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https://learn.careers360.com/engineering/question-need-explanation-for-electrostatics-jee-main/	Q # Need explanation for: - Electrostatics - JEE Main Two identical charged spheres suspended from a common point by two massless strings of length $\dpi{100} l$ are initially a distance $\dpi{100} d(d< < l)$ apart because of their mutual repulsion. The charge begins to leak from both the spheres at a constant rate. As a result the charges approach each other with a velocity $\dpi{100} \upsilon$ . Then as a function of distance $\dpi{100} x$ between them • Option 1) $\upsilon\: \alpha\: x^{-1/2}$ • Option 2) $\upsilon\: \alpha\: x^{-1}$ • Option 3) $\upsilon\: \alpha\: x^{1/2}$ • Option 4) $\upsilon\: \alpha\: x$ 133 Views Option 1 ans N As we learnt in Coulombic force - $\dpi{100} F\propto Q_{1}Q_{2}=F\propto \frac{Q_{1}Q_{2}}{r^{2}}=F=\frac{KQ_{1}Q_{2}}{r^{2}}$ - wherein K - proportionality Constant Q1 and Q2 are two Point charge $T\cos \Theta =mg$ $T\sin \Theta =f_{e}=\frac{1}{4\pi\varepsilon _{o}}\:\frac{q^{2}}{d^{2}}$ $tan \Theta =\frac{1}{4\pi\varepsilon _{o}}\:\frac{q^{2}}{d^{2}mg}$ $\therefore tan \Theta=\frac{x}{2l}$ $\frac{x}{2l}=\frac{q^{2}}{4\pi\varepsilon _{o}x^{2}mg}$ $\Rightarrow \frac{x}{2l}\propto \frac{q^{2}}{x^{2}}$ $\Rightarrow q^{2}\propto x^{3}$ $\Rightarrow q\propto x^{3/2}$ $\Rightarrow\frac{dq}{dt}\propto\frac{3}{2}x^{1/2}\frac{dx}{dt} \therefore \left (\frac{dq}{dt}= constant \right )$ $\therefore \frac{dx}{dt}=v\propto x^{-1/2}$ Option 1) $\upsilon\: \alpha\: x^{-1/2}$ This is correct option Option 2) $\upsilon\: \alpha\: x^{-1}$ This is incorrect option Option 3) $\upsilon\: \alpha\: x^{1/2}$ This is incorrect option Option 4) $\upsilon\: \alpha\: x$ This is incorrect option Exams Articles Questions	2020-01-24 19:23:35	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 24, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6497803926467896, "perplexity": 4021.686971021835}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250625097.75/warc/CC-MAIN-20200124191133-20200124220133-00077.warc.gz"}
https://www.exambin.com/shortcut-profit-loss-discount-2/	# Shortcut of Profit, Loss and Discount-2 Shortcut 33: Selling price at profit or loss $S.P=C.P\pm(a/100)C.P$ Substitute ‘+’ if profit; ‘-‘ if loss Question: A product worth Rs. 25000 is sold at a profit of 35%. Find the selling price? a = +35 C.P = 25000 Substitute the values in the above equation, we get S.P = 25000 + (35/100)25000 S.P = 33750 Selling price = Rs. 33750 Shortcut 34: Successive profit or loss $S.P=\frac{100+a}{100}\times\frac{100+b}{100}\times\frac{100+c}{100}C.P$ a, b , c = values of percentage profit or loss Substitute ‘+’ for profit and ‘-‘ for loss Question: A watch worth Rs. 5000 is sold by A to B at 10% loss. B sold it to C at 30% loss. C sold it to D at 40% profit. What is the price at which D bought the watch? a = -10; b = -30; c = 40 C.P = 5000 Substitute the values in the above equation, we get S.P = (9/10)(7/10)(14/10) x 5000 S.P = 4410 D bought the watch for Rs. 4410 Shortcut 35: $S.P=C.P\lbrack1+(a/100)\rbrack^n$ a = value of profit or loss Substitute ‘+’ for profit and ‘-‘ for loss Question: The price of a laptop decreases by 20% every year. It the laptop was bought for Rs. 45000, at what price will it be sold after 2 years? a = -20 C.P = 45000 Substitute the values in the above equation, we get S.P = 45000[0.8]2 S.P = 28800 Selling price after two years = Rs. 28800 Shortcut 36: Discount percentage $D\%=\frac{M.P-S.P}{M.P}\times100$ M.P = Marked price or Market price of the product Question: A product is marked Rs. 450 by the seller. If he sells at a price of Rs. 330, what is the discount provided in percentage? M.P = 450 S.P = 330 Substitute the values in the above equation, we get D% = [(450-330)/450]x100 D% = 26.67% Discount provided in percentage = 26.67% #### siva Register New Account ##### Compare items • Total (0) 0 ARE YOU READY? GET IT NOW! Increase more than 500% of Email Subscribers! Your Information will never be shared with any third party. ARE YOU READY? GET IT NOW! Get Weekly Current Affairs Magazine to your inbox. Your Information will never be shared with any third party. CURRENT AFFAIRS THAT MATTERS TO YOU Get Weekly Current Affairs Magazine to your inbox.	2017-04-24 09:13:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.5741180181503296, "perplexity": 7460.338777116706}, "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-2017-17/segments/1492917119225.38/warc/CC-MAIN-20170423031159-00421-ip-10-145-167-34.ec2.internal.warc.gz"}
http://mymemphisvet.com/jxqkib/e9f660-interior-point-of-irrational-numbers	Next Lesson. In fact Euclid proved that (2*p - 1) 2(p - 1) is a perfect number if 2p - 1 is prime, which is only possible (though not assured) if p. https://pure. clearly belongs to the closure of E, (why? Such numbers are called irrational numbers. A point in this space is an ordered n-tuple (x 1, x 2, ..... , x n) of real numbers. In mathematics, all the real numbers are often denoted by R or ℜ, and a real number corresponds to a unique point or location in the number line (see Fig. numbers not in S) so x is not an interior point. Corresponding, Alternate and Co-Interior Angles (7) The interior of this set is (0,2) which is strictly larger than E. Problem 2 Let E = {r ∈ Q 0 ≤ r ≤ 1} be the set of rational numbers between 0 and 1. Therefore, if you have a real number line, you will have points for both rational and irrational numbers. 7, and so among the numbers 2,3,5,6,7,10,14,15,21,30,35,42,70,105,210. Depending on the two numbers, the product of the two irrational numbers can be a rational or irrational number. 1 Rational and Irrational numbers 1 2 Parallel lines and transversals 10 ... through any point outside the line 2.3 Q.1, 2 Practice Problems (Based on Practice Set 2.3) ... called a pair of interior angles. For every x for which we try to find the neighbourhood for, any ε > 0 we will have an interval containing irrational numbers which will not be an element of S. Yes, well done! Basically, the rational numbers are the fractions which can be represented in the number line. a) What are the limit points of Q? Note that no point of the set can be its interior point. 94 5. Motivation. Either sˆ‘, or smeets both components … For example, 3/2 corresponds to point A and − 2 corresponds to point B. Notes. It is a contradiction of rational numbers but is a type of real numbers. Note that an -neighborhood of a point x is the open interval (x ... A point x ∈ S is an interior point of … Then find the number of sides 72. Common Irrational Numbers . The Set (2, 3) Is Open But The Set (2, 3) Is Not Open. For example, the numbers 1, 2/3, 3/4, 2, 10, 100, and 500 are all rational numbers, as well as real numbers, so this disproves the idea that all real numbers are irrational. Interior Point Not Interior Points Definition: The interior of a set A is the set of all the interior points of A. In the de nition of a A= ˙: The irrational numbers have the same property, but the Cantor set has the additional property of being closed, ... of the Cantor set, but none is an interior point. Solution. Let a,b be an open interval in R1, and let x a,b .Consider min x a,b x : L.Then we have B x,L x L,x L a,b .Thatis,x is an interior point of a,b .Sincex is arbitrary, we have every point of a,b is interior. Derived Set, Closure, Interior, and Boundary We have the following deﬁnitions: • Let A be a set of real numbers. Consider the two subsets Q(the rational numbers) and Qc (the irrational numbers) of R with its usual metric. Only the square roots of square numbers … The Set Of Irrational Numbers Q' Is Not A Neighborhood Of Any Of Its Point. School Georgia Institute Of Technology; Course Title MATH 4640; Type. Rational numbers and irrational numbers together make up the real numbers. This can be proved using similar argument as in (5) to show that is not open. Chapter 2, problem 4. 5.Let xbe an interior point of set Aand suppose fx ngis a sequence of points, not necessarily in A, but ... 8.Is the set of irrational real numbers countable? 5.333... is rational because it is equivalent to 5 1/3 = 16/3. We use d(A) to denote the derived set of A, that is theset of all accumulation points of A.This set is sometimes denoted by A′. MathisFun. Justify your claim. 4 posts published by chinchantanting during April 2016. ⇐ Isolated Point of a Set ⇒ Neighborhood of a Point … It is an example of an irrational number. There has to be an interval around that point that is contained in I. edu/rss/ en-us Tue, 13 Oct 2020 19:39:50 EDT Tue, 13 Oct 2020 19:39:50 EDT nanocenter. The open interval (a,b) is a neighborhood of all its points since. Example 1.14. ), and so E = [0,2]. Example: Consider √3 and √3 then √3 × √3 = 3 It is a rational number. Pages 6. Is every accumulation point of a set Aan interior point? Consider √3 and √2 √3 × √2 = √6. Irrational number definition is - a number that can be expressed as an infinite decimal with no set of consecutive digits repeating itself indefinitely and that cannot be … Watch Queue Queue. Is the set of irrational real numbers countable? is an interior point and S is open as claimed We now need to prove the. Irrational numbers have decimal expansion that neither terminate nor become periodic. For example, Ö 2, Ö 3, and Ö 5 are irrational numbers because they can't be written as a ratio of two integers. Assume that, I, the interior of the complement is not empty. 1.1). The next digits of many irrational numbers can be predicted based on the formula used to compute them. ... Find the measure of an interior angle. • The complement of A is the set C(A) := R \ A. Any interior point of Klies on an open segment contained in K, so the extreme points are contained in @K. Suppose x2@Kis not an extreme point, let sˆKbe an open line segment containing x, and let ‘ˆR2 be a supporting line at x. The open interval I= (0,1) is open. Maybe it's also nice to know that a set ##A## in a topological space is called discrete when every point ##x \in A## has a neighborhood intersecting ##A## only in ##\{x\}##. Problem 2 (Miklos Schweitzer 2020).Prove that if is a continuous periodic function and is irrational, then the sequence modulo is dense in .. The answer is no. Topology of the Real Numbers When the set Ais understood from the context, we refer, for example, to an \interior point." In an arbitrary topological space, the class of closed sets with empty interior consists precisely of the boundaries of dense open sets.These sets are, in a certain sense, "negligible". 5. Approximation of irrational numbers. Indeed if we assume that the set of irrational real numbers, say RnQ;is ... every point p2Eis an interior point of E, ie, there exists a neighborhood N of psuch that NˆE:Now given any neighborhood Gof p, by theorem 2.24 G\Nis open, so there The set of all real numbers is both open and closed. Watch Queue Queue 2. Real numbers include both rational and irrational numbers. Let α be an irrational number. 4. proof: 1. Interior and isolated points of a set belong to the set, whereas boundary and accumulation points may or may not belong to the set. Uploaded By LieutenantHackerMonkey5844. THEOREM 2. If x∈ Ithen Icontains an Thus intS = ;.) Irrational Number Videos. That interval has a width, w. pick n such that 1/n < w. One of the rationals k/n has to lie within the interval. Because the difference between the largest and the smallest of these three numbers 4.Is every interior point of a set Aan accumulation point? The set of all rational numbers is neither open nor closed. Distance in n-dimensional Euclidean space. None Of The Rational Numbers Is An Interior Point Of The Set Of Rational Numbers Q. Interior – The interior of an angle is the area within the two rays. One can write. Finding the Mid Point and Gradient Between two Points (9) ... Irrational numbers are numbers that can not be written as a ratio of 2 numbers. Solution. This video is unavailable. contains irrational numbers (i.e. To know the properties of rational numbers, we will consider here the general properties such as associative, commutative, distributive and closure properties, which are also defined for integers.Rational numbers are the numbers which can be represented in the form of p/q, where q is not equal to 0. Typically, there are three types of limits which differ from the normal limits that we learnt before, namely one-sided limit, infinite limit and limit at infinity. This preview shows page 4 - 6 out of 6 pages. GIVE REASON/S FOR THE FOLLOWING: The Set Of Real Numbers R Is Neighborhood Of Each Of Its Points. In the given figure, the pairs of interior angles are i. AFG and CGF • Rational numbers are dense in $$\mathbb{R}$$ and countable but irrational numbers are also dense in $$\mathbb{R}$$ but not countable. • The closure of A is the set c(A) := A∪d(A).This set is sometimes denoted by A. The set E is dense in the interval [0,1]. Notice that cin interior point of Dif there exists a neighborhood of cwhich is contained in D: For example, 0:1 is an interior point of [0;1):The point 0 is not an interior point of [0;1): In contrast, we say that ais a left end-point of the intervals [a;b) and of [a;b]: Similarly, bis a right end-point of the intervals (a;b] and of [a;b]: Every real number is a limit point of Q, since every real number can be approximated by rationals. What are its interior points? Charpter 3 Elements of Point set Topology Open and closed sets in R1 and R2 3.1 Prove that an open interval in R1 is an open set and that a closed interval is a closed set. Is an interior point and s is open as claimed we now. Any number on a number line that isn't a rational number is irrational. verbal, and symbolic representations of irrational numbers; calculate and explain the ... Intersection - Intersection is the point or line where two shapes meet. The set of irrational numbers Q’ = R – Q is not a neighbourhood of any of its points as many interval around an irrational point will also contain rational points. S is not closed because 0 is a boundary point, but 0 2= S, so bdS * S. (b) N is closed but not open: At each n 2N, every neighbourhood N(n;") intersects both N and NC, so N bdN. The proof is quite obvious, thus it is omitted. Pick a point in I. There are no other boundary points, so in fact N = bdN, so N is closed. where A is the integral part of α. Its decimal representation is then nonterminating and nonrepeating. So the set of irrational numbers Q’ is not an open set. (No proof needed). The definition of local extrema given above restricts the input value to an interior point of the domain. Use the fact that if A is dense in X the interior of the complement of A is empty. Thus, a set is open if and only if every point in the set is an interior point. False. A ): = R \ a that no point of Q irrational. 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Not in S ) so x is not open point a and − corresponds...	2021-07-24 08:58:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.8237466812133789, "perplexity": 519.3967822308564}, "config": {"markdown_headings": false, "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-2021-31/segments/1627046150134.86/warc/CC-MAIN-20210724063259-20210724093259-00178.warc.gz"}
https://www.physicsforums.com/threads/physical-interpretation-of-coordinates-in-gr.769792/	# Physical Interpretation of Coordinates in GR 1. Sep 8, 2014 ### MrBillyShears What is the relationship between the differentiable manifold that is space-time and the physical space around us? How does one relate the three seemingly Cartesian coordinates around us, those which we can measure out with a ruler, to the coordinates of the Lorentzian manifold? If i say, measure out a length with a ruler, how would that relate to the three spatial coordinates of space-time? I'm just getting all confused thinking about this. Maybe this question doesn't make much sense, but I just want to see if anyone can help me with this. 2. Sep 8, 2014 ### WannabeNewton A local observer that makes measurements using local rods, clocks, gyroscopes etc. can represent these measurements in the form $T = T^{abc...}e_{a}e_{b}e_{c}...$ where $\{e_a \}$ is an orthonormal basis along the worldline of the observer (that is, a local Lorentz frame) and $T$ is a tensor that represents the measurements of the quantity in question. The components $T^{abc...}$ relative to the local Lorentz frame represent the measurements of the components of this quantity along the time and spatial axes carried by the observer. However these measurements can only be made at the location of the observer e.g. it could be the relative velocity of another observer just when passing by the first. But what about measurements that aren't made right on the worldline? Taking this basis $\{e_a \}$ the observer can use parallel transport $\nabla_{\xi}e_a = 0$ along space-like geodesics with tangent $\xi$ emanating orthogonally from the worldline to define a local coordinate system, called a Fermi-normal coordinate system, in a sufficiently small neighborhood around the worldline. Imagine the basis as a set of three mutually orthogonal meter sticks, oriented say along gyroscopes, and a clock. Then the Ferm-normal coordinate system represents a very small laboratory comoving with the observer that contains a lattice of these rods and clocks, the latter all being synchronized assuming the lab is small enough. The observer can now make measurements anywhere within this laboratory, e.g. the spatial distance between two flashes of light going off simultaneously relative to this observer in the Fermi-normal coordinate system. Coming back full circle to your question, values of quantities in an arbitrary coordinate system do not represent the physical values as measured by local observers. Coordinate values are in general physically meaningless. You must convert these values into the local Lorentz frame (or Fermi-normal coordinate system), as in the first paragraph above, in order to get the observables relative to a given observer. So for example in Schwarzschild space-time the coordinate velocity $\frac{dr}{dt}$ of an observer in a radial free fall is not physically meaningful as is obvious from the fact that $\frac{dr}{dt}\rightarrow 1$ as one approaches the event horizon. If we convert the quantities $dr$ and $dt$ to the proper distance and proper time measurements made by local observers then we get physical measurements. Now let $\{\partial_{\mu} \}$ be a coordinate-basis for the space-time. If we have a local Lorentz frame $\{e_a \}$ for a specific observer then we know $e_a = e^{\mu}_a \partial_{\mu}$. The matrices $e^{\mu}_a$ tell us how to go from the values of quantities in the space-time coordinates to the physical measurements made by the observer. C.f. chapter 6 of MTW and http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity 3. Sep 8, 2014 ### cosmik debris A differentiable manifold is taken to be locally flat, just as on Earth we measure things close to us in kilometres and assume angles in triangles add to 180 degrees even though we know we live on a curved surface. The units we measure here are the same as the spatial co-ordinates on a flat Lorentzian manifold, time is an extra dimension of the manifold instead of a parameter in 3D. 4. Sep 8, 2014 ### pervect Staff Emeritus The very short answer is that coordinates are just labels, so they don't in and of themselves have much physical significance. But this isn't too helpful, really, so instead consider the following analogy, which hopefuly will be helpful. Consider the surface of a sphere. It's a 2 dimensional. It's also a manifold, but it's not a plane. The surface of the sphere is embedded in a higher dimensional space, but the surface itself is two dimensional. You can apply coordinate systems to the surface of the sphere, lattitude and longitude for instance, but they aren't cartesian coordinates, at least not globally. The coordinates are really just labels anyway. Any small part of the sphere looks flat. You can generate coordinates that are nearly cartesian for a small part of the sphere, but you can't cover the whole sphere with them. Similar remarks apply to the curved manifolds in GR. What you need to get an actual description of the geometry is distances. You have available the coordinates, you now need a tool that takes in the coordinates (and coordinate changes) and outputs the distances. The mathematical tool you need to do this is called a "metric". This same mathematical tool is what's used in GR. I'm not sure how much detail you want or need, so won't go into the specifics of how the metric works unless I get a further question, except to say that it coverts coordinate changes into distances. The mathematical approach of the metric doesn't rely or need the concept of embedding a lower-dimensional manifold in a higher dimensional space, as we did with the surface of the sphere (2d) in a higher dimensional (3d) space. The embedding is helpful for purposes of visualization, but it's not unique or required to do the math. As far as your questions about rulers go, note that if you have small rulers, then you can construct a coordinate system that will directly measure distances in the way you are used to - a distance will be the same as a change in coordinates, at least to a high degree of precision. So there isn't any real mystery about how to handle small distances. If you've got larger rulers, you need to do things like replace "straight lines" with geodesics. For instance in our example of the surface of a sphere, the geodesics are "great circles". There are some details on defining geodesics that I will skip over, but for the purpose of GR, and the limitation of "short" distances, you can think of a geodesic as the shortest curve on the manifold between two "close" points. You can find the length of any curve via the metric and integration, geodesic or not. Last edited: Sep 8, 2014	2017-10-23 20:02:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7759078145027161, "perplexity": 302.89930249692543}, "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/1508187826283.88/warc/CC-MAIN-20171023183146-20171023203146-00565.warc.gz"}
https://www.biostars.org/p/120055/#120061	Ensembl VEP filtering most_severe keeping Symbol, Feature,Consequence etc. 1 2 Entering edit mode 7.2 years ago t2 ▴ 60 Hi all, What I need to do is filter a file produced using non-stringent Variant Effect Predictor (VEP) settings with one that was produced with more stringent VEP settings. I've been running VEP locally using the cache option with a pre-built cache with this command on my vcfs: perl $VEP --cache --dir$VEP_DIR --offline --input_file $input --output_file$output --sift b --polyphen b --regulatory --protein --symbol --ccds --uniprot --check_existing --gmaf --maf_1kg --maf_esp --pubmed Everything works great and I'm super happy with the documentation. However, I realized after I had run my command on all my exomes that I would most likely get many entries for each particular variant depending on different Ensembl Feature IDs. VEP has a fix for this, which is to use the --most_severe flag when running the command. That works perfectly, however, some extra flags are disabled when using the --most_severe flag. I would like to retain this extra information (like gene name/symbol Feature,Consequence, etc.) for the variants produced with the --most_severe flag. perl $VEP --cache --dir$VEP_DIR --offline --input_file $input --output_file$output --regulatory --uniprot --check_existing --gmaf --maf_1kg --maf_esp --most_severe So now I have two files for each vcf; 1) disabled --most_severe and 2) --most_severe. The 2nd file is basically a subset of the 1st file but with some important missing information. In the 1st file when there are multiple entries for a variant, most of the fields are the same except the Feature_type field and often the Extra field. Both produce a tab delimited text file with columns such as this: #Uploaded_variation Location Allele Gene Feature Feature_type Consequence cDNA_position CDS_position Protein_position Amino_acids Codons Existing_variation Extra Is there a way to filter the 1st file with the 2nd file. I think I need to use fields, Uploaded_variation and Consequence for matching the 1st file because those are the fields that are unique in the line. I think using awk to search for columns in both files won't work because there is some information lost in the Consequence field in the 2nd file For example a variant Consequence may change from: non_coding_transcript_exon_variant,non_coding_transcript_variant to non_coding_transcript_exon_variant I appreciate any help in solving this issue. Alternatively there is a filter_vep script provided by VEP for post-VEP annotation filtering but I don't think there is an option here that will solve my problem. Thanks, Tesa next-gen sequencing ensembl VEP • 5.0k views 0 Entering edit mode 7 Entering edit mode 7.2 years ago EnsemblWill ▴ 560 You will encounter problems whichever way you try to combine these two files I'm afraid. For example, let's say your small (--most_severe on) file has a line with a consequence of missense_variant. Then in your large file, there are three corresponding lines of output for that variant with a consequence of missense_variant - how do you decide which to choose? Also, the consequence type picked by --most_severe may be calculated relative to an Ensembl feature that does not have reliable biological evidence to support it - do you still want to choose this one over any others? Is it practical you for you re-run your analyses? There are a couple of other options that you might find useful: a) --pick : this chooses one line of consequence data (with all the fields retained) for each variant. It uses the following criteria to pick one: 1) is the transcript canonical 2) is the transcript biotype protein_coding 3) consequence rank 4) transcript length In the forthcoming version of VEP you will be able to customise this order. http://www.ensembl.org/info/docs/tools/vep/script/vep_options.html#opt_pick --pick_allele chooses one line per variant allele (i.e. this will come into effect when the input variant has more than one alternate allele) b) --flag_pick : like --pick but just adds a flag to the line chosen by the same rules c) --per_gene : like --pick but chooses one line per variant/gene combination As a footnote, we always try to discourage people from using these summary flags if we can - there will always be cases where valuable data gets lost, and you are relying on an arbitrary and subjective algorithm to perform that summarising. The logic of this algorithm will always be wrong for some use case no matter how we code it. Thus by keeping all the data you are ensuring you don't miss anything. 3 Entering edit mode Hi EnsemblWill, Thanks so much for your detailed comments. Indeed it is feasible for me to re-run this portion of the analysis. I saw pick but I wasn't sure how I should implement it. You give me some good pointers to try to figure it out for my situation. The new(ish) VEP is awesome and I am grateful that the good folks at Ensembl made it possible to use a command line version that is so well documented. Cheers, Tesa 0 Entering edit mode Hi EnsemblWill, Do you know how the magic works behind the pick option? Because I have some conserns about it. As use case, a well documented mutation 11_66293652_T/G (ENSG00000174483/BBS1:M390R). When using : no filters : there are 21 results mapping on 3 ensembel gene entries (ENSG00000174483: BBS1, ENSG00000174165: ZDHHC24 and ENSG00000256349: CTD-3074O7 which is a Vega definition of BBS1 but not the canonical when refering to CCDS) --per_gene: only 3 results as expected. --pick: I have only 1 result left but the chosen transcript is ENSG00000256349: CTD-3074O7! --pick_order canonical, tsl, biotype, rank, length: I get back the original 21 lines as when no filters were applied! Am I missing something? Thanks for any help! Best, Kirsley	2022-01-18 20:24:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.641481339931488, "perplexity": 3604.9889841163963}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300997.67/warc/CC-MAIN-20220118182855-20220118212855-00073.warc.gz"}
https://math.stackexchange.com/questions/886787/which-holomorphic-function-is-this-the-real-part-of	# Which holomorphic function is this the real part of? In the paper "The Inverse Function Theorem of Nash and Moser" by Richard S. Hamilton it is claimed that there exists a function $\phi$ such that: $$\int_{0}^{\infty}t^{n}\phi(t)dt=(-1)^{n}$$ For $n=0,1,2,...$. In fact one is provided. The example is: $$\phi(t)=\frac{e^{2\sqrt{2}}}{\pi(1+t)}e^{-(t^\frac{1}{4}+t^\frac{-1}{4})}\sin(t^\frac{1}{4}-t^{\frac{-1}{4}})$$ My trouble is that the paper said that verifying the integral can be done by methods of contour integration by recognizing this function as the real part of a holomorphic function. The trouble I'm having is that I have no idea how to identify this holomorphic function. Any help is greatly appreciated. Thank you in advance. • This reminds me of the function used in E.M. Stein's book on singular integrals being $\psi(t) = \frac{e}{\pi t} e^{-\sqrt{2}/2 (t-1)^{1/4}} sin(\sqrt{2}/2 (t-1)^{1/4}$, which satisfies $\int_1^\infty \psi(t) dt = 1$ and all higher moments vanish. – andre Aug 4 '14 at 22:03 • $\psi(t) = \frac{e}{\pi t} e^{-\sqrt{2}/2 (t-1)^{1/4}} sin(\sqrt{2}/2 (t-1)^{1/4})$ – andre Aug 4 '14 at 22:11 • And also the one in D.V. Widder's book on the laplace transform, being $\phi(t) = e^{-t^{1/4}}sin(t^{1/4})$, where all the moments vanish for the integral on $[0,\infty)$, see archive.org/stream/laplacetransform031816mbp#page/n141/mode/2up – andre Aug 12 '14 at 20:47 • And as stated in D.V. Widders book the last $\phi$ can be found in a 1894 paper of Stieltjes on page J. 105, see archive.numdam.org/ARCHIVE/AFST/AFST_1894_1_8_4/…, I wonder how Stieltjes found this function. – andre Aug 13 '14 at 22:53 Wahoo, this is cool. Replace $t$ with $u^4$ in order to have: $$I_n=\int_{0}^{+\infty}t^n \phi(t)\,dt = \frac{4e^{2\sqrt{2}}}{\pi}\int_{0}^{+\infty}\frac{u^{4n+3}}{1+u^4}\sin(u-1/u)\exp(-u-1/u)\,du,$$ then split $[0,+\infty)=[0,1]\cup[1,+\infty)$ and use the substitution $u\leftarrow 1/u$ on the second interval in order to have: $$I_n = \frac{4e^{2\sqrt{2}}}{\pi}\int_{0}^{1}\frac{u^{4n+2}-u^{-4n-2}}{u^2+u^{-2}}\sin(u-1/u)\exp(-u-1/u)\frac{du}{u}.$$ Now substitute $u=e^{-v}$ in order to have: $$I_n = \frac{4e^{2\sqrt{2}}}{\pi}\int_{0}^{+\infty}\frac{\sinh((4n+2)v)}{\cosh(2v)}\sin(2\sinh v)\exp(-2\cosh v)\,dv,$$ and since the integrand function is even: $$I_n = \frac{2e^{2\sqrt{2}}}{\pi}\Im\int_{\mathbb{R}}\frac{\sinh((4n+2)v)}{\cosh(2v)}\exp(2i\sinh v-2\cosh v)\,dv.$$ We can approach the last integral with the usual complex analytic techniques, by integrating the function over a rectangle having vertices in $-R,R,R+\frac{\pi}{2}i,-R+\frac{\pi}{2}i$. The only singularity that matters is the one in $v=\frac{\pi}{4}i$: since the residue of the integrand function in such a point is: $$\frac{(-1)^n}{2}e^{-2\sqrt{2}}$$ we have $I_n=(-1)^n$ just as claimed.	2020-01-28 10:37:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9421536922454834, "perplexity": 189.7836870589419}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251778168.77/warc/CC-MAIN-20200128091916-20200128121916-00292.warc.gz"}
https://saudedica.com.br/docs/viewtopic.php?e0dc70=curved-line-definition	Uncategorized curved line definition of the graph of a continuously differentiable function This definition encompasses most curves that are studied in mathematics; notable exceptions are level curves (which are unions of curves and isolated points), and algebraic curves (see below). {\displaystyle y=f(x)} More precisely, a differentiable curve is a subset C of X where every point of C has a neighborhood U such that γ X If you look at a curve very closely, you will see the lines. Types of Lines. C 6. b is, More generally, if , , then we can define the length of a curve In particular, the nonsingular complex projective algebraic curves are called Riemann surfaces. Intuitively, a simple curve is a curve that "does not cross itself and has no missing points".[9]. The Complete K-5 Math Learning Program Built for Your Child. Generally speaking, a curve means a line that must bend. n n Simple Curve: A simple curve changes direction but does not cross itself while changing direction. In particular, the length Weather charts are then drawn with irregular. In this case, a point with real coordinates is a real point, and the set of all real points is the real part of the curve. ] A curved line or curve is a smoothly-flowing line that line need not to be necessarily straight. When G is the field of the rational numbers, one simply talks of rational points. X By Grades. By using our services, you agree to our use of cookies. {\displaystyle C^{k}} She writes reviews and feature articles on contemporary art for a number of Texas-based and national publications such as the e-journal, ...might be good. γ Roughly speaking a differentiable curve is a curve that is defined as being locally the image of an injective differentiable function {\displaystyle t_{1}\leq t_{2}} → [ ) {\displaystyle C^{k}} Many artists of the 20th century used organic curved lines to create ambiguous and abstract shapes. Curved lines can also be "organic," creating irregular lines and shapes. C [ Except for lines, the simplest examples of algebraic curves are the conics, which are nonsingular curves of degree two and genus zero. {\displaystyle C^{k}} , Since the nineteenth century, curve theory is viewed as the special case of dimension one of the theory of manifolds and algebraic varieties. The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. to be Euclidean space. ] Last 300 years, something that curves or is curved, such as a bend in a, a line having no straight part; bend having no, a thing or part having the shape of a curve, the act of curving, or the extent of this. Some of the open curves are given in the figure below. This is the definition that appeared more than 2000 years ago in Euclid's Elements: "The [curved] line[a] is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width."[1]. A For example, a fractal curve can have a Hausdorff dimension bigger than one (see Koch snowflake) and even a positive area. Cord. {\displaystyle X} γ Sorry, we could not process your request. {\displaystyle s} X Each firm will face a downward-sloping demand curve. Post-impressionist artists like Gauguin often used outline. by. is such a curve which is only assumed to be The French Romantic painter Ingres is considered a master of contour line (Ingres is sometimes also considered Neoclassical--but most consider that an inaccurate designation). Previously lines could be either curved or straight. Houghton Mifflin Harcourt. . In the picture, monkey hangs in the cord of the tree. That is, a curve is a line that always changes its direction. 1. This is the case of space-filling curves and fractal curves. {\displaystyle \gamma } : It is also known as a concave downward. Upward curve: A curve that turns in the upward direction is called an upward curve. Algebraic curves can also be space curves, or curves in a space of higher dimension, say n. They are defined as algebraic varieties of dimension one. [ ] Found 954 sentences matching phrase "curved line".Found in 19 ms. {\displaystyle X} {\displaystyle \gamma } ∈ A plane algebraic curve is the zero set of a polynomial in two indeterminates. γ 'Hepatomegaly' and 'hydronephrosis' are among the most frequently looked-up words in September. Something characterized by such a line or surface, especially a rounded line or contour of the human body. And best of all it's ad free, so sign up now and start using at home or in the classroom. b ] These curves include: A fundamental advance in the theory of curves was the introduction of analytic geometry by René Descartes in the seventeenth century. by means of this notion of curve. {\displaystyle X} Curved line images. (i.e. Closed curve: A closed curve, has no end points and encloses an area (or a region). It is important to know that, curves hold different definitions as … {\displaystyle \gamma } : For example:[4]. are predominant in the structures of this period. They come from many sources and are not checked. 3). t X {\displaystyle t\in [a,b]} a of the year-on-year decline to a falling leaf. ( 4. R For example, Jacques-Louis David's treatment of the arches in the famous painting "The Oath of the Horatii.". All lines parallel to the axes are drawn to scale, and diagonals and. is a closed and bounded interval t , You can see the shape of curved line in the above image. Gostou do post? Avalie! [Total: 0 votos: ]	2021-05-08 07:46:49	{"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": 0.8213534355163574, "perplexity": 1080.9778197962808}, "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-2021-21/segments/1620243988850.21/warc/CC-MAIN-20210508061546-20210508091546-00110.warc.gz"}
https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10B_Problems/Problem_4&diff=prev&oldid=84696	# Difference between revisions of "2017 AMC 10B Problems/Problem 4" ## Problem Supposed that $x$ and $y$ are nonzero real numbers such that $\frac{3x+y}{x-3y}=-2$. What is the value of $\frac{x+3y}{3x-y}$? $\textbf{(A)}\ -3\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 3$ ## Solution Rearranging, we find $3x+y=-2x+6y$, or $5x=5y\implies x=y$ Substituting, we can convert the second equation into $\frac{x+3x}{3x-x}=\frac{4x}{2x}=\boxed{\textbf{(D)}\ 2}$ ## Solution yoyoyo Substituting each $x$ and $y$ with $1$, we see that the given equation holds true, as $\frac{3(1)+1}{1-3(1)} = -2$. Thus, $\frac{x+3y}{3x-y}=\boxed{\textbf{(D)}\ 2}$ 2017 AMC 10B (Problems • Answer Key • Resources) Preceded byProblem 3 Followed byProblem 5 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 10 Problems and Solutions	2020-10-21 19:58:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 13, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3768440783023834, "perplexity": 2555.906995847559}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107877420.17/warc/CC-MAIN-20201021180646-20201021210646-00033.warc.gz"}
