that a combination of the errors discussed above resulted in the probe measuring snow depths $$1~cm$$ greater than the GPR on average. The final plot displays the errors in depth spatially. Roughly, by eye, it appears that areas with low travel-time ($$\sim4~ns$$, southeast quadrant) overestimate the depth, perhaps due to smearing introduced by the non-localized inverse distance weighting algorithm. # Calculate the Correlation r = np.corrcoef(dfProbe.depth,dfProbe.depthGPR) print('The correlation is', round(r[0,1],2)) # Calculate the Mean Error bias = np.mean(dfProbe.error) print('The bias is', round(bias,2), 'cm') # Calculate the Mean Absolute Error mae = np.mean(np.abs(dfProbe.error)) # Calculate the Root Mean Squared Error rmse = np.sqrt(np.mean((dfProbe.error)*2)) print('The rmse is', round(rmse,2), 'cm') # Compute the Regression Line m, b = np. polyfit(dfProbe.depth,dfProbe.depthGPR, 1) # Plot the Correlation plt.figure(0) plt.plot(dfProbe.depth,dfProbe.depthGPR,'o') plt.plot(dfProbe.depth, mdfProbe.depth + b,'k') plt.xlabel('Probe Depth [cm]') plt.ylabel('GPR Depth [cm]') # Plot a Histogram of the Errors plt.figure(1) plt.hist(dfProbe.error, density=True, bins=10, edgecolor='black') # density=False would make counts plt.ylabel('PDF') plt.xlabel('Error [cm]'); # Get the Matplotlib Axes object from the dataframe object, color points by snow depth value ax = dfProbe.plot(column='error', legend=True, cmap='PuBu') # Use non-scientific notation for x and y ticks ax.ticklabel_format(style='plain', useOffset=False) # Set the various plots x/y labels and title. ax.set_title('Error [cm] (Probed Depth - GPR Depth)') ax.set_xlabel('Easting [m]') ax.set_ylabel('Northing [m]') The correlation is 0.64 The bias is 1.29 cm The rmse is 10.62 cm Text(60.245147830207486, 0.5, 'Northing [m]') # Close the session to avoid hanging transactions session.close()	2023-03-25 18:01:22	{"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": 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": 0.2884770929813385, "perplexity": 13010.637077712787}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945368.6/warc/CC-MAIN-20230325161021-20230325191021-00362.warc.gz"}
http://docs.h2o.ai/h2o/latest-stable/h2o-r/docs/reference/h2o.assign.html	Makes a copy of the data frame and gives it the desired the key. h2o.assign(data, key) ## Arguments data An H2OFrame object The key to be associated with the H2O parsed data object	2019-10-14 04:07:43	{"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": 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": 0.238676518201828, "perplexity": 2817.7674059609444}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986649035.4/warc/CC-MAIN-20191014025508-20191014052508-00194.warc.gz"}