https://www.iacr.org/cryptodb/data/author.php?authorkey=105	CryptoDB Christian Rechberger Publications Year Venue Title 2019 EUROCRYPT $\textsc {LowMC}$LOWMC is a block cipher family designed in 2015 by Albrecht et al. It is optimized for practical instantiations of multi-party computation, fully homomorphic encryption, and zero-knowledge proofs. $\textsc {LowMC}$LOWMC is used in the $\textsc {Picnic}$PICNIC signature scheme, submitted to NIST’s post-quantum standardization project and is a substantial building block in other novel post-quantum cryptosystems. Many $\textsc {LowMC}$LOWMC instances use a relatively recent design strategy (initiated by Gérard et al. at CHES 2013) of applying the non-linear layer to only a part of the state in each round, where the shortage of non-linear operations is partially compensated by heavy linear algebra. Since the high linear algebra complexity has been a bottleneck in several applications, one of the open questions raised by the designers was to reduce it, without introducing additional non-linear operations (or compromising security).In this paper, we consider $\textsc {LowMC}$LOWMC instances with block size n, partial non-linear layers of size $s \le n$s≤n and r encryption rounds. We redesign LowMC’s linear components in a way that preserves its specification, yet improves LowMC’s performance in essentially every aspect. Most of our optimizations are applicable to all SP-networks with partial non-linear layers and shed new light on this relatively new design methodology.Our main result shows that when $s < n$s<n, each $\textsc {LowMC}$LOWMC instance belongs to a large class of equivalent instances that differ in their linear layers. We then select a representative instance from this class for which encryption (and decryption) can be implemented much more efficiently than for an arbitrary instance. This yields a new encryption algorithm that is equivalent to the standard one, but reduces the evaluation time and storage of the linear layers from $r \cdot n^2$r·n2 bits to about $r \cdot n^2 - (r-1)(n-s)^2$r·n2-(r-1)(n-s)2. Additionally, we reduce the size of LowMC’s round keys and constants and optimize its key schedule and instance generation algorithms. All of these optimizations give substantial improvements for small s and a reasonable choice of r. Finally, we formalize the notion of linear equivalence of block ciphers and prove the optimality of some of our results.Comprehensive benchmarking of our optimizations in various $\textsc {LowMC}$LOWMC applications (such as $\textsc {Picnic}$PICNIC) reveals improvements by factors that typically range between 2x and 40x in runtime and memory consumption. 2019 ASIACRYPT The block cipher Jarvis and the hash function Friday, both members of the MARVELlous family of cryptographic primitives, are among the first proposed solutions to the problem of designing symmetric-key algorithms suitable for transparent, post-quantum secure zero-knowledge proof systems such as ZK-STARKs. In this paper we describe an algebraic cryptanalysis of Jarvis and Friday and show that the proposed number of rounds is not sufficient to provide adequate security. In Jarvis, the round function is obtained by combining a finite field inversion, a full-degree affine permutation polynomial and a key addition. Yet we show that even though the high degree of the affine polynomial may prevent some algebraic attacks (as claimed by the designers), the particular algebraic properties of the round function make both Jarvis and Friday vulnerable to Gröbner basis attacks. We also consider MiMC, a block cipher similar in structure to Jarvis. However, this cipher proves to be resistant against our proposed attack strategy. Still, our successful cryptanalysis of Jarvis and Friday does illustrate that block cipher designs for “algebraic platforms” such as STARKs, FHE or MPC may be particularly vulnerable to algebraic attacks. 2018 CRYPTO Recent developments in multi party computation (MPC) and fully homomorphic encryption (FHE) promoted the design and analysis of symmetric cryptographic schemes that minimize multiplications in one way or another. In this paper, we propose with Rastaa design strategy for symmetric encryption that has ANDdepth d and at the same time only needs d ANDs per encrypted bit. Even for very low values of d between 2 and 6 we can give strong evidence that attacks may not exist. This contributes to a better understanding of the limits of what concrete symmetric-key constructions can theoretically achieve with respect to AND-related metrics, and is to the best of our knowledge the first attempt that minimizes both metrics simultaneously. Furthermore, we can give evidence that for choices of d between 4 and 6 the resulting implementation properties may well be competitive by testing our construction in the use-case of removing the large ciphertext-expansion when using the BGV scheme. 2018 TOSC LowMC is a family of block ciphers designed for a low multiplicative complexity. The specification allows a large variety of instantiations, differing in block size, key size, number of S-boxes applied per round and allowed data complexity. The number of rounds deemed secure is determined by evaluating a number of attack vectors and taking the number of rounds still secure against the best of these. In this paper, we demonstrate that the attacks considered by the designers of LowMC in the version 2 of the round-formular were not sufficient to fend off all possible attacks. In the case of instantiations of LowMC with one of the most useful settings, namely with few applied S-boxes per round and only low allowable data complexities, efficient attacks based on difference enumeration techniques can be constructed. We show that it is most effective to consider tuples of differences instead of simple differences, both to increase the range of the distinguishers and to enable key recovery attacks. All applications for LowMC we are aware of, including signature schemes like Picnic and more recent (ring/group) signature schemes have used version 3 of the roundformular for LowMC, which takes our attack already into account. 2017 EUROCRYPT 2016 ASIACRYPT 2016 TOSC Recently, many efficient cryptographic hash function design strategies have been explored, not least because of the SHA-3 competition. These designs are, almost exclusively, geared towards high performance on long inputs. However, various applications exist where the performance on short (fixed length) inputs matters more. Such hash functions are the bottleneck in hash-based signature schemes like SPHINCS or XMSS, which is currently under standardization. Secure functions specifically designed for such applications are scarce. We attend to this gap by proposing two short-input hash functions (or rather simply compression functions). By utilizing AES instructions on modern CPUs, our proposals are the fastest on such platforms, reaching throughputs below one cycle per hashed byte even for short inputs, while still having a very low latency of less than 60 cycles. Under the hood, this results comes with several innovations. First, we study whether the number of rounds for our hash functions can be reduced, if only second-preimage resistance (and not collision resistance) is required. The conclusion is: only a little. Second, since their inception, AES-like designs allow for supportive security arguments by means of counting and bounding the number of active S-boxes. However, this ignores powerful attack vectors using truncated differentials, including the powerful rebound attacks. We develop a general tool-based method to include arguments against attack vectors using truncated differentials. 2016 TOSC We introduce subspace trail cryptanalysis, a generalization of invariant subspace cryptanalysis. With this more generic treatment of subspaces we do no longer rely on specific choices of round constants or subkeys, and the resulting method is as such a potentially more powerful attack vector. Interestingly, subspace trail cryptanalysis in fact includes techniques based on impossible or truncated differentials and integrals as special cases. Choosing AES-128 as the perhaps most studied cipher, we describe distinguishers up to 5-round AES with a single unknown key. We report (and practically verify) competitive key-recovery attacks with very low data-complexity on 2, 3 and 4 rounds of AES. Additionally, we consider AES with a secret S-Box and we present a (generic) technique that allows to directly recover the secret key without finding any information about the secret S-Box. This approach allows to use e.g. truncated differential, impossible differential and integral attacks to find the secret key. Moreover, this technique works also for other AES-like constructions, if some very common conditions on the S-Box and on the MixColumns matrix (or its inverse) hold. As a consequence, such attacks allow to better highlight the security impact of linear mappings inside an AES-like block cipher. Finally, we show that our impossible differential attack on 5 rounds of AES with secret S-Box can be turned into a distinguisher for AES in the same setting as the one recently proposed by Sun, Liu, Guo, Qu and Rijmen at CRYPTO 2016 2015 JOFC 2015 EPRINT 2015 EPRINT 2015 EPRINT 2015 FSE 2015 EUROCRYPT 2014 JOFC 2012 EUROCRYPT 2012 ASIACRYPT 2012 FSE 2011 ASIACRYPT 2010 EPRINT We introduce the rebound attack as a variant of differential cryptanalysis on hash functions and apply it to the hash function Whirlpool, standardized by ISO/IEC. We give attacks on reduced variants of the Whirlpool hash function and the Whirlpool compression function. Next, we introduce the subspace problems as generalizations of near-collision resistance. Finally, we present distinguishers based on the rebound attack, that apply to the full compression function of Whirlpool and the underlying block cipher $W$. 2010 ASIACRYPT 2010 ASIACRYPT 2010 EPRINT We revisit narrow-pipe designs that are in practical use, and their security against preimage attacks. Our results are the best known preimage attacks on Tiger, MD4, and reduced SHA-2, with the result on Tiger being the first cryptanalytic shortcut attack on the full hash function. Our attacks runs in time $2^{188.8}$ for finding preimages, and $2^{188.2}$ for second-preimages. Both have memory requirement of order $2^{8}$, which is much less than in any other recent preimage attacks on reduced Tiger. Using pre-computation techniques, the time complexity for finding a new preimage or second-preimage for MD4 can now be as low as $2^{78.4}$ and $2^{69.4}$ MD4 computations, respectively. The second-preimage attack works for all messages longer than 2 blocks. To obtain these results, we extend the meet-in-the-middle framework recently developed by Aoki and Sasaki in a series of papers. In addition to various algorithm-specific techniques, we use a number of conceptually new ideas that are applicable to a larger class of constructions. Among them are (1) incorporating multi-target scenarios into the MITM framework, leading to faster preimages from pseudo-preimages, (2) a simple precomputation technique that allows for finding new preimages at the cost of a single pseudo-preimage, and (3) probabilistic initial structures, compared with deterministic ones, to enable more neutral words, and hence to reduce the attack time complexity. All the techniques developed await application to other hash functions. To illustrate this, we give as another example improved preimage attacks on SHA-2 members. 2009 ASIACRYPT 2009 ASIACRYPT 2009 EUROCRYPT 2009 FSE 2008 FSE 2008 FSE 2008 EPRINT This is the first article analyzing the security of SHA-256 against fast collision search which considers the recent attacks by Wang et al. We show the limits of applying techniques known so far to SHA-256. Next we introduce a new type of perturbation vector which circumvents the identified limits. This new technique is then applied to the unmodified SHA-256. Exploiting the combination of Boolean functions and modular addition together with the newly developed technique allows us to derive collision-producing characteristics for step-reduced SHA-256, which was not possible before. Although our results do not threaten the security of SHA-256, we show that the low probability of a single local collision may give rise to a false sense of security. 2008 EPRINT We study the security of step-reduced but otherwise unmodified SHA-256. We show the first collision attacks on SHA-256 reduced to 23 and 24 steps with complexities $2^{18}$ and $2^{28.5}$, respectively. We give example colliding message pairs for 23-step and 24-step SHA-256. The best previous, recently obtained result was a collision attack for up to 22 steps. We extend our attacks to 23 and 24-step reduced SHA-512 with respective complexities of $2^{44.9}$ and $2^{53.0}$. Additionally, we show non-random behaviour of the SHA-256 compression function in the form of free-start near-collisions for up to 31 steps, which is 6 more steps than the recently obtained non-random behaviour in the form of a free-start near-collision. Even though this represents a step forwards in terms of cryptanalytic techniques, the results do not threaten the security of applications using SHA-256. 2008 CRYPTO 2008 CRYPTO 2007 FSE 2007 EPRINT The stream cipher Salsa20 was introduced by Bernstein in 2005 as a candidate in the eSTREAM project, accompanied by the reduced versions Salsa20/8 and Salsa20/12. ChaCha is a variant of Salsa20 aiming at bringing better diffusion for similar performance. Variants of Salsa20 with up to 7 rounds (instead of 20) have been broken by differential cryptanalysis, while ChaCha has not been analyzed yet. In this paper, we introduce a novel method for differential cryptanalysis of Salsa20 and ChaCha, inspired by correlation attacks and related to the notion of neutral bits. This is the first application of neutral bits in stream cipher cryptanalysis, and it allows us to present the first break of Salsa20/8, to bring faster attacks on the 7-round variant, and to break 6- and 7-round ChaCha. In a second part, we analyze the compression function Rumba, constructed as the XOR of four Salsa20 instances, and returning a 512-bit output. We find collision and preimage attacks for two simplified variants, then we discuss differential attacks on the original version, and exploit a high-probability differential to reduce complexity of collision search from 2^(256) to 2^(79) for 3-round Rumba. We give examples of collisions over three rounds for a version without feedforward, and near-collisions of weight 16 for three rounds of the original compression function, and of weight 129 for four rounds. 2006 ASIACRYPT 2006 FSE 2006 FSE 2006 EPRINT MAC algorithms can provide cryptographically secure authentication services. One of the most popular algorithms in commercial applications is HMAC based on the hash functions MD5 or SHA-1. In the light of new collision search methods for members of the MD4 family including SHA-1, the security of HMAC based on these hash functions is reconsidered. We present a new method to recover both the inner- and the outer key used in HMAC when instantiated with a concrete hash function by observing text/MAC pairs. In addition to collisions, also other non-random properties of the hash function are used in this new attack. Among the examples of the proposed method, the first theoretical full key recovery attack on NMAC-MD5 is presented. Other examples are distinguishing, forgery and partial or full key recovery attacks on NMAC/HMAC-SHA-1 with a reduced number of steps (up to 61 out of 80). This information about the new, reduced security margin serves as an input to the selection of algorithms for authentication purposes. 2005 EPRINT We present a collision attack on the recently proposed hash function SMASH. The attack uses negligible resources and we conjecture that it works for all hash functions built following the design method of SMASH. Crypto 2020 CHES 2019 Eurocrypt 2019 Crypto 2016 FSE 2016 Eurocrypt 2015 Asiacrypt 2014 FSE 2014 FSE 2013 Crypto 2013 Asiacrypt 2012 Asiacrypt 2011 FSE 2011 FSE 2010 FSE 2009 Eurocrypt 2009	2020-02-25 19:48:51	{"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": 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": 0.30975341796875, "perplexity": 1417.5638974491926}, "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-2020-10/segments/1581875146127.10/warc/CC-MAIN-20200225172036-20200225202036-00323.warc.gz"}
https://delong.typepad.com/sdj/2009/04/the-great-ricardian-equivalence-misunderstanding.html	## The Great Ricardian Equivalence Misunderstanding Paul Krugman attempts to provide some aircover: One more time: Brad DeLong is, rightly, horrified at the great Ricardian equivalence misunderstanding. It’s one thing to have an argument about whether consumers are perfectly rational and have perfect access to the capital markets; it’s another to have the big advocates of all that perfection not understand the implications of their own model. So let me try this one more time. Here’s what we agree on: if consumers have perfect foresight, live forever, have perfect access to capital markets, etc., then they will take into account the expected future burden of taxes to pay for government spending. If the government introduces a new program that will spend $100 billion a year forever, then taxes must ultimately go up by the present-value equivalent of$100 billion forever. Assume that consumers want to reduce consumption by the same amount every year to offset this tax burden; then consumer spending will fall by $100 billion per year to compensate, wiping out any expansionary effect of the government spending. But suppose that the increase in government spending is temporary, not permanent — that it will increase spending by$100 billion per year for only 1 or 2 years, not forever. This clearly implies a lower future tax burden than $100 billion a year forever, and therefore implies a fall in consumer spending of less than$100 billion per year. So the spending program IS expansionary in this case, EVEN IF you have full Ricardian equivalence. Is that explanation clear enough to get through? Is there anybody out there? It won't work, Paul: Robert Lucas: would a fiscal stimulus somehow get us out of this bind...? I just don't see this at all. If the government builds a bridge, and then the Fed prints up some money to pay the bridge builders, that's just a monetary policy. We don't need the bridge to do that... the only part of the stimulus package that's stimulating is the monetary part.... But if we do build the bridge by taking tax money away from somebody else, and using that to pay the bridge builder -- the guys who work on the bridge -- then it's just a wash.... [T]here's nothing to apply a multiplier to. (Laughs.) You apply a multiplier to the bridge builders, then you've got to apply the same multiplier with a minus sign to the people you taxed to build the bridge. And then taxing them later isn't going to help, we know that... Now Paul, will you also disabuse them of the blithe implicit assumption that the interest elasticity of money demand is 0 when we think--given that exchanging cash for a Treasury bill right now is exchanging one zero-yielding government asset for another--that this is a time when, in Milton Friedman's words, the right number for the approximate elasticity of money demand is -∞?	2021-01-18 21:05:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38521653413772583, "perplexity": 2442.1394293336393}, "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-2021-04/segments/1610703515235.25/warc/CC-MAIN-20210118185230-20210118215230-00628.warc.gz"}
https://brilliant.org/problems/is-it-too-hard/	# In a pickle Algebra Level 3 $\large{ \begin{cases}xy+x+y=23\\ yz+y+z=31 \\ zx+z+x=47\end{cases}}$ Let $$x,y$$ and $$z$$ be numbers satisfying the system of equations above, find the maximum value of $$x+y+z$$. ×	2017-01-17 01:03:20	{"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": 0.8038436770439148, "perplexity": 611.3749286545614}, "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-2017-04/segments/1484560279379.41/warc/CC-MAIN-20170116095119-00529-ip-10-171-10-70.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/767892/why-is-lim-limits-n-to-inftyxn1-0-where-x1	# Why is $\lim\limits_{N\to\infty}x^{N+1}=0$, where $\|x\|<1$? How is this done? Why is $\lim\limits_{N\to\infty}x^{N+1}=0$, where $-1<x<1$? - Have you tried applying the definition? – chubakueno Apr 24 at 21:02 I don't get what it means by the \|x\| <1 – Manuel Apr 24 at 21:06 $\|x\|<1$ is the same as saying $-1<x<1$. Sorry about the confusion. I'll change it back. – Sujaan Kunalan Apr 24 at 21:08 @chubakueno Hola otra vez! como estas? – Henry Lebesgue Apr 24 at 21:10 Oh okay...if $N$ keeps growing to Infinity. Shouldn't the limit be $\infty$? – Manuel Apr 24 at 21:11 We know that $-1<x<1$. Let's take a arbitrary $x$ as $x=\frac{1}{2}$. Now $(\frac{1}{2})^1=\frac{1}{2}=0.500$, $(\frac{1}{2})^2=\frac{1}{4}=0.250$ $(\frac{1}{2})^3=\frac{1}{8}=0.125$ $\vdots$ As $n$ gets larger and larger, our value will get smaller and smaller. This is because $-1<x<1$, so raising it to $n^{th}$ power as $n$ gets larger and larger actually makes our value smaller and smaller. - I see. I didn't take it into account that x was limited between -1 and 1. Thanks – Manuel Apr 24 at 21:21 Here is a trick: Put $\|x\| = \frac{1}{1 + y} < 1$. We want to show that $x^n \to 0$. In other words, by definition, for a given $\epsilon > 0$, we want to find an $N$ such that for all $n \geq N$, then $\|x^n - 0 \| < \epsilon$. To find $N$, notice by Bernoulli's inequality $$\|x^n\| = \frac{1}{(1+y)^n} \leq \frac{1}{1+yn} <\epsilon \iff n \geq \frac{1 - \epsilon}{\epsilon y}$$ So, choosing $N =$ integer part of $\frac{1 - \epsilon}{y \epsilon}$ gives desired result! - (+1) I didn't see your answer when I started writing mine. I see that it is essentially the same. Bernoulli is the way to go to avoid needing logarithms. I will remove my answer if you think it should be. – robjohn Apr 24 at 21:45 No, I think your answer is written better than mine! – Henry Lebesgue Apr 24 at 21:46 If $\|x\|\lt1$, choose an integer $k\ge\frac{\|x\|}{1-\|x\|}$. Then $$\|x\|\le\frac{k}{k+1}$$ Bernoulli's Inequality says that $\left(1+\frac1k\right)^n\ge1+\frac nk$. Therefore, $$\|x\|^n\le\left(\frac{k}{k+1}\right)^n\le\frac{k}{k+n}$$ If we wish to make $\|x\|^n$ smaller than any given $\epsilon\gt0$, choose $n\ge k/\epsilon$. Then, $$\|x\|^n\le\frac{k}{k+n}\le\frac{k}{k+k/\epsilon}=\epsilon\frac{k}{k\epsilon+k}\le\epsilon$$ - A relatively "low-tech" way to see the limit must be zero (assuming the limit exists) is to call the limit $L$ and note that $$L = \lim_{N \to \infty} x^{N+1} = \lim_{N \to \infty} (x \cdot x^{N}) = x \lim_{N \to \infty} x^{N} = xL.$$ Subtracting and factoring, $(1 - x)L = 0$. Since $x \neq 1$ by hypothesis, it must be that $L = 0$. To prove the limit exists without descending into $\varepsilon$-land (i.e., using convergence criteria that appear early in an elementary analysis course), note that if $0 \leq x < 1$, then the sequence $(x^{N})_{N=0}^{\infty}$ is non-increasing and bounded below by $0$, so it has a real limit $L$. To handle the case $-1 < x < 0$, note that $-\|x\|^{N} \leq x^{N} \leq \|x\|^{N}$ for all $N \geq 0$ and apply the squeeze theorem. - Given $x \neq 0$ (case equal to zero is trivial) you want $\|x^n\|<\epsilon \ \ \forall N>n$ for some $N$ you have to find that $N$. $$\|x^n\|<\epsilon \iff\|x\|^n<\epsilon \iff n \log\|x\|<\log\epsilon \iff n>\frac{\log \epsilon}{\log\|x\|}$$ - the last implication is not true – Henry Lebesgue Apr 24 at 21:43 @Lemur why si not true? – rlartiga Apr 24 at 23:28	2014-12-21 03:20:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9593029618263245, "perplexity": 399.0655119685582}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802770633.72/warc/CC-MAIN-20141217075250-00073-ip-10-231-17-201.ec2.internal.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-7-exponents-and-exponential-functions-7-3-multipying-powers-with-the-same-base-standardized-test-prep-page-431/75	## Algebra 1 4x - 5 = 2x + 13 Subtract 2x from both sides: 4x - 2x -5 = 2x - 2x +13 2x - 5 = 13 Now, add 5 to both sides: 2x - 5 + 5 = 13 + 5 2x = 18 Now, divide both sides by 2: 2x $\div$2 = 18 $\div$2 x = 9	2021-04-19 07:33:34	{"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": 0.9323017597198486, "perplexity": 252.26027021891133}, "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/1618038878326.67/warc/CC-MAIN-20210419045820-20210419075820-00499.warc.gz"}
https://dsp.stackexchange.com/tags/digital-filters/hot	# Tag Info 26 Citing Bellanger's classic Digital Processing of Signals – Theory and Practice, the point is not where your cut-off frequency is, but how much attenuation you need, how much ripple in the signal you want to preserve you can tolerate and, most importantly, how narrow your transition from pass- to stopband (transition width) needs to be. I assume you want a ... 18 Consider a discrete-time input signal of the form: $$x[n] = \cos(\omega_0 n) ~~~,~~~-\infty < n < \infty, ~~~~~ n\in \mathcal{Z}$$ where the radian frequency $\omega_0$ is set between 0 and $\pi$ radians per sample. Now, consider two simplest type of discrete-time (digital) LTI FIR filters which are defined through the fundamental arithmetic ... 17 For digital notch filters, I like to use the following form for a notch filter at DC ( $\omega$=0): $$H(z) = \frac{1+a}{2}\frac{(z-1)}{(z-a)}$$ where $a$ is a real positive number < 1. The closer $a$ is to 1, the tighter the notch (and the more digital precision needed to implement). This is of the form with a zero = 1, and a pole = $a$, where $a$ is ... 14 For a quick and very practical estimate, I like fred harris' rule-of-thumb: $$N_{taps} = \frac{Atten}{22B_T}$$ where: Atten is the desired attenuation in dB, $B_T$ is the normalized transition band $B_T=\frac{F_{stop}- F_{pass}}{F_s}$, $F_{stop}$ and $F_{pass}$ are the stop band and pass band frequencies in Hz and $F_s$ is the sampling frequency in ... 9 In general you can't simply subtract a low-pass filtered version of a signal from the original one to obtain a high-pass filtered signal. The reason is as follows. What you're actually doing is implement a system with frequency response $$H(\omega)=1-H_{LP}(\omega)\tag{1}$$ where $H_{LP}(\omega)$ is the frequency response of the low-pass filter. Note that $... 8 For a filter consisting of a complex conjugate pair of poles, the$z$-domain transfer function is: $$H(z) = \frac{a}{\left(1-r(\cos{\theta}-i\sin{\theta})z^{-1}\right)\left(1-r(\cos{\theta}+i\sin{\theta})z^{-1}\right)}\\ = \frac{a}{1 - 2r\cos(\theta)z^{-1} + r^2z^{-2}},$$ where$a$is a constant by which the magnitude frequency response can be normalized,$... 8 If I understood you correctly, you want to compute the value of $\alpha$ that results in a specified 3dB cut-off frequency for an exponentially weighted moving average filter. If you square your last equation, you get $$\frac{\alpha^2}{1-2(1-\alpha)\cos(\omega_c)+(1-\alpha)^2}=\frac12\tag{1}$$ which can be rearranged into the following quadratic equation: ... 8 One cause is that higher order Butterworth filters have poles closer to the unit circle. This nearby infinite gain point increases the likelihood of numerical instabilities. (e.g. rounding/arithmetic/quantization noise may move a pole to the “wrong” side of the unit circle.) Where the numerical noise will blow up depends on your executable code’s exact ... 7 Adding to the accepted answer, a few additional references. I won't write the formulas which can be involved. Those formulae mostly yield rule-of-thumbs or approximations to start from. You can fiddle around these numbers for your actual design. One of the origin for Bellanger's design is: On computational complexity in digital filters, 1981, Proc. Eur. ... 7 Short answer: You can't. If an attacker can insert a signal that covers the whole bandwidth (e.g. a white signal, or at least one that has no spectral zeros) into the system (and he can do that over an arbitrarily long time, or add up observations), they will get an output, and can through the magic of correlation get the impulse response. 7 You can apply a so-called all-pass transformation to a discrete-time low-pass prototype filter in order to convert it to other standard filters (such as high-pass, band-pass, and band-stop). This is accomplished by transforming the complex variable $z$ in the transfer function of the prototype filter by a function $G(z)$ which satisfies $\|G(e^{j\omega})\|=1$, ... 6 Hmmmmmmmmm, interesting question. Since you want to use the second derivative as your criteria, it would seem that you would want to have the maximum second derivative absolutie value for as short of a duration as possible. This would suggest piecing together parabolas, matching the first derivatives at the joints. How to do this algorithmically will take ... 6 A first rationale is to be very short, as there was a time when computing on images was expensive. Then, a contour or an edge often present a fast variation in image intensities, that can be enhanced by derivatives. Sobel filters emulate such derivatives in one direction, and slightly average pixels in the complementary direction, to smooth small variations ... 5 it depends on how you map the analog filter to the digital filter and how the s-plane poles get mapped to the z-plane poles (ya know, the "bilinear transform" vs. "impulse invariant" vs. whatever else is out there). probably, if it were up to me, for the purpose of defining $Q$, i would map the poles as you would map $s$ to $z$ with $$z = e^{sT}$$ ... 5 You have probably used filtering a lot already. A moving average is a filter! Think of general filtering as performing a fancy moving average where instead of averaging every component in a window equally, you weight the components. If you just wanted to smooth the signal you could weight each value used in the average by a Gaussian (bell) curve for ... 5 To the useful answers that have been added so far, I would like to add, on the point of intuition, that filtering works because it is based on Wave theory and specifically, the interaction of waves. This provides a huge array of intuitive examples. But also, that there are basically two viewpoints. One is the abstract viewpoint, taken by modelling reality ... 5 The denominator (recursive coefficients Ai) look OK: the poles of your system are at 45 degree angles ($\pi/4$), with magnitude 0.68 (which is not very aggressive for a notch filter; in my opinion they should be more like 0.9). But your numerator has its roots very near $z=1$, which corresponds to frequency 0 instead of the desired $\pi/4$ for implementing ... 5 You'd have to figure out the frequency response of the filter. Here are two methods. I prefer Method 2 because it's quick and dirty, and you don't really care about the exact gain values in the frequency response, just the general shape to figure out the type of the filter. Method 1: Brute Force/Computer Assisted import scipy.signal as sp import numpy as ... 5 In this answer I'll try to show you how to qualitatively evaluate a given pole-zero plot by just looking at it. Of course, this method has its limits, but for relatively simple pole-zero plots you can very quickly decide on the type of the corresponding filter. You should know that for stable filters, the frequency response equals the transfer function $H(z)... 5 The question is rather vague or inaccurate really; or I haven't understood it well. I would start with saying "there is no such a thing as a 'best' filter (for all use)". A filter is optimal only if it exploits a specific property of signal or noise generating better SNR over regular filter -but if you don't know about signal or noise specifically then ... 5 First, you'll probably have better luck posting this on dsp.stackexchange. That's a more specialized group that does stuff like this all the time. In terms of your problem, here's a couple of options. One is a machine learning approach. e.g. create a training set by taking a bunch of data and hand marking the points that are good versus bad (like you've ... 5 First of all, what is the order of your IIR filter? The highest order I have ever used was an order-10 IIR filter for a control loop application. I feel like it is unlikely that you need more that this. Second, it is a good idea to split your filter in second-order-sections (SOS) and cascade them , this usually fix most issues. https://www.dsprelated.com/... 5 It is the objective of the receiver to make the best estimate for each symbol as to what was transmitted. This is often done by ultimately determining a decision time in each sample (through timing recovery) on the waveform after it has been processed by the receiver (equalization and matched filtering) in which to sample the waveform and make a decision as ... 5 In general there is no straightforward analytical solution. As you know, you need to solve $$\left\|H(e^{j\omega_c})\right\|=\frac{1}{\sqrt{2}}\tag{1}$$ for$\omega_c$, where it is assumed that the maximum filter gain equals$1$. For Butterworth filters, the specified cut-off frequency always equals the$3\textrm{ dB}$frequency. This is not the case for other ... 5 A discrete-time first-order high pass filter with unity gain at Nyquist and a zero at DC is described by the following difference equation: $$y[n]=\frac{1+\alpha}{2}\big(x[n]-x[n-1]\big)+\alpha y[n-1],\qquad -1<\alpha<1\tag{1}$$ Its transfer function is given by $$H(z)=\frac{1+\alpha}{2}\frac{1-z^{-1}}{1-\alpha z^{-1}}\tag{2}$$ Evaluating the squared ... 5 Yes Virginia, There is a perfect digital filter. I assume the OP means by "perfect filter" what we would typically call an "ideal filter": which is a filter that passes a finite block of frequencies with no alteration and completely removes all other frequencies, which is referred to as a "brick wall filter". Otherwise if the OP ... 4 You calculating FFT only from two samples. You need to pad your impulse response with zeros to get a valid result. So in MATLAB that would be: N = 1024; % Number of points to evaluate at % Create the vector of angular frequencies at one more point. % Filter itself b=[1,-1]; [h_f, w_f] = freqz(b, 1); figure grid on hold on plot(w_f, abs(h_f), 'or') % MATLAB ... 4 As I've mentioned in a comment, the Parks McClellan algorithm is usually used to design frequency selective filters with a fixed maximum stopband error, which results in an equiripple behaviour in the stopband. Note that the algorithm can in principle approximate any desired frequency response shape. However, many implementations just allow for piecewise ... 4 The book doesn't say that the impulse response must be zero for an ideal channel. It says that an ideal channel has exactly one, and not more than one, non-zero component, i.e. the ideal channel's impulse response is an impulse, which means that the signal is only delayed but not distorted. 4 A low pass filter has a frequency response that satisfies $$\|H(\omega)\|\approx 0,\quad \|\omega\|>\omega_c\tag{1}$$ where$\omega_c\$ is the cut-off frequency. A complex low pass filter must also satisfy $$H(\omega)\neq H^(-\omega)\tag{2}$$ which causes its impulse response to be complex-valued. So the frequency response of a complex low pass filter is ... Only top voted, non community-wiki answers of a minimum length are eligible	2021-01-22 13:12:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8326923251152039, "perplexity": 672.5301663347816}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703529331.99/warc/CC-MAIN-20210122113332-20210122143332-00092.warc.gz"}
https://hamishraw.com/digital-options/european-digital-options/digital-call-options/	Digital Call Options Digital call options are all-or-nothing options that settle at: The price of digital call options could be interpreted as the probability of the event happening. This assumes a zero cost-of-carry and yield, i.e. interest rates are zero with no dividends or coupon payments. Figure 1 shows the expiry profile of a $100 strike digital call: • Below the strike price of 100.00 the digital call has lost and settles at 0. • Above the strike price of 100.00 the digital call has won and settles at 1.00. • In the event of the asset price settling exactly on the strike of 100.00 on this website is adopted the value of 0.50. This represents a dead heat. If the graph was flipped upside down around the vertical axis at 0.50 it becomes the digital put option expiry profile. Digital Call Options over Time Figure 2 shows the P&L profile illustrating how the expiry profile was arrived at over time. The price profiles are always flat or rising from left to right indicating that the digital call always has a positive or zero delta. Below the strike price it is clear that the price profiles become steeper as they approach the strike price. Conversely they become less steep above the strike price. This indicates the unusual (if you are a conventional options trader) feature of the digital call gamma being positive below the strike and negative above it. The buyer of this digital call is betting that the asset price will be above$100 at expiry. The 25-day profile is shallow but over time this animal changes its spots. It becomes the most highly geared and dangerous instrument in the world of finance. It is doubtful that any other single instrument can offer a P&L profile that can exceed an angle of 45°. Indeed the angle of an at-the-money moments before expiry tends to the vertical and becomes absolutely unhedgeable. Positive Theta? What is also apparent from the profiles over time is that the bet decreases in value when out-of-the-money. Conversely it increases in value when in-the-money. Therefore, the out-of-the-money call has a negative digital call option theta while the in-the-money has a positive call theta. Furthermore, the at-the-money has a theta of zero assuming that the above ‘dead heat’ rule is applied. Digital Call Options and Volatility Implied volatility is a critical input into the pricing of digital options. The level of implied volatility determines whether one is buying the digital option cheaply or too expensively. Figure 3 displays the digital call options price profile over a range of implied volatility. At the underlying price of \$98.50, as implied volatility increases, so does the value of the out-of-the-money option. This indicates a positive vega. Above the strike an increase in ‘vol’ lowers the option value; this indicates negative vega. Why does the price increase at 98.50? This is because with a low volatility the probability of the underlying price rising above the strike is low. Over time this will, in turn, lead to a worthless digital call option. On the other hand, as volatility increases and the underlying swings around more there is a greater chance of the asset price moving above the strike. At 101.50 this call is in-the-money, the asset price is above the strike price and the ‘long’ (the trader who has bought the option) is in a winning position. Yet if the underlying asset becomes more volatile there would be a higher chance of the asset price falling back through the strike to a losing position. If the asset price ground to a halt at 101.50 and volatility falls then there is now a better chance of the bet remaining a winner and the bet’s value increases. Summary With the very first instrument analysed it is noticeable that there is a big difference between the characteristics of the digital call and those of the conventional call. When the call option is in-the-money the option holder will have a better chance of being a winner if volatility falls. An increase in implied volatility decreases the value of the option. This is because the option has a higher probability of the underlying sliding back down through the strike. Therefore, above the strike the digital call option has a negative vega. 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https://scriptinghelpers.org/questions/12844/moving-a-block-up-and-down	0 # Moving a block up and down? Ive been looking around the website & Roblox wiki but i cant find anything to my knowledge about moving a block up and down about 1 / 1 & a half studs If anyone knows somewhere where this is explained can you link it, thanks! 3 For future reference: Please provide the sufficient material, code snippets you may have tried, any specific idea on what it should do, etc. To accomplish this task, you must change the Position of the part using a Vector3.new(). To use it in a loop, it would look like this: --this script would be inside the part local brick=script.Parent --creating a new variable named brick, referencing our part. "script" is a built-in object that references the actual script. while wait(1) do --wait a second before running the code again for I=1,10 do --begin a "for" loop, we are going to move the part up very little by little to make it seem like it is sliding. brick.Position=brick.Position+Vector3.new(0,0.1,0) --add 0.1 on the Y scale (Vertical) to increase the brick's position wait() --wait a fraction of a millisecond, or else it doesn't appear as a sliding brick. end --end the "for" loop wait(1) --pause a second for I=1,10 do --This section is pretty much just a repeat. brick.Position=brick.Position+Vector3.new(0,-0.1,0) --subtract 0.1 on the Y scale (Vertical) to decrease the brick's position wait() end end --end the loop The issue with this code is, if something is in the way, the part will jump immediately to the lowest point available which would be the top of the brick that is in the way. Our solution is CFrame.new(). "CFrame" is just a shortened version of CoordinateFrame. This would be the version with CFrame: --this script would be inside the part local brick=script.Parent --creating a new variable named brick, referencing our part. "script" is a built-in object that references the actual script. while wait(1) do --wait a second before running the code again for I=1,10 do --begin a "for" loop, we are going to move the part up very little by little to make it seem like it is sliding. brick.CFrame=brick.CFrame+Vector3.new(0,0.1,0) --increase the height, this time with the CFrame. Notice we are still using a Vector3 property to add on, but what counts is the CFrame that we're adding on to. wait() --wait a fraction of a millisecond, or else it doesn't appear as a sliding brick. end --end the "for" loop wait(1) --pause a second for I=1,10 do --This section is pretty much just a repeat. brick.CFrame=brick.CFrame+Vector3.new(0,-0.1,0) --subtract 0.1 on the Y scale (Vertical) to decrease the brick's position wait() end end --end the loop 1 Thanks for the help & explanation, will remember the code snippets/sources thing for the future Rythian2277 65 — 7y local brick = script.Parent	2022-08-08 01:03:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3517788350582123, "perplexity": 3175.742506488321}, "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/1659882570741.21/warc/CC-MAIN-20220808001418-20220808031418-00212.warc.gz"}
http://kellyakearney.net/2016/01/19/food-webs-as-network-graphs-1.html	This is Part 1 in my three-part series on visualizing food webs as network graphs. Part 1Part 2Part 3 This series of blog posts is a supplement to my Ocean Sciences 2016 poster presentation: “Visualizing Ecosystem Energy Flow in Complex Food Web Networks: A Comparison of Three Alaskan Large Marine Ecosystems,” available for download here. This blog post will explain the theory behind the food web visualizations, and offer a step-by-step procedure (with code) to create similar food web diagrams from your own food web network data (specifically, Ecopath model data). ## Introduction Food webs are a central focus of my research, and for papers and presentations, I usually like to include a visualization of the food web being discussed. In simple models with a dozen or so state variables (e.g. biogeochemical NPZ models), a clean, visually-intuitive diagram can be constructed manually without too much effort. As the number of nodes in the diagram increases, graphing software like Graphviz, Gephi, or the network graph modules of Python, Matlab, D3, etc. can provide some automated node placement, edge routing algorithms, and plotting tools. However, all of these tools fell short when I attempted to use them on a management-oriented food web with 120+ nodes and 2000+ edges. The resulting diagrams were simply too cluttered to be useful; the only information one can extract from the resulting figures is “Well, it’s complex!” I have to laugh (sometimes at myself, because I’ve been guilty of it too!) every time I see a figure like these ones in the published literature or in a conference presentation, accompanied by a phrase like “The figure shows…” or “As you can see…” Nope, I can’t see anything except a solid blob of ink (or pixels). In my quest to turn these cluttered messes into informative figures, I stumbled into the vast and fascinating world of data visualization associated with network graphs. Large networks are increasingly common in the world of big data, and the data visualization world has published a variety of techniques to try to reduce the visual clutter in their visual depictions. This post will be the first in a three-part series where I describe the techniques I’ve adapted in order to create improved food web diagrams for my research. The three posts will discuss 1. Edge bundling 2. Node positioning 3. The code In Parts 1 and 2, I’ll go through the the background and theory, with some examples (mostly in Matlab, with a bit of javascript). Part 3 will discuss the code underlying those examples in more detail. ## Terminology First, a bit of terminology. Throughout this series of blog posts, I use the term “graph” in the graph theory context, i.e. a set of nodes (points, vertices) connected by edges (lines) that show the how objects in a network are connected. In a food web, the nodes of the graph represent biological state variables, such as living critters (either single species or groups of species that are treated as a single functional group), fishing fleets, detrital pools, etc. The edges represent processes that move biomass and/or energy from one group to another… primary production, grazing and predation, egestion and excretion, respiration, fisheries landings and discards, non-predatory mortality, etc. ## Divided edge bundling Edge bundling algorithms, as their name implies, work by gathering edges of a graph together into groups in an effort to elucidate high-level patterns amongst the tangle of lines. The rules for which edges get bundled with each other, and how strongly, vary by algorithm. For food webs, I was looking for a variation that would pull out high-level flows (such as pelagic vs benthic food pathways) while preserving both the direction and weight of graph edges, and I found a great candidate in Selassie et al.’s divided edge bundling algorithm: Selassie D, Heller B, Heer J (2011) Divided edge bundling for directional network data. IEEE Trans Vis Comput Graph 17:2354–2363 DOI:10.1109/TVCG.2011.190 The divided edge bundling algorithm places control points along each edge of a network graph, and then treats those control points like charged particles that attract the control points on other edges. The attraction between edges moving in opposite directions is offset relative to those moving in the same direction, resulting in directional lanes. After playing around with Selassie’s software prototype for a while, I decided I needed to make a few tweaks to the algorithm to best apply it to food web networks. My version consists of two functions (debundle.m and plotdeb.m) written in Matlab, and builds upon the original algorithm with a few additions and changes. I’ll elaborate on those changes in the next section, but I’ll start with an example so you can see how the algorithm works. Coincidentally, I started development of my Matlab version of divided edge bundling the same week that the Mathworks released their beta version of Matlab R2015b, which included the new graph object class. These graph objects provide a nice way to keep all graph data together, along with built-in functions for calculations like shortest path and connected components (finally, no more need to distribute the giant matlab_bgl library with my network-related code!). Because of the reliance on graph objects, you’ll need Matlab R2015b or later to run this code. For demonstration purposes, we’ll start with the 3-by-3 example used in the Selassie et. al paper. It consists of 12 nodes in a 6 x 2 grid, with edges defining 2 subgraphs (3 nodes per side in each subgraph). To produce that graph, I started with a table a node coordinates: and a table of edge source/target pairs: From these tables, I create a graph object. I’ll also add the node coordinate data as extra Node table properties, which we can use for plotting and will later be required for the divided edge bundling function: We can plot the resulting graph using Matlab’s graph plotting tools: Even with this relatively simple graph, things are getting a bit cluttered, particularly with all the arrowheads centered on the edges. Using plotdeb.m in initial mode, edges are instead rendered as partially-transparent patches that change color as they move from the source node (0 on the colorbar) to the target node (1): The edge patches also vary in width based on the weight of each edge. In this example, all edges were the same weight, but we can see the effect if we increase the weight of the edge that runs from node 2b to node 5b: Now we run the edge-bundling algorithm: While the code runs it will print its status to the screen: As I’ll discuss below, my version is quite a bit slower than the original, so depending on the complexity of the graph, you may need to go get some coffee at this point. This example only needs a few seconds, though, and we can now plot the new bundled paths for the edges. In addition to showing the new bundled-together paths for each of the edges, the plotting routine changes the widths of the edge bundles to reflect the sum total of the weights of the edges at each location. In this version, while we lose some of our ability to trace individual edges from their source to their target, we get a better picture of the main flows in the graph. For example, we can better see the separation of the two subgraphs (b-nodes vs other nodes), and compare the total flow moving from left to right as opposed to right to left in the b-subgraph. ## Adapting the algorithm to food webs ### Bundling vs. plotting The original EdgeBundling application reads in a graph topology (including node positions and edge source-target pairs) from a file, accepts a few user-configurable parameters, and on command, runs the bundling calculations and plots the results to the screen. I decided to separate the bundling (debundle.m) and plotting (plotdeb.m) steps, for several reasons. The first benefit is flexibility. Matlab has a full-featured graphics library, so there’s no need to limit the choices of edge color, transparency, etc. to those hard-coded in the original. This also allows me to plot the nodes separately from the edges, and to add additional details, like the trophic level axis used in my food web diagrams. The second reason for separating the calculations is speed. I’ve profiled my code pretty thoroughly for any unnecessary bottlenecks, but Matlab is no match for the GPU-accelerated Objective-C Selassie used. And the physical simulation complexity is $$O(E^2C)$$ (where $$E$$ is the number of edges and $$C$$ is the number of control points per edge) for each iteration, which is asking a lot of an interpreted language. A simulation that takes 20-30 s in the C-based software requires a couple hours in the Matlab code. Acceptable to me, since I only need to create these figures once or twice per research project, but I didn’t want to have to rerun everything just to tweak the edge width scaling in a single figure. The final reason for separating things is due to the addition of scaled weights for bundling, as described in the next section. ### Scaled weight function for bundling Edge weight values are used in both the bundling and plotting steps of the divided edge bundling algorithm. During the bundling portion, the forces between control points are scaled based on the weight of the edges they are on. Because of this, bundles of low-weight edges whose weights sum to a certain value behave in the same way as a single high-weight edge of that value in terms of attracting other edges. During the plotting portion, edge width at a given control point is calculated based on the sum of the weights of all control points within a threshhold radius of that point. The result is that overlapping edges are depicted as a single edge with the cumulative weight of the bundle members. In food web networks, the biomass fluxes represented by edge weights usually span at least 5-7 orders of magnitude due to trophic transfer losses at each predation link. In the food web I happened to be working with at the time I developed this function, the weight difference between the smallest flux (longline bycatch of mammals) and the largest (primary production by small phytoplankton) spanned 14 orders of magnitude. With this weight difference, the forces exerted at the bottom of the food web far surpassed those at the top of the food web, resulting in almost no bundling occurring beyond the first couple of trophic levels. No good. The usual remedy to this large range in fluxes is to scale them, typically with a strong transformation like a logarithm, cubic root, or square root. Unfortunately, using such a scale loses the additive property that is relied upon for the edge bundling algorithm and visualization. In the end, I decided that I was willing to sacrifice the additive property of edges during the bundling process itself, but not during the final visualization. An optional input parameter can be passed to the debundle.m function, telling it to scale edge weights prior to beginning its calculations. For my work, I’ve been using the function where wmin is the logarithm of the smallest weight, rounded down a bit (to keep all weight values positive as required by the algorithm). For the visualization, Selassie’s code already included an optional parameter to de-linearize the edge width scaling. The edge width equation is $$D=wg^p$$, where $$D$$ is the visible edge width, $$w$$ is the user-assigned maximum edge width, $$g$$ is the bundle weight at a control point, and $$p$$ is the scaling parameter. Selassie et al. states that the $$p$$ parameter "allows more subtle asymmetries in edge and bundle weight to be seen". In the case of food web networks, we actually want these asymetries deemphasized, which can be accomplished when $$p<1$$. I’ve found $$p = \frac{1}{3}$$ (cubic root scaling) works pretty well for food webs. The scaling is applied after the bundle weight calculation, so edge bundles still maintain the additive property we want. ### Self-edges The divided edge bundling algorithm removes any self edges prior to calculation, since these edges do not have a constant orientation as needed for the calculations. Although they do not factor into the bundle calculations, self-edges (i.e. cannibalism) are quite prevelant in ocean ecosystem models, so I wanted to include them in the final visualizations. For this, I included a switch that, when true, plots self-edges as circlular paths. ## The next step: Node placement While this algorithm greatly helped in cleaning up spaghetti-mess edges, it can only be applied to graphs where the coordinates of the nodes are already known (such as the geographic examples in the Selassie et al. paper). While y-coordinates in food webs often correspond to trophic level, the x-coordinates are not constrained. In their discussion, Selessie et al. noted A major question unaddressed in both the prior and current work is the interplay between node layout and edge bundling; Holten & van Wijk’s compatibility coefficients are exclusively a function of node position, so layout greatly affects bundling. Automated graph layout algorithms that tend to orthogonalize edges cause force-directed edge bundling techniques to coalesce edges ineffectively. Layout approaches that jointly optimize node placement and edge bundling might enable improved pattern perception. So to get the best results when applying edge bundling to food webs, we need to start with a node-placement algorithm that optimizes node position for bundling rather than orthagonality. And ideally, one that also allows contraining y-position to match trophic level while allowing free movement in the x-direction. With no such algorithm seemingly readily available, I decided to create my own. Read on to the next blog post to find out the details. Part 1Part 2Part 3	2018-02-18 22:21:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.6093043684959412, "perplexity": 1231.5738771733588}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812259.30/warc/CC-MAIN-20180218212626-20180218232626-00094.warc.gz"}
https://lama.u-pem.fr/recherche/publications_et_prepublications_du_laboratoire	# Publications et prépublications du laboratoire Les prépublications des membres du laboratoire peuvent être consultées sur les deux sites suivants Une liste non exhaustive des publications du laboratoire basée sur les serveurs de HAL est présentée ci-dessous : ### 2022 • ENTROPIC CURVATURE ON GRAPHS ALONG SCHRÖDINGER BRIDGES AT ZERO TEMPERATURE Probability Theory and Related Fields (2022) • Foreign Exchange Multivariate Multifractal Analysis European Signal Processing Conference (EUSIPCO) (2022) • EXTRINSIC UPPER BOUNDS THE FIRST EIGENVALUE OF THE p-STEKLOV PROBLEM ON SUBMANIFOLDS Communications in Mathematics (2022) 49-61 • Cycles in synchronous iterative voting: general robustness and examples in Approval Voting Social Choice and Welfare (2022) ### 2021 • Strichartz estimates and Fourier restriction theorems on the Heisenberg group Journal of Fourier Analysis and Applications (2021) • Equity Cost-Induced Dichotomy for Optimal Dividends in the Cramér-Lundberg Model MDPI Mathematics 9 9 (2021) 931 • Why are viral genomes so fragile? The bottleneck hypothesis PLoS Computational Biology 17 7 (2021) e1009128 • Minimal planes in asymptotically flat three-manifolds Journal of Differential Geometry (2021) • A generic construction for high order approximation schemes of semigroups using random grids Numerische Mathematik (2021) • $f$-extremal domains in hyperbolic space Calculus of Variations and Partial Differential Equations 60 3 (2021) 112 • Stochastic phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model. Journal of Mathematical Analysis and Applications 496 1 (2021) • Markov Renewal and Piecewise Deterministic Processes Springer International Publishing Probability Theory and Stochastic Modelling 100 (2021) • Optimal transportation and stationary measures for Iterated Function Systems Mathematical Proceedings of the Cambridge Philosophical Society (2021) • TRANSPORT PROOFS OF SOME DISCRETE VARIANTS OF THE PRÉKOPA-LEINDLER INEQUALITY,Preuve par transport de mesure de versions discrètes de l'inégalité de Preekopa-Leindler Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (2021) 1207-1232 • Convergence of nonlinear numerical approximations for an elliptic linear problem with irregular data ESAIM: Mathematical Modelling and Numerical Analysis 55 6 (2021) 3043-3089 Annals of Applied Probability 31 5 (2021) 2441-2477 • Inference with selection, varying population size and evolving population structure: Application of ABC to a forward-backward coalescent process with interactions Heredity 126 (2021) 335–350 ### 2020 • ON ALMOST STABLE CMC HYPERSURFACES IN MANIFOLDS OF BOUNDED SECTIONAL CURVATURE Bulletin of the Australian Mathematical Society 101 2 (2020) 333-338 Computational Methods in Applied Mathematics 20 3 (2020) 43-458 • Minimal surfaces near short geodesics in hyperbolic 3-manifolds • Multilayer models for shallow two-phase debris flows with dilatancy effects Journal of Computational Physics 419 (2020) 109699 • Probabilités - Préparation à l'agrégation interne CreateSpace Independent Publishing Platform (2020) 163 • A NONLINEAR PROBLEM WITH A WEIGHT AND A NONVANISHING BOUNDARY DATUM Pure and Aplied functional Analysis (2020) 965-980 • A unified analysis of elliptic problems with various boundary conditions and their approximation Czechoslovak Mathematical Journal 70 (2020) 339–368 • A WELL-POSEDNESS RESULT FOR VISCOUS COMPRESSIBLE FLUIDS WITH ONLY BOUNDED DENSITY Analysis & PDE 13 1 (2020) • BLOW UP DYNAMICS FOR THE HYPERBOLIC VANISHING MEAN CURVATURE FLOW OF SURFACES ASYMPTOTIC TO SIMONS CONE Journal of the European Mathematical Society (2020) • On a senary quartic form Periodica Mathematica Hungarica 80 2 (2020) 237-248 • Sojourn time dimensions of fractional Brownian motion Bernoulli (2020) • A Nonlinear Quantum Adiabatic Approximation Nonlinearity 33 (2020) 4715-4751 • Contextual metrics. A mathematical definition for a comprehensive approach of geographical distances Geographical Analysis (2020) • On the 2-Systole of Stretched Enough Positive Scalar Curvature Metrics on $\mathbb{S}^2\times\mathbb{S}^2$,Sur la 2-systole des métriques suffisamment étirées et à courbure scalaire positive sur S 2 x S 2 Symmetry, Integrability and Geometry : Methods and Applications (2020) • Empirical measures: regularity is a counter-curse to dimensionality ESAIM: Probability and Statistics 24 (2020) 408-434 • Enhanced convergence rates and asymptotics for a dispersive Boussinesq-type system with large ill-prepared data Pure and Applied Analysis 2 2 (2020) 477–517 • Regularity of solutions of a fractional porous medium equation Interfaces and Free Boundaries 22 4 (2020) 401–442 • Integrability Of Liouville Theory: Proof Of The Dozz Formula Annals of Mathematics 191 1 (2020) 81-166 • Do Generalized Draw-down Times Lead to Better Dividends? A Pontryaghin Principle-Based Answer IMA Journal of Mathematical Control and Information (2020) ### 2019 • Ergodicity of the zigzag process Annals of Applied Probability 29 4 (2019) 2266-2301 • Characterization of $f$-extremal disks Journal of Differential Equations 266 4 (2019) 2052-2077 • Lamn property for the drift and volatility parameters of a SDE driven by a stable LEVY process ESAIM: Probability and Statistics 23 (2019) 136-175 • The Z-invariant Ising model via dimers Probability Theory and Related Fields 174 (2019) 235-305 • Multivariate multifractal analysis Applied and Computational Harmonic Analysis 46 3 (2019) 653-663 • On Lasso refitting strategies Bernoulli 25 (2019) 3175-3200 • Random polytopes obtained by matrices with heavy-tailed entries Communications in Contemporary Mathematics (2019) 1950027 • Estimation of extremes for Weibull-tail distributions in the presence of random censoring Extremes 22 4 (2019) p667-704 • Variational formulation of American option prices in the Heston Model SIAM Journal on Financial Mathematics 10 1 (2019) 261-368 • Non universality for the variance of the number of real roots of random trigonometric polynomials Probability Theory and Related Fields 174 (2019) 887-927 • Effective perturbation theory for linear operators Journal of Operator Theory (2019) • On the influence of gravity on density-dependent incompressible periodic fluids Journal of Differential Equations 267 2 (2019) 1510-1559 • A short proof of a conjecture of Erd\"os proved by Moreira, Richter and Robertson Discrete Analysis (2019) 19 • Numerical stability of a hybrid method for pricing options International Journal of Theoretical and Applied Finance (2019) 1950036 • Bounded solutions for an ordinary differential system from the Ginzburg-Landau theory Proceedings of the Royal Society of Edinburgh: Section A, Mathematics (2019) • Surfaces with planar curvature lines: discretization, generation and application to the rationalization of curved architectural envelopes Automation in Construction 106 (2019) 102880 • On Fourier coefficients of modular forms of half integral weight at squarefree integers Mathematische Zeitschrift 293 (2019) 789-808 • ON A SUM INVOLVING THE EULER TOTIENT FUNCTION Indagationes Mathematicae 30 4 (2019) 536-541 • Estimation of the extreme value index in a censorship framework: asymptotic and finite sample behaviour Journal of Statistical Planning and Inference 202 (2019) 31-56 • ON COMPACT ANISOTROPIC WEINGARTEN HYPERSURFACES IN EUCLIDEAN SPACE Archiv der Mathematik 113 2 (2019) 213-224 • Upper bounds for the function solution of the homogenuous 2D Boltzmann equation with hard potential Annals of Applied Probability (2019) • Hörmander functional calculus on UMD lattice valued Lp spaces under generalised Gaussian estimates Journal d'analyse mathématique (2019) • A PONTRYAGHIN MAXIMUM PRINCIPLE APPROACH FOR THE OPTIMIZATION OF DIVIDENDS/CONSUMPTION OF SPECTRALLY NEGATIVE MARKOV PROCESSES, UNTIL A GENERALIZED DRAW-DOWN TIME Scandinavian Actuarial Journal 9 (2019) • MONK -- Outlier-Robust Mean Embedding Estimation by Median-of-Means ICML 2019 - 36th International Conference on Machine Learning (2019) • Total variation distance between stochastic polynomials and invariance principles Annals of Probability 47 (2019) 3762 - 3811 • Furstenberg Systems of Bounded Multiplicative Functions and Applications International Mathematics Research Notices (2019) • On the shortest distance between orbits and the longest common substring problem Advances in Mathematics 344 (2019) 311 - 339 • Leveraging Labeled and Unlabeled Data for Consistent Fair Binary Classification NeurIPS 2019 - 33th Annual Conference on Neural Information Processing Systems (2019) • Unified convergence analysis of numerical schemes for a miscible displacement problem Foundations of Computational Mathematics 19 2 (2019) 333–374 • Metrical results on the distribution of fractional parts of powers of real numbers Proceedings of the Edinburgh Mathematical Society 62 2 (2019) 505-525 • Tube estimates for diffusions under a local strong Hörmander condition Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques 55 4 (2019) 2320--2369 • Ping-pong configurations and circular orders on free groups Groups, Geometry, and Dynamics 13 4 (2019) 1195–1218 • Analyse fréquentielle du signal Images des Mathématiques (2019) • Explicit expression of the microscopic renormalized energy for a pinned Ginzburg-Landau functional Journal of Elliptic and Parabolic Equations (2019) • Effective Berry-Esseen and concentration bounds for Markov chains with a spectral gap Annals of Applied Probability 29 3 (2019) 1778-1807 • One more proof of the Alexandrov–Fenchel inequality Comptes Rendus Mathématique 357 8 (2019) 676-680 • Apprendre Autrement : le point de vue d’un mathématicien sur la création d’un parcours de licence La gazette des mathématiciens (2019) • Multivariate scale-free temporal dynamics: From spectral (Fourier) to fractal (wavelet) analysis Comptes Rendus. Physique 20 5 (2019) 489-501 • Polytopes of Maximal Volume Product Discrete and Computational Geometry 62 (2019) 583-600 • Diophantine approximation and run-length function on β-expansions Journal of Number Theory 202 (2019) 60 - 90 • THE LITTLEWOOD-PALEY THEORY : A COMMON THREAD OF MANY WORKS IN NONLINEAR ANALYSIS • Convexity Preserving Contraction of Digital Sets The 5th Asian Conference on Pattern Recognition (ACPR 2019) (2019) 611-624 • Entropy, Lyapunov exponents, and rigidity of group actions Ensaios Matemáticos 33 (2019) 1-197 • Unbounded Largest Eigenvalue of Large Sample Covariance Matrices: Asymptotics, Fluctuations and Applications Linear Algebra and its Applications 577 (2019) • Multifractal formalisms for multivariate analysis Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475 2229 (2019) 20190150 • POLYAKOV'S FORMULATION OF 2d BOSONIC STRING THEORY Publications mathematiques de l' IHES 130 1 (2019) 111-185 • Exploring Allosteric Pathways of a V-Type Enzyme with Dynamical Perturbation Networks Journal of Physical Chemistry B 123 16 (2019) 3452-3461 • On a certain non-split cubic surface Science China Mathematics 62 12 (2019) 2435-2446 • Minimal graphs over Riemannian surfaces and harmonic diffeomorphisms American Journal of Mathematics 141 5 (2019) 1149--1177 • A Generalized Multifractal Formalism for the Estimation of Nonconcave Multifractal Spectra IEEE Transactions on Signal Processing 67 1 (2019) 110-119 ### 2018 • The convexification effect of Minkowski summation EMS Surveys in Mathematical Sciences 5 (2018) 1-64 • Multifractal analysis of the Birkhoff sums of Saint-Petersburg potential Fractals 26 3 (2018) 13 pages • Errata to the paper “Minimization of a Ginzburg-Landau type energy with potential having a zero of infinite order” Differential Integral Equations 31 (2018) 157-159. • Moments of the Hermitian Matrix Jacobi process Journal of Theoretical Probability 31 3 (2018) 1759-1778	2023-01-28 06:14:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.42965102195739746, "perplexity": 5464.031018462861}, "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/1674764499524.28/warc/CC-MAIN-20230128054815-20230128084815-00339.warc.gz"}
https://robotics.stackexchange.com/questions/12102/how-path-following-works-analytically-for-way-points/12107	# How path following works analytically for way points? First of all sorry for the confusing question title. I am also confused about the concept. I have implemented a quadcopter and its controller. Controller finds the rotor speeds based on the position and yaw angle references. The thing that I don't understand is, let's say I want the vehicle to climb 5m up and then 5m left. At this point, I think I need to create a vector containing the reference values. By the way, the model is discretized by some deltaT time interval so the reference vector. This does not coincide and well behave according to the dynamics of the vehicle. Let's say, the reference input for altitude is 5m until 5sec and 0 for [5,10]sec. But it is not guaranteed that the vehicle will reach to 5m altitude in 5sec. Thus, my intention is, that reference vector shouldn't rely on the time. Therefore, my perception is to use some if condition to check if the vehicle is reached for the first waypoint and then, register the next one. Which is a simple if-else statement? This makes me think what is the mathematical or analytical background in this. Is it just the following of the line between two waypoints by geometrical analysis like Line of Sight guidance law things. Can you give me some insight about the concept which makes me confused? You have a waypoint, which is some $<x,y,z>_\mbox{waypoint}$ coordinate. You should have some internal position estimate, $<x,y,z>_\mbox{quadcopter}$. If the waypoint coordinates are $x_w,y_w,z_w$, and the quadcopter coordinates are $x_q,y_q,z_q$, then you can calculate your distance to the waypoint as: $$r = \sqrt{(x_q-x_w)^2 + (y_q-y_w)^2 + (z_q-z_w)^2}\\$$ Now you have a singular value that indicates how close you are to the waypoint. You can also check speeds by looking at the quadcopter's speed on each axis - $$\dot{s} = \sqrt{\dot{x}_q^2 + \dot{y}_q^2 + \dot{z}_q^2} \\$$ You can check now that you're at target along two criteria. For example, maybe your position tracking has some degree of overshoot, and you'd like to be sure that the quadcopter has stabilized at the waypoint before continuing on to the next point. Maybe the first waypoint is at the entrance to a duct or pipe, and the second waypoint is in the duct/pipe. So, then you'd want to make sure that your distance to the waypoint, $r$, is small, and that your vehicle speed $\dot{s}$ is also small. If your position error $r$ is big but your speed $\dot{s}$ is small, then maybe you're at the previous waypoint and about to start moving. If your position error $r$ is small but your speed $\dot{s}$ is big, then you're close to your waypoint but you're moving quickly and are about to overshoot it. You might not care, or you might care a lot. It depends on the type of maneuver you're trying to complete and the allowable deviation from your waypoint-to-waypoint path is. If you overshoot the first waypoint by a lot and then immediately proceed to the second, then you're not actually taking the desired path to the second waypoint. Last piece of advice I could offer is to make sure that your waypoint coordinates and quadcopter coordinates are in the same frame of reference. You wouldn't want a quadcopter in real GPS coordinates trying to track a relative waypoint coordinate - your quadcopter might autonomously take off for the intersection of the prime meridian and the equator. • I am still not sure how to target and follow one waypoint after another one. You are just saying how I can find my state based on the position and velocity error. Am I wrong? Apr 11 '17 at 18:58 • @freezer - Maybe I'm not understanding your problem correctly. I thought your problem was that you had no way of knowing if you had arrived at the current waypoint. You said you were using a timing system that was prone to errors; you said, Therefore, my perception is to use some if condition to check if the vehicle is reached for the first waypoint. The condition you check to see if you've reached a waypoint is proximity, and that's the formula I've given: the straight-line distance from your quadcopter to the waypoint. I cautioned that you may want to check approach speed, too. Apr 11 '17 at 20:20 • @freezer - If your question isn't "am I at the first waypoint" but instead, "how do I navigate from waypoint A to waypoint B", then that's a difference question altogether. Maybe you're asking about sequencing? You can store the sequence of waypoints in a list and "pop" off the next waypoint in the list when your current position is equal to the current waypoint (as defined by an $r$ value at or below some threshold you set). Again, though, I'm not really sure what you're asking about at this point. Apr 11 '17 at 20:23 • I am actually asking both. Actually, I think there may be a way to connect all the points and fly over them continuously. In waypoint checks, I need to check what you offer in the answer and need to use if conditions whether the conditions are satisfied or not. Right? Then I need to register the next wp when proximity is enough? Sorry for the confusion, I am really confused too and have lots of questions. I am seeking to find a book or article to clarify this actually. Apr 12 '17 at 20:34 Most people just create a queue of waypoints on the path and go through one at a time. There are more complex path execution schemes that are more continuous but your application I doubt this is required. If your interested in looking for more complex schemes you can use B-Splines(https://en.wikipedia.org/wiki/B-spline) between waypoints. You hit the nail on the head when you brought up that you might not be at the target location when you think you will be there. Usually people get around that by having a good controller and recomputing the path as needed.	2021-10-15 22:46:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5006957650184631, "perplexity": 459.6719781339367}, "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-2021-43/segments/1634323583087.95/warc/CC-MAIN-20211015222918-20211016012918-00370.warc.gz"}
https://glassmirror.ca/l-s-ugcm/a10be4-grade-12-calculus-summary	PDF Format (798 KB) Plain Text Format (392 KB) Native Languages, Grades 11 and 12. MPM2D - Grade 10 Math. Below are the various download categories for Grades 10 to 12 Maths & Science learner and teacher materials. Ontario Curriculum Course Code: MCV4U. MPM2D - Grade 10 Math. A rectangle’s width and height, when added, are 114mm. MPM1D - Grade 9 Math. Business Siyavula's open Mathematics Grade 12 textbook, chapter 3 on Finance covering Summary Next. MCV4Ub. In the first minute of its journey, i.e. the integral. Grade 12 Mathematics Exam Practice Book . "Calculus arithmetic" also has X-Ray (split apart) and Time-Lapse (glue together). A function f(x) is said to be continuous at a particular point x = a, if the following three conditions are satisfied – f(a) is defined $$lim_{x \to a}f(x)$$ exists $$lim_{x \to a^-}f(x) = lim_{x \to a^+}f(x) =f(a)$$ Continuity and Differentiability. ... 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Includes: - Functions (Log) - Patterns, Sequences & Series - Finance - Pre-Calculus - Differential Calculus - Probability and Counting. Self Study Guides Grade 10 - 12. from 09:00 till 09:01 it travels a distance of 7675 metres. Find out more about our partnered and sponsored work and potential avenues to get involved. The main purpose is to develop the skills needed to continue on with the study of calculus. A mathematics study and revision guide (in English) for grade 12 based on the new CAPS syllabus. Grade 12 Mathematics Study Guide/ Studiegids Mind the Gap. The ages of the final 23 players selected by … Improve your math knowledge with free questions in "Describe function transformations" and thousands of other math skills. Average gradient or average rate of change: Gradient at a point or instantaneous rate of change: Notation. There are also fully worked out memos. These documents are intended to serve as resources for teachers and learners. Early Childhood SGEN K-12 LIP Programs SISC College AY 2019-2020 College AY 2020-2021 Southville FLEX K12 International Baccalaureate Graduate School Stanford Lexile Books FFL Anthologies Online Distance Learning Unit 1 Review Video . Differentiate the expression, let the derivative equal zero and solve the equation. All Siyavula textbook content made available on this site is released under the terms of a A stationary point can either be a local maximum, a local minimum or a point of inflection. Practise now to improve your marks. The Grade 10 and 11 Solution Study Guides will become vital as revision material for Grade 12 learners, especially during their final exams. MCF3M - Grade 11 Functions and Applications. by this license. All downloads are listed and hyperlinked and the actual files are stored on DropBox. 6.8 Summary . Here is a list of all of the maths skills students learn in grade 12! to personalise content to better meet the needs of our users. We think you are located in The downloads are categorised into three main groups relating to the type of device namely Computer, Tablet and Smart Phone. x. 2. MCF3M - Grade 11 Functions and Applications. Is this correct? We use this information to present the correct curriculum and MPM1D1. MCV4Uc+ MDM4U - Grade 12 Data Management. Lesson 1: Sequences and Series. All you need to know for Calculus - Grade 12 Mathematics. This course builds on students' previous experience with functions and their developing understanding of rates of change. 2. April 2005 in Großbritannien, Australien und Neuseeland sowie am folgenden Tag in den Vereinigten Staaten erstmals gezeigt. This course is a continuation of the concepts you have studied in previous years, as well as an introduction to new topics. You will put to use many of the skills that Below is the suggested sequence of course unit delivery as well as the recommended number of hours to complete the respective unit. This course helps you develop the skills, ideas, and confidence you will need to continue studying mathematics in the future. m = (2 - -1) / (0 - -2) = 3 / 2 : slope of line through the center and the point of … MCV4Ub2. MAT0511 Access to Mathematics assignment one 2019. United States. Creative Commons Attribution License. MCV4U - Grade 12 Calculus and Vectors. This course builds on students’ previous experience with functions and their developing understanding of rates of change. At each time t, the position vector R(t) locates the moving body: MPM1D1. MHF4U - Grade 12 Advanced Functions. Effective speeds over small intervals 1. MCV4Ur. It builds upon the pre-calculus topics you were introduced to in Grade 11 Pre-Calculus Mathematics. Thomas A. Blakelock High School ~ 1160 Rebecca Street ~ Oakville, Ontario ~ L6L 1Y9 ~ (905) 827 1158 This course relies heavily on topics covered in previous grades. Calculus Grade 12. The final exam takes place on January 22, 2019 @ 9:00 am in Room 213. Calculus in 10 minutes: New viewpoints lead to insights: Calculus explains X-Ray and Time-Lapse vision exist, they are opposites (splitting apart, gluing together) and any pattern can be analyzed.Arithmetic gives us add/subtract, multiply/divide, exponents/roots. The vector 2i +4j + 8k is constant. These geometry problems are presented here to help you think and learn how to solve problems. TOPIC: DATA HANDLING . i See more info. Home‎ > ‎ Grade 12 Advanced Functions. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. Australia. In the first minute of its journey, i.e. We use this information to present the correct curriculum and Mathematics Grade 12; Differential Calculus; Summary; Previous. It is a function of the parameter t, which often represents time. Finding the stationary points: let $$f'(x) = 0$$ and solve for $$x$$. Siyavula Practice guides you at your own pace when you do questions online. Application on area, volume and perimeter A. Discussions. MDM4U1-N. LAU. Get ready for school reopening. How to obtain maximum benefit from these resources . from 09:00 till 09:01 it travels a distance of 7675 metres. MCV4U Calculus and Vectors. Version 10642 Download 29.86 MB File Size 2 File Count December 24, 2018 Create Date August 12, 2020 Last Updated File Action; Gr 12 Wiskunde Studiegids(MTG).pdf: Download : MTG-Maths-24-Feb-2015.pdf: Download . iv Grade 12 Pre-Calculus Mathematics. Sketching. Mathematics; 373. Great for Exam/ Test Revision Notes. Mathematics Grade 12; Differential Calculus; Summary; Previous. They provide notes, examples, problem-solving exercises with solutions and examples of practical activities. CAMI Mathematics: :: : Grade 12. Derivative Rules. ISBN: … Grade 12 Preparatory Exam and Memo November 2019 Overberg District P2. Related Notes. y x x y x y Perimeter x y ∴ = − =+ = + = + 156 156 312 2 2 2 2. Topics include: Analytical Geometry, Proportionality, Euclidean Geometry, Measurement, Statistics and Trigonometry. MCV4U Calculus and Vectors, Grade 12, University Preparation MCV4U - Calculus and Vectors Grade 12 builds on students’ previous experience with functions and their developing understanding of rates of change. Sign up here. Grade 12 Mathematics Study Guide/ Studiegids Mind the Gap Past papers and memos. The gradient of the curve and the tangent to the curve at stationary points is zero. CAMI Mathematics: :: : Grade 12 12.5 Calculus12.5 Calculus 12.5 Practical application 12.5 Practical application A. The Exam Practice Book is CAPS compliant and follows the national examination guidelines structure. The limit of a function exists and is equal to $$L$$ if the values of $$f(x)$$ get closer to $$L$$ from both sides as $$x$$ gets closer to $$a$$. MPM 2D3. 77 - 86. Courses. Test yourself and learn more on Siyavula Practice. The derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. The Arts. math. Search Main menu Skip to primary content Skip to secondary content • Home • Average Calculator • Submit • About • About Us • Contact Us Post navigation MCV4U – Grade 12 Calculus & Vectors – Cartesian Vectors Test Object 1 Grade 12 – Calculus and Vectors Cartesian Vectors Test Cartesian Vectors • Unit Vectors: are i = [1, 0], j = [0,1] have magnitude 1 and tails at origin. Summary on Calculus. to personalise content to better meet the needs of our users. Effective speeds over small intervals 1. Join thousands of learners improving their maths marks online with Siyavula Practice. Description. To learn more on calculus class 11 and calculus class 12, visit our BYJU’S page to get a proper definition with examples. Improve your skills with free problems in 'Domain and range' and thousands of other practice lessons. Sign up to improve your marks . 1 IGCSE Grade 11 and Grade 12 Math Study Notes. High marks in maths are the key to your success and future plans. Curve. Die Dreharbeiten endeten im August 2004 und der Film wurde am 28. MCV4Ur. These SmartGirl study notes include: Proofs for trigonometry identities Proofs for the sin, cos & area rules Proofs for grade 11 & 12 Euclidean geometry calculus course. Analytical Geometry, Functions, Polynomials and Calculus. Calculate the dimensions of a rectangle with a perimeter of 312 m for which the area, V, is at a maximum. Many of the examples in the Grade 11 and 12 Study guides are similar to the questions put in the National Senior Certificate examination. Unit 3. Identify functions A.3. Hi Everyone, As you know, I'm no longer at school. Make sure you know how to: Calculate the average gradient of a curve using the formula. List of articles in category English Tests for Grade 12; Title; English Grade 12 - The Same Meaning Sentences Test 01 English Grade 12 - The Same Meaning Sentences Test 02 English Grade 12 - The Same Meaning Sentences Test 03 English Grade 12 - The Same Meaning Sentences Test 04 English Grade 12 - The Same Meaning Sentences Test 05 A complete set of Class Notes, Handouts, Worksheets, PowerPoint Presentations, and Practice Tests. MALATI materials: Introductory Calculus, Grade 12 5 3. ICS2O Small. Application on area, volume and perimeter [ and perimeter [ and perimeter [5.7.3.3; 5.7.3.45.7.3.3; 5.7.3.45.7.3.3; 5.7.3.4]]]] 1. MCV4Uc+ MDM4U - Grade 12 Data Management. Please have a look. Applications of differential calculus. MCV4U - Grade 12 Calculus and Vectors. (-2 , -1) : center of circle. Embedded videos, simulations and presentations from external sources are not necessarily covered 16. All worksheets are free and follow the South African curriculum. Includes limits gradient graphs differentiation etc. Calculus Summary Calculus has two main parts: differential calculus and integral calculus. MPM 2D1. Summary; Calculus and Vectors, Grade 12, University Preparation. For complete details of targeted expectations within each unit and activity, please see each Unit Overview found in the MCV4U course profile. Summary Of Units And Timelines For Grade 12 Calculus and Vectors MCV4U. Let us help you to study smarter to achieve your goals. End of chapter exercises. 1.1 IGCSE Grade 11 and Grade 12 Math Study Notes- Boolean Algebra; 1.2 IGCSE Grade 11 and Grade 12 Math Study Notes- Codes in Everyday; 1.3 IGCSE Grade 11 and Grade 12 Math Study Notes- Critical Path Analysis; 1.4 IGCSE Grade 11 and Grade 12 Math Study Notes- Difference Equation 1; 1.5 IGCSE Grade 11 and Grade 12 Math Study … by this license. New CAPS worksheets to arrive in 2014. In Australia, the twelfth grade is referred to as Year 12.In New South Wales, students are usually 16 or 17 years old when they enter Year 12 and 17–18 years during graduation (end of year).A majority of students in Year 12 work toward getting an ATAR (Australian Tertiary Admission Rank). MCR3U - Grade 11 Functions. (PDF). The derivative of a difference is equal to the difference of the derivatives. Pre-calculus 12 is designed for students who have a particular interest in mathematics, or who have career aspirations in the fields of engineering, mathematics, the sciences, economics, and some business programs. Between 09:01 and 09:02 it … Find the gradient of a linear function A.4. Creative Commons Attribution License. Video. Mathematics, Grades 11 and 12, 2007. Business Mathematics Exam Practice Book includes exam papers and memoranda written by expert teachers and examiners to prepare learners for exams. Summary; Grade 12 Calculus and Vectors - Mr. Peddle. Don't get left behind. Self Study Guides for Grades 10 - 12. This grade 12 worksheet on calculus consists of finding the first and second derivatives using the short-cut method finding the point of inflection and the concavity of the graph drawing the cubic graph by finding the x and y intercepts finding the stationary or turning points finding the point of inflection and finally, optimization of […] Within this PDF find summaries taken from Grade 10 to Grade 12 covering all you need to revise for the Mathematics Paper 2 exam. GHCI Grade 12 Calculus & Vectors: Home Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8 Calendar Exam ... (Brief summary of concepts learned in Unit 3) - Chapter 3 Multiple Choice Review (McGraw-Hill Ryerson) Review Practices: - eBook: Chapter 3 Review on p. 204 #1 to 16 - eBook: Chapter 3 Practice Test on p. 206 #1 to 18. Between 09:01 and 09:02 it … 1 of 7. Calculus 38 – 47 18 Linear Programming 48 - 55 19 Trigonometry 56 – 60 Data Handling 3 - 18 2D Trigonometry 3D Trigonometry 61 - 76. Home. Worksheets for grade 12 students free to teachers to download once they are registered on the site. Finding the stationary points: let $$f'(x) = 0$$ and solve for $$x$$. End of chapter exercises. MCR 3U PDL. Twelfth grade, senior year, or grade 12 is the final year of secondary school in most of North America. All you need to know to pass Calculus! We think you are located in Site news. MFM2P - Grade 10 Math - Applied. Average gradient or average rate of change: Gradient at a point or instantaneous rate of change: The derivative of a constant is equal to zero. Early Childhood SGEN K-12 LIP Programs SISC College AY 2019-2020 College AY 2020-2021 Southville FLEX K12 International Baccalaureate Graduate School Stanford Lexile Books FFL Anthologies Online Distance Learning A. Assignments, Tests and more. Grade 12 Mathematics Exam Practice Book . We use this information to present the correct curriculum and to personalise content to better meet the needs of our users. An in-depth summary of the content covered in Grade 12 Mathematics (IEB) for paper 1. Graph a linear function A.5. Perimeter =2x +2y and the Area =xy. Differential calculus studies the derivative and integral calculus studies (surprise!) End of chapter exercises. Lesson 2: Sequences and Series: Sigma Notation and Sum to infinity. MATHEMATICS Grade 12. Applications of derivatives. MCF3M1-02. A quick, easy to use A5 summarize before exams, Calculus Summary, contact me for more.... Request a Tutor ... Study Notes > Calculus Grade 12. Checklist. Functions. All of the resources hosted by the La Citadelle web site are free to visit, test, study or learn. Embedded videos, simulations and presentations from external sources are not necessarily covered QUESTION 1 . Many of the skills that you have already learned will be put to use as you solve problems and learn new skills along the way. Page1. Site news. Do not give up quickly if … CHAPTER 12 Motion Along a Curve I [ 12.1 The Position Vector I-, This chapter is about "vector functions." Contents. MCR3U. Fun maths practice! In this lesson on Sequences and Series we focus on quadratic sequences, arithmetic sequences, geometric sequences and simultaneous equations. MEMOMEMO. X-kit Achieve! math. Welcome to Grade 12 Pre-Calculus Mathematics! This PDF contains summaries of all Grade 12 Math topics covered for the IEB final exams. Claire N. Contact this tutor . IEB standard calculus summary from my own class notes. Yes, I retired, but I will still be doing some tutoring, and obviously will also be available to help some of my former students with your questions during this chaotic time (if you so desire). Per Anhalter durch die Galaxis (Originaltitel: The Hitchhiker’s Guide to the Galaxy) ist ein britischer Science-Fiction-Film, der auf dem gleichnamigen Buch von Douglas Adams basiert. Application on area, volume and perimeter 1. All Siyavula textbook content made available on this site is released under the terms of a Next. MDM4U1-N. LAU. Application on area, volumeA. An object starts moving at 09:00 (nine o'clock sharp) from a certain point A. Related. Courses. MFM2P - Grade 10 Math - Applied. MCV4Ub. Home. This course covers the Grade 12 Term 2 topics, i.e. Learner's book and teacher's guide Students will solve problems involving geometric and algebraic representations of vectors and representations of lines and planes in three-dimensional space; … ICS2O Small. They cover all the major topics in the South African NCS curriculum such as calculus, algebra, functions, trigonometry. Summary Summary Math Lit Notes (Grade 11-12) These notes were typed out to help you out! This includes the whole IEB Maths Literacy syllabus These Notes are for sure worth the time and money! IEB standard calculus summary from my own class notes. The vector R(t) = ti + t2j+ t3k is moving. MPM1D1 2017. Unit 1 . Grade 12 Pre-Calculus Mathematics. TabletClass Math http://www.tabletclass.com learn the basics of calculus quickly. Meaning of the derivative in context: Applications of derivatives Straight … The derivative of a sum is equal to the sum of the derivatives. Domain and range A.2. United Kingdom. Is this correct? MPM1D - Grade 9 Math. The focus of these videos is to promote real understanding, reasoning and confidence in mathematical thinking. Integral Calculus Study of Calculus Book is CAPS compliant and follows the national examination guidelines structure national examination structure... To know for Calculus - Probability and Counting 1 IGCSE Grade 11 Mathematics. Compliant and follows the national examination guidelines structure, ideas, and Practice tests below will give you a idea. Apart ) and solve for \ ( f ' grade 12 calculus summary x ) 0\... A local … Grade 12 Math Clinic Study Guide Grade 12 is the suggested sequence course... To your success and future plans 12.5 Calculus12.5 Calculus 12.5 Practical application 12.5 Practical application a Calculus Calculus. Simultaneous equations a rectangle ’ s width and height, when added are! Marks in maths are the key to your success and future plans R ( t ) 0\. Know, I 'm no longer at school Attribution License introduced to in Grade 12 Mathematics Study and revision (. Curriculum such as Calculus, algebra, functions, Trigonometry to solve problems rates of change: Notation Analytical! Guides you at your own pace when you do questions online and hyperlinked and the tangent the... You were introduced to in Grade 11 and Grade 12, University Preparation the site rectangle a. Device namely Computer, Tablet and Smart Phone sum of the maths skills learn! Heavily on topics covered in previous years, as you know how to solve problems their final exams the purpose. 10 - 12 covered by this License marks online with Siyavula Practice 2019 @ 9:00 am in Room.! Often represents time gets closer to year of SECONDARY school in most of North America include. O'Clock sharp ) from a certain point a in this lesson on Sequences and simultaneous equations cover. Mr. Peddle Science learner and teacher 's Guide ( in English ) for paper 1, &. Course profile solutions are presented 12 geometry problems are presented here to help you and... Guides will become vital as revision material for Grade 12 Mathematics this License, problem-solving exercises with solutions and of... Commons Attribution License Describe function transformations '' and thousands of learners improving their maths marks with... Ieb standard Calculus Summary Calculus has two main parts: Differential Calculus ; Summary previous... Compliant and follows the national examination guidelines structure journey, i.e sides as gets closer to from both sides gets... Is at a point of inflection, University Preparation, is at a maximum & Science learner and teacher Guide! They provide Notes, Handouts, worksheets, PowerPoint presentations, and tests. Give you a good idea of what to expect on the Exam Practice Book Exam. 12 geometry problems are presented 10 and 11 Solution Study Guides Grade 10 to 12 maths & Science and. The formula and follows the national examination guidelines structure learn how to: Calculate the average gradient or rate! Change: Notation players selected by … MCV4U Calculus and Vectors develop the skills, ideas, Practice... This information to present the correct curriculum and to personalise content to better meet needs... Exam & sol ; Test revision Notes in this lesson on Sequences Series. Rectangle with a Perimeter of 312 m for which the area, V, is at a maximum Log. Simultaneous equations 12 geometry problems are presented Probability and Counting the needs of our users ) Native Languages, 11. Focus on quadratic grade 12 calculus summary, geometric Sequences and simultaneous equations the gradient of the.! Targeted expectations within each unit and activity, please see each unit Overview found in first... 12 ; Differential Calculus ; Summary ; Grade 12 ; Differential Calculus (! Previous Grades present the correct curriculum and to personalise content to better meet the needs of our users not covered. Functions ( Log ) - Patterns, Sequences & Series - Finance - Pre-Calculus - Differential Calculus ; ;. Arithmetic Sequences, arithmetic Sequences, geometric Sequences and simultaneous equations Practical activities it is function! Major topics in the first minute of its journey, i.e between 09:01 09:02..., simulations and presentations from external sources are not necessarily covered by this License Siyavula textbook made. Year of SECONDARY school in most of North America which often represents time ages of the derivatives and the... They are registered on the Exam or a point or instantaneous rate of change 12 Math Study! Calculus Summary from my own class Notes, Handouts, worksheets, PowerPoint presentations, and confidence in mathematical.... + = + 156 156 312 2 2 2 2 = − =+ = =. The respective unit Handouts, worksheets, PowerPoint presentations, and confidence in mathematical thinking of! Content to better meet the needs of our users examination guidelines structure und der Film am... On quadratic Sequences, arithmetic Sequences, arithmetic Sequences, arithmetic Sequences, arithmetic Sequences, geometric and... Summary of Units and Timelines for Grade 12 Mathematics rectangle with a Perimeter of m! Memo November 2019 Overberg District P2 multiplied by a function exists and is equal to the curve and the to... 11 Pre-Calculus Mathematics topics include: Analytical geometry, Proportionality, Euclidean,! Mathematics paper 2 Exam click on any link SENIOR year, or Grade 12 Preparatory and! To start practising, just click on any link they provide Notes, examples, problem-solving with... Know, I 'm no longer at school first minute of its journey, i.e 9:00 am Room! Dreharbeiten endeten im August 2004 und der Film wurde am 28 ; Differential Calculus Grade! Registered on the site sure you know how to: Calculate the average gradient or average of! Intervention PROGRAMME Mathematics Grade 12 ; Differential Calculus ; Summary ; Grade 12 geometry problems detailed! Surprise! on students ’ previous experience with functions and their developing understanding rates! Teachers to download once they are registered on the new CAPS syllabus the final Exam takes place on 22... Marks in maths are the grade 12 calculus summary to your success and future plans our users ideas..., algebra, functions, Trigonometry Handouts, worksheets, PowerPoint presentations, and tests. Endeten im August 2004 und der Film wurde am 28 Guides Grade 10 and 11 Study. Problems in 'Domain and range ' and thousands of other Practice Lessons Calculus ; ;. A Creative Commons Attribution License the function and the tangent to the sum of the parameter t, which represents. Into three main groups relating to the curve and the actual files are stored on DropBox Calculus12.5! Citadelle web site are free and follow the South African NCS curriculum such Calculus. Of other Practice Lessons both sides as gets closer to on students ’ previous with! ( x\ ) from external sources are not necessarily covered by this.! The concepts you have studied in previous years, as you know, I no..., and Practice tests ) - Patterns, Sequences & Series - Finance - Pre-Calculus - Differential -... Revision Guide ( in English ) for paper 1 ( surprise! PDF ) you studied! Under the terms of a constant multiplied by the derivative of a Creative Attribution! With a Perimeter of 312 m for which the area, V, is at point! Online with Siyavula Practice by this License ' previous experience with functions and their developing understanding rates!, Proportionality, Euclidean geometry, Proportionality, Euclidean geometry, Measurement, Statistics and Trigonometry to both. Parameter t, which often represents time transformations '' and thousands of other Math.! Years, as well as an introduction to new topics to prepare learners exams... For Grade 12 learners, especially during their final exams free questions in Describe function transformations '' and of... This PDF find summaries taken from Grade 10 and 11 Solution Study Guides will become vital revision! & sol ; Test revision Notes are free to visit, Test, or... Maths & grade 12 calculus summary learner and teacher 's Guide ( PDF ) of get closer to from sides... Follow the South African curriculum 2019 @ 9:00 am in Room 213 derivative of the concepts you studied. To formulate an expression that contains only one variable a maximum all you need to on. - Pre-Calculus - Differential Calculus ; grade 12 calculus summary ; Grade 12 covering all you need to revise for the paper! Within this PDF find summaries taken from Grade 10 and 11 Solution Study Guides Grade 10 to 12... Questions in Describe function transformations '' and thousands of other Math skills not necessarily covered by this.. … Summary ; Calculus and Vectors MCV4U students learn in Grade 12 Mathematics ( IEB for! Object starts moving at 09:00 ( nine o'clock sharp ) from a certain point a to download once are., or Grade 12 local maximum, a local maximum, a local maximum, a local maximum, local... Mathematical thinking am 28 an in-depth Summary of Units and Timelines for Grade 12 12.5 Calculus12.5 Calculus 12.5 Practical a! Is a continuation of the function Log ) - Patterns, Sequences & -... 12 12.5 Calculus12.5 Calculus 12.5 Practical application a derivative and integral Calculus studies the derivative of a Creative Attribution... Guide ( PDF ) expect on the new CAPS syllabus 2 Exam a Mathematics Study Guide/ Mind!, Grade 12 Preparatory Exam and Memo November 2019 Overberg District P2 798 KB ) Native Languages, Grades and! Textbook content made available on this site is released under the terms of Creative! 9:00 am in Room 213 MCV4U Calculus and integral Calculus Pre-Calculus - Differential Calculus ; Summary ;.... The needs of our users ( surprise! Calculate the average gradient or average rate of change: gradient a. Expression that contains only one variable found in the first minute of its journey i.e! Find summaries taken from Grade 10 - 12 geometry problems with detailed solutions are presented Exam... Targeted expectations within each unit Overview found in the South African NCS curriculum as...	2021-02-25 10:42:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.257289856672287, "perplexity": 2963.7490654971657}, "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/1614178350942.3/warc/CC-MAIN-20210225095141-20210225125141-00071.warc.gz"}
https://ubpdqnmathematica.wordpress.com/2017/04/20/relaxation-oscillator/	Home > Uncategorized > Relaxation Oscillator ## Relaxation Oscillator Professor Strogatz Cornell course video lectures on “Nonlinear Dynamics and Chaos” are now online. This post pays homage to lecture 10 on analysis of the Van der Pol equaltion: $\ddot{x}+x+\mu \dot{x}(x^2-1)=0$ In the following the case for $\mu =2,\mu =10$ and the examination of the $\mu >>1$. It nicely illustrates the two time scales referred to, the slow then fast parts of the oscillation. The phase portrait is the $x, y$ plane where $y$ is the the Lienard transformed and rescaled version of the equation. So the system is: $\dot{x}=\mu(y-(x^3/3-x))$ $\dot{y}=-x/\mu$	2017-11-19 21:13:05	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 7, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6779253482818604, "perplexity": 1702.3764371448904}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934805809.59/warc/CC-MAIN-20171119210640-20171119230640-00637.warc.gz"}
https://wikihost.nscl.msu.edu/talent/doku.php?id=help&rev=1399898645	# Talent ### Sidebar Nuclear Talent Course 7 NT4A help This is an old revision of the document! # TALENT ## Training in Advanced Low Energy Nuclear Theory: HELP Most things are pretty obvious. As a first step, go to your pre-populated page and click “Edit”, look at the syntax, and go from there. Below some non obvious things that are commonly needed (this is only meant as a very short guide to the essentials) For more details see the official, more comprehensive DokuWiki Manual and DokuWiki Syntax ## Basic Text Formatting DokuWiki supports bold, italic, underlined and monospaced texts. Of course you can combine all these. DokuWiki supports bold, //italic//, __underlined__ and ''monospaced'' texts. Of course you can __//''combine''//__ all these. You can use subscript and superscript, too. You can use <sub>subscript</sub> and <sup>superscript</sup>, too. You can mark something as deleted as well. You can mark something as <del>deleted</del> as well. Paragraphs are created from blank lines. If you want to force a newline without a paragraph, you can use two backslashes followed by a whitespace or the end of line. This is some text with some linebreaks Note that the two backslashes are only recognized at the end of a line or followed by a whitespace \\this happens without it. This is some text with some linebreaks\\ Note that the two backslashes are only recognized at the end of a line\\ or followed by\\ a whitespace \\this happens without it. You should use forced newlines only if really needed. DokuWiki supports multiple ways of creating links. ### External DokuWiki supports multiple ways of creating links. External links are recognized addresses like this one: <andi@splitbrain.org> are recognized, too. ### Internal Internal links are created by using square brackets. You can either just give a pagename or use an additional link text. 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So don't rely on section linking too much. ### Interwiki DokuWiki supports Interwiki links. These are quick links to other Wikis. For example this is a link to Wikipedia's page about Wikis: Wiki. DokuWiki supports [[doku>Interwiki]] links. These are quick links to other Wikis. For example this is a link to Wikipedia's page about Wikis: [[wp>Wiki]]. You can also use an image to link to another internal or external page by combining the syntax for links and images (see below) like this: [[http://www.php.net\|{{wiki:dokuwiki-128.png}}]] Please note: The image formatting is the only formatting syntax accepted in link names. The whole image and link syntax is supported (including image resizing, internal and external images and URLs and interwiki links). ## Footnotes You can add footnotes 1) by using double parentheses. You can add footnotes ((This is a footnote)) by using double parentheses. ## Sectioning You can use up to five different levels of headlines to structure your content. If you have more than three headlines, a table of contents is generated automatically – this can be disabled by including the string ~~NOTOC~~ in the document. ==== Headline Level 3 ==== == Headline Level 5 == By using four or more dashes, you can make a horizontal line: ## Media Files You can include external and internal images, videos and audio files with curly brackets. Optionally you can specify the size of them. Real size: Resize to given width: Resize to given width and height2): Resized external image: Real size: {{wiki:dokuwiki-128.png}} Resize to given width: {{wiki:dokuwiki-128.png?50}} Resize to given width and height: {{wiki:dokuwiki-128.png?200x50}} Resized external image: {{http://de3.php.net/images/php.gif?200x50}} By using left or right whitespaces you can choose the alignment. {{ wiki:dokuwiki-128.png}} {{wiki:dokuwiki-128.png }} {{ wiki:dokuwiki-128.png }} Of course, you can add a title (displayed as a tooltip by most browsers), too. {{ wiki:dokuwiki-128.png \|This is the caption}} ### Supported Media Formats DokuWiki can embed the following media formats directly. Image gif, jpg, png Video webm, ogv, mp4 Audio ogg, mp3, wav Flash swf If you specify a filename that is not a supported media format, then it will be displayed as a link instead. ### Fallback Formats Unfortunately not all browsers understand all video and audio formats. To mitigate the problem, you can upload your file in different formats for maximum browser compatibility. For example consider this embedded mp4 video: {{video.mp4\|A funny video}} When you upload a video.webm and video.ogv next to the referenced video.mp4, DokuWiki will automatically add them as alternatives so that one of the three files is understood by your browser. Additionally DokuWiki supports a “poster” image which will be shown before the video has started. That image needs to have the same filename as the video and be either a jpg or png file. In the example above a video.jpg file would work. ## Lists Dokuwiki supports ordered and unordered lists. To create a list item, indent your text by two spaces and use a * for unordered lists or a - for ordered ones. • This is a list • The second item • You may have different levels • Another item 1. The same list but ordered 2. Another item 1. Just use indention for deeper levels 3. That's it * This is a list * The second item * You may have different levels * Another item - The same list but ordered - Another item - Just use indention for deeper levels - That's it Also take a look at the FAQ on list items. #### Plugins We have installed a few plugins that are useful for our purposes. 1) This is a footnote 2) when the aspect ratio of the given width and height doesn't match that of the image, it will be cropped to the new ratio before resizing ## Help Discussion Enter your comment. Wiki syntax is allowed:	2022-01-25 03:18:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3904631733894348, "perplexity": 5629.619040776518}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304749.63/warc/CC-MAIN-20220125005757-20220125035757-00638.warc.gz"}
https://mathematica.stackexchange.com/questions/94302/nintegrate-option-for-fourierseries	# NIntegrate option for FourierSeries I would like to use FourierSeries but it is very slow. My guess is that it is because it uses symbolic integration: (* function for the example ) f[t_] = Cos[t] + 0.2Sin[2t] + Piecewise[{{.1, 1 < Mod[t, 2 Pi] < 2}}, 0]; ( using FourierSeries ) FourierSeries[f[t], t, 3]; // AbsoluteTiming ( using NIntegrate, manually ) Total[Table[Exp[Ikt]/(2 Pi)NIntegrate[f[t]Exp[-Ikt],{t, -Pi, Pi}], {k, -3, 3}]]; // AbsoluteTiming 10.129225 ( for FourierSeries ) 1.923198 ( with NIntegrate ) I am sure there is an easy way to force FourierSeries to use NIntegrate, but I don't know how. Why don't you use NFourierSeries? << FourierSeries f[t_] = Cos[t] + 0.2Sin[2*t] + Piecewise[{{.1, 1 < Mod[t, 2 Pi] < 2}}, 0]; NFourierSeries[f[t], t, 3]; // AbsoluteTiming {3.39228, Null} This time is measured on my laptop! • For a very simple reason: I did not know NFourierSeries existed. That's exactly what I was looking for. Computation time is very close to that of NIntegrate (tested for k from 3 to 20), but it's simpler. I did not find NFourierSeries from the documentation. I think it is because it requires the package FourierSeries. What would have been a good approach to possibly find it? – anderstood Sep 9 '15 at 15:06 • Look at the documentation "guide/StandardExtraPackages" – user31001 Sep 9 '15 at 15:19 • @anderstood With "NFourierSeries" you can get warning messages. Please insert then "AccuracyGoal". – user31001 Sep 9 '15 at 16:03	2020-02-17 18:05:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5059943199157715, "perplexity": 4419.870593482395}, "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-2020-10/segments/1581875143079.30/warc/CC-MAIN-20200217175826-20200217205826-00147.warc.gz"}
https://open.kattis.com/problems/ambush	Kattis # Horse in Ambush When Farmer Oskar doesn’t watch his cows closely enough, they tend to wander off into the forest to hunt for horse spies. To catch an enemy agent horse, the cows lure it into a cow trail and set up fences at the endpoints of the trail, which are $L$ meters apart. Two previously hidden cows then reveal themselves, having concealed themselves in nearby bushes – and the chase is on! The cows are located at positions $A$ and $B$ meters from the left endpoint. One particular horse (positioned $P$ meters from the left endpoint) who they attempted to catch remembered to bring her walkie-talkie and have now called for backup. The backup will arrive to rescue her very soon, so she wonders for how long she can hold the cows off. Horse chasing happens in $1$ minute steps. During each minute, the following steps happen in order: 1. The cows choose to move either $0$ or $1$ meters, in any direction. 2. The horse jumps either $0$, $1$, or $2$ meters in any direction. However, if a cow occupies the same position as the horse, she can only jump a single meter since she now must jump over the cow. 3. If there is a cow in the same position as the horse, the cows capture her. How long will it take for the horse to be captured, assuming both she and the cows move optimally? ## Input The first and only line of input contains four integers $1 \le L \le 1\, 000$, $0 \le A \le L$, $0 \le B \le L$, $0 \le P \le L$ separated by spaces, the meanings of which are described in the statement. The numbers $A$, $B$ and $P$ are all distinct. ## Output A single integer, the number of minutes before the cows can catch the horse no matter how the horse moves. Sample Input 1 Sample Output 1 5 4 3 2 3 Sample Input 2 Sample Output 2 5 4 2 3 3	2020-05-25 20:48:36	{"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": 0.43033331632614136, "perplexity": 1590.8717802929782}, "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-2020-24/segments/1590347389355.2/warc/CC-MAIN-20200525192537-20200525222537-00271.warc.gz"}
https://brilliant.org/discussions/thread/gravitation-3/	# Gravitation We all know that the gravitational field inside a thin spherical shell is zero. There are different methods to prove this. I came across one such odd yet amazing method . I would like everyone to try proving it by this method .I will put my proof in a couple of days. Zero force inside a sphere. Show that the gravitational force inside a spherical shell is zero by showing that the pieces of mass at the ends of the thin cones in the figure give canceling forces at point P (any point inside the sphere). Note by Ayush Shridhar 4 years, 7 months ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. MarkdownAppears as italics or _italics_ italics bold or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting. 2 \times 3 $2 \times 3$ 2^{34} $2^{34}$ a_{i-1} $a_{i-1}$ \frac{2}{3} $\frac{2}{3}$ \sqrt{2} $\sqrt{2}$ \sum_{i=1}^3 $\sum_{i=1}^3$ \sin \theta $\sin \theta$ \boxed{123} $\boxed{123}$ Sort by: - 4 years, 7 months ago same can you prove the same thing for an ellipsoid? - 4 years, 7 months ago I'll try that also. Thanks for enlightening. - 4 years, 7 months ago Ellipsoid will be having the same proof. There'll be a change in diagram. Rest all will be the same thing. - 4 years, 7 months ago The tougher task is doing the same thing for an ellipsoid. I am trying to prove that! Might take a couple of days. - 4 years, 7 months ago	2020-09-18 14:28:31	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "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": 0.9884337782859802, "perplexity": 2481.099110650957}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400187899.11/warc/CC-MAIN-20200918124116-20200918154116-00078.warc.gz"}
http://openstudy.com/updates/515f3e09e4b0115bc14d4e65	## kymber Group Title LaTeX (pretty words) tutorial! (❍ᴥ❍ʋ) one year ago one year ago 1. kymber Group Title $$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Huge\mathsf{Basics!}$$ $$\huge\mathsf{☞}$$ $$\tt{Put\ your\ commands\ and\ messages\ inside\ this:}$$ $$message$$. $$\qquad\tt{\ so\ that\ would\ look\ like:}$$ $$message$$. $$\huge\mathsf{☞}$$ $$\tt{Put\ a\ tilde}$$ ~ $$\tt{wherever\ you\ want\ a\ space.}$$ $$\qquad\tt{so}$$ $$sentence~with~spaces$$ $$\tt{will\ give\ you:}$$ $$sentence~with~spaces$$. 2. kymber Group Title $$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Huge\mathsf{Sizes!}$$ $$\large\mathsf{☯}$$ $$\large\tt{To~ change~ sizes~ do}$$ $$\size{message}$$. $$\large\mathsf{☯}$$ $$\large\tt{So}$$ $$\large{sacapuntas}$$ $$\tt\large{is}$$ $$\large{sacapuntas}$$. $$\large\mathsf{⇢}$$ $$\tiny{tiny}$$ $$\large\mathsf{⇢}$$ $$\scriptsize{scriptsize}$$ $$\large\mathsf{⇢}$$ $$\small{small}$$ $$\large\mathsf{⇢}$$ $$\normalsize{normalsize}$$ $$\large\mathsf{⇢}$$ $$\large{large}$$ $$\large\mathsf{⇢}$$ $$\Large{Large}$$ $$\large\mathsf{⇢}$$ $$\LARGE{LARGE}$$ $$\large\mathsf{⇢}$$ $$\huge{huge}$$ $$\large\mathsf{⇢}$$ $$\Huge{Huge}$$ $$\normalsize\tt{Remember,~some~of~them~are~case~sensitive!}$$ 3. kymber Group Title $$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Huge\mathsf{Colors!}$$ $$\Large\mathsf{⚡}$$ $$\large\tt{To~ make~ colors~ do}$$ $$\color{color}{message}$$. $$\Large\mathsf{⚡}$$ $$\large\tt{So}$$ $$\color{blue}{sacapuntas}$$ $$\large\tt{would~ be}$$ $$\color{blue}{sacapuntas}$$. $$\Large\color{maroon}{\tt{Maroon}}$$ $$\Large\color{brown}{\tt{Brown}}$$ $$\Large\color{red}{\tt{Red}}$$ $$\Large\color{orangered}{\tt{OrangeRed}}$$ $$\Large\color{salmon}{\tt{Salmon}}$$ $$\Large\color{orange}{\tt{Orange}}$$ $$\Large\color{goldenrod}{\tt{Goldenrod}}$$ $$\Large\color{gold}{\tt{Gold}}$$ $$\Large\color{yellow}{\tt{Yellow}}$$ $$\Large\color{greenyellow}{\tt{GreenYellow}}$$ $$\Large\color{yellowgreen}{\tt{YellowGreen}}$$ $$\Large\color{Olive}{\tt{Olive}}$$ $$\Large\color{lime}{\tt{Lime}}$$ $$\Large\color{springgreen}{\tt{SpringGreen}}$$ $$\Large\color{green}{\tt{Green}}$$ $$\Large\color{forestgreen}{\tt{ForestGreen}}$$ $$\Large\color{Seagreen}{\tt{SeaGreen}}$$ $$\Large\color{teal}{\tt{Teal}}$$ $$\Large\color{turquoise}{\tt{Turquoise}}$$ $$\Large\color{aqua}{\tt{Aqua~or~cyan}}$$ $$\Large\color{aquamarine}{\tt{Aquamarine}}$$ $$\Large\color{cadetblue}{\tt{CadetBlue}}$$ $$\Large\color{cornflowerblue}{\tt{CornflowerBlue}}$$ $$\Large\color{royalblue}{\tt{RoyalBlue}}$$ $$\Large\color{midnightblue}{\tt{MidnightBlue}}$$ $$\Large\color{navy}{\tt{Navy}}$$ $$\Large\color{blue}{\tt{Blue}}$$ $$\Large\color{purple}{\tt{Purple}}$$ $$\Large\color{blueviolet}{\tt{BlueViolet}}$$ $$\Large\color{darkorchid}{\tt{DarkOrchid}}$$ $$\Large\color{Magenta}{\tt{Magenta~or~fuchsia}}$$ $$\Large\color{orchid}{\tt{Orchid}}$$ $$\Large\color{plum}{\tt{Plum}}$$ $$\Large\color{grey}{\tt{Grey}}$$ $$\Large\color{tan}{\tt{Tan}}$$ $$\large\mathsf{⚡}$$ $$\large\tt{To\ use\ both\ a\ size\ and\ a\ color\ it's}$$ $$\size\color{color}{message}$$. $$\large\mathsf{⚡}$$ $$\large\tt{So}$$ $$\LARGE\color{teal}{sacapuntas}$$ $$\large\tt{would\ output}$$ $$\LARGE\color{teal}{sacapuntas}$$. 4. kymber Group Title $$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Huge\mathsf{Fonts!}$$ $$\tt{To\ change\ the\ font,\ put:}$$ $$\font{message}$$. $$\tt{so}$$ $$\mathsf{sacapuntas}$$ $$\tt{will\ give\ you:}$$ $$\mathsf{sacapuntas}$$. $$\huge\mathtt{Mathtt\ or\ tt}$$ $$\huge\sf{Mathsf\ or\ sf}$$ $$\huge\mathrm{Mathrm\ or\ rm}$$ $$\huge\mathbf{Mathbf\ or\ bf}$$ $$\huge\mathcal{Mathcal\ or\ cal}$$ $$\huge\mathbb{Mathbb}$$ $$\Huge\frak{Mathfrak~or~frak}$$ $$\huge\it{Mathit~or~it}$$ $$\huge\mathscr{Mathscr\ or\ scr}$$ $$\huge\boldsymbol{Boldsymbol}$$ $$\huge\text{Text}$$ $$\tt{To\ use\ a\ size,\ color,\ and\ font,\ type}$$ $$\size\color{color}{\font{message}}$$. $$\tt{So}$$ $$\large\color{teal}{\mathsf{Sacapuntas}}$$ $$\tt{will~ look~ like}$$ $$\large\color{teal}{\mathsf{Sacapuntas}}$$. 5. kymber Group Title $$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Huge\mathsf{Symbols!}$$ $$\large\sf{Check\ out\ this\ site\ for\ a\ list\ of\ symbols:}$$ $$\Large{➱}$$ http://detexify.kirelabs.org/symbols.html $$\large\sf{or\ go\ here\ if\ you're\ looking\ for\ a\ specific\ symbol:}$$ $$\Large{➱}$$ http://detexify.kirelabs.org/classify.html 6. kymber Group Title Make sure you join the $$\LaTeX$$ practicing group! http://openstudy.com/study#/groups/latex%20practicing!%20%3A%29 Also, all the $$\LaTeX$$ I used in this tutorial is here https://gist.github.com/Kymber/ec72626d366e04687f93 if anyone wants to look at it. 7. .Sam. Group Title $\color{red}{\huge\mathtt{I}} \color{orange}{ \huge\sf{m}} \color {yellow}{ \huge\mathrm{p}} \color {green}{\huge\mathbf{r}} \color {blue}{ \huge\mathcal{e} } \color {indigo}{\huge\mathbb{s}} \color{violet}{ \Huge\frak{s}} \color{cyan}{\huge\it{i}} \color{magneta}{ \huge\mathscr{v}} \color{purple}{ \huge\boldsymbol{e}} \Huge \color{blue}{\text{~}}$ 8. Gabylovesyou Group Title how do u put different colors? 9. kymber Group Title $$\color{red}{\huge\tt{T}}\color{orange}{\huge\tt{r}}\color{gold}{\huge\tt{y}}$$ $$\color{forestgreen}{\huge\tt{t}}\color{teal}{\huge\tt{h}}\color{blue}{\huge\tt{i}}\color{blueviolet}{\huge\tt{s}}$$ $$\color{red}{\huge\tt{T}}\color{orange}{\huge\tt{r}}\color{gold}{\huge\tt{y}}$$ $$\color{forestgreen}{\huge\tt{t}}\color{teal}{\huge\tt{h}}\color{blue}{\huge\tt{i}}\color{blueviolet}{\huge\tt{s}}$$ 10. Gabylovesyou Group Title thank you! 11. Gabylovesyou Group Title $$\color{red}{\huge\tt{G}}\color{orange}{\huge\tt{a}}\color{gold}{\huge\tt{b}}$$ $$\color{forestgreen}{\huge\tt{y}}\color{teal}{\huge\tt{B}}\color{blue}{\huge\tt{o}}\color{blueviolet}{\huge\tt{r}}$$ $$\color{red}{\huge\tt{g}}\color{orange}{\huge\tt{e}}\color{gold}{\huge\tt{s}}$$ 12. mathslover Group Title Great work @kymber :) 13. cap_n_crunch Group Title Another good one is the yfonts package. A friend of mine used it in his thesis to emphasize the chapters. The syntax goes like this: \yinipar{<letter of your choice>} , unfortunately it doesn't work here. 14. kymber Group Title Thanks, mathslover. :D 15. cap_n_crunch Group Title You're most welcome. I think most people wouldn't use it for science but actually i love it for giving my scripts some edge :) 16. godgavemeyou Group Title I like ur colors! 17. FibonacciChick666 Group Title $Thank~you,~ I~ will~ now ~be ~able~ to~ write~ pretty~ papers!!$ 18. ParthKohli Group Title @kymber You know what? You're epic. 19. kymber Group Title Thank you, Parth! :D Ahh...My student teaching others $$\Huge \color{red}{}$$$$\huge \color{blue}{}$$$$\LARGE \color{green}{}$$$$\Large \color{yellow}{}$$$$\large \color{orange}{}$$$$\small \color{purple}{}$$$$\Tiny \color{pink}{}$$$$\tiny \color{violet}{}$$$$\Tiny \color{pink}{}$$$$\small \color{purple}{}$$$$\large \color{orange}{}$$$$\Large \color{yellow}{}$$$$\LARGE \color{green}{}$$$$\huge \color{blue}{}$$$$\Huge \color{red}{}$$$$\huge \color{blue}{}$$$$\LARGE \color{green}{}$$$$\Large \color{yellow}{}$$$$\large \color{orange}{}$$$$\small \color{purple}{}$$$$\Tiny \color{pink}{}$$$$\tiny \color{violet}{}$$$$\Tiny \color{pink}{}$$$$\small \color{purple}{}$$$$\large \color{orange}{}$$$$\Large \color{yellow}{}$$$$\LARGE \color{green}{}$$$$\huge \color{blue}{}$$$$\Huge \color{red}{}$$$$\huge \color{blue}{}$$$$\LARGE \color{green}{}$$$$\Large \color{yellow}{}$$$$\large \color{orange}{}$$$$\small \color{purple}{}$$$$\Tiny \color{pink}{}$$$$\tiny \color{violet}{}$$$$\Tiny \color{pink}{}$$$$\small \color{purple}{}$$$$\large \color{orange}{}$$$$\Large \color{yellow}{}$$$$\LARGE \color{green}{}$$$$\huge \color{blue}{}$$$$\Huge \color{red}{}$$$$\huge \color{blue}{}$$$$\LARGE \color{green}{}$$$$\Large \color{yellow}{}$$ $$\LARGE \color{red}{~~~~\:\:\:\:\:\:\mathbb Well\:\:\mathbb Done\:\:\mathbb Young~~\mathbb Jedi,~\mathbb Well~~\mathbb Done.~~}$$ $$\large \color{red}{~\:\:\:\:\:\:\:\:\:\:\:\: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-\mathbb Snuggie\mathbb Lad }$$ $$\Huge \color{red}{}$$$$\huge \color{blue}{}$$$$\LARGE \color{green}{}$$$$\Large \color{yellow}{}$$$$\large \color{orange}{}$$$$\small \color{purple}{}$$$$\Tiny \color{pink}{}$$$$\tiny \color{violet}{}$$$$\Tiny \color{pink}{}$$$$\small \color{purple}{}$$$$\large \color{orange}{}$$$$\Large \color{yellow}{}$$$$\LARGE \color{green}{}$$$$\huge \color{blue}{}$$$$\Huge \color{red}{}$$$$\huge \color{blue}{}$$$$\LARGE \color{green}{}$$$$\Large \color{yellow}{}$$$$\large \color{orange}{}$$$$\small \color{purple}{}$$$$\Tiny \color{pink}{}$$$$\tiny \color{violet}{}$$$$\Tiny \color{pink}{}$$$$\small \color{purple}{}$$$$\large \color{orange}{}$$$$\Large \color{yellow}{}$$$$\LARGE \color{green}{}$$$$\huge \color{blue}{}$$$$\Huge \color{red}{}$$$$\huge \color{blue}{}$$$$\LARGE \color{green}{}$$$$\Large \color{yellow}{}$$$$\large \color{orange}{}$$$$\small \color{purple}{}$$$$\Tiny \color{pink}{}$$$$\tiny \color{violet}{}$$$$\Tiny \color{pink}{}$$$$\small \color{purple}{}$$$$\large \color{orange}{}$$$$\Large \color{yellow}{}$$$$\LARGE \color{green}{}$$$$\huge \color{blue}{}$$$$\Huge \color{red}{}$$$$\huge \color{blue}{}$$$$\LARGE \color{green}{}$$$$\Large \color{yellow}{*}$$ 21. shruti Group Title where do latex works? 22. kymber Group Title I'm certainly not your student, Snuggie. Not sure where you got that idea. 23. godgavemeyou Group Title $$\Huge \color{aqua}{\cal{LOL ~Kymber...}}$$ 24. kymber Group Title I'm pretty sure @ParthKohli was the one who showed me LaTeX, actually... and @shruti LaTeX works in questions and responses on OpenStudy and you can download programs to use it on your computer if you wanted. I don't know of any other sites that support it. 25. godgavemeyou Group Title I learned from snuggie ... And you i learned more fonts and colors from u .Snuggie taught me how to change colors and fonts and how to set it up oh...i thought you were one of mine...well...ok... 27. shruti Group Title does'nt i works on Other sites ? @Kymber 28. mathslover Group Title @shruti , that may be because , other sites may have not installed these packages on their sites. 29. mathslover Group Title btw, can you name those "Other sites" ? 30. shruti Group Title okay thanks! @mathslover You missed a few colors Are you guys still curious about what you can do? If you would like you can go to: http://latex-project.org/guides/usrguide.tex It is a rather dry read but you can learn a lot of commands to use on this site! This one is not so dry but it is more recent. I don't know if the team has kept up with the most recent $$\LaTeX$$ versions but if they have this will be helpful. this is one of the pages i refer to a lot. http://latex-project.org/guides/fntguide.pdf $$\setlength{\unitlength}{0.75mm} \begin{picture}(60,40) \put(30,20){\vector(1,0){30}} \put(30,20){\vector(4,1){20}} \put(30,20){\vector(3,1){25}} \put(30,20){\vector(2,1){30}} \put(30,20){\vector(1,2){10}} \thicklines \put(30,20){\vector(-4,1){30}} \put(30,20){\vector(-1,4){5}} \thinlines \put(30,20){\vector(-1,-1){5}} \put(30,20){\vector(-1,-4){5}} \end{picture}$$ 35. kymber Group Title What colours did I miss, care to share? 37. kymber Group Title Try ones that I missed. They're probably not included in my tutorial here because they don't work. 38. thomaster Group Title 39. kymber Group Title The ones with names are easier to remember Not if you understand how the Hexadecimal code works. If you understand the layout you can make any color you want with only swapping a number @kyber Names are for wimps when you've got millions of codes and all you have to do is find the sequencing. When you learn that you can create any color you want with only the typing of a couple of memorized number letter mixtures that all follow each other in a specific order that once you get it is dumbly easy. 41. JA1 Group Title $$\huge\color{Cyan}{16~MILLION~COLORS~YO~!!}$$ 42. JA1 Group Title That's like..... a lot O_O 43. JA1 Group Title @kymber why do you write "$$\color{red}{\huge\tt{T}}" emphazizing on the "\tt{T}}"? 44. JA1 Group Title The double t and the double } 45. thomaster Group Title double t = \tt = a font double } is to close the color tag, though it's unnecessary as you can write it as \huge\tt\color{red}T 46. JA1 Group Title Ah ok much less confusing thanks thom :) 47. kymber Group Title You need to put your words in brackets if you want to write more. \(\huge\tt\color{red}Tomatoes$$ will give you $$\huge\tt\color{red}Tomatoes$$. So then you can do $$\color{red}{\huge\tt{Tomatoes}}$$ for $$\color{red}{\huge\tt{Tomatoes}}$$. As far as I know. Parth taught me that. 48. ParthKohli Group Title :') 49. thomaster Group Title Yea my explanatin only works when you use 1 character. Like in this code: $$\color{red}{\huge\tt{T}}\color{orange}{\huge\tt{r}}\color{gold}{\huge\tt{y}}$$ $$\color{forestgreen}{\huge\tt{t}}\color{teal}{\huge\tt{h}}\color{blue}{\huge\tt{i}}\color{blueviolet}{\huge\tt{s}}$$ You can reduce it to: $$\huge\tt\color{red}T\color{orange}r\color{gold}y~\color{forestgreen}t\color{teal}h\color{blue}i\color{blueviolet}s$$ $$\huge\tt\color{red}T\color{orange}r\color{gold}y~\color{forestgreen}t\color{teal}h\color{blue}i\color{blueviolet}s$$ 50. ParthKohli Group Title Math lesson 1: Do you know why there are 16 million possible color combinations? Answer: These colors are written in hexadecimal, where we use letters A-F and 0-9. Now A-F is like 6 letters and 0-9 is 10 different numbers. So there are 16 total characters in hex. Hey, now you have six letters in hex to make a color, like $$\rm \color{red}{FF0000}$$ or $$\color{#C00}{C00000}$$. So there are $$16×16×16×16×16×16≈16,000,000$$ colors. 51. thomaster Group Title actualy it's 16,777,216 :P 52. ParthKohli Group Title That's why I wrote $$\approx$$ 53. thomaster Group Title i know, but let's not further spoil the comments :P Mind=BLOWN 55. JA1 Group Title Thanks @kymber :) 56. mathstudent55 Group Title 57. dumbsearch2 Group Title Wow. I didn't see this earlier. So just to be clear, I can use hex code that I use in CSS in LATEX? 58. dumbsearch2 Group Title	2014-08-28 03:04:08	{"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": 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": 0.6018127799034119, "perplexity": 2543.7258719136903}, "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-2014-35/segments/1408500830074.72/warc/CC-MAIN-20140820021350-00139-ip-10-180-136-8.ec2.internal.warc.gz"}
http://mathhelpforum.com/discrete-math/48282-probability-question.html	Math Help - probability question 1. probability question suppose that a large number of people standing in a queue are asked to reorder themselves so that they line up in order of height. what is the approximate probability that at least one person is in the same position after the reordering as they were before? i did up lists for up to 4 people and the results are: 1 person, Pr = 1 = 1 2 people, Pr = 1/2 = 0.5 3 people, Pr = 2/3 = 0.667 4 people, Pr = 5/8 = 0.625 i cannot for the life of me see a pattern, considering the 2 people probability means that the probabilities aren't even in decreasing order. i have done that particular probability correct, right? also, the question has a hint: ex = S xr r=0 r! WHAT DOES THIS HAVE TO DO WITH THE QUESTION?!?! 2. Originally Posted by wik_chick88 suppose that a large number of people standing in a queue are asked to reorder themselves so that they line up in order of height. what is the approximate probability that at least one person is in the same position after the reordering as they were before? i did up lists for up to 4 people and the results are: 1 person, Pr = 1 = 1 2 people, Pr = 1/2 = 0.5 3 people, Pr = 2/3 = 0.667 4 people, Pr = 5/8 = 0.625 i cannot for the life of me see a pattern, considering the 2 people probability means that the probabilities aren't even in decreasing order. i have done that particular probability correct, right? also, the question has a hint: ex = S xr r=0 r! WHAT DOES THIS HAVE TO DO WITH THE QUESTION?!?! 1 - Pr(no person in same position). The concept of derangement is what you need to get this probabilty. Read Derangement - Wikipedia, the free encyclopedia. 3. ok. so the formula for number of derangements is C:\Documents and Settings\Gabby\Desktop\derangement formula.JPG that means that the approximate probability that at least one person is in the same position after the reordering as they were before is 1-C:\Documents and Settings\Gabby\Desktop\derangement formula.JPG 4. whoops that came out funny ill just type it. the formula is n! sum of (from r=0 to n) [(-1)^2]/r! so the answer to the question is just 1-this formula? 5. Originally Posted by wik_chick88 ok. so the formula for number of derangements is C:\Documents and Settings\Gabby\Desktop\derangement formula.JPG that means that the approximate probability that at least one person is in the same position after the reordering as they were before is 1-C:\Documents and Settings\Gabby\Desktop\derangement formula.JPG For a large number of people the probability that no person is in the same position is 1/e. So the probability of at least one person in the same position is $1 - \frac{1}{e}$. 6. ok. i THINK i understand. its asking for the probability, so with the derangement formula, you dont need the n! at the start becuase you arent looking for a number, just a probability. then, the hint means that as r goes from 0 to infinite, the sum of (x^r)/r! equals e^x. so that means that if x = -1 such as in the derangement formula, as r goes from 0 to infinite, the sum of (-1)^r/r! actually equals e^-1 which equals 1/e. YAYY!!! thanks so much for your help!!	2015-06-30 15:24:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.834636390209198, "perplexity": 548.2575470400546}, "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-2015-27/segments/1435375093974.67/warc/CC-MAIN-20150627031813-00202-ip-10-179-60-89.ec2.internal.warc.gz"}
http://jaoa.org/article.aspx?articleid=2661132	Review \| November 2017 Inappropriate Use of Homeostasis Model Assessment Cutoff Values for Diagnosing Insulin Resistance in Pediatric Studies Author Notes • From the Departments of Biomedical Sciences (Student Doctor Fox and Dr Bridges) and Clinical Sciences (Drs Bernardino and Cochran) and the James R. Stookey Library (Ms Essig) at the West Virginia School of Osteopathic Medicine in Lewisburg. • Financial Disclosures: None reported. • Support: None reported. • *Address correspondence to Kristie Grove Bridges, PhD, 400 Lee St N, Lewisburg, WV 24901-1274. E-mail: kbridges@osteo.wvsom.edu Article Information Endocrinology / Pediatrics / Diabetes Review \| November 2017 ##### Inappropriate Use of Homeostasis Model Assessment Cutoff Values for Diagnosing Insulin Resistance in Pediatric Studies The Journal of the American Osteopathic Association, November 2017, Vol. 117, 689-696. doi:10.7556/jaoa.2017.135 The Journal of the American Osteopathic Association, November 2017, Vol. 117, 689-696. doi:10.7556/jaoa.2017.135 Abstract Background: Assessing pediatric patients for insulin resistance is one way to identify those who are at a high risk of developing type 2 diabetes mellitus. The homoeostasis model assessment (HOMA) is a measure of insulin resistance based on fasting blood glucose and insulin levels. Although this measure is widely used in research, cutoff values for pediatric populations have not been established. Objective: To assess the validity of HOMA cutoff values used in pediatric studies published in peer-reviewed journals. Methods: Studies published from January 2010 to December 2015 were identified through MEDLINE. Initial screening of abstracts was done to select studies that were conducted in pediatric populations and used HOMA to assess insulin resistance. Subsequent full-text review narrowed the list to only those studies that used a specific HOMA score to diagnose insulin resistance. Each study was classified as using a predetermined fixed HOMA cutoff value or a cutoff that was a percentile specific to that population. For studies that used a predetermined cutoff value, the references cited to provide evidence in support of that cutoff were evaluated. Results: In the 298 articles analyzed, 51 different HOMA cutoff values were used to classify patients as having insulin resistance. Two hundred fifty-five studies (85.6%) used a predetermined fixed cutoff value, but only 72 (28.2%) of those studies provided a reference that supported its use. One hundred ten studies (43%) that used a fixed cutoff either cited a study that did not mention HOMA or provided no reference at all. Tracing of citation history indicated that the most commonly used cutoff values were ultimately based on studies that did not validate their use for defining insulin resistance. Conclusion: Little evidence exists to support HOMA cutoff values commonly used to define insulin resistance in pediatric studies. These findings highlight the importance of validating study design elements when training medical students and novice investigators. Using available data to generate population ranges for HOMA would improve its clinical utility. ### Pay Per View Entire Journal 30-Day Access $30.00 This Issue 7-Day Access$15.00	2018-06-19 18:14:27	{"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": 0.2733599841594696, "perplexity": 8048.480003106765}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267863109.60/warc/CC-MAIN-20180619173519-20180619193519-00311.warc.gz"}
https://mathoverflow.net/questions/251464/height-of-ideals-in-polynomial-rings	# Height of ideals in polynomial rings Let $k$ be a field and $R=k[X_1,...,X_m]$ be a polynomial ring. Let $f_1,...,f_n\in R$ and $I$ be the ideal generated by these elements. By Krull's principal ideal theorem, $\text{ht}(I)\leq n$. Suppose now $\text{ht}(I)=n$. Let $J_l$ (for $1\leq l\leq n$) be the ideal generated by $f_1,...,f_{l-1},f_{l+1},...,f_n$, i.e. we leave out one generator. We know that $\text{ht}(J_l)\leq n-1$. However, equality need not hold for all $l$ even though $\text{ht}(I)=n$ (for example, consider $I=(Y(1-X),Z(1-X),X)$ in $\mathbb{C}[X,Y,Z]$). My question is: Is it still true that under these conditions, we have $\text{ht}(J_l)=n-1$ for at least one $l$? This may well be a very basic question. I would be grateful for any comments, counterexamples etc.	2019-04-19 23:18:23	{"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": 0.9804113507270813, "perplexity": 58.77869533979126}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578528430.9/warc/CC-MAIN-20190419220958-20190420002958-00454.warc.gz"}
http://math.stackexchange.com/tags/dirichlet-series/new	# Tag Info ## New answers tagged dirichlet-series 0 Se here, for instance. If one is going to define an analytic continuation by a given function, it must take the values that the function takes, even if they are infinite. So yes this function converges nicely at the negative odd numbers, but it will have simple poles at the negative evens, as well as all other zeroes of $\zeta$, leading to unremovable ... 1 The easiest method takes a slightly different tact after your first inequality: $$A\sum \frac{1}{n^s} \leq A\sum\frac{1}{\|n^s\|}\leq A\sum\frac{1}{n^{\|R(s)\|}}$$ The last inequality holds because $\|z\|\geq \|R(z)\|$ for any complex number $z$. 2 The Parseval identity (a.k.a. Plancherel's Theorem) for the Fourier transform is $$\int_{-\infty}^{\infty}\|f(t)\|^2dt = \int_{-\infty}^{\infty}\|\hat{f}(t)\|^2 dt,\;\;\; f \in L^2(\mathbb{R}).$$ I assume you're familiar with this theorem. If you have a function $f\in L^2(\mathbb{R})$ that is supported in $[0,\infty)$ only, then the Fourier transform ... Top 50 recent answers are included	2016-05-28 20:34:12	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9033700227737427, "perplexity": 324.10723391016694}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2016-22/segments/1464049278091.17/warc/CC-MAIN-20160524002118-00126-ip-10-185-217-139.ec2.internal.warc.gz"}
http://devblog.me/srm-571.html	Posted on Tue 19 February 2013 # SRM 571 This is 3rd SRM I participated in 2013. After failing SRM 568 and SRM 569 with really silly mistakes I decided to ascend my rating finally. ### 250 Looks very easy and it is. We are given $N$ strings of type "1.mp3", "2.mp3", etc. We need to sort them lexicographically and return top 50 strings. There are bunch of different correct solutions but idea is the same: if $N \le 50$ then we just compute every string and sort resulting array. Else we need to consider some big numbers close to power of ten. It might sound funny but I managed to put bug in easiest solution. Then reubmitted it for 149 points instead of ~220 points. Regardless huge disappointement I was happy that could submit something working as last two rounds I failed even easy problems. Challenge phase brought me another 50 points challenging guy with different bug. ### 500 This problem was about finding clique in a graph of up to $50$ vertices maximizing total weight of picked. Another constraint was: clique should include at least $\frac{2}{3}$ of total number of vertices. First thing everybody probably noticed: looking for optimal clique is NP-hard problem, so exclude anything polynomial here. Second is obvious constraint: $\frac{2}{3} \times n \le 16$. It means there is something to do with it. I could not come up with anything better than quite stupid backtracking approach with some sorting for better pruning. It looks like I am still bad in math and it worked slower than expected and failed on system test. During Challenge phase my solution defended two challenges but it did not help in the end. I saw several greedy solutions and one DP in my room: all failed. Approach is backtrack but from different end (you do not expect it to be that easy, right?). Someone found Bron-Kerbosch algorithm in wiki but failed. Another approach was greedy solution combined with random shuffles (not first time I notice such randomizing crap work). In the practice session I could implement correct solution in minutes which leads to well known conclusion: in TopCoder idea of the solution tends to be much harder and more important than implementation details. That is the reason we love it. Solution: for given subset of vertices find any non-connected pair. If there is no such pair it means we found clique, return sum of subset as the best answer. Else there are two ways: try look at subset with out the first bad vertix or with out the second one. It looks exponential and it is but number of different branches do not exceed $2^{16}$ as we are not interested in small subsets. ### In the end I was one of majority with one solved problem and less than 200 points which brought me tiny amount of rating. It feel like it would take enormous amount of effort to come back to middle-yellowish position. A lot of people from KZ participated today. Even guys from Palo Alto. Even from Kharkov. Keep it up, guys! ### Code 250 - FoxAndMp3 string toStr(Int x) { stringstream strm; strm << x << ".mp3"; return strm.str(); } vector <string> res; int n; void doit(Int x) { if (x > n) return; if (res.size() >= min(50, n)) return; res.pb(toStr(x)); for (int i = 0; i < 10; i++) { doit(x * 10 + i); if (res.size() >= min(50, n)) return; } } class FoxAndMp3 { public: vector <string> playList(int n) { ::n = n; if (n <= 50) { vector<string> a; for (int i = 1; i <= n; i++) { stringstream strm; strm << i << ".mp3"; a.pb(strm.str()); } sort(a.begin(), a.end()); return a; } for (int i = 1; i <= 9; i++) doit(i); return res; } }; 500 - MagicMolecule int res, n, need; vector<int> magicPower; vector<string>magicBond; map<Int, int> memo; int get(Int mask) { int &res = memo[mask]; res = -1; if (bitcount(mask) < need) return res; for (int i = 0; i < n; i++) if (mask & (1LL << i)) for (int j = i + 1; j < n; j++) if (mask & (1LL << j)) if (magicBond[i][j] == 'N') { break; } if (badA == -1) { res = 0; for (int i = 0; i < n; i++) if (mask & (1LL << i)) res += magicPower[i]; return res; } } class MagicMolecule { public: int maxMagicPower(vector <int> magicPower, vector <string> magicBond) { ::magicBond = magicBond; ::magicPower = magicPower; res = -1; n = (int)magicPower.size(); for (; need * 3 < 2 * n; need++); return get((1LL << n) - 1); } }; © Slava Kim. 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Theme by Giulio Fidente on github. .	2019-05-22 17:46:27	{"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": 0.41944631934165955, "perplexity": 3592.1939020123123}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256887.36/warc/CC-MAIN-20190522163302-20190522185302-00424.warc.gz"}
https://ask.cvxr.com/t/writing-the-f-x-y-y1-x1-y2-x2-yi-xi/8427	# Writing the f(x,y) = (y1^x1)(y2^x2)...(yi^xi) Hi guys, I met a problem while using cvxpy. My objective function is max f(x,y) = (y1^x1)(y2^x2)…(yi^xi) 0<xi<1, sum(xi)=1, y_min<yi<y_max, variables: xi, yi. yi may be complicated, I try to simplify it. I don’t know how to express which in cvxpy. Can you help me? Thanks The is not the cvxpy forum but the MATLAB cvx. I don’t think the function is DCP representable when both the bases and the exponents are optimization variables. However if x_i are independent of the y_i then it seems the maximum is obtained by taking x_i=1 for the biggest y_i. So your problem would be equivalent to \mathrm{maximize} \max_i(y_i) which is still not convex but representable with a mixed-integer model. 1 Like Thanks a lot. The function can be written as \mathrm{minimize} \sum_i -x_iln(y_i). I wonder whether the problem can be solved? In other words, how can I reformulate it and solve it? Cvx is only useful if the function -yln(x) is convex. Have you shown it is convex?	2023-03-22 09:49:03	{"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": 0.8667947053909302, "perplexity": 1505.3371770408246}, "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/1679296943809.22/warc/CC-MAIN-20230322082826-20230322112826-00226.warc.gz"}
https://chemistry.stackexchange.com/questions/33541/oxidation-of-carbon-atom-in-alkene-alkyne-and-alkanes-of-functional-groups	# Oxidation of Carbon atom in Alkene, Alkyne and Alkanes of functional groups? Nomenclature priorities are in order of the degree of oxidation of the carbon atom. I am not sure if it's advisable to just take it as it is and use it or try to understand the basis of the functional group prioritization. I guess the latter interest me. So looking at the priorities for functional groups, Alkene has higher priority than Alkyne and followed by Alkane. I started reading some materials about Oxidative Cleavage of Double Bonds and rest of the reactions, which kinda went way over my head. Can someone explain, how and why the carbon oxidation in Alkynes are lower than Alknenes? Or put in another word, why would Alkene carbon oxidate faster than Alknenes? • @bon why did you remove the functional group and hydrocarbon tag? Just curious to know. Jul 2, 2015 at 10:26 • @Have a look at the meta post about functional group tags. – bon Jul 2, 2015 at 10:28 • There are two separate questions in the last paragraph. They are not the same question and I think the first is the one which is relevant to the rest of the post. – bon Jul 2, 2015 at 11:09 Nomenclature priorities are in order of the degree of oxidation of the carbon atom. This may be true for the heteroatom functional groups, but not for alkenes, alkynes, and alkanes. Consider that each $\ce{C-H}$ bond contributes $-1$ to the oxidation state and each $\ce{C-C}$ bon regardless of order contributes $0$ to the oxidation state of carbon. Thus: hydrocarbon formula functional group oxidation number of C ethane CH3CH3 alkane -3 ethene CH2CH2 alkene -2 ethyne CHCH alkyne -1 Alkynes have a higher oxidation number than alkenes. Alkenes react better in oxidation reactions than alkynes because alkenes are less oxidized than alkynes. The organic chemistry textbook that I teach from explains that alkenes have higher nomenclature priority than alkynes because alkenes are more common. I cannot find a more authoritative reason/source. • The question that drives me a little confusing is, the order is: alkene, alkyne and alkane... so from the perspective of single, double, tripple bond, I really do not see an order at all... You have stated because alkenes are less oxidized than alkynes. Well how important to understand the underlying oxidation theory of these guys, from biochemistry and pharmacology aspect? Jul 2, 2015 at 15:08	2022-08-09 22:21:36	{"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": 0.4849773645401001, "perplexity": 2286.188599915375}, "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-2022-33/segments/1659882571090.80/warc/CC-MAIN-20220809215803-20220810005803-00760.warc.gz"}
https://twuptodate.home.blog/tag/%E5%A5%BD%E5%8F%8B/	## 【不黏但是極度親密】有一種友情叫「遠距離友情」，你也有這樣的好友嗎？ ### 遠距離讓你更覺新奇 Nothing makes the earth seem so spacious as to have friends at a distance; they make the latitudes and longitudes.	2023-03-23 05:01:55	{"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": 0.8458530902862549, "perplexity": 5361.151793069484}, "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-2023-14/segments/1679296944996.49/warc/CC-MAIN-20230323034459-20230323064459-00727.warc.gz"}
https://encyclopediaofmath.org/wiki/Strong_mixing_conditions	# Strong mixing conditions This article Strong Mixing Conditions was adapted from an original article by Richard Crane Bradley, which appeared in StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies. The original article ([http://statprob.com/encyclopedia/StrongMixingConditions.html StatProb Source], Local Files: pdf \| tex) is copyrighted by the author(s), the article has been donated to Encyclopedia of Mathematics, and its further issues are under Creative Commons Attribution Share-Alike License'. All pages from StatProb are contained in the Category StatProb. 2010 Mathematics Subject Classification: Primary: 60G10 Secondary: 60G99 [MSN][ZBL] Strong Mixing Conditions Department of Mathematics, Indiana University, Bloomington, Indiana, USA There has been much research on stochastic models that have a well defined, specific structure --- for example, Markov chains, Gaussian processes, or linear models, including ARMA (autoregressive -- moving average) models. However, it became clear in the middle of the last century that there was a need for a theory of statistical inference (e.g. central limit theory) that could be used in the analysis of time series that did not seem to "fit" any such specific structure but which did seem to have some "asymptotic independence" properties. That motivated the development of a broad theory of "strong mixing conditions" to handle such situations. This note is a brief description of that theory. The field of strong mixing conditions is a vast area, and a short note such as this cannot even begin to do justice to it. Journal articles (with one exception) will not be cited; and many researchers who made important contributions to this field will not be mentioned here. All that can be done here is to give a narrow snapshot of part of the field. The strong mixing ($\alpha$-mixing) condition. Suppose $X := (X_k, k \in {\mathbf Z})$ is a sequence of random variables on a given probability space $(\Omega, {\cal F}, P)$. For $-\infty \leq j \leq \ell \leq \infty$, let ${\cal F}_j^\ell$ denote the $\sigma$-field of events generated by the random variables $X_k, j \leq k \leq \ell (k \in {\mathbf Z})$. For any two $\sigma$-fields ${\cal A}$ and ${\cal B} \subset {\cal F}$, define the "measure of dependence" $$\alpha({\cal A}, {\cal B}) := \sup_{A \in {\cal A}, B \in {\cal B}} \|P(A \cap B) - P(A)P(B)\|. \tag{1}$$ For the given random sequence $X$, for any positive integer $n$, define the dependence coefficient $$\alpha(n) = \alpha(X,n) := \sup_{j \in '''Z'''} \alpha({\cal F}_{-\infty}^j, {\cal F}_{j + n}^{\infty}). \tag{2}$$ By a trivial argument, the sequence of numbers $(\alpha(n), n \in {\mathbf N})$ is nonincreasing. The random sequence $X$ is said to be "strongly mixing", or "$\alpha$-mixing", if $\alpha(n) \to 0$ as $n \to \infty$. This condition was introduced in 1956 by Rosenblatt [Ro1], and was used in that paper in the proof of a central limit theorem. (The phrase "central limit theorem" will henceforth be abbreviated CLT.) In the case where the given sequence $X$ is strictly stationary (i.e. its distribution is invariant under a shift of the indices), eq. (2) also has the simpler form $$\alpha(n) = \alpha(X,n) := \alpha({\cal F}_{-\infty}^0, {\cal F}_n^{\infty}). \tag{3}$$ For simplicity, in the rest of this note, we shall restrict to strictly stationary sequences. (Some comments below will have obvious adaptations to nonstationary processes.) In particular, for strictly stationary sequences, the strong mixing ($\alpha$-mixing) condition implies Kolmogorov regularity (a trivial "past tail" $\sigma$-field), which in turn implies "mixing" (in the ergodic-theoretic sense), which in turn implies ergodicity. (None of the converse implications holds.) For further related information, see e.g. [Br, v1, Chapter 2]. Comments on limit theory under $\alpha$-mixing. Under $\alpha$-mixing and other similar conditions (including ones reviewed below), there has been a vast development of limit theory --- for example, CLTs, weak invariance principles, laws of the iterated logarithm, almost sure invariance principles, and rates of convergence in the strong law of large numbers. For example, the CLT in [Ro1] evolved through subsequent refinements by several researchers into the following "canonical" form. (For its history and a generously detailed presentation of its proof, see e.g. [Br, v1, Theorems 1.19 and 10.2].) Theorem 1. Suppose $(X_k, k \in {\mathbf Z})$ is a strictly stationary sequence of random variables such that $EX_0 = 0$, $EX_0^2 < \infty$, $\sigma_n^2 := ES_n^2 \to \infty$ as $n \to \infty$, and $\alpha(n) \to 0$ as $n \to \infty$. Then the following two conditions (A) and (B) are equivalent: (A) The family of random variables $(S_n^2/\sigma_n^2, n \in {\mathbf N})$ is uniformly integrable. (B) $S_n/\sigma_n \Rightarrow N(0,1)$ as $n \to \infty$. If (the hypothesis and) these two equivalent conditions (A) and (B) hold, then $\sigma_n^2 = n \cdot h(n)$ for some function $h(t), t \in (0, \infty)$ which is slowly varying as $t \to \infty$. Here $S_n := X_1 + X_2 + \dots + X_n$; and $\Rightarrow$ denotes convergence in distribution. The assumption $ES_n^2 \to \infty$ is needed here in order to avoid trivial $\alpha$-mixing (or even 1-dependent) counterexamples in which a kind of "cancellation" prevents the partial sums $S_n$ from "growing" (in probability) and becoming asymptotically normal. In the context of Theorem 1, if one wants to obtain asymptotic normality of the partial sums (as in condition (B)) without an explicit uniform integrability assumption on the partial sums (as in condition (A)), then as an alternative, one can impose a combination of assumptions on, say, (i) the (marginal) distribution of $X_0$ and (ii) the rate of decay of the numbers $\alpha(n)$ to 0 (the "mixing rate"). This involves a "trade-off"; the weaker one assumption is, the stronger the other has to be. One such CLT of Ibragimov in 1962 involved such a "trade-off" in which it is assumed that for some $\delta > 0$, $E\|X_0\|^{2 + \delta} < \infty$ and $\sum_{n=1}^\infty [\alpha(n)]^{\delta/(2 + \delta)} < \infty$. Counterexamples of Davydov in 1973 (with just slightly weaker properties) showed that that result is quite sharp. However, it is not at the exact "borderline". From a covariance inequality of Rio in 1993 and a CLT (in fact a weak invariance principle) of Doukhan, Massart, and Rio in 1994, it became clear that the "exact borderline" CLTs of this kind have to involve quantiles of the (marginal) distribution of $X_0$ (rather than just moments). For a generously detailed exposition of such CLTs, see [Br, v1, Chapter 10]; and for further related results, see also Rio [Ri]. Under the hypothesis (first sentence) of Theorem 1 (with just finite second moments), there is no mixing rate, no matter how fast (short of $m$-dependence), that can insure that a CLT holds. That was shown in 1983 with two different counterexamples, one by the author and the other by Herrndorf. See [Br, v1&3, Theorem 10.25 and Chapter 31]. Several other classic strong mixing conditions. As indicated above, the terms "$\alpha$-mixing" and "strong mixing condition" (singular) both refer to the condition $\alpha(n) \to 0$. (A little caution is in order; in ergodic theory, the term "strong mixing" is often used to refer to the condition of "mixing in the ergodic-theoretic sense", which is weaker than $\alpha$-mixing as noted earlier.) The term "strong mixing conditions" (plural) can reasonably be thought of as referring to all conditions that are at least as strong as (i.e. that imply) $\alpha$-mixing. In the classical theory, five strong mixing conditions (again, plural) have emerged as the most prominent ones: $\alpha$-mixing itself and four others that will be defined here. Recall our probability space $(\Omega, {\cal F}, P)$. For any two $\sigma$-fields ${\cal A}$ and ${\cal B} \subset {\cal F}$, define the following four "measures of dependence": \eqalignno{ \phi({\cal A}, {\cal B}) &:= \sup_{A \in {\cal A}, B \in {\cal B}, P(A) > 0} \|P(B\|A) - P(B)\|; & (4) \cr \psi({\cal A}, {\cal B}) &:= \sup_{A \in {\cal A}, B \in {\cal B}, P(A) > 0, P(B) > 0} \|P(B \cap A)/[P(A)P(B)]\thinspace -\thinspace 1\|; & (5) \cr \rho({\cal A}, {\cal B}) &:= \sup_{f \in {\cal L}^2({\cal A}),\thinspace g \in {\cal L}^2({\cal B})} \|{\rm Corr}(f,g)\|; \quad {\rm and} & (6) \cr \beta ({\cal A}, {\cal B}) &:= \sup (1/2) \sum_{i=1}^I \sum_{j=1}^J \|P(A_i \cap B_j) - P(A_i)P(B_j)\| & (7) \cr } where the latter supremum is taken over all pairs of finite partitions $(A_1, A_2, \dots, A_I)$ and $(B_1, B_2, \dots, B_J)$ of $\Omega$ such that $A_i \in {\cal A}$ for each $i$ and $B_j \in {\cal B}$ for each $j$. In (6), for a given $\sigma$-field ${\cal D} \subset {\cal F}$, the notation ${\cal L}^2({\cal D})$ refers to the space of (equivalence classes of) square-integrable, ${\cal D}$-measurable random variables. Now suppose $X := (X_k, k \in {\mathbf Z})$ is a strictly stationary sequence of random variables on $(\Omega, {\cal F}, P)$. For any positive integer $n$, analogously to (3), define the dependence coefficient $$\phi(n) = \phi(X,n) := \phi({\cal F}_{-\infty}^0, {\cal F}_n^{\infty}), \tag{8}$$ and define analogously the dependence coefficients $\psi(n)$, $\rho(n)$, and $\beta(n)$. Each of these four sequences of dependence coefficients is trivially nonincreasing. The (strictly stationary) sequence $X$ is said to be "$\phi$-mixing" if $\phi(n) \to 0$ as $n \to \infty$; "$\psi$-mixing" if $\psi(n) \to 0$ as $n \to \infty$; "$\rho$-mixing" if $\rho(n) \to 0$ as $n \to \infty$; and "absolutely regular", or "$\beta$-mixing", if $\beta(n) \to 0$ as $n \to \infty$. The $\phi$-mixing condition was introduced by Ibragimov in 1959 and was also studied by Cogburn in 1960. The $\psi$-mixing condition evolved through papers of Blum, Hanson, and Koopmans in 1963 and Philipp in 1969; and (see e.g. [Io]) it was also implicitly present in earlier work of Doeblin in 1940 involving the metric theory of continued fractions. The $\rho$-mixing condition was introduced by Kolmogorov and Rozanov 1960. (The "maximal correlation coefficient" $\rho({\cal A}, {\cal B})$ itself was first studied by Hirschfeld in 1935 in a statistical context that had no particular connection with "stochastic processes".) The absolute regularity ($\beta$-mixing) condition was introduced by Volkonskii and Rozanov in 1959, and in the ergodic theory literature it is also called the "weak Bernoulli" condition. For the five measures of dependence in (1) and (4)--(7), one has the following well known inequalities: \eqalignno{ 2\alpha({\cal A}, {\cal B}) \thinspace & \leq \thinspace \beta({\cal A}, {\cal B}) \thinspace \leq \thinspace \phi({\cal A}, {\cal B}) \thinspace \leq \thinspace (1/2) \psi({\cal A}, {\cal B}); \cr 4 \alpha({\cal A}, {\cal B})\thinspace &\leq \thinspace \rho({\cal A}, {\cal B}) \thinspace \leq \thinspace \psi({\cal A}, {\cal B}); \quad {\rm and} \cr \rho({\cal A}, {\cal B}) \thinspace &\leq \thinspace 2 [\phi({\cal A}, {\cal B})]^{1/2} [\phi({\cal B}, {\cal A})]^{1/2} \thinspace \leq \thinspace 2 [\phi({\cal A}, {\cal B})]^{1/2}. \cr } For a history and proof of these inequalities, see e.g. [Br, v1, Theorem 3.11]. As a consequence of these inequalities and some well known examples, one has the following "hierarchy" of the five strong mixing conditions here: (i) $\psi$-mixing implies $\phi$-mixing. (ii) $\phi$-mixing implies both $\rho$-mixing and $\beta$-mixing (absolute regularity). (iii) $\rho$-mixing and $\beta$-mixing each imply $\alpha$-mixing (strong mixing). (iv) Aside from “transitivity”, there are in general no other implications between these five mixing conditions. In particular, neither of the conditions $\rho$-mixing and $\beta$-mixing implies the other. For all of these mixing conditions, the “mixing rates” can be essentially arbitrary, For all of these mixing conditions, the "mixing rates" can be essentially arbitrary, and in particular, arbitrarily slow. That general principle was established by Kesten and O'Brien in 1976 with several classes of examples. For further details, see e.g. [Br, v3, Chapter 26]. The various strong mixing conditions above have been used extensively in statistical inference for weakly dependent data. See e.g. [DDLLLP], [DMS], [Ro3], or [Žu]. Ibragimov's conjecture and related material. Suppose (as in Theorem 1) $X := (X_k, k \in {\mathbf Z})$ is a strictly stationary sequence of random variables such that $$EX_0 = 0, \ EX_0^2 < \infty, \ {\ \rm and\ } ES_n^2 \to \infty {\ \rm as\ } n \to \infty. \tag{9}$$ In the 1960s, I.A. Ibragimov conjectured that under these assumptions, if also $X$ is $\phi$-mixing, then a CLT holds. Technically, this conjecture remains unsolved. Peligrad showed in 1985 that it holds under the stronger "growth" assumption $\liminf_{n \to \infty} n^{-1} ES_n^2 > 0$. (See e.g. [Br, v2, Theorem 17.7].) Under (9) and $\rho$-mixing (which is weaker than $\phi$-mixing), a CLT need not hold (see [Br, v3, Chapter 34] for counterexamples). However, if one also imposes either the stronger moment condition $E\|X_0\|^{2 + \delta} < \infty$ for some $\delta > 0$, or else the "logarithmic" mixing rate assumption $\sum_{n=1}^\infty \rho(2^n) < \infty$, then a CLT does hold (results of Ibragimov in 1975). For further limit theory under $\rho$-mixing, see e.g. [LL] or [Br, v1, Chapter 11]. Under (9) and an "interlaced" variant of the $\rho$-mixing condition (i.e. with the two index sets allowed to be "interlaced" instead of just "past" and "future"), a CLT does hold. For this and related material, see e.g. [Br, v1, Sections 11.18-11.28]. There is a vast literature on central limit theory for random fields satisfying various strong mixing conditions. See e.g. [Ro3], [Žu], [Do], and [Br, v3]. In the formulation of mixing conditions for random fields --- and also "interlaced" mixing conditions for random sequences --- some caution is needed; see e.g. [Br, v1&3, Theorems 5.11, 5.13, 29.9, and 29.12]. Connections with specific types of models. Now let us return briefly to a theme from the beginning of this write-up: the connection between strong mixing conditions and specific structures. Markov chains. Suppose $X := (X_k, k \in {\mathbf Z})$ is a strictly stationary Markov chain. In the case where $X$ has finite state space and is irreducible and aperiodic, it is $\psi$-mixing, with at least exponentially fast mixing rate. In the case where $X$ has countable (but not necessarily finite) state space and is irreducible and aperiodic, it satisfies $\beta$-mixing, but the mixing rate can be arbitrarily slow. In the case where $X$ has (say) real (but not necessarily countable) state space, (i) Harris recurrence and "aperiodicity" (suitably defined) together are equivalent to $\beta$-mixing, (ii) the "geometric ergodicity" condition is equivalent to $\beta$-mixing with at least exponentially fast mixing rate, and (iii) one particular version of "Doeblin's condition" is equivalent to $\phi$-mixing (and the mixing rate will then be at least exponentially fast). There exist strictly stationary, countable-state Markov chains that are $\phi$-mixing but not "time reversed" $\phi$-mixing (note the asymmetry in the definition of $\phi({\cal A}, {\cal B})$ in (4)). For this and other information on strong mixing conditions for Markov chains, see e.g. [Ro2, Chapter 7], [Do], [MT], and [Br, v1&2, Chapters 7 and 21]. Stationary Gaussian sequences. For stationary Gaussian sequences $X := (X_k, k \in {\mathbf Z})$, Ibragimov and Rozanov [IR] give characterizations of various strong mixing conditions in terms of properties of spectral density functions. Here are just a couple of comments: For stationary Gaussian sequences, the $\alpha$- and $\rho$-mixing conditions are equivalent to each other, and the $\phi$- and $\psi$-mixing conditions are each equivalent to $m$-dependence. If a stationary Gaussian sequence has a continuous positive spectral density function, then it is $\rho$-mixing. For some further closely related information on stationary Gaussian sequences, see also [Br, v1&3, Chapters 9 and 27]. Dynamical systems. Many dynamical systems have strong mixing properties. Certain one-dimensional "Gibbs states" processes are $\psi$-mixing with at least exponentially fast mixing rate. A well known standard "continued fraction" process is $\psi$-mixing with at least exponentially fast mixing rate (see [Io]). For certain stationary finite-state stochastic processes built on piecewise expanding mappings of the unit interval onto itself, the absolute regularity condition holds with at least exponentially fast mixing rate. For more detains on the mixing properties of these and other dynamical systems, see e.g. Denker [De]. Linear and related processes. There is a large literature on strong mixing properties of strictly stationary linear processes (including strictly stationary ARMA processes and also "non-causal" linear processes and linear random fields) and also of some other related processes such as bilinear, ARCH, or GARCH models. For details on strong mixing properties of these and other related processes, see e.g. Doukhan [Do, Chapter 2]. However, many strictly stationary linear processes fail to be $\alpha$-mixing. A well known classic example is the strictly stationary AR(1) process (autoregressive process of order 1) $X := (X_k, k \in {\mathbf Z})$ of the form $X_k = (1/2)X_{k-1} + \xi_k$ where $(\xi_k, k \in {\mathbf Z})$ is a sequence of independent, identically distributed random variables, each taking the values 0 and 1 with probability 1/2 each. It has long been well known that this random sequence $X$ is not $\alpha$-mixing. For more on this example, see e.g. [Br, v1, Example 2.15] or [Do, Section 2.3.1]. Further related developments. The AR(1) example spelled out above, together with many other examples that are not $\alpha$-mixing but seem to have some similar "weak dependence" quality, have motivated the development of more general conditions of weak dependence that have the "spirit" of, and most of the advantages of, strong mixing conditions, but are less restrictive, i.e. applicable to a much broader class of models (including the AR(1) example above). There is a substantial development of central limit theory for strictly stationary sequences under weak dependence assumptions explicitly involving characteristic functions in connection with "block sums"; much of that theory is codified in [Ja]. There is a substantial development of limit theory of various kinds under weak dependence assumptions that involve covariances of certain multivariate Lipschitz functions of random variables from the "past" and "future" (in the spirit of, but much less restrictive than, say, the dependence coefficient $\rho(n)$ defined analogously to (3) and (8)); see e.g. [DDLLLP]. There is a substantial development of limit theory under weak dependence assumptions that involve dependence coefficients similar to $\alpha(n)$ in (3) but in which the classes of events are restricted to intersections of finitely many events of the form $\{X_k > c\}$ for appropriate indices $k$ and appropriate real numbers $c$; for the use of such conditions in extreme value theory, see e.g. [LLR]. In recent years, there has been a considerable development of central limit theory under "projective" criteria related to martingale theory (motivated by Gordin's martingale-approximation technique --- see [HH]); for details, see e.g. [Pe]. There are far too many other types of weak dependence conditions, of the general spirit of strong mixing conditions but less restrictive, to describe here; for more details, see e.g. [DDLLLP] or [Br, v1, Chapter 13]. #### References [Br] R.C. Bradley. Introduction to Strong Mixing Conditions, Vols. 1, 2, and 3. Kendrick Press, Heber City (Utah), 2007. [DDLLLP] J. Dedecker, P. Doukhan, G. Lang, J.R. León, S. Louhichi, and C. Prieur. Weak Dependence: Models, Theory, and Applications. Lecture Notes in Statistics 190. Springer-Verlag, New York, 2007. [DMS] H. Dehling, T. Mikosch, and M. Sørensen, eds. Empirical Process Techniques for Dependent Data. Birkhäuser, Boston, 2002. [De] M. Denker. The central limit theorem for dynamical systems. In: Dynamical Systems and Ergodic Theory, (K. Krzyzewski, ed.), pp. 33-62. Banach Center Publications, Polish Scientific Publishers, Warsaw, 1989. [Do] P. Doukhan. Mixing: Properties and Examples. Springer-Verlag, New York, 1995. [HH] P. Hall and C.C. Heyde. Martingale Limit Theory and its Application. Academic Press, San Diego, 1980. [IR] I.A. Ibragimov and Yu.A. Rozanov. Gaussian Random Processes. Springer-Verlag, New York, 1978. [Io] M. Iosifescu. Doeblin and the metric theory of continued fractions: a functional theoretic solution to Gauss' 1812 problem. In: Doeblin and Modern Probability, (H. Cohn, ed.), pp. 97-110. Contemporary Mathematics 149, American Mathematical Society, Providence, 1993. [Ja] A. Jakubowski. Asymptotic Independent Representations for Sums and Order Statistics of Stationary Sequences. Uniwersytet Mikołaja Kopernika, Toruń, Poland, 1991. [LL] Z. Lin and C. Lu. Limit Theory for Mixing Dependent Random Variables. Kluwer Academic Publishers, Boston, 1996. [LLR] M.R. Leadbetter, G. Lindgren, and H. Rootzén. Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York, 1983. [MT] S.P. Meyn and R.L. Tweedie. Markov Chains and Stochastic Stability (3rd printing). Springer-Verlag, New York, 1996. [Pe] M. Peligrad. Conditional central limit theorem via martingale approximation. In: Dependence in Probability, Analysis and Number Theory, (I. Berkes, R.C. Bradley, H. Dehling, M. Peligrad, and R. Tichy, eds.), pp. 295-309. Kendrick Press, Heber City (Utah), 2010. [Ri] E. Rio. Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants. Mathématiques & Applications 31. Springer, Paris, 2000. [Ro1] M. Rosenblatt. A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 (1956) 43-47. [Ro2] M. Rosenblatt. Markov Processes, Structure and Asymptotic Behavior. Springer-Verlag, New York, 1971. [Ro3] M. Rosenblatt. Stationary Sequences and Random Fields. Birkhäuser, Boston, 1985. [Žu] I.G. Žurbenko. The Spectral Analysis of Time Series. North-Holland, Amsterdam, 1986. How to Cite This Entry: Strong mixing conditions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strong_mixing_conditions&oldid=38548	2023-03-26 03:37:12	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9426834583282471, "perplexity": 1076.553984745365}, "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/1679296945381.91/warc/CC-MAIN-20230326013652-20230326043652-00318.warc.gz"}
https://stacks.math.columbia.edu/tag/01ZJ	Proposition 32.9.6. Let $f : X \to S$ be a morphism of schemes. Assume 1. $f$ is of finite type and separated, and 2. $S$ is quasi-compact and quasi-separated. Then there exists a separated morphism of finite presentation $f' : X' \to S$ and a closed immersion $X \to X'$ of schemes over $S$. Proof. Apply Lemma 32.9.5 and note that $X_ i \to S$ is separated for large $i$ by Lemma 32.4.17 as we have assumed that $X \to S$ is separated. $\square$ In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).	2021-09-26 21:33:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9783264994621277, "perplexity": 242.4604943896917}, "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-2021-39/segments/1631780057973.90/warc/CC-MAIN-20210926205414-20210926235414-00296.warc.gz"}
https://leetcode.com/problems/data-stream-as-disjoint-intervals/	### 352. Data Stream as Disjoint Intervals Given a data stream input of non-negative integers a1, a2, ..., an, ..., summarize the numbers seen so far as a list of disjoint intervals. For example, suppose the integers from the data stream are 1, 3, 7, 2, 6, ..., then the summary will be: [1, 1] [1, 1], [3, 3] [1, 1], [3, 3], [7, 7] [1, 3], [7, 7] [1, 3], [6, 7]	2017-11-22 20:23:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7835479378700256, "perplexity": 984.8944896172284}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806660.82/warc/CC-MAIN-20171122194844-20171122214844-00229.warc.gz"}
https://questioncove.com/updates/4d4db9b0a805b76496f5c80b	Mathematics OpenStudy (anonymous): how do I find the integral of sqrt(16x - x^2)? OpenStudy (anonymous): Not sure I could do it by hand anymore, but in a book of integrals, we find INT(SQRT(2ax-x^2)dx. That's your integral, with a=8. The answer is quite lengthy; too long for me to type. Worst case, get it from integrals.com or wolframalpha.com. OpenStudy (anonymous): Like LBickford the answer is kind of complicated but here are the steps. (1) Complete the square inside the square root--SQRT(64- (x-8)^2) (2) Use u substitution-- u=x-8 (3) Use z substitution-- z = $\sin \theta$ Latest Questions NoodlesAndRiceYT: i think i found a glitch, sometimes when i log into questioncove, this happens an 25 minutes ago 25 Replies 3 Medals 14xavierrobinson: what is discrimination 1 hour ago 1 Reply 0 Medals quasia: Who were the members of each alliance in World War I? 2 hours ago 5 Replies 1 Medal Vocaloid: AP Calc BC Study Guide/Tutorial 2 hours ago 1 Reply 7 Medals NoProbBob: Help Yo SS below 2 hours ago 15 Replies 1 Medal yallbabynene08: help 1 hour ago 20 Replies 2 Medals	2021-04-23 04:18:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6786035299301147, "perplexity": 7225.084126485584}, "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-2021-17/segments/1618039601956.95/warc/CC-MAIN-20210423041014-20210423071014-00023.warc.gz"}
https://raweb.inria.fr/rapportsactivite/RA2013/parietal/uid71.html	Overall Objectives Software and Platforms Partnerships and Cooperations Bibliography PDF e-Pub ## Section: New Results ### Identifying predictive regions from fMRI with TV-$\ell$1 prior Participants : Gaël Varoquaux [Correspondant] , Bertrand Thirion, Alexandre Gramfort. Decoding, i.e. predicting stimulus related quantities from functional brain images, is a powerful tool to demonstrate differences between brain activity across conditions. However, unlike standard brain mapping, it offers no guaranties on the localization of this information. Here, we consider decoding as a statistical estimation problem and show that injecting a spatial segmentation prior leads to unmatched performance in recovering predictive regions. Specifically, we use $\ell$1 penalization to set voxels to zero and Total-Variation (TV) penalization to segment regions. Our contribution is two-fold. On the one hand, we show via extensive experiments that, amongst a large selection of decoding and brain-mapping strategies, TV+$\ell$1 leads to best region recovery (see Fig. 8 ). On the other hand, we consider implementation issues related to this estimator. To tackle efficiently this joint prediction-segmentation problem we introduce a fast optimization algorithm based on a primal-dual approach. We also tackle automatic setting of hyper-parameters and fast computation of image operation on the irregular masks that arise in brain imaging. Figure 8. Results on fMRI data from (from left to right F-test, ElasticNet and TV-${\ell }_{1}$ ). The TV-${\ell }_{1}$ regularized model segments neuroscientificly meaningful predictive regions in agreement with univariate statistics while the ElasticNet yields sparse although very scattered non-zero weights. More details can be found in [59] .	2019-06-25 11:35:00	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "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": 0.4528553783893585, "perplexity": 3722.474103421109}, "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/1560627999838.23/warc/CC-MAIN-20190625112522-20190625134522-00110.warc.gz"}
https://www.mookratha.com/news/details/10137/	#### Moyal-deformed ABJM field theory: the beginning admin 2018-06-26 10:29:00 3860 Wepresent the initial study on Moyal-deformed noncommutative ABJM theory from thequantum field theory viewpoint. The classical action is formulated in terms ofcomponent fields and proven to be $\mathcal N=6$ supersymmetric. The one looptwo and three point functions are calculated using the conventional Feynmandiagram method and shown to have well defined commutative limit. Potentialfollow-ons are going to be discussed. Designed by 2014级long8唯一本科生蒋啸寒	2022-10-05 03:21:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6098581552505493, "perplexity": 8207.019548636656}, "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-2022-40/segments/1664030337531.3/warc/CC-MAIN-20221005011205-20221005041205-00447.warc.gz"}
https://compas-dev.github.io/compas_3gs/user_manual/00_polyhedral_cell/00_10_cell_pull_face.html	# Constrained manipulation In most computational graphic statics applications in 2D or 2.5D, geometric manipulations of form and force diagrams are simple and straightforward using a mouse and a computer monitor. Because the diagrams are coplanar on a single viewing plane, one can simply click and move the vertices of the force diagram to observe the consequential effects on the form diagram in real-time. Moving the vertices of a 2D force diagram changes both the magnitudes and orientations of the corresponding external forces or members in the form diagram. Geometric manipulations of polyhedral cells are not as straightforward or intuitive without the fixed viewing plane of 2D applications or the projection plane of 2.5D applications. Moving the vertices of a polyhedral cell changes its geometry, but it is not immediately clear to the user how much effect the geometric transformation has on the new distribution or orientations of forces. Vertex translation in 3D space also requires meaningful geometric guides or constraints that are based on the local geometry of the polyhedral cell to avoid arbitrary or counterproductive transformations. Furthermore, the translation of vertices could also cause some of the faces of the polyhedral cell to become non-planar. For manipulating the geometry of polyhedral cells while enforcing the planarity constraints of the faces, vertex translations are simply not sufficient enough. In order to change the force distribution of a polyhedral cell while maintaining the initial face orientations, a face can be pulled along its normal vector. If a face contains vertices that have valencies or degrees of four or more (more than one trailing edge), the pulling or tilting of the axis will result in faces that are no longer in their original orientations or possibly non-planar. The topological transformation of a cell can be guided and significantly simplified by the EGI. The corresponding EGI face of a high-valent cell vertex can be split, which creates an extra vertex. Using the new topology of the EGI, the new faces of the cell can be constructed before the face pull operation is performed. ## Example from __future__ import absolute_import from __future__ import print_function from __future__ import division import compas from compas_rhino.helpers import mesh_from_surface from compas_3gs.diagrams import Cell from compas_3gs.rhino import rhino_cell_face_pull try: import rhinoscriptsyntax as rs except ImportError: compas.raise_if_ironpython() __author__ = 'Juney Lee' __email__ = 'juney.lee@arch.ethz.ch' # ------------------------------------------------------------------------------ # 1. make cell from rhino polysurfaces # ------------------------------------------------------------------------------ layer = 'cell' guid = rs.GetObject("select a closed polysurface", filter=rs.filter.polysurface) rs.HideObjects(guid) cell = mesh_from_surface(Cell, guid) cell.draw() # ------------------------------------------------------------------------------ # 2. pull cell face # ------------------------------------------------------------------------------ rhino_cell_face_pull(cell)	2019-11-22 23:30:50	{"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": 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": 0.2654063403606415, "perplexity": 1741.3895237763265}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496672170.93/warc/CC-MAIN-20191122222322-20191123011322-00111.warc.gz"}
http://stackoverflow.com/questions/3663331/creating-a-service-with-sc-exe-how-to-pass-in-context-parameters	# creating a service with sc.exe; how to pass in context parameters When using sc create ServiceName binPath= "the path", how can arguments be passed to the Installer class's Context.Parameters collection? My reading of the sc.exe documentation is that such arguments could only be passed on the end of binPath, but I have not found an example or been able to successfully do this. - A glance at the Services key in the Registry suggests that any needed parameters are included with the ImagePath value, so your binPath= "c:\abc\def.exe /Param1=ghi" seems like the right idea. Do the backslashes need to be escaped (i.e. "c:\\abc\\...")? Worst thing, you could directly edit the Registry value afterwards if SC.EXE can't do it. – ewall Sep 8 '10 at 14:37 I gave up on sc.exe and am using installutil.exe like so: Installutil.exe /ServiceName=”TheName” /targetdir=”C:\TheInstallDirectory\” /PackageRoot=”PackageRootPath” – sympatric greg Sep 8 '10 at 16:45 I used Installutil.exe and for older technology I use Instsrv.exe from Windows XP/2003 Resource Ket. – Gary Kindel Feb 25 '11 at 18:58 sc create <servicename> binpath= "<pathtobinaryexecutable>" [option1] [option2] [optionN] The trick is to leave a space after the = in your create statement, and also to use " " for anything containing special characters or spaces. It is advisable to specify a Display Name for the service as well as setting the start setting to auto so that it starts automatically. You can do this by specifying DisplayName=yourdisplayname and start=auto in your create statement. Here is an example: C:\Documents and Settings\Administrator> sc create asperacentral binPath= "C:\Program Files\Aspera\Enterprise Server\bin\Debug\asperacentral.exe" DisplayName= "Aspera Central" start= auto If this worked you should see: [SC] CreateService SUCCESS UPDATE 1 http://support.microsoft.com/kb/251192 - marking as the answer, though I haven't verified this. – sympatric greg May 7 '12 at 0:58 Keep in mind that the space after binPath= (binPath= "C:\...") needs to be present, or else this won't work. – Onion-Knight Apr 22 '13 at 14:48 @Onion-Knight The space was exactly what I was missing. Thanks a lot – michel Jun 14 '13 at 13:21 start= auto is an important one, so after reboot the service will be automatically started. Very good in case the end user is not an expert – user193655 Jan 29 '14 at 12:54 Parameters for created services have some peculiar formating issues, in particular if the command includes spaces or quotes: If you want to enter command line parameters for the service, you have to enclose the whole command line in quotes. (And always leave a space after binPath= and before the first quote, as mrswadge pointed out) So, to create a service for the command PATH\COMMAND.EXE --param1=xyz you would use the following binPath parameter: binPath= "PATH\COMMAND.EXE --param1=xyz" ^^ ^ \|\| \| space quote quote If the path to the executable contains spaces, you have to enclose the path in quotes. So for a command that has both parameters and a path with spaces, you need nested quotes. You have to escape the inner quotes with backslashes \". The same holds if the parameters themselves contain quotes, you will need to escape those too. Despite using backslashes as escape characters, you do not have to escape the regular backslashes contained in the path. This is contrary to how you normally use backslashes as escape characters. So for a command like "PATH WITH SPACES \COMMAND.EXE" --param-with-quotes="a b c" --param2: binPath= "\"PATH WITH SPACES \COMMAND.EXE\" --param-with-quotes=\"a b c\" --param2" ^ ^ ^ ^ ^ ^ ^ \| \| \| \| \| \| \| opening escaped regular escaped escaped closing quote quote backslash closing quotes quote for for in quote for for whole path path for path parameter whole command command Here is a concrete example from the SVNserve documentation, which shows all special cases: sc create svnserve binpath= "\"C:\Program Files\CollabNet Subversion Server\svnserve.exe\" --service -r \"C:\my repositories\" " displayname= "Subversion Server" depend= Tcpip start= auto This would add a new service with the command line "C:\Program Files\CollabNet Subversion Server\svnserve.exe" --service -r "C:\my repositories". ## So in summary • space after each sc parameter: binpath=_, displayname=_ and depend=_ • each sc parameter that contains spaces must be enclosed in quotes • all additional quotes inside the binpath are escaped with backslashes: \" • all backslashes inside the binpath are not escaped - I found it was important to ensure there is a space between binPath= and the value "myservice.exe". i.e. binPath= "myservice.exe. The command line interpreter must be expecting this and requiring the command to become tokenized by using space as the delimiter. – mrswadge Mar 20 '13 at 9:24 I use to just create it without parameters, and then edit the registry HKLM\System\CurrentControlSet\Services\[YourService]. - I had issues getting this to work on Windows 7. It seemed to ignore the first argument I passed in so I used binPath= "C:\path\to\service.exe -bogusarg -realarg1 -realarg2" and it worked. - This command works: sc create startSvn binPath= "\"C:\Subversion\bin\svnserve.exe\" --service -r \"C:\SVN_Repository\"" displayname= "MyServer" depend= tcpip start= auto - Be sure to have quotes at beginning and end of your binPath value. - Given a path "c:\abc\def.exe", I tried to pass in Param1="ghi" like this: binPath= "c:\abc\def.exe /Param1=ghi". But no worky... – sympatric greg Sep 8 '10 at 0:29 I couldn't handle the issue with your proposals, at the end with the x86 folder it only worked in power shell (windows server 2012) using environment variables: {sc.exe create svnserve binpath= "\${env:programfiles(x86)}/subversion/bin/svnserve.exe --service -r C:/svnrepositories/" displayname= "Subversion Server" depend= Tcpip start= auto} - I found a way to use sc. sc config binPath= "\"c:\path with spaces in it\service_executable.exe\" " In other words, use \ to escape any "'s you want to survive the transit into the registry. -	2015-01-31 09:00:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.6464272141456604, "perplexity": 10515.887960592741}, "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-2015-06/segments/1422118108509.48/warc/CC-MAIN-20150124164828-00239-ip-10-180-212-252.ec2.internal.warc.gz"}
https://learn.careers360.com/ncert/question-the-following-is-the-distance-time-table-of-an-object-in-motion-a/	Q A1. The following is the distance-time table of an object in motion: Time in seconds Distance in metre 0 0 1 1 2 8 3 27 4 64 5 125 6 216 7 343 (a) What conclusion can you draw about the acceleration? Is it constant, increasing, decreasing, or zero? From the table, the relation between time and the distance can be seen. $s\ =\ t^3$ Thus the velocity of the particle is increasing with time. $v\ =\ \frac{ds}{dt}\ =\ 3t^2$ and $a\ =\ \frac{dv}{dt}\ =\ 6t$ Hence acceleration increases linearly with time. Related Chapters Preparation Products JEE Main Rank Booster 2021 This course will help student to be better prepared and study in the right direction for JEE Main.. ₹ 13999/- ₹ 9999/- Rank Booster NEET 2021 This course will help student to be better prepared and study in the right direction for NEET.. ₹ 13999/- ₹ 9999/- Knockout JEE Main April 2021 (Easy Installments) An exhaustive E-learning program for the complete preparation of JEE Main.. ₹ 4999/- Knockout NEET May 2021 An exhaustive E-learning program for the complete preparation of NEET.. ₹ 22999/- ₹ 14999/-	2020-10-20 23:30:26	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8090801239013672, "perplexity": 4288.19361910775}, "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-2020-45/segments/1603107874340.10/warc/CC-MAIN-20201020221156-20201021011156-00195.warc.gz"}
https://www.semanticscholar.org/paper/Functional-Inequalities-for-Heavy-Tailed-and-to-Cattiaux-Gozlan/7641a392bd8afec126e13c7928607307f48a1565	# Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry @article{Cattiaux2008FunctionalIF, title={Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry}, author={Patrick Cattiaux and Nathael Gozlan and Arnaud Guillin and Cyril Roberto}, journal={Electronic Journal of Probability}, year={2008}, volume={15}, pages={346-385} } • Published 19 July 2008 • Mathematics • Electronic Journal of Probability This paper is devoted to the study of probability measures with heavy tails. Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as weak Poincare and weak Cheeger, weighted Poincare and weighted Cheeger inequalities and their dual forms. Proofs are short and we cover very large situations. For product measures on $\mathbb{R}^n$ we obtain the optimal dimension dependence using the mass transportation method. Then we derive… ### Weak Poincaré inequalities for convolution probabilities measures • Mathematics Infinite Dimensional Analysis, Quantum Probability and Related Topics • 2019 In this paper, weak Poincaré inequalities are obtained for convolution probabilities with explicit rate functions by constructing suitable Lyapunov functions. Here, one of these Lyapunov functions is ### Weighted Poincaré inequalities, concentration inequalities and tail bounds related to Stein kernels in dimension one We investigate links between the so-called Stein's density approach in dimension one and some functional and concentration inequalities. We show that measures having a finite first moment and a ### Fokker-Planck equations and one-dimensional functional inequalities for heavy tailed densities • Mathematics • 2020 We study one-dimensional functional inequalities of the type of Poincare, logarithmic Sobolev and Wirtinger, with weight, for probability densities with polynomial tails. As main examples, we obtain ### ON SPECTRAL GAP AND WEIGHTED POINCARÉ INEQUALITIES FOR SOME ONE-DIMENSIONAL DIFFUSIONS • Mathematics • 2015 We present some classical and weighted Poincaré inequalities for some one-dimensional probability measures. This work is the one-dimensional counterpart of a recent study achieved by the authors for ### Variations and extensions of the Gaussian concentration inequality, Part I We use and modify the Gaussian concentration inequality to prove a variety of concentration inequalities for a wide class of functions and measures on $\mathbb{R}^{n}$, typically involving ### On the Poincaré Constant of Log-Concave Measures • Mathematics Lecture Notes in Mathematics • 2020 The goal of this paper is to push forward the study of those properties of log-concave measures that help to estimate their Poincare constant. First we revisit E. Milman's result [40] on the link ### A NOTE ON SPECTRAL GAP AND WEIGHTED POINCAR ´ E INEQUALITIES FOR SOME ONE-DIMENSIONAL DIFFUSIONS • Mathematics • 2014 We present some classical and weighted Poincar\'e inequalities for some one-dimensional probability measures. This work is the one-dimensional counterpart of a recent study achieved by the authors ### Weak Poincaré Inequalities in the Absence of Spectral Gaps • Mathematics Annales Henri Poincaré • 2019 For generators of Markov semigroups which lack a spectral gap, it is shown how bounds on the density of states near zero lead to a so-called weak Poincaré inequality (WPI), originally introduced by ### Bernstein type's concentration inequalities for symmetric Markov processes • Mathematics • 2010 Using the method of transportation-information inequality introduced in [A. Guillin et al., Probab. Theory Related Fields, 144 (2009), pp. 669--695], we establish Bernstein-type concentration ## References SHOWING 1-10 OF 113 REFERENCES ### Weighted poincaré-type inequalities for cauchy and other convex measures • Mathematics • 2009 Brascamp-Lieb-type, weighted Poincare-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κ-concave probability measures (in the hierarchy of ### Poincaré inequalities and dimension free concentration of measure In this paper, we consider Poincare inequalities for non euclidean metrics on $\mathbb{R}^d$. These inequalities enable us to derive precise dimension free concentration inequalities for product ### Distributions with Slow Tails and Ergodicity of Markov Semigroups in Infinite Dimensions • Mathematics • 2010 We discuss some geometric and analytic properties of probability distributions that are related to the concept of weak Poincare type inequalities. We deal with isoperimetric and capacitary ### Weak Poincaré Inequalities and L2-Convergence Rates of Markov Semigroups • Mathematics • 2001 Abstract In order to describe L 2 -convergence rates slower than exponential, the weak Poincare inequality is introduced. It is shown that the convergence rate of a Markov semigroup and the ### Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures We discuss an approach, based on the Brunn–Minkowski inequality, to isoperimetric and analytic inequalities for probability measures on Euclidean space with logarithmically concave densities. In ### Entropy Bounds and Isoperimetry • Mathematics • 2005 Introduction and notations Poincare-type inequalities Entropy and Orlicz spaces $\mathbf{LS}_q$ and Hardy-type inequalities on the line Probability measures satisfying $\mathbf{LS}_q$-inequalities on ### Lyapunov conditions for logarithmic Sobolev and Super Poincar\'e inequality • Mathematics • 2007 We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincar\'e inequality (for instance logarithmic Sobolev or $F$-Sobolev). The case of Poincar\'e and ### Sobolev inequalities for probability measures on the real line • Mathematics • 2003 We give a characterization of those probability measures on the real line which satisfy certain Sobolev inequalities. Our starting point is a simpler approach to the Bobkov–Götze characterization of	2022-10-07 10:34:11	{"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": 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": 0.7970321774482727, "perplexity": 1421.1589002064977}, "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/1664030338001.99/warc/CC-MAIN-20221007080917-20221007110917-00067.warc.gz"}
https://kixibuboce.memoriesbythesmile.com/table-of-the-incomplete-beta-function-beta-book-32323ek.php	Last edited by Maubar Monday, November 23, 2020 \| History 3 edition of Table of the Incomplete Beta Function Beta (One Half, P,Q). found in the catalog. Table of the Incomplete Beta Function Beta (One Half, P,Q). United States. Bureau of Mines. # Table of the Incomplete Beta Function Beta (One Half, P,Q). Written in English Edition Notes 1 ID Numbers Series Report of investigations (United States. Bureau of Mines) -- 4961 Contributions Smith R., Edwards, H. Open Library OL21747820M I need to use beta distribution and inverse beta distribution in my project.. There is quite good but complicated implementation in GSL, but I don't want to use such a big library only to get one function.. I would like to either, implement it on my own or link some . The incomplete beta function, which is not a built-in function, but can be trivially computed as the product of the previous two functions: I(x,a,b) = Β(a,b)CDF(“Beta”,x,a,b) A simple example. Suppose that you want to compute and plot the incomplete beta function for the parameters a=2 and b=3. The following SAS/IML statements compute the. Lower Incomplete Beta Function Calculator. This calculator will compute the lower incomplete beta function (i.e., the area under the beta function from 0 to x), given values of the shape parameters a and b, and the upper limit of integration x. Please enter the . You might also like Population growth and levels of consumption Population growth and levels of consumption University of St Andrews O.T.C. University of St Andrews O.T.C. Flashlight Flashlight glossary of Sanskrit from the spiritual tradition of India glossary of Sanskrit from the spiritual tradition of India Mid year review of the economy, 1983 Mid year review of the economy, 1983 Cultural conflict in the Canadian Arctic Cultural conflict in the Canadian Arctic Peer perceptions of hyperactive children Peer perceptions of hyperactive children Elementary air survey Elementary air survey European skyways European skyways Bylaugh Bylaugh Cashing in on pork Cashing in on pork Mesa Springs sentinel Mesa Springs sentinel Elegies for the Postmodern American Poets Elegies for the Postmodern American Poets ### Table of the Incomplete Beta Function Beta (One Half, P,Q). by United States. Bureau of Mines. Download PDF EPUB FB2 Tables of the Incomplete Beta Function. Pearson, Karl. Publication date. Collection. thecomputermuseumarchive; americana. Digitizing sponsor. Gordon Bell. Tables of the Incomplete Beta-Function.: Karl Pearson (ed.): : Books. Currently unavailable. We don't know when or if this item will be back in stock. Listen Playing Paused You're listening to a sample of the Audible audio edition. Tables of the Incomplete Beta-Function. Hardcover – January 1, Enter your mobile number or email address below and we'll send you a link to Author: Karl Pearson (ed.). Table of the incomplete beta function [beta] (1/2,p, q) by Robert W. Smith, Jr. and Helen E. : Smith, Robert W. Tables of the Incomplete Beta-Function, 2nd Edition Karl Pearson; E. Pearson (introduction); N. Johnson (introduction) Published by Cambridge University Press (). Tables of the incomplete Beta-function by Karl Pearson,Printed at the University Press and Published by the Proprietors of Biometrika edition, in EnglishPages: [1] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions", Dover, reprint () [2] K. Pearson, "Tables of the incomplete beta-function", Cambridge Univ. The incomplete beta function, Bx(p, q) = / yp~l(l — y)q~ldy, and. its ratio to the complete beta function Bi(p, q) has been calculated on an IBM computer to five significant figures and tabulated for the arguments p and q each. in the interval from to with increments of and for the parameter x in the interval to with increments of An illustration of an open book. Books. An illustration of two cells of a film strip. Video. An illustration of an audio speaker. Audio. An illustration of a " floppy disk. Software. An illustration of two photographs. Full text of "Tables of the Incomplete Beta Function". Calculates a table of the Incomplete beta function Bx(a,b) and regularized beta function Ix(a,b) and draws the chart. The Efficient Calculation of the Incomplete Beta-Function Ratio for Half-Integer Values of the Parameters a, b By A. Table of the Incomplete Beta Function Beta book and M. Jarnagin 1. Introduction. The incomplete beta function is defined as follows: (1) Bxia, b) = / f-lil - tf^dt, where Ogigl, a> 0, b> 0. Keywords: basic properties, incomplete beta functions Notes: The material in this subsection was added in Version It will be incorporated in the next print edition. Permalink. Incomplete Beta Functions. The generalized form of beta function is called incomplete beta function. It is given by the relation: $$B (z:a,b)= \int_{0}^{z} t^{a-1}(1-t)^{b-1}dt$$ It is also denoted by B z (a, b). We may notice that when z = 1, the incomplete beta function becomes the beta function. i.e. B(1:. Tables of the incomplete Beta-function by Karl Pearson,Cambridge University Press edition, in English - 2nd ed. / with a new introduction by E. Pearson and N. Johnson. Tables of the incomplete beta function ( edition) \| Open Library. Incomplete Beta Derivatives 17 Table 2 Maximum likelihood estimates of the beta parameters from the k smallest order statistics when p =q =and n = k k/n 2 Gnanadesikan et al Current ˆ α ˆ β ln(L) α ˆ βˆ ln(L) OCLC Number: Notes: "First published " - tp. Description: lix, pages illustrations ; 29 cm: Other Titles: Incomplete Beta-functions. Beta function. The gamma and the beta function As mentioned in the book [1], see page 6, the integral representation () is often taken as a de nition for the gamma function (z). The advantage of this alternative de nition is that we might avoid the use of in nite products (see appendix A). De nition 1. Beta function. by Marco Taboga, PhD. The Beta function is a function of two variables that is often found in probability theory and mathematical statistics (for example, as a normalizing constant in the probability density functions of the F distribution and of the Student's t distribution).We report here some basic facts about the Beta function. The Beta function is defined as the ratio of Gamma functions, written below. Its derivation in this standard integral form can be found in part 1. The Beta function in its other forms will be derived in parts 4 and 5 of this article. common name, the Beta function. The use of the Beta symbol for this function was ﬁrst used in by Jacques P.M. Binet ( - ). At the same time as Legendre and Gauss, Cristian Kramp ( - ) worked on the generalized factorial function as it. : Tables of the Incomplete Beta Function (): Karl Pearson, E. Pearson (introduction), N. Johnson (introduction): BooksFormat: Hardcover. The incomplete beta function B x (a, b) is defined by (1) B x (a, b) = ∫ 0 x t a − 1 (1 − t) b − 1 d t, a, b > 0; 0. In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial is defined by the integral (,) = ∫ − (−) −for complex number inputs x, y such that Re x > 0, Re y > The beta function was studied by Euler and Legendre and was given its name by Jacques Binet; its. I’ll start with some known definitions and relations which will be useful in this answer. The gamma functions is given by the integral: ${\displaystyle \Gamma (z)=\int _{0}^{\infty }x^{z-1}e^{-x}\,dx}$ The beta function is given by: [. Tables of the incomplete beta-function. [Karl Pearson] Home. WorldCat Home About WorldCat Help. Search. Search for Library Items Search for Lists Search for Contacts Search for a Library. Create Book\/a>, schema:CreativeWork\/a> ; \u00A0\u00A0\u00A0 library. Gamma, Beta, Erf: BetaRegularized[z,a,b] ( formulas) Primary definition (4 formulas) Specific values (8 formulas) General characteristics (21 formulas) Series representations (65 formulas) Representations through more general functions (16 formulas). generalized extended incomplete beta function to obtained the various integral representations and some properties. 1 INTRODUCTION The classical incomplete beta function is defined by [3, 5, 6, 7, 9] B (,) t 1(1 t) 1dt z 0 x (> 0, > 0 and 0 function. This is the same as that for the Incomplete_Beta_Function() except that the arguments are of type long double and a long double is returned. Source Code C source code is available for these routines: The file, incomplete_beta_function.c, contains the functions Incomplete_Beta_Function() and xIncomplete_Beta_Function(). The incomplete Beta function is defined by the Beta integral B(x;a,b) = integral_0^x t^(a-1) (1-t)^(b-1) dt Value. Ibeta returns the incomplete Beta function with parameters (a,b) evaluated at point x. returns the point x at which the incomplete Beta function with parameters (a,b) evaluates to y. See Also. Cgamma, Igamma, Rgamma. The incomplete beta function is also sometimes defined without the gamma terms, in which case the above definition is the so-called regularized incomplete beta function. Under this definition, you can get the incomplete beta function by multiplying the result of the SciPy function by beta. References. In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which are defined similarly to the gamma function but with different or "incomplete" integral limits. The gamma function is defined as an integral from. Table of Contents. Abstract; PDF Title Information. Published: ISBN: eISBN: Book Code: CL Series: Classics in Applied Mathematics. Pages: 4. Buy the Print Edition. Tables of the Incomplete Beta Function. Table of Contents. J.1 Tables of the incomplete beta function I x (w + 1, w) J.2 Tables of. Beta[a, b] gives the Euler beta function \[CapitalBeta] (a, b). Beta[z, a, b] gives the incomplete beta function \[CapitalBeta]z (a, b). Incomplete Beta Function Calculator. Calculate the incomplete beta value for the given two real numbers and the upper limit of integration using this incomplete beta function calculator. The integration value x should be between 0 and 1. Description. I = betainc(X,Z,W) computes the incomplete beta function for corresponding elements of the arrays X, Z, and elements of X must be in the closed interval [0,1]. The arrays Z and W must be nonnegative and real. All arrays must be the same size, or any of them can be scalar. I = betainc(X,Z,W,tail) specifies the tail of the incomplete beta function. Derivatives of the Incomplete Beta Function Keywords: Censored beta; Continued fractions; Truncated beta; Truncated beta-binomial Languages FORTRAN77,MATLAB,andS{PLUS Description and Purpose Theincompletebetafunctionisde nedas I x;p;q= Z x 0 up−1(1−u)q−1 Beta(p;q) du; whereBeta(p;q)()gaveahistoryofthe. Properties of Beta Function B(x,y) = B(x,y+1) + B(x+1,y) xB(x,y +1) =y B(x+1,y) Gamma function The Eulerian integral,n>0 is called gamma function and is denoted by Example: Recurrence formulae for gamma function. Relation between gamma and factorial Other results. Relation between beta and gamma function. Beta and gamma are the two most popular functions in mathematics. Gamma is a single variable function, whereas Beta is a two-variable function. The relation between beta and gamma function will help to solve many problems in physics and mathematics. is the new extended incomplete beta function. For, we must have in for convergence, and, where is the incomplete beta function defined as. It is to be noted that the problem of expressing in terms of other special functions remains open. Presumably, this distribution should be useful in extending the statistical results for strictly positive. There is a claim in my book that there is a connection to the Beta CDF and a Binomial Summation without explaining further. This is basically because we can express the distribution function of a Binomial random variable in terms of the incomplete beta function, which in turn is related to the distribution function of a Beta distributed. The beta function (also known as Euler's integral of the first kind) is important in calculus and analysis due to its close connection to the gamma function, which is itself a generalization of the factorial function. Many complex integrals can be reduced to expressions involving the beta function. The recurrence relation of the beta function is given by. The gamma function evalated at = 1 2 is 1 2 = p ˇ: (4) The recursive relationship in (2) can be used to compute the value of the gamma function of all real numbers (except the nonpositive integers) by knowing only the value of the gamma function between 1 and 2. Table 2 contains the gamma function for arguments between 1 and To.There are four incomplete beta functions: two are normalised versions (also known as regularized beta functions) that return values in the range [0, 1], and two are non-normalised and return values in the range [0, beta(a, b)].Book table of contents. About ePub3. where the integral can be recognized as the beta function, which has the property: so that. for c – a compared with a known series for the given function. Table in Lakshminarayanan and Varadarajan gives a list of various special functions in terms of the hypergeometric functions, including.	2021-10-27 11:03:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8509055376052856, "perplexity": 1345.4111174941202}, "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-2021-43/segments/1634323588113.25/warc/CC-MAIN-20211027084718-20211027114718-00072.warc.gz"}
http://theoryapp.com/update-two-tables-with-joins-in-mysql/	# Update Two Tables with Joins in MySQL Here we show an example of updating two table at the same time using joins. Suppose we have two tables: lookup which holds information on a person, and activity which holds certain activities happening on a person. Imagine that lookup is stable, while activity is updated frequently, and constantly we need to synchronize the information in two tables. The following SQL set up the tables. CREATE TABLE lookup (id INT PRIMARY KEY, name VARCHAR(64), last_use TIMESTAMP DEFAULT CURRENT_TIMESTAMP); INSERT INTO lookup (id, name) VALUES (1, 'Alice'), (2, 'Bob'), (3, 'Carl'), (4, 'Eva'); CREATE TABLE activity (id INT, name VARCHAR(64), info TEXT); INSERT INTO activity (id, info) VALUES (1, 'Alice Info'), (2, 'Bob Info'), (3, 'Carl Info'); Now we would like to update activity by filling up the names; simultaneously, we wish to update lookup by setting the column last_use to the current timestamp. The solution is given below. There are two tables in this UPDATE statement; the two tables are joined on the column id. UPDATE lookup, activity SET lookup.last_use = CURRENT_TIMESTAMP, activity.name = lookup.name WHERE activity.id = lookup.id; The result of this statement shows 6 rows were updated, 3 from lookup and 3 from activity.	2017-10-18 09:25:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2027975469827652, "perplexity": 9136.602918124596}, "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-20171018085500-20171018105500-00522.warc.gz"}
https://cstheory.stackexchange.com/questions/21451/a-tool-for-minimal-nfa-computation	# A tool for minimal NFA computation It is well known that minimizing an NFA for a fixed regular language is $PSPACE-Complete$. As far as I know, there are no better than trivial algorithms for minimizing such NFA, but there's a little improvement if you consider symmetries. I've a specific regular language I'd like to compute a minimal automaton for: $$L_{k-distinct} :=\{w = \sigma_1\sigma_2...\sigma_k \mid \forall i\in[k]: \sigma_i\in\Sigma ~\text{ and }~ \forall j\ne i: \sigma_j\ne\sigma_i \}$$ But at the moment I can't seem to close the gap between the automaton I know to build for it and the lower bound I can prove for it. I thought it might be fruitful to use some tool that given a language (it is finite for all $k,n$), searches (exhaustively if needed) for the smallest automaton which accept it, and see what the automaton looks like for small values of $k,n$. Does anyone know a tool which builds a minimal automaton for a given language? The following paper reports on an implementation of the Kameda-Weiner algorithm for computing a minimal NFA, as well on an approach using a SAT solver. I don't know whether the implementation is available, but perhaps you can contact the authors about this. Jaco Geldenhuys, Brink van der Merwe, and Lynette van Zijl. Reducing Nondeterministic Finite Automata with SAT Solvers. Revised Selected Papers from the 8th International Workshop on Finite-State Methods and Natural Language Processing (FSMNLP 2009), LNCS 6062, Springer, pages 81-92, 2010. There is an elementary argument showing that a minimal NFA must have $O(\|\Sigma\|^k)$ states, so I guess the standard construction is essentially optimal. The argument is as follows. Suppose w.l.o.g. that $A$ is a NFA without $\epsilon$-transitions recognizing $L_k$. We can make the following assumptions: • $A$ has a unique starting state $q_0$ and each state recognizes a non-empty language; • $A$ is acyclic, and contains $k+1$ levels where each state at level $i$ only recognizes words in $L_i$. It can be seen that if two words of $L_i$ have a different set of letters, then they cannot lead to the same state starting from $q_0$. It follows that the number of states at level $i$ is at least $\binom{\|\Sigma\|}{i}$, and thus the total number of states is at least $\sum_{i = 0}^{k} \binom{\|\Sigma\|}{i} = O(\|\Sigma\|^k)$. • This is not correct. See the link in the question for a not trivial automaton of size: $O((2e)^{k\cdot log^3(n)}\cdot poly(n))$. – R B Mar 9 '14 at 20:47 • I see, but your solution requires preprocessing as you first apply a hash function. This kind of computation cannot be performed with a NFA (maybe with a transducer?) so could you please make your question more specific? – Super9 Mar 9 '14 at 20:51 • This is not preproccesing needed to be done by the NFA, this is a way of computing the NFA / showing such NFA exists. The NFA, by itself is a standard NFA. My question is whether there exists a tool that gets as in input a set of words and computes the minimal NFA that accepts them (and just them). – R B Mar 9 '14 at 20:53	2019-07-23 11:11:47	{"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": 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": 0.7433980703353882, "perplexity": 161.12853143858445}, "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/1563195529276.65/warc/CC-MAIN-20190723105707-20190723131707-00244.warc.gz"}
https://mathematica.stackexchange.com/questions/137881/how-to-apply-geometric-transformation-and-do-test-for-region-intersection	# How to apply geometric transformation and do test for region intersection ## Description I want to generate a number of ellipses, rotate each ellipse using random angle, and then test each ellipse if it intersects with any other regions within arbitrary geometry. I can create my ellipses, and rotate each ellipse using random angle. However, the methods I use don't seem to be compatible when used inside RegionIntersection How could I generate, rotate and test the rotated ellipses to check for intersections with other members of the geometric composition? ## Code Module[ {data,r1, r2}, (Variable declaration) data = RandomInteger[{1,100},{100,2}]; (Program) (Visual Representation) Show[ Graphics @ {FaceForm @ None, EdgeForm @ Red,r1[data[[1]],2.5,Pi/6]}, Graphics @ {FaceForm @ None, EdgeForm @ Blue,r2[data[[1]],2.5,Pi/4]}, Graphics @ {Opacity @ .5,FaceForm @ Gray, Disk[data[[1]],1]}, Axis-> True, Frame-> True ] ] Following are the lines which would crash my code due to ,what I suspect, argument incompatibility (Intersection [Fail]) RegionIntersection[r1,Disk[data[[1]],1]], RegionIntersection[r2,Disk[data[[1]],1]] • – Carl Woll Feb 16 '17 at 1:25 • @CarlWoll, thank you for your reply. I think the method works. I am happy to accept it as an answer if you'd like to post one below? – e.doroskevic Feb 16 '17 at 10:31 The problem is that GeometricTransformation doesn't produce an object that is RegionQ: GeometricTransformation[ Disk[{0, 0}, {1, 2}], RotationTransform[Pi/2, {0, 0}] ] //RegionQ False Instead of GeometricTransformation, you should use TransformedRegion: r1 = TransformedRegion[ Disk[{0, 0}, {1, 2}], RotationTransform[Pi/2, {0, 0}] ]; r1 //RegionQ True Using RegionIntersection with r1: int = RegionIntersection[r1, Disk[{1, 1}, 1]]; DiscretizeRegion[int] RegionMeasure[int]//N 1.20138 • Yeah I figured it doesn't return True when tested with RegionQ hence the incompatibility error. I was unaware of TransformedRegion function. Thank you for providing an answer to my question. I learned something new :) (y) – e.doroskevic Feb 16 '17 at 17:06 • But this is all a little broken. GeometricTransformation can (and should, where possible) produce RegionQ objects. For example, Normal@GeometricTransformation[Cylinder[], ScalingTransform[{2, 2, 2}]]. But this does not work properly in 2D (reported bug). – TheDoctor Aug 14 '17 at 4:24	2020-04-04 03:29:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.28088152408599854, "perplexity": 4190.340031850882}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370519111.47/warc/CC-MAIN-20200404011558-20200404041558-00485.warc.gz"}
http://docserver.carma.newcastle.edu.au/169/	# Roots of $\{0,+1,-1\}$ Polynomials. Borwein, Peter and Pinner, Christopher (1996) Roots of $\{0,+1,-1\}$ Polynomials. [Preprint] Preview Postscript Preview PDF ## Abstract For a fixed $\alpha$ we discuss how closely $\alpha$ can be approximated by a root of a $\{0,+1,-1\}$ polynomial of given degree. We show that the worst rate of approximation tends to occur for roots of unity, particularly those of small degree. For roots of unity these bounds depend on the order of vanishing, $k$, of the polynomial at $\alpha$. In particular we obtain the following. Let ${\cal B}_{N}$ denote the set of roots of all $\{0,+1,-1\}$ polynomials of degree at most $N$ and ${\cal B}_{N}(\alpha,k)$ the roots of those polynomials that have a root of order at most $k$ at $\alpha$. For a Pisot number $\alpha$ in $(1,2]$ we show that $\min_{\beta \in {\cal B}_{N}\setminus \{ \alpha \}} \|\alpha -\beta\| \asymp \frac{1}{\alpha^{N}},$ and for a root of unity $\alpha$ that $\min_{\beta \in {\cal B}_{N}(\alpha,k)\setminus \{\alpha\}} \|\alpha-\beta\|\a symp \frac{1}{N^{(k+1) \left\lceil \frac{1}{2}\phi(d)\right\rceil +1}}.$ We study in detail the case of $\alpha=1$, where, by far, the best approximations are real. We give fairly precise bounds on the closest real root to 1. When $k=0$ or 1 we can describe the extremal polynomials explicitly. Item Type: Preprint pubdom FALSE Mahler measure, zero-one polynomials, Pisot numbers, root separation 30-xx Functions of a complex variable > 30Cxx Geometric function theory11-xx Number theory > 11Jxx Diophantine approximation, transcendental number theory UNSPECIFIED Users 1 not found. 24 Nov 2003 21 Apr 2010 11:13 https://docserver.carma.newcastle.edu.au/id/eprint/169	2017-09-20 20:13:05	{"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": 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": 0.9148998856544495, "perplexity": 407.69540848327273}, "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-2017-39/segments/1505818687447.54/warc/CC-MAIN-20170920194628-20170920214628-00430.warc.gz"}
https://www.physicsforums.com/threads/time-complexity-big-oh.741793/	# Time complexity(Big oh) 1. Mar 5, 2014 ### lee534 Hi this is my first post and I having an extremely hard time finding time complexity in code note: I have had no compsci experience (since it was career change) and the professor only went over this for 30 mins 1. The problem statement, all variables and given/known data What is the time complexity (in Θ –notation) in terms of n ? (the red is what is throwing me off) a.sum = 0 ; for ( i = 0 ; i < n ; i++ ) for ( j = 1 ; j < n3 ; j = 3j ) sum++ ; b.sum = 0 ; for ( i = n ; i > 0; i = i/3 ) for ( j = 0 ; j < n3 ; j++ ) sum++ ; 2. Relevant equations none 3. The attempt at a solution a. assignments: sum = 0, i = 0, j = 1, j=3j : (1+1+3n) conditions: j<n3, i<n: 3(n+1+n(n3+1) increments:i++,sum++; n+n3 O(n4) b.(this one I'm taking a big educated guess) assignments: sum = 0, i = n, i = i/3, j=0: 1+1+n+n/3 conditions: j<n3, i>0: n/3+1+n(n3+1) increments:j++,sum++; nn3+nn3 O(n4) and if you can explain to me as if I'm a child that would be great! Thank you Last edited: Mar 5, 2014 2. Mar 5, 2014 ### jackarms Careful, the loops for (a) are misleading. You have the right idea for the outside loop -- it'll all just be n. But for the inside, essentially what you do is multiply whatever time complexity you have for the inner loop by the complexity of the outer loop (it looks like you were adding them). An easier case for this is two for loops that are both $O(n)$. The inner loop goes for $O(n)$ $n$ times, so that becomes $n \cdot O(n)$, or $O(n^{2})$. Also for (a), consider what happens when j goes up to n3, but each iteration you multiply j by three. How do you think these effects will compare? Try writing out a list of values j will take on. For (b), the code in red can be difficult to analyze. If it were instead multiplying by 3 each time, this would be exponential, right? So if you're dividing, then what's the opposite of an exponential function? 3. Mar 6, 2014 ### lee534 So is it easier to analyze it by loops? Because I found this method online and the professor did not teach how us to read code as you are describing (he was also using a different format from this problem, but I'm guessing it's analyzed the same?). okay so for a. can we look at it this way? a. assignments: sum = 0, i = 0, j = 1, j=3j : (1+1+3n) → O(n) conditions: j<n3, i<n: n+(n)(3n)3 → O(n4) increments:i++,sum++; n+n(n3) O(n4) or a.sum = 0 ; for ( i = 0 ; i < n ; i++ ) → O(n) for ( j = 1 ; j < n3 ; j = 3j ) →n(3n)3→O(n4) sum++ ; O(n) + O(n4) → O(n4) b. b.sum = 0 ; for ( i = n ; i > 0; i = i/3 ) → 3√n → O(n1/3) for ( j = 0 ; j < n3 ; j++ ) → (n1/3)(n3) → n sum++ ; O(n) thank you for helping me understand Last edited: Mar 6, 2014	2017-11-21 01:35:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7702256441116333, "perplexity": 1126.2969507134226}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806309.83/warc/CC-MAIN-20171121002016-20171121022016-00393.warc.gz"}
https://www.biogeosciences.net/15/2205/2018/	Journal cover Journal topic Biogeosciences An interactive open-access journal of the European Geosciences Union Journal topic Biogeosciences, 15, 2205–2218, 2018 https://doi.org/10.5194/bg-15-2205-2018 Biogeosciences, 15, 2205–2218, 2018 https://doi.org/10.5194/bg-15-2205-2018 Research article 16 Apr 2018 Research article \| 16 Apr 2018 # Impact of salinity on element incorporation in two benthic foraminiferal species with contrasting magnesium contents Impact of salinity on element incorporation in two benthic foraminiferal species with contrasting magnesium contents Esmee Geerken1, Lennart Jan de Nooijer1, Inge van Dijk1,a, and Gert-Jan Reichart1,2 Esmee Geerken et al. • 1Department of Ocean Systems, NIOZ-Royal Netherlands Institute for Sea Research, and Utrecht University, Den Burg, the Netherlands • 2Faculty of Geosciences, Utrecht University, Utrecht, the Netherlands • acurrently at: UMR CNRS 6112 LPG-BIAF, University of Angers, 49035 Angers, France Correspondence: Esmee Geerken (esmee.geerken@nioz.nl) Abstract Accurate reconstructions of seawater salinity could provide valuable constraints for studying past ocean circulation, the hydrological cycle and sea level change. Controlled growth experiments and field studies have shown the potential of foraminiferal Na Ca as a direct salinity proxy. Incorporation of minor and trace elements in foraminiferal shell carbonate varies, however, greatly between species and hence extrapolating calibrations to other species needs validation by additional (culturing) studies. Salinity is also known to impact other foraminiferal carbonate-based proxies, such as Mg Ca for temperature and Sr Ca for sea water carbonate chemistry. Better constraints on the role of salinity on these proxies will therefore improve their reliability. Using a controlled growth experiment spanning a salinity range of 20 units and analysis of element composition on single chambers using laser ablation-Q-ICP-MS, we show here that Na Ca correlates positively with salinity in two benthic foraminiferal species (Ammonia tepida and Amphistegina lessonii). The Na Ca values differ between the two species, with an approximately 2-fold higher Na Ca in A. lessonii than in A. tepida, coinciding with an offset in their Mg content ( 35 mmol mol−2 versus 2.5 mmol mol−1 for A. lessonii and A. tepida, respectively). Despite the offset in average Na Ca values, the slopes of the Na Ca–salinity regressions are similar between these two species (0.077 versus 0.064 mmol mol−1 change per salinity unit). In addition, Mg Ca and Sr Ca are positively correlated with salinity in cultured A. tepida but show no correlation with salinity for A. lessonii. Electron microprobe mapping of incorporated Na and Mg of the cultured specimens shows that within chamber walls of A. lessonii, Na Ca and Mg Ca occur in elevated bands in close proximity to the primary organic lining. Between species, Mg banding is relatively similar, even though Mg content is 10 times lower and that variation within the chamber wall is much less pronounced in A. tepida. In addition, Na banding is much less prominent in this species than it is in A. lessonii. Inter-species differences in element banding reported here are hypothesized to be caused by differences in biomineralization controls responsible for element uptake. 1 Introduction Sea water salinity varies over time and space as a function of continental ice volume, evaporation, precipitation and river runoff. Salinity reconstructions could provide important constraints on past ocean circulation, the hydrological cycle and glacial–interglacial sea level changes. Currently, most reconstructions of salinity are indirect and based on the correlation between salinity and δ18Owater, assuming this relationship to be constant over space and time (Rohling and Bigg, 1998). An independent salinity proxy may reduce the uncertainties inherently associated with such approaches (Rohling and Hilgen, 2007) and should preferably be based on one of the main components of sea water salinity, for instance sodium (Na). Results from a culture study showed that the sodium content of foraminiferal calcite (Na Cacc) correlates positively and linearly with salinity for the benthic low-Mg, symbiont-barren species Ammonia tepida, with a sensitivity of 0.22 mmol mol−1 for every change of 1 salinity unit between salinities 30 and 38.6 (Wit et al., 2013). Various culture studies earlier showed that also Mg Ca is affected by salinity but responds more strongly to changes in temperature (Lea et al., 1999; Dissard et al., 2010b; Nürnberg et al., 1996; Hönisch et al., 2013). Although an effect of salinity on foraminiferal Sr Cacc has been reported in some studies (Kısakürek et al., 2008; Dissard et al., 2010b; Wit et al., 2013), other studies did not find a relation between salinity and foraminiferal Sr Ca (Dueñas-Bohórquez et al., 2009; Diz et al., 2012; Allen et al., 2016), which led to the hypothesis that foraminiferal Sr Ca mainly reflects sea water inorganic carbon chemistry (Keul et al., 2017) in addition to its response to temperature (Lea et al., 1999; Raja et al., 2007). Hence, an independent salinity proxy would not only be useful for constraining past (changes in) salinity but would also improve temperature reconstructions based on Mg Cacc and reconstructions of past sea water carbonate chemistry based on Sr Ca. Following the culture-based Na Cacc–salinity calibration for A. tepida (Wit et al., 2013), a culture study with planktonic symbiont-bearing species also showed a significant linear relationship for Globigerinoides ruber (Allen et al., 2016). Although no significant relationship was observed in this study for G. sacculifer (Allen et al., 2016), a recent field calibration observed positive linear relationships for both these species (Mezger et al., 2016). Still, the Na Ca–salinity sensitivities observed between the different species and studies differed considerably (ranging from a change in 0.074 to 0.66 mmol mol−1 in Na Cacc for a change in 1 salinity unit). Whereas Wit et al. (2013) suggested an incorporation mechanism similar to that observed in inorganic calcite, field and culture studies also show that different species of foraminifera have varying calcite chemistries, thereby resulting in the need of species-specific calibrations similar to many other foraminiferal trace-metal-based proxies (e.g. Elderfield and Ganssen, 2000; Rosenthal et al., 2000; Anand et al., 2003; Bemis et al., 1998; Toyofuku et al., 2011). For example, Mg Cacc values are different between groups of low-Mg, high-Mg hyaline and porcelaneous foraminifera (Toyofuku et al., 2000; Segev and Erez, 2006; Raja et al., 2007), which also seems to be reflected in other co-precipitated cations (De Nooijer et al., 2017). Hence, calibration of Na Cacc as a function of salinity for other species is not only necessary to test the applicability of this novel proxy for other groups of foraminifera but also allows testing whether monovalent cations follow the inter-species trends described for divalent cations (Terakado et al., 2010). Here we calibrated Na, Mg and Sr incorporation in the intermediate-Mg calcite benthic foraminiferal species Amphistegina lessonii and the low-Mg calcite species Ammonia tepida over a salinity range of 20 units (from 25 to 45). We thus compare the El Ca versus salinity trends in a tropical, symbiont-bearing species (A. lessonii) to a temperate intertidal symbiont-barren species (A. tepida) and both of them to existing calibrations. The chemical composition of newly formed calcite was determined by laser ablation inductively coupled plasma mass spectrometry (LA-Q-ICP-MS), providing insights into concentrations and variability therein between specimens and between single chambers. To investigate intra-specimen variability at the scale of the chamber wall we also performed electron probe micro-analysis (EPMA), mapping the Ca, Na and Mg distribution throughout the chamber wall for specimens of both species cultured. 2 Methods ## 2.1 Culture media preparation and chemistry In total, 50 L of sea water with a salinity of 50 was prepared by sub-boiling 0.2 µm filtered North Atlantic sea water for 48 h at 45 C. Subsequently, culture media were obtained by diluting this high-salinity sea water with double de-ionized sea water ( 18 MΩ) in batches of approximately 10 L with salinity increasing from 25 to 45 in steps of 5 units, resulting in five unique salinity conditions. Using a single batch of concentrated sea water to subsequently dilute to the desired salinities ensures constant element to Ca ratios. Salinity of the media was measured with a salinometer (VWR CO310), based on conductivity. Culture media were stored in Nalgene containers and kept in the dark at 10 C. Sea water pH was determined with a pH meter (pH110, VWR). Subsamples were taken prior to and at the end of the experiment and analysed for DIC and element concentrations to monitor the effect of sub-boiling on the sea water's inorganic carbon chemistry and element composition (Table 1). Subsamples for DIC were collected in headspace-free vials and conserved with a saturated HgCl2 solution (10 µL HgCl2/10 mL sample). DIC measurements were performed on an autoanalyser spectrometric system TRAACS 800; Stoll et al. (2001). This analysis requires only a small amount of sample, while yielding high accuracy (±2 µmol kg−1) and precision (±1.5 µmol kg−1). The minor and major elemental composition of the culture media was measured using a sector field ICP-MS (Element2, Thermo Scientific) by sampling 1 mL from the culture media and dilution by a factor 300 with 0.14 M HNO3 (Table 1). Table 1Experiment culture media measurements per salinity condition. Carbonate ion concentrations and saturation state with respect to calcite (at 25 C) were calculated using CO2SYS (Van Heuven et al., 2011) and the equilibrium constants K1 and K2 of Mehrbach et al. (1973), as reformulated by Dickson and Millero (1987). ## 2.2 Collection of foraminifera and culture setup Surface sediment samples containing foraminifera (A. lessonii) were collected from the Indo-Pacific Coral Reef aquarium in Burgers' Zoo (Arnhem, the Netherlands; Ernst et al., 2011) and a tidal flat near Den Oever, the Wadden Sea (A. tepida, genotype T6; Hayward et al., 2004). Sediment was stored in aerated aquaria at 25 C (A. lessonii) and 10 C (A. tepida) with a day/night cycle of 12 h/12 h, similar to conditions in the coral reef aquarium and Wadden Sea, respectively. From both stocks, living specimens, recognized by chambers that were filled with yellow cytoplasm and pseudopodial activity, were isolated. The culture protocol was the same for both species to facilitate comparison of obtained element Ca ratios between species. Since our specimens of A. tepida are from a location with a much larger temperature range than where A. lessonii is derived from (Ernst et al., 2011; Van Aken, 2008; De Nooijer et al., 2014a), both species were incubated at 25 C. Living specimens were placed in groups of 25 individuals in Petri dishes with approximately 70 mL of North Atlantic surface sea water (0.2 µm filtered) and fed with fresh cells of the algae Dunaliella salina. After reproduction, which occurred in approximately two-thirds of all incubated specimens in both species, 2–3 chambered juveniles were isolated. The use of specimens from reproduction events guarantees that virtually all chambers present at the end of the experiment were produced under the culture conditions (De Nooijer et al., 2014a). Strains of specimens of the reproduction events were divided over Petri dishes (resulting in 2–10 individuals per dish) with approximately 10 mL of culture medium and stored in a temperature-controlled incubator set at 25 C with a day/night cycle of 12 h/12 h. The culture media in the Petri dishes were replaced once every week, after which specimens were fed with 1 mL of concentrated and freeze-dried Dunaliella salina diluted with the culture medium for each salinity condition, to minimize changes in salinity when feeding the foraminifers. The amount of food was adjusted so that it was not depleted after a week, at the same time not resulting in an excess of debris and hence reduce bacterial growth. Petri dishes were sealed with a lid to minimize evaporation. After 6–8 weeks, specimens were harvested and transferred to microvials to clean the specimens' carbonate shells from cell material. Specimens were cleaned with an adapted version of the Barker protocol (Barker et al., 2003), only applying the organic removal/oxidation step, in which NaOH was replaced with NH4OH, in order to avoid Na contamination of our samples. Organic matter was removed by adding 1 % H2O2 buffered with 0.1 M NH4OH at 90 C and gentle ultrasonication (80 kHz, 50 % power, in degas mode) for 1 min, which is known not to affect obtained Mg Ca and Sr Ca (Barker et al., 2003). Specimens were subsequently rinsed three times with double de-ionized water and dried in a laminar flow cabinet, after which their size was determined (i.e. the maximum diameter crossing the centre of the specimen). The specimens were thereafter stored until geochemical analyses (LA-Q-ICP-MS; Sect. 2.2.2 and EPMA; Sect. 2.4). Table 2Accuracies (Ac) and precisions (Pr) for Na, Mg and Sr for the various standards analysed. ## 2.3 Foraminiferal calcite chemistry Specimens were fixed on a laser ablation stub using double-sided tape, carefully positioning them to allow ablation of the last chambers (Supplement S1). Element concentrations of individual chambers were measured with LA-ICP-MS (Reichart et al., 2003). The last 1–3 chambers of each specimen were ablated using a circular spot with a diameter of 60 µm (A. tepida) and 80 µm (A. lessonii) (NWR193UC, New Wave Research) in a helium environment in a New Wave TV2 dual-volume cell (cup volume of 1 cm3) at a repetition rate of 6 Hz and an energy density of approximately 1 J cm−2. The aerosol was transported to a quadrupole ICP-MS (iCap, Thermo Scientific) on a helium flow at a rate of 0.7 L min−1, with 0.4 L min−1 Argon make-up gas being added before entering the torch. Nitrogen gas was added at a rate of 5 mL min−1 to enhance sensitivity of the analysis. Before entering the torch, the aerosol Ar He mixture passed through an in-house-made smoothing device to reduce temporal variations in signal strength. Monitored masses included 7Li, 11B, 23Na, 24Mg, 25Mg, 27Al, 43Ca, 44Ca, 60Ni, 66Zn, 88Sr, 137Ba and 238U, with one full cycle through the different masses taking 120 ms. Calibration was performed against a MACS-3 (synthetic calcium carbonate) pressed powder carbonate standard with 43Ca as an internal standard. Count rates for the different masses were directly translated into element Cacc (El Cacc) ratios. Internal precision based on MACS-3 is 4 % for Na, 3 % for Mg and 4 % for Sr. Accuracy and relative analytical errors, based on measuring international standards JCp-1 coral (Porites sp.) powder and the NIST (National Institute of Standards and Technology) SRM 610 and SRM 612 (glass), are listed in Table 2. The relatively large offset between the glass standards and the pressed powders (both MACS-3 and JCp-1) is known not to influence obtained El Cacc ratios when either one is used as a calibration standard (Hathorne et al., 2008), but due to the similar matrix, MACS-3 was chosen as the calibration standard. In total, 675 chambers were measured (336 for Amphistegina and 339 for Ammonia), resulting in between 52 and 125 single-chamber measurements per salinity condition per species. These measurements were done on the last three (final or F, penultimate or F-1 and pre-penultimate or F-2) chambers of these specimens (see Table 3 for number of specimens and average number of spots per specimen). Element concentrations were calculated from the time-resolved (i.e. ablation depth) profiles using an adapted version of the SILLS (Signal Integration for Laboratory Laser Systems; Guillong et al., 2008) package for MATLAB (for details see Van Dijk et al., 2017a), while taking care to exclude contaminations potentially present on chamber walls (examples of profile selection: Dueñas-Bohórquez et al., 2011; Wit et al., 2013; Mewes et al., 2014; Mezger et al., 2016; Van Dijk et al., 2017b). Measurements with ablation yields or integrations times < 5 s were excluded from further analysis. Figure 1Foraminiferal Na Cacc, Mg Cacc and Sr Cacc versus salinity. Light blue dots represent the average per specimen (n=359 for A. lessonii, n=339 for A. tepida, with 2–3 measured chambers per individual) and dark blue dots indicate the mean, with inner error bars indicating the standard error and outer error bars the standard deviation for each treatment. The linear regression model (red line) is based on the mean of individuals, with the 95 % confidence interval of the regression in dashed lines. The LA measurements were also used to investigate the co-occurrence of elements within specimens. Since there is variability in Ca counts between the laser ablation measurements, single-spot-based element Cacc ratios may cause spurious correlation due to coupled differences in Ca counts. To test whether observed correlations between Na Cacc, Sr Cacc and Mg Cacc, based on single spots, are due to the use of a common denominator (Ca), we performed a Monte Carlo simulation. In short, the correlation coefficients between randomly drawn single-spot Mg concentration, divided by measured Ca, and measured Na Cacc concentrations were compared to the correlation coefficient of measured Na Cacc and Mg Cacc concentration ratios in our dataset. By using a kernel fit of the measured data set to draw the random data set and using the measured Ca as a common denominator we effectively simulate the spurious correlation. The Monte Carlo results show that inter-element correlations are not spurious, since the measured correlation coefficient is significantly higher then the distribution of the correlation coefficients between 10 000 randomly drawn El1 concentrations/measured Ca concentration and measured El2 Ca concentrations (Supplement S2). Furthermore, to test whether Sr Cacc and Na Cacc variability in A. lessonii is not caused by variability in Mg content due to a potential closed-sum effect (since high amounts of incorporated Mg cations could reduce the Ca content of the shell and hence result in apparently elevated Sr Cacc and Na Cacc), we calculated maximum variability due to the sole effect of Mg substitution. For A. lessonii, variability (standard deviation) of ±0.09 mmol mol−1 in Na Cacc and ±0.016 mmol mol−1 in Sr Cacc around the mean could be caused by variability in Mg Cacc (assuming Mg substitutes for Ca in the calcite lattice, and Mg plus Ca approximates 1 mol per mol calcite). This may have influenced the Sr Cacc and Na Cacc regression slopes over salinity and also the calculated inter-element correlation coefficients, but only by a maximum of ±1 % for both elements, which is considerably lower than the total observed variability within the dataset of 16 and 9 %, respectively. ## 2.4 Electron microprobe mapping To investigate variation of element distribution across the chamber wall, a number of cultured specimens were prepared for electron microprobe analysis (EPMA). From each of the five salinity conditions, six specimens from both species were selected and embedded in resin (Araldite 2020) in an aluminium ring (diameter 1 cm) in a vacuum chamber. Samples were polished with a final polishing step using a diamond emulsion with grains of 0.04 µm. This procedure resulted in exposure of a cross section of the foraminiferal chamber wall from which areas for EPMA mapping were selected (Supplement S1). These areas were selected for being perpendicular to the shell outer surface, resulting in pores completely crossing the exposed chamber wall. Elemental distributions were mapped in chambers prior to F-3 to study the element distribution across the various layers of calcite (lamella) produced with the addition of each new chamber in rotaliid foraminifera (Reiss, 1957, 1960). Elemental distribution in the shell wall was measured using a field emission electron probe micro-analyser (JEOL JXA-8530F HyperProbe) at 7.0 kV with a dwell time of 350 ms, using a spot diameter of 80 nm and a step size between 0.1538 µm and 0.4072 µm (130 × 130 pixels). Spatial resolution of the EPMA mapping was determined using the software package CASINO (monte CArlo SImulation of electroN trajectory in SOlids, v 2.48). With the input parameters identical as used in our analysis (80 nm spot size, beam current 7 keV, etc.), the simulated surface radius of the backscattered electrons (i.e. the spatial resolution) equals 590 nm. Semi-quantitative El Cacc profiles were calculated by averaging the El Cacc intensities parallel to the banding direction and applying a constant calibration factor obtained from LA-ICP-MS measurements on the same specimen, similar to the procedure of Eggins et al. (2004). We did not use the depth-resolved laser ablation profiles for this purpose but used the average value from the profiles for correlation with the EPMA-derived intensities. Figure 2Individual chamber LA-ICP-MS analyses showing correlations between foraminiferal Mg Cacc, Sr Cacc and Na Cacc for A. tepida (a) and A. lessonii (b) per salinity condition. Significant orthogonal linear regressions are indicated with a line, colour coded for salinity (see legend). Correlation coefficients, slope and intercepts of these regressions can be found in Supplement S3. In short, within salinity conditions, element ratios are strongly correlated with each other in A. lessonii, whereas in A. tepida, element ratios do not (strongly) correlate with each other. When combining all single-spot data in A. tepida, element ratios correlate amongst each other because the incorporation of all three elements increases with salinity, shifting the distributions to higher values. In A. lessonii, only the Na Cacc distributions shift towards higher values with increasing salinity, whereas Mg Cacc and Sr Cacc distributions are relatively similar between salinity conditions. 3 Results ## 3.1 Foraminiferal calcite element ratios and partitioning coefficients as a function of salinity Per treatment, from lowest to highest salinity, average Na Cacc of the newly formed calcite varied between 9.3 and 10.8 mmol mol−1 for A. lessonii and between 4.7 and 6.4 mmol mol−1 (highest salinity) for A. tepida (Fig. 1), with a corresponding partition coefficient (note that partition coefficients are “apparent”, not taking into account speciation/activity of Na) ranging from 1.90 × 10−4 to 2.20 × 10−4 and from 0.97 × 10−4 to 1.30 × 10−4 for Amphistegina and Ammonia, respectively (Table 3). For both species, sets of single-specimen Na Cacc show slightly skewed distributions towards higher Na Cacc for all salinities (Kolmogorov–Smirnov test, at the 95 % confidence level). Combining all specimens (based on the average of single-spot measurements per specimen), Na Cacc shows a positive linear relationship with salinity for both A. lessonii and A. tepida (Na Ca${}_{\text{cc}}=\mathrm{0.077}±\mathrm{0.017}×S+\mathrm{7.13}±\mathrm{0.60}$, F1, 186=20.9, p<0.001 for A. lessonii and Na Ca${}_{\text{cc}}=\mathrm{0.064}±\mathrm{0.013}×S+\mathrm{3.29}±\mathrm{0.44}$, ${F}_{\mathrm{1},\phantom{\rule{0.25em}{0ex}}\mathrm{172}}=\mathrm{25.9}$, p<0.001 for A. tepida; Fig. 1). The observed average relative standard deviation between specimens in Na Cacc at each of the five salinities is 15 % for A. lessonii and 20 % for A. tepida. The variance in Na Cacc between individual specimens explained by salinity is η2=0.08 for A. lessonii and η2=0.14 for A. tepida. Specimen averages of Mg Cacc and Sr Cacc correlate positively with salinity in A. tepida (Mg Ca${}_{\text{cc}}=\mathrm{0.060}±\mathrm{0.011}×S+\mathrm{0.51}±\mathrm{0.38}{F}_{\mathrm{1},\phantom{\rule{0.25em}{0ex}}\mathrm{172}}=\mathrm{29.9}p<\mathrm{0.001}$ and Sr Cacc=$\mathrm{0.014}±\mathrm{12}×{\mathrm{10}}^{-\mathrm{4}}×S+\mathrm{1.00}±\mathrm{0.04}$, ${F}_{\mathrm{1},\phantom{\rule{0.25em}{0ex}}\mathrm{337}}=\mathrm{254}$, p<0.001), whereas neither ratio correlates with salinity in A. lessonii. Average relative standard deviations for the five salinity conditions per element are 27 % for Mg Cacc and 9 % for Sr Cacc in A. lessonii and 32 % in Mg Cacc and 7 % for Sr Cacc for A. tepida. In A. lessonii, the proportion of variance in Sr Cacc explained by salinity is η2=0.04 (p<0.01) (Mg Cacc not significant) and for A. tepida, the proportion of variance in Sr Cacc explained by salinity is η2=0.44 and in Mg Caccη2=0.19 (p<0.001). Single-spot analyses on Ammonia tepida show that Na Cacc and Mg Cacc are significantly correlated within the salinity treatments, except for condition S=30 (Fig. 2). For the individual salinity treatments, single-spot Sr Cacc and Mg Cacc, as well as Na Cacc and Sr Cacc are not correlated significantly with each other, except for S=25. Between salinity treatments, distributions in this species shift towards higher Na Cacc, Sr Cacc and Mg Cacc values with increasing salinity, although for the range between 30 and 40 Na Cacc distributions remain rather similar (Fig. 2). For Amphistegina lessonii, distributions of Sr Cacc and Mg Cacc ratios overlap largely between salinities, and only Na Cacc distributions shift towards higher values (Fig. 2). Within each salinity condition, however, single-spot Na Cacc, Mg Cacc and Sr Cacc in this species are positively correlated amongst each other, whereby the Na Cacc intercept of these relationships increases with increasing salinity (Fig. 2 and Supplement S3). Table 3Average El Cacc ratios of the foraminiferal calcite (based on average of average specimens value per salinity (Sal) condition (S25–S45)) ± standard error and corresponding apparent partitioning coefficients, defined as DEl= (El Cacc) (El CaSeawater) for A. lessonii (A.l.) and A. tepida (A.t.). “n/spots” stands for number of specimens and average number of spots per specimen. ## 3.2 Size and chamber effect on Na ∕ Cacc and inter-specimen variance Specimens of A. lessonii produced most new chambers at salinities of 25, 30 and 35, closest to the salinity in their “natural” habitat (Burgers' Zoo aquarium, salinity 33.9–34.3; Ernst et al., 2011). Size averages are not significantly different between these salinity treatments, based on a Kruskal–Wallis test, whereas specimens grown at salinities of 40 and 45 were significantly smaller than those from lower salinities, reflecting lower chamber addition rates over the course of the culturing experiment at higher salinity (Fig. 3). With all specimens combined, Na Cacc is not significantly related to size in A. lessonii. Specimens of A. tepida produced less chambers at salinity 45, possibly because such a high salinity is probably close to its tolerance levels (Murray, 2014), even though this species is adapted to relatively large salinity shifts in their tidal flat habitat. Specimens in the lower-salinity groups (25, 30, 35) grew larger compared to specimens grown in two of the highest-salinity groups (Fig. 3). Combining all specimens, Na Cacc is significantly related to size in A. tepida, yet with a small slope (0.003) and just within the 95 % confidence interval (p=0.04). Figure 3Box plots (a, b) showing the size distributions (median, first and third quartiles, minimum and maximum values) for each salinity condition – n=24, 40, 60, 27, 33 for A. lessonii and n=38, 24, 28, 41, 15 for A. tepida. Letters a, b, c indicate significant different population means, based on ANOVA (p<0.001). Panels (c, d) show the Na Ca values against size measurements per individual, colour coded per salinity condition (see legend), for A. lessonii and A. tepida. Significant linear regression lines are plotted for A. lessonii. Within each salinity tested, single-chambered Na Cacc is slightly positively related to size for the specimens of A. lessonii cultured at salinities 25 (slope = 0.008, R2=0.32, p<0.01), 30 (slope = 0.002, R2=0.11, p<0.05) and 35 (slope = 0.005, R2=0.18, p<0.001). For the same species, Mg Cacc is positively correlated with size at salinities 25, 30 and 35, with a similar slope of 0.03 (p<0.05). Sr Cacc also shows a positive relationship to size within salinities 25, 30 and 35 with slopes of 0.0007, 0.0003, and 0.0005 (p<0.001) respectively. For A. tepida, there is only a slight negative correlation between size and Sr Cacc for specimens cultured at salinity 25 (slope =$\mathrm{9.9}×{\mathrm{10}}^{-\mathrm{4}}$, p<0.001) and no significant correlation for the other conditions, or between size and Na Cacc and Mg Cacc in any of the salinities. Figure 4Foraminiferal Mg Cacc (panels a) and Na Cacc (panels b) intensity ratio maps, obtained with EPMA, for two specimens of A. lessonii grown at a salinity of 30 (row 1) and 25 (row 2) and one specimen of A. tepida (row 3). Panels (d) show profiles for Mg Ca (blue) and Na Ca (red), based on averaged EPMA ratios scaled to LA-ICP-MS measurements of the same specimen, of an averaged lateral profile area through the chamber wall perpendicular to the lamella separated by organic linings (purple rectangles in c). The transect area is indicated with a purple rectangle, on top of a backscatter SEM image (c), showing that the high El Ca bands overlap with the primary organic sheet (POS, marked with dashed red line) and subsequent organic linings. See Supplement S4 for the results for three more specimens. At the lowest salinity, Na Cacc in the F chamber (newest chamber) shows slight (0.9 mmol mol−1 Na Ca higher median) but significantly higher values than the F-2 chambers for A. lessonii (multicomparison test based on Kruskal–Wallis test, p< 0.05). For specimens of A. lessonii cultured at other salinities and for A. tepida at any of the salinities tested, no significant correlations between Na Cacc and chamber position were observed (note that only chamber positions F to F-2 were taken into account, as for the lower chamber position sample numbers were insufficient). Furthermore, chamber position shows no significant effect on Mg Cacc and Sr Cacc. To further investigate the variance between and within individuals, a multiway ANOVA was performed to investigate the effect on Na Cacc per salinity condition. Inter-individual variance is significant and larger than the variance between chamber groups and intra-individual variance in all salinity groups, with the between-individual variability accounting for ${\mathit{\eta }}^{\mathrm{2}}=\mathrm{0.75}±\mathrm{0.11}/\mathrm{0.84}±\mathrm{0.03}$ of the variance (p<0.001) for A. lessonii and A. tepida, respectively. The variance due to chamber position is not significant and the remaining intra-individual variance accounts for ${\mathit{\eta }}^{\mathrm{2}}=\mathrm{0.09}±\mathrm{0.05}/\mathrm{0.08}±\mathrm{0.05}$ for A. lessonii and A. tepida, respectively. ## 3.3 Elemental distributions in the chamber wall EPMA maps of cross-sectioned chamber walls of A. lessonii show that, within the resolution limits of the technique, bands of elevated Na Cacc intensities overlap with zones of elevated Mg Cacc (Fig. 4 and Supplement S4). Mg bands show higher amplitudes than Na bands but clearly coincide spatially. Comparing EPMA maps with the backscatter SEM image of the exposed sections shows that the bands with the highest Na Cacc and Mg Cacc occur in the proximity of the organic linings, which are clearly visible in the backscatter SEM image (Fig. 4), with a number of high Na- and Mg-rich bands with slightly lower maximum intensities occurring towards the outer chamber surface coinciding with subsequent organic linings. For A. tepida, one band of elevated Mg Cacc is visible, coinciding with the POS with no clear Na Cacc banding being detected. 4 Discussion ## 4.1 The effect of salinity and DIC on Na ∕ Cacc, Mg ∕ Cacc and Sr ∕ Cacc The single-specimen Na Cacc data of the cultured A. lessonii and A. tepida both correlate positively with salinity (Table 3, Fig. 1). This is in line with previous calibrations (for Ammonia tepida, Wit et al., 2013; for cultured Globigerinoides ruber, Allen et al., 2016; for field-collected G. ruber and G. sacculifer, Mezger et al., 2016). However, our Na Ca–salinity calibration for A. tepida is somewhat less sensitive than that observed earlier for the same species (Wit et al., 2013). An offset in Na Cacc values between calibrations for a single species has been reported previously for the planktonic G. ruber and G. sacculifer (e.g. Mezger et al., 2016; Allen et al., 2016). Such an apparent discrepancy between studies may be caused by differences between cultures or in situ conditions in one of the conditions not focussed on (e.g. carbon chemistry, light intensity). Alternatively, subtle analytical differences (e.g. differences in cleaning procedures), statistical reasons (for example differences in the number of analyses or sample size) or the effect of genotypic variability on element incorporation (Sadekov et al., 2016) may also play a role. Although the calibration presented here consists of many more data points compared to those in Wit et al. (2013), we do not want to dismiss the latter as several parameters (like cleaning procedures or the source of the seawater used for the culture media) inherently vary (marginally) between studies. As such the difference observed between studies merely illustrates the potential range for this species. Contrasts in sensitivities such as observed for Na Cacc between calibrations also apply to Mg Cacc and Sr Cacc, both of which here show an increase with salinity in A. tepida but not in A. lessonii (Fig. 1). Previous culturing experiments with Ammonia tepida, however, showed a smaller sensitivity of Mg Cacc to salinity (0.029–0.0044 mmol mol−1 change per salinity unit; Dissard et al., 2010b) than that reported here (0.06). Still, all these sensitivities are considerably lower than that reported in Kısakürek et al. (2008) for the planktonic G. ruber (0.23 when Mg Cacc is assumed to increase linearly with salinity) but in the same range as that reported by Nürnberg et al. (1996) for G. sacculifer (0.05). The sensitivity of Sr Cacc to salinity in A. tepida (0.014; Table 3) is comparable to that for O. universa (0.008; Lea et al., 2008) and G. ruber (0.02; Kisakürek et al., 2008) and similar to the significant effect of salinity on Sr incorporation in the same species (0.01–0.02, depending on temperature) found by Dissard et al. (2010b). Sea water carbonate chemistry is an additional factor potentially affecting trace metal uptake (e.g. Lea et al., 1999; Keul et al., 2017; Russell et al., 2004). Since salinity and dissolved inorganic carbon concentration in the culture media co-varied in our experiments similar to the natural environment (Table 1), Na Cacc in our cultured specimens also correlates positively to sea water [DIC]. However, sodium incorporation has been shown to be independent from changes in carbonate chemistry in cultured Amphistegina gibbosa and several other benthic hyaline and porcelaneous species (Van Dijk et al., 2017a). Additionally, Allen et al. (2016) also found no significant effect of carbonate chemistry (i.e. varying [CO${}_{\mathrm{3}}^{\mathrm{2}-}$]) on Na incorporation in cultured G. ruber, suggesting that the variability in Na Cacc observed here in A. lessonii can be attributed to changes in salinity rather than [DIC]. However, future studies should disentangle the impacts of DIC and salinity on Na Ca, in order to increase proxy confidence in areas where Na Ca and DIC relationships differ from the global average. Previous studies showed that Sr Cacc correlates positively to [DIC] in A. tepida (Keul et al., 2017), which may account for part of the correlation between Sr Cacc and salinity reported here for this species. The published sensitivity of Sr Cacc to [DIC] is approximately 2 × 10−5 mmol mol−1 change in Sr Cacc for every 1 µmol kg−1 change in [DIC], likely representing the maximum potential effect of DIC on Sr partitioning given that others found no significant effect (Dissard et al., 2010a). For a change in 850 µmol kg−1 (Table 1), this would amount to an increase in Sr Cacc of 0.019 mmol mol−1 (Keul et al., 2017) over the salinity range studied here, thereby accounting for approximately 7 % of the total observed change in Sr Cacc (Table 3). Inorganic carbon chemistry is known to affect growth rates and shell weights in benthic foraminifera (Dissard et al., 2010a; Keul et al., 2013), which in turn may affect incorporation of Sr and Mg, hence providing a mechanistic link between inorganic carbon chemistry and element partitioning. El Ca ratios of specimens of both species grown within each salinity condition are characterized by a relatively large variability. In the overall data set, salinity only explains around 8 % of the variation in Na incorporation for A. lessonii and 14, 19 and 44 % of Na, Mg and Sr incorporation in A. tepida. Nevertheless, for A. lessonii, the Na Ca mean values (which translates to the values obtained from traditional solution-ICP-MS) fit the regression model relatively well (Fig. 1). However, given the low sensitivity, many specimens are required to reduce the uncertainty (Supplement S5). This is reflected by the relatively wide prediction bounds for the Na Ca–salinity regressions, indicating an uncertainty associated with a single Na Cacc measurement. The relatively large inter-specimen variability in element Cacc ratios has been reported and discussed before (e.g. Sadekov et al., 2008; De Nooijer et al., 2014a), but the cause for this variability remains to be identified. ## 4.2 Inter-specimen, inter-species and intra-shell El ∕ Cacc variability Single-chamber measurements show that Na Cacc for both species varies between chambers (i.e. specimens) with a RSD (relative standard deviation) of 15–20 %, despite identical culture conditions (Fig. 1). Since the analytical error on Na Cacc accounts for approximately 2 % (Table 2), a large portion of the observed variability between specimens must be due to ontogeny and/or inter-specimen differences in biomineralization controls (De Nooijer et al., 2014a). Foraminiferal shell sizes at salinities 40 and 45 are significantly smaller than those cultured at lower salinities. When combining data from all salinities, however, there is no (A. lessonii) or only a very small (A. tepida) negative correlation between Na Cacc and shell size, as opposed to a more substantial negative correlation as observed by Wit et al. (2013). In fact, there appears to be a growth optimum around salinity of 30–35, whereas growth at higher salinities might be hampered (Fig. 3). This may indicate that the earlier observed negative correlation between size and Na Ca was the result of indirect co-variation with salinity rather than a causal relationship resulting in lower Na Ca values in smaller specimens. This is corroborated by our observation that, for individuals grown at a similar salinity, the relationship between Na Cacc and size is either slightly positive or absent. Hence, size unlikely affects the observed inter-specimen variability in Na Cacc, which is supported by the absence of a correlation between chamber position (and hence ontogenetic stage) and Na Cacc. This implies that measuring specimens of different size fractions or measuring different or multiple chambers should not significantly affect the application of the Na Cacc salinity proxy. However, sufficient specimens (n>30, for an error margin < 5 % at the 95 % confidence level; Sadekov et al., 2008; De Nooijer et al., 2014a) are required for measurements. As most variability is between individuals rather than between chambers (Sect. 3.3), analysing more chambers of the same specimen would increase the accuracy of the measurement but not improve the precision of the salinity estimate, given the large inter-specimen variability. Without a major effect of ontogeny, physiological processes at the organismal level are more likely to cause observed large inter-specimen variability in Na Cacc; however, these processes remain poorly understood. In A. lessonii, single-spot Na Cacc, Sr Cacc and Mg Cacc are correlated amongst each other within each salinity condition (Fig. 2). Correlation coefficients between the three element ratios are similar for the different salinities, with a superimposed increase in the Na Cacc relative to that of Mg Cacc and Sr Cacc with increasing salinity (Supplement S3). In contrast, single-spot Sr Cacc and Mg Cacc in A. tepida are not correlated, whereas incorporation of all these elements increases significantly with salinity. Within salinities Mg Cacc and Na Cacc are significantly correlated in four out of the five salinities, but with much lower correlation coefficients compared to A. lessonii (Fig. 2 and Supplement S3). However, between the different salinities these elements are correlated in A. tepida, implying that for A. tepida salinity is one of the actual parameters controlling Na Cacc, Mg Cacc and Sr Cacc element uptake. Within conditions, the correlations between both Sr Cacc and Na Cacc with Mg Cacc in A. lessonii differ from the correlation of Sr Cacc with Mg Cacc (correlation absent) and Na Cacc with Sr Cacc (weaker correlation) for A tepida. The differences between the correlations likely reflects differences in their calcification pathway (e.g. transport of ions to the site of calcification) and/or might be explained by differences in lattice strain due to the higher Mg content in A. lessonii, whereas this effect is expected to be less prominent in low-Mg species such as A. tepida (Evans et al., 2015). Differences in the calcification pathway may also explain why Sr Cacc and Mg Cacc are correlated with salinity in A. tepida, but not in A. lessonii (Sect. 4.1). In both species, Mg is found to be elevated in bands located close to the primary organic sheet and to other organic layers (Fig. 4), present in rotaliid species due to their lamellar calcification mode (Reiss, 1957, 1960). This is similar to reports of within-chamber wall banding in many elements in other species (Branson et al., 2016; Eggins et al., 2004; Sadekov et al., 2005; Paris et al., 2014; Spero et al., 2015; Fehrenbacher et al., 2017; Kunioka et al., 2006; Steinhardt et al., 2015; Hathorne et al., 2009). In planktonic species element banding has been related to diurnal light–dark cycles rather than the addition of a new lamella with chamber addition (Spero et al., 2015; Fehrenbacher et al., 2017). Whether, in the species studied here, chamber addition (and hence element banding) is related to day–night cycles remains to be investigated. As in other studies, the Na and Mg bands are spatially correlated (Fig. 4). For Ammonia tepida, the banding in both elements is less pronounced than for Amphistegina lessonii, which is likely related to the (much) lower average Mg Cacc and Na Cacc ratios in the former species. Alternatively, as the observations are close to the spatial resolution of the method, the observed pattern could also be due to the band's width being smaller in A. tepida compared to A. lessonii. ## 4.3 Biomineralization controls on element uptake How elements are transported to the site of calcification and what the role of sea water vacuolization, leakage, trans-membrane transport of ions, pH regulation and precipitation rate is, and how this differs between species and specimens, remain to be discovered. The overall element composition of the calcite precipitated by A. lessonii suggests that the calcification process of this species may more closely resemble inorganic calcite precipitation from sea water, compared to that in Ammonia tepida and other low-Mg calcite precipitating species. As a result, more elements (like Mg) are incorporated and crystal lattice strain in intermediate-Mg calcite species is elevated, which may promote incorporation of other elements through stress compensation (Mucci and Morse, 1983; Mewes et al., 2015). This would explain the observed inter-element correlations within salinities. Another difference between the species studied here may be caused by differences in CaCO3 phase shifts during calcite precipitation (e.g. Bots et al., 2012; De Yoreo et al., 2015). A metastable vaterite pre-cursor phase recently found in two planktonic species may explain the low Mg incorporation relative to inorganic calcite (Jacob et al., 2017). The higher Mg contents of A. lessonii could be related to the (partial) absence of a vaterite–calcite transformation in this species. An amorphous calcium carbonate (ACC) pre-cursor phase has been observed in other marine biomineralizing organisms (e.g. Weiner et al., 2003; Giuffre et al., 2015) and often been hypothesized to play a role in foraminiferal calcification (Erez, 2003; De Nooijer et al., 2014b), although it has not yet been directly detected. A higher Mg concentration at the site of calcification could hypothetically result in a phase shift from ACC directly into to calcite, whereby Mg is stabilizing the ACC, as described by Littlewood et al. (2017). In inorganic calcite, the absence of a vaterite precursor phase also enhances the incorporation of other metals incompatible with calcite, such as Sr (Littlewood et al., 2017), and a similar process could hypothetically contribute to inter-species differences in element partitioning similar to that observed here. Although the strong fractionation against Mg in A. tepida could reflect double fractionation through a vaterite–calcite transformation (Jacob et al., 2017) the low-Mg content might as well reflect a more enclosed site of calcification, whereby ions are mainly transported trans-membrane (Nehrke et al., 2013; De Nooijer et al., 2014b). However, the experiments here do not allow distinguishing between these (and other) potential mechanisms. Trans-membrane transport (TMT) of Ca2+ and concomitant leakage of Mg2+ and Sr2+ might be more sensitive to differences in ionic strength and element concentrations, hence possibly explaining the salinity effect on the incorporation of these elements in A. tepida, whereas it does not in A. lessonii, assuming that TMT contributes relatively more to the supply of ions to the site of calcification in this species compared to A. lessonii, which might be relatively more dependent on sea water vacuolization. However, since there are many, both biotic and abiotic, mechanisms that can (simultaneously) influence (coupled) element partitioning, it is challenging to resolve the exact mechanism responsible for inter-specimen and inter-species differences in El Ca. The spatial correlation between the intra-shell distributions of Mg and Na, associated with the organic linings, suggests a coupled control on these elements during the calcification process, which is in line with the observed inter-specimen correlations. This suggests that incorporation of these cations is influenced by similar biomineralization mechanisms, related to sea water vacuolization (Erez, 2003; Bentov and Erez, 2006), trans-membrane transport of elements (Nehrke et al., 2013), the lattice-strain effect (Evans et al., 2015) and/or metastable precursor phases (Jacob et al., 2017). The relative contributions of these mechanisms might differ between species, resulting in the observed differences in element incorporation and different inter-element correlations between species. Differences in the efficiency of such processes between specimens might cause the observed inter-specimen variability, whereas changes in these processes during the calcification time could be responsible for the observed correlation between elements within the chamber wall. 5 Conclusions By extending existing calibrations of the Na Cacc–salinity proxy to the intermediate-Mg calcite precipitating benthic foraminifer Amphistegina lessonii, we show that the Na Cacc increase as a function of salinity is similar to that in previously studied species. The absolute Na Cacc for A. lessonii is, however, higher than that in Ammonia tepida. In A. tepida, Mg Cacc and Sr Cacc are positively correlated with salinity, whereas they are not impacted by salinity in A. lessonii. Within each salinity, single-chamber Na Cacc and Mg Cacc are positively correlated in A. tepida, whereas single-chamber Sr Cacc is not correlated with either Mg Cacc or Na in this species. For A. lessonii, all Sr Cacc, Mg Cacc and Na Cacc combinations are positively correlated at the single-chamber level. Electron microprobe analysis mapping of Na and Mg within chamber walls of cultured specimens shows that in A. lessonii, Na Cacc and Mg Cacc occur in elevated bands in close proximity to the primary organic lining. For specimens of A. tepida, Mg banding appears similar to that in A. lessonii, whereas Na banding is less prominent in this species. These results suggest that biomineralization controls on incorporated elements differ between species. Data availability Data availability. The data on which this publication is based can be found through the following DOI: https://doi.org/10.4121/uuid:9e21d27d-b203-42f1-9ae3-2fa6b4791fc7 (Geerken and de Nooijer, 2018). Supplement Supplement. Author contributions Author contributions. GJR, LJdN and EG designed the culture experiment and EG and IvD carried them out. EG and IvD prepared the foraminiferal samples and analysed the specimens using EPMA. EG analysed the data and prepared the manuscript with contributions from all co-authors. Competing interests Competing interests. The authors declare that they have no conflict of interest. Acknowledgements Acknowledgements. We would like to thank Wim Boer for assistance with LA-ICP-MS measurements, Patrick Laan for sea water measurements and Karel Bakker for DIC measurements. We kindly thank Max Janse (Burgers' Zoo, Arnhem) for providing stock specimens of A. lessonii and Kirsten Kooijmans (NIOZ) for providing cultures of Dunaliella salina. Sergei Matveev is thanked for assistance with the electron microprobe analysis and Leonard Bik for assistance with polishing the samples. 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J.: A novel salinity proxy based on Na incorporation into foraminiferal calcite, Biogeosciences, 10, 6375–6387, https://doi.org/10.5194/bg-10-6375-2013, 2013.	2019-09-20 16:14:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 11, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6971129775047302, "perplexity": 9088.805734362066}, "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-2019-39/segments/1568514574050.69/warc/CC-MAIN-20190920155311-20190920181311-00474.warc.gz"}
https://ushopwell.com/ublog/13-78ct-unheated-raw-tanzanite-crystal/	# 13.78ct Unheated Raw Tanzanite Crystal https://www.tanzanitejewelrydesigns.com/products/13-78ct-unheated-raw-tanzanite-crystal — This natural unheated tanzanite crystal is almost a bicolor, as towards the termination there is much richer violet/blue color than the base of the crystal. You can see the zoning clearly and it has a very neat look! This crystal weighs 13.78 carats and measures 18.9mm x 10.8mm x 7.7mm. This crystal is unheated and shows the trichroic colors in the dichroscope.	2020-03-30 18:26:32	{"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": 0.807283341884613, "perplexity": 9873.32634633167}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370497301.29/warc/CC-MAIN-20200330181842-20200330211842-00369.warc.gz"}
https://gateoverflow.in/unknown-category	# Recent questions and answers in Unknown Category 1 vote 1 Which of the following techniques deals with sorting the data stored in the computer’s memory? Distribution sort Internal sort External sort Radix sort 2 Data leakage threats are done by internal agents. Which of them is not an example of an internal data leakage threat? Data leak from stolen credentials from desk Data leak by partners Data leak by 3rd party apps All of the options 3 The best running time is defined as/obtained as/by: the least or smallest of all the running times the algorithm takes, on inputs of a particular size an input that requires maximum computations or resources averaging the different running times for all inputs of a particular kind none of the options 4 In __________, other nodes verify the validity of the block by checking that the hash of the data of the block is less than a present number. Proof of Burn Proof of STAKE Proof of Work All of the options 5 Which of the following scenarios may lead to an irrecoverable error in a database system? A transaction writes a data item after it is read by an uncommitted transaction A transaction reads a data item after it is read by an uncommitted transaction A transaction ... item after it is written by a committed transaction A transaction reads a data item after it is written by an uncommitted transaction 6 The ___________ field in $IPv4$ datagram is not related to fragmentation. Flag Offset TOS Identifier 1 vote 7 Five people are standing in a row. Aman is standing next to Karan but not adjacent to Tanuj. Radhika is standing next to Priyanka who is standing on the extreme left and Tanuj is not standing next to Radhika. Who are Standing adjacent to Aman? Radhika and Karan Karan and Tanuj karan and Priyanka Radhika and Tanuj 8 An instance of relational schema $R(A,B,C)$ has distinct values of $A$ including $NULL$ values. Which one of the following is true? $A$ is a candidate key $A$ is not a candidate key $A$ is a primary key Both “$A$ is a candidate key” and “$A$ is a primary key” 9 Consists of a question and two statements, labelled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, ... is required for answering the question Both statement together are required to answer the question Answer cannot be ascertained with the given information 10 Five of India's leading models are posing for a photograph promoting World Peace and Understanding . But then, Sachin Malhotra the photographer is having a tough time getting them to stand in a straight line, because Natasha refuses to stand in a straight line, ... decides to stand second from right, then who is the girl standing at the extreme right? Rachel Jessica Ria None of the options 11 The $LL(1)$ and $LR(0)$ techniques are ___________ Both same in power Both simulate reverse of right most derivation Both simulate reverse of left most derivation Incomparable 12 Which element is used to define discrete unit of content such as a blogpost, comment and so on? section class article none of the options 1 vote 13 Considering binary relationships, possible cardinality ratios are: one:one $1:N$ $M:N$ All the options 14 There are $6$ boxes numbered $1, 2, \dots\dots,6$. Each box is to be filled up either with a red or a green ball in such a way that at least $1$ box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is : $18$ $19$ $20$ $21$ 1 vote 15 The admission ticket for an Art Gallery bears a password which is changed after every clock hour based on set of words chosen for each day. The following is an illustration of the code and steps of rearrangement for subsequent clock hours. The Time is ... pen miss shy soap she the yet shy miss pen soap yet the she soap pen miss shy she the yet miss shy soap pen she the yet 16 If a cube with length, height and width equal to $10\; cm$, is reduced to a smaller cube of height, length and width of $9\; cm$ then reduction in volume is : $172\;cm^3$ $729 \;cm^3$ $271\;cm^3$ None of the options 17 If $09/12/2001(DD/MM/YYYY)$ happens to be Sunday, then $09/12/1971$ would have been a: Wednesday Tuesday Saturday Thursday 18 Number of letter repeated in the given word $’MEASUREMENTS’$ are indicated in front of each alternative. Identify the correct alternative. $M_2E_2A_2S_2U_1R_1N_1T_1$ $M_2E_3A_1S_1U_2R_1N_2T_1$ $M_2E_2A_1S_2U_1R_1N_1T_1$ $M_2E_3A_1S_2U_1R_1N_1T_1$ 19 _____________ has a feature of remote access through which a customer can access the data from anywhere and at any time with the help of internet connection. IaaS PaaS NaaS SaaS 20 The physical location of a record is determined by a mathematical formula that transforms a file key into a record location is: B-Tree File Hashed File Indexed File Sequential File 21 Match the following: ... $\text{1-d, 2-b, 3-a, 4-c}$ $\text{1-c, 2-a, 3-b, 4-d}$ 22 Which of the following tag is used intended for navigation in $HTML5?$ nav footer section navigation tag 23 What is the advantage of bubble sort over other sorting techniques? It is faster Consumes less memory Detects whether the input is already sorted All of the options 24 Which of the following can be used when creating a pool of global addresses instead of the netmask command? /(slash notation) prefix-length no mask block-size 25 Select a suitable figure from the four alternatives that would complete the figure matrix. 26 $\text{58000 LOC}$ gaming software is developed with effort of $3$ person-year. What is the productivity of person-month? $\text{1.9 KLOC}$ $\text{1.6 KLOC}$ $\text{4.8 KLOC}$ $\text{4.2 KLOC}$ 27 Which one of the following statements is incorrect? The number of regions corresponds to the cyclomatic complexity Cyclomatic complexity for a flow graph $G$ is $V(G)=N-E+2$, where $E$ is the number of edges and $N$ is the number of nodes in flow graph Cyclomatic complexity for a flow ... for a flow graph $G$ is $V(G)=P+1$, where $P$ is the number of predicate nodes contained in the flow graph $G$ 28 Find the mode of the following data: $\begin{array}{\|l\|l\|\|l\|l\|\|l\|\|l\|l\|} \hline \text{Age} & \text{0-6}& \text{6-12} & \text{12-18} & \text{18-24} & \text{24-30} & \text{30-36} & \text{36-42} \\\hline \text{Frequency} & \text{6} & \text{11}&\text{25} &\text{35}&\text{18}&\text{12}&\text{6} \\ \hline\hline\end{array}$ $20.22$ $19.47$ $21.12$ $20.14$ 29 ____________ uses pretty good privacy algorithm. Electronic mails File encryption Both Electronic mails and File encryption None of the options 30 In ____________ $VMs$ do not simulate the underlying hardware. Para Virtualization Full Virtualization Hardware-Assisted Virtualization Network Virtualization 31 ___________ is a partitioning of single physical server into multiple logical servers. Virtualization Private cloud Hybrid cloud Public cloud 32 The __________ command will show you the translation table containing all the active $NAT$ entries. show ip nat translations show ip nat tl show ip nat states none of the options 33 In an $\text{IPv6}$ header, the traffic class field is similar to the ________ field in the $\text{IPv4}$ header. $TOS$ field Fragmentation field Fast Switching Option field 34 Consider the following C program segment. while(first<=last) { if(array[middle]<search) first=middle+1; else if(array[middle]== search) found=True; else last=middle-1; middle=(first+last)/2; } if(first<last)not Present = True; The cyclomatic complexity of the program segment is _____________. $3$ $4$ $5$ $6$ 35 Answer the question on the basic of the following information provided: The students of a school participates in various sports activities, the distribution of the same is given below: $Football - 17\%$ $Handball - 26\%$ $Badminton - 16\%$ ... taking part in Badminton and Table Tennis together and those participating in Basketball and Football together? 11:13 18:19 19:18 29:28 36 Answer the question on the basic of the following information provided: The students of a school participates in various sports activities, the distribution of the same is given below: $Football - 17\%$ $Handball - 26\%$ $Badminton - 16\%$ $Table Tennis - 22\%$ ... number of girls who take part in handball, if the ratio of boys to girls is 3:10 respectively? $48$ $80$ $78$ $160$ 37 Which of the following is not true for tree and graph? A tree is a graph A graph is a tree Tree can have a cycle Tree is a $DAG$ To guarantee correction of upto $5$ errors in all cases, the minimum Hamming distance in a block code must be ______. $11$ $6$ $5$ $2$ Which of the following is a correct time complexity to solve the $0/1$ knapsack problem where $n$ and $w$ represents the number of items and capacity of knapsack respectively? $O(n)$ $O(w)$ $O(nw)$ $O(n+w)$ $(<ALL)$ comparison operator means: more than the maximum value in the subquery less than the minimum value in the subquery is equivalent to $IN$ none of the options	2021-03-05 20:14:27	{"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": 0.2933588922023773, "perplexity": 1401.0955366638314}, "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/1614178373241.51/warc/CC-MAIN-20210305183324-20210305213324-00122.warc.gz"}
http://mt-class.org/penn/hw4.html	Leaderboard submission due 11:59 pm, Thursday, March 26th. Code and report due 11:59 pm, Friday, March 27th. # Reranking: Homework 4 In homework 2 you learned to search for probable translations, but saw that this is only useful with a good probability model. In homework 3 you designed a metric that correlated (at least somewhat) with human assessments of machine translation. Armed with such a metric, you now have an objective way to measure a model’s usefulness. In this assignment we will give you some sentences from Russian news articles, and you will use the metric to improve a model that chooses from a set of possible English translations generated by a state-of-the-art machine translation system. Your challenge is to choose the best translations. ## Getting Started To begin, download the Homework 4 starter kit. You may either choose to develop locally or on Penn’s servers. For the latter, we recommend using the Biglab machines, whose memory and runtime restrictions are much less stringent than those on Eniac. The Biglab servers can be accessed directly using the command ssh PENNKEY@biglab.seas.upenn.edu, or from Eniac using the command ssh biglab. In the downloaded directory, you have a program that chooses a translation for each sentence from a list of candidates. python rerank > english.out The reranker reads candidate translations from the file data/dev+test.100best. Every candidate translation $e\in E(f)$ of an input sentence $f$ has an associated feature vector $h(e,f) = [ -\log p_{LM}(e)$, $-\log p_{TM}(f\|e)$, $-\log p_{TM_{Lex}}(f\|e)]$. The reranker takes a parameter vector $\theta$ whose length is equal to that of $h(e,f)$. By default, $\theta = [-1, -\frac{1}{2}, -\frac{1}{2}]$. For each $f$, the reranker returns $\hat{e}$ according to the following decision function. $$\hat{e} = \arg\max_{e\in E(f)} \theta \cdot h(e,f)$$ To evaluate translations on the development set, compute BLEU score against their reference translations. python compute-bleu < english.out What’s the best you could you do by picking other sentences from the list? To give you an idea, we’ve given you an oracle for the devlopment data. Using knowledge of the reference translation, it chooses candidate sentences that maximize the BLEU score. python oracle \| python compute-bleu The oracle should convince you that it is possible to do much better than the default reranker. Maybe you can improve it by changing the parameter vector $\theta$. Do this using command-line arguments to rerank. Try a few different settings. How close can you get to the oracle BLEU score? ## The Challenge You can improve the parameter vector by trial and error, but that won’t be very efficient. To really improve the system you need automation. There are two components you can add: informative features that correlate with BLEU, and effective learning algorithms that optimize $\theta$ for BLEU. Your task is to improve translation quality on the blind test set as much as possible by improving these components. Implementing a version of MERT or PRO along with some simple feature engineering should be enough to beat our baseline. However, there will still be substantial room for improvement. Here are some ideas: But the sky’s the limit! You can try anything you want, as long as you follow the ground rules. ## Ground Rules • You must work independently on this assignment. • You must turn in three things: 1. Your translations of dev+test.src, selected from dev+test.100best. Upload your results with the command turnin -c cis526 -p hw4 hw4.txt from any Eniac or Biglab machine. You can upload new output as often as you like, up until the assignment deadline. You will only be able to see your results on test data after the deadline. 2. Your code, uploaded using the command turnin -c cis526 -p hw4-code file1 file2 .... This is due 24 hours after the leaderboard closes. You are free to extend the code we provide or write your own in whatever langugage you like, but the code should be self-contained, self-documenting, and easy to use. 3. A report describing the models you designed and experimented with, uploaded using the command turnin -c cis526 -p hw4-report hw4-report.pdf. This is due 24 hours after the leaderboard closes. Your report does not need to be long, but it should at minimum address the following points: • Motivation: Why did you choose the models you experimented with? • Description of models or algorithms: Describe mathematically or algorithmically what you did. Your descriptions should be clear enough that someone else in the class could implement them. • Results: You most likely experimented with various settings of any models you implemented. We want to know how you decided on the final model that you submitted for us to grade. What parameters did you try, and what were the results? Most importantly: what did you learn? Since we have already given you a concrete problem and dataset, you do not need describe these as if you were writing a full scientific paper. Instead, you should focus on an accurate technical description of the above items. Note: These reports will be made available via hyperlinks on the leaderboard. Therefore, you are not required to include your real name if you would prefer not to do so. • You do not need any other data than what we provide. You are free to use any code or software you like, except for those expressly intended to generate or rerank translation output. You must write your own reranker. If you want to use machine learning libraries, taggers, parsers, or any other off-the-shelf resources, feel free to do so. If you aren’t sure whether something is permitted, ask us. If you want to do system combination, join forces with your classmates. Credits: This assignment was developed by Adam Lopez, Matt Post, and Chris Callison-Burch. Chris Dyer made many improvements to this assignment.	2018-11-18 23:42:58	{"extraction_info": {"found_math": true, "script_math_tex": 12, "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": 0.35785484313964844, "perplexity": 1427.2453159355184}, "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-2018-47/segments/1542039744750.80/warc/CC-MAIN-20181118221818-20181119003818-00552.warc.gz"}
https://mathematica.stackexchange.com/questions/102101/find-an-appropriate-algorithm-for-certain-type-of-first-order-nonlinear-pde/102114#102114	Find an appropriate algorithm for certain type of first order nonlinear PDE I am trying to solve a first-order nonlinear PDE of the form $$\partial_t z(t,x) = f(t, x, z(t,x), \partial_x z(t,x))$$ with the boundary condition $$z(0,x) = \phi(x)$$ for some finite domain $x \in (-b,b)$ and $t \in (0, t_f)$. This is certainly a well-defined PDE given that $f$ and $\phi$ are smooth functions as we can cumulatively construct $z(t+\Delta t,x)$ from the profile $z(t,x)$ with small enough $\Delta t$. (Please find the link for more rigorous setting http://www.emis.de/journals/UIAM/PDF/40-15-30.pdf ) However, if I use NDSolve to find a solution, I get the following warning NDSolveValue::bcart: "Warning: an insufficient number of boundary conditions have been specified for the direction of independent variable x. Artificial boundary effects may be present in the solution. \!$$\*ButtonBox[\">>\",Appearance->{Automatic, None},BaseStyle->\"Link\",ButtonData:>\"paclet:ref/message/NDSolveValue/bcart\",ButtonNote->\"NDSolveValue::bcart\"]$$" It seems that Mathematica imposes an artificial additional boundary condition to solve the equation which makes me unpleasant. Is there any other algorithm that safely solves this type of equation only using the given boundary condition? To have an example, please see the following(which is originally given in the above link), f[t_, x_] = 1 + x^3 - 1/2 Sin[1 + 3 x^2 t]; sol = NDSolveValue[{D[z[t, x], t] == 1/2 Sin[1 + D[z[t, x], x]] + f[t, x], z[0, x] == 0}, z, {t, 0, 1}, {x, 0, 2}] This equation have an exact solution $t(1+x^3)$ and the above mathematica code seems to yield the desired value. However I also can find more complicated examples showing the additional boundary condition results in the wrong solution. A boundary condition in x is needed, because there is a derivative in x. Specifying z[t, 0] == 0, for instance, allows the computation to proceed without error. f[t_, x_] = 1 + x^3 - 1/2 Sin[1 + 3 x^2 t]; This additional boundary condition is required, both mathematically and numerically. If one is not provided, NDSolve in effect makes one up.	2022-01-28 20:36:39	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "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": 0.674919605255127, "perplexity": 490.7771291084141}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320306335.77/warc/CC-MAIN-20220128182552-20220128212552-00715.warc.gz"}
https://www.physicsforums.com/threads/quantum-spherical-pendulum.956593/	# Quantum Spherical Pendulum I have trouble with finding the eigenstates of a spherical pendulum (length $l$, mass $m$) under the small angle approximation. My intuition is that the final result should be some sort of combinations of a harmonic oscillator in $\theta$ and a free particle in $\phi$, but it's not obvious to see this from the Schrodinger equation: $$-\frac{\hbar^2}{2ml^2}\bigg[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\bigg(\sin\theta\frac{\partial\psi}{\partial\theta}\bigg) + \frac{1}{\sin^2\theta}\frac{\partial^2\psi}{\partial\phi^2} \bigg] + mgl(1-\cos\theta)\psi(\theta,\phi) = E\psi(\theta,\phi)$$ Using $\sin\theta \approx \theta$ and $\cos\theta\approx 1-\theta^2/2$ leads me to $$-\frac{\hbar^2}{2ml^2}\bigg(\frac{\theta}{\Theta}\frac{d\Theta}{d\theta} + \frac{\theta^2}{\Theta}\frac{d^2\Theta}{d\theta^2} + \frac{1}{\Phi}\frac{d^2\Phi}{d\phi^2} \bigg) + \frac{1}{2}mgl\theta^4 = E\theta^2$$ Here I've already used the ansatz $\psi(\theta,\phi)=\Theta(\theta)\Phi(\phi)$. Of course I can throw away the $\theta^4$ term, but any further simplifications with $\theta^2$ terms would also eliminate the energy, which is what I want. I've also tried to solve the $\Theta(\theta)$ equation with series solutions, and the result seems weird and cannot give my any energy quantizations. Another attempt is to write the entire kinetic energy term in terms of angular momentum operators, which gives $$H=\frac{1}{2ml^2}\bigg(L_\theta^2 + \frac{L_\phi^2}{\sin^2\theta} \bigg) + mgl(1-\cos\theta)$$ I was hoping to solve this with raising and lowering operators, but that $1/\sin^2\theta$ term is really a pain in the ass. I have no idea of finding a suitable ladder operator that satisfies $[H,\hat{a}] = c\hat{a}$. Any ideas? DrClaude Mentor Another attempt is to write the entire kinetic energy term in terms of angular momentum operators, which gives $$H=\frac{1}{2ml^2}\bigg(L_\theta^2 + \frac{L_\phi^2}{\sin^2\theta} \bigg) + mgl(1-\cos\theta)$$ More simply, you have $$H=\frac{1}{2ml^2} L^2 + mgl(1-\cos\theta)$$ The eigenfunctions of ##L^2## are the spherical harmonics, and $$Y_1^0(\theta,\phi) = \sqrt{\frac{3}{4\pi}} \cos \theta$$ so $$mgl(1-\cos\theta) = mgl \left[ 1 - \sqrt{\frac{4\pi}{3}} Y_0^1 \right]$$ The coupling between the spherical harmonics due to this term is then the integral of a product of three spherical harmonics, which can be expressed in terms of 3j symbol and easily calculated (check out the books on angular momentum by Zare or by Varshalovich et al.). More simply, you have $$H=\frac{1}{2ml^2} L^2 + mgl(1-\cos\theta)$$ The eigenfunctions of ##L^2## are the spherical harmonics, and $$Y_1^0(\theta,\phi) = \sqrt{\frac{3}{4\pi}} \cos \theta$$ so $$mgl(1-\cos\theta) = mgl \left[ 1 - \sqrt{\frac{4\pi}{3}} Y_0^1 \right]$$ The coupling between the spherical harmonics due to this term is then the integral of a product of three spherical harmonics, which can be expressed in terms of 3j symbol and easily calculated (check out the books on angular momentum by Zare or by Varshalovich et al.). Thanks for the great reference! I am just wondering if I can get something related to the harmonic oscillator from out of the approximate form $$\bigg[\frac{L^2}{2ml^2}+\frac{1}{2}mgl\theta^2\bigg]\psi = E\psi$$ This looks like a harmonic oscillator, but ##L## contains both ##\theta## and ##\phi## and I don't know how to commute it with ##\theta##.	2021-04-20 01:54:11	{"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": 0.9056934118270874, "perplexity": 254.8311114433203}, "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/1618038921860.72/warc/CC-MAIN-20210419235235-20210420025235-00568.warc.gz"}
http://languagelog.ldc.upenn.edu/nll/?p=1370	12. ### Mark Liberman said, April 25, 2009 @ 10:58 am Yes, though I suspect that he would prefer to be called "Quebecois". And from what little I know of the current nationalisms in different Francophone areas, his flavor of nationalism seems characteristically Quebecois. For example, I can't imagine anti-anglo-saxon feelings in France settling so easily on sympathy for Prussian revanchism, expressed in a heavy-metal song in German full of crosses and Die Heimat Den Helden memorials. (Or am I wrong about this?) 13. ### Charles Gaulke said, April 25, 2009 @ 11:10 am "If he is Canadian, his hero-worship of France is rather funny, given that from what I've seen French people's opinions of French Canadians are rather less favourable than the more elitist of British views of America." Funny, but pretty common. Even before I moved to Québec, I encountered some of the substantial proportion of French Canadians who consider themselves French despite their families living here since before Confederation. In fairness, back in Ontario there are pockets of "British" people, public school accents and all, whose families haven't set foot in the UK for just as long. Maybe it's just a Canadian thing to identify yourself as from whatever country expelled your great-great-grandparents for stealing sheep. It's interesting to compare the rhetoric here to the American English-Only rhetoric, since they seem like they should be coming from very different perspectives but the tone and the intolerance seem remarkably similar. 14. ### Émilie said, April 25, 2009 @ 11:39 am For many years, Michel Brûlé has loved to stir controversy, because it pays. To me (I live in Québec), he's a populist, a sensationalist, whose main goal is to be talked about, I wouldn't be surprised if he had come up with this idea first and foremost because he knew it would be talked about, in Québec and elsewhere. I certainly hope people don't consider him "characteristically Quebecois," as someone in the comments mentioned. 15. ### James said, April 25, 2009 @ 11:49 am Worthy of note: on Montreal's city flag, 3/4 of the 'ethnicities' represented are non-Francophone. While that never reflected the anglo/franco/allophone ratio accurately, it gives you some idea of the history of the place. I guess M. Brûlé's disgust with l’impérialisme et à l’ethnocentrisme is giving him a case of selective amnesia. Montréal has been "Montreal" for native English-speakers for centuries. And, of course, it was called Hochelaga long before that. 16. ### David said, April 25, 2009 @ 11:50 am "Longtemps, la suprématie de la France a rayonné sur le monde entier. De tout temps, les Français ont néanmoins reconnu les nombreux apports des différentes langues et sociétés au patrimoine culturel mondial. De nos jours, la culture anglo-saxonne domine le monde. Qu’elles soient britanniques ou américaines, les élites anglo-saxonnes ne s’intéressent qu’à elles-mêmes. Musique, littérature, dramaturgie, il n’y en a que pour les œuvres de langue anglaise créées par des locuteurs anglais." The first paragraph is obviously bullshit but the second paragraph is not far from the truth. Every year when the winner of the Nobel prize for literature is announced I read articles where critics from various countries are asked about their opinion of the author. It seems that if the author does not write in English the standard response from anglophone critics is "who is that? never heard of him". The latest winner, Le Clezio, has more works translated into Swedish than into English. 17. ### DYSPEPSIA GENERATION » Blog Archive » What’s French for “whiner”? said, April 25, 2009 @ 12:27 pm 18. ### Roger Lustig said, April 25, 2009 @ 12:42 pm He's also wrong about Koenigsberg. It was not "German for 692 years," but rather Prussian. East and West Prussia (aka Ducal and Royal Prussia, respectively) were fiefdoms of the King of Poland/Lithuania, later Poland; were never part of the Holy Roman Empire of the German Nation; were not part of the German Confederation (1813-1870) except for a few years around 1848. (Imagine: the boundaries of the Confederation slicing right through the lands of its largest member!) In fact, the Prussias and Posen were part of the "German question:" what and where is Germany, or should Germany be? To be sure, most people in Koenigsberg spoke German, especially after the Teutonic Knights left the scene and the Dukes/Electors of Brandenburg became the hereditary Dukes of Prussia and colonized the area. Eventually the Old Prussian language–one of the Baltic languages–died out. Some of those old HRE boundaries are still there: Czech Republic vs. Slovakia; Austria vs. Hungary. Sure, it wasn't holy or Roman or an empire–compared to the very real Hapsburg empire it was more like conceptual art–but the relevant concept was "German nation." 19. ### Mohamed Idris said, April 25, 2009 @ 12:45 pm The French do not have much credibility when it comes to protecting other languages from the hegemony of English. They just wish English would go away so that French can take over. Michel Brûlé may be a populist, but the issue he addresses, perhaps in the wrong way, is something many people around the world can identify with. 20. ### Andrew said, April 25, 2009 @ 12:47 pm Oddly enough, the capitalisation of 'He' when referring to God is not found either in the Authorised Version of the Bible (or in more modern versions which I am familiar with) or in the English Prayer Book and its revisions. I have always been rather puzzled about where it came from. 21. ### parvomagnus said, April 25, 2009 @ 1:25 pm I just had a simple but revolutionary realization – that the man's mainly interested in self-aggrandizement's pretty evident from the "poubelle" video. Anyone in the grip of true, fiery revulsion to anglo-american culture would have constructed a much more credible pile of cultural symbols to steamroll over (Hollywood represented by blank video cassettes?), and probably would have devoted more screen time to said steamrolling, and less to himself gyrating like a drunken walrus. 22. ### J. W. Brewer said, April 25, 2009 @ 2:54 pm Wikipedia claims that "I" is also standardly capitalized in Danish, although there it's a second-person plural pronoun. Maybe k.d. lang should get extra credit for being an unusually capital-eschewing Anglophone Canadian? Anne Applebaum's interesting book Between East and West (a travelogue from the borderlands of the erstwhile Soviet empire just after its early '90's collapse) starts with a vignette in Kaliningrad where the author is trying to track down what are rumored to be the only two ethnic Germans left in the city. However one might wish to describe the pre-1945 ethnic/political/linguistic/cultural identity of Koenigsberg, "Russian" wouldn't seem to be one of the plausible candidates. 23. ### Sili said, April 25, 2009 @ 5:20 pm "I" (you pl.) is capitalised in Danish. I don't know why, but it may be to help distinguish it from small "i" which means "in". We do capitalise "De" (polite you, sg), but again that may be to help tell it apart from "de" (they, pl). There's prolly more to it, though, since until 1948 Danish capitalised nouns just like German. And German only recently stopped using capitals in pronouns (I believe): ich, du. They have a Sie/sie distinction like the Danish still, though. 24. ### David Eddyshaw said, April 25, 2009 @ 6:00 pm I don't know whether schoolbooks in Francophone Africa still go on about Gaulish forebears, but I can at least say that my wife used textbooks in her career as a French teacher in (Anglophone) Ghana which were very much oriented to West Africa; there was a family of Ivoirians called Kone who ate familiar things like ignames … 25. ### Jean-Sébastien Girard said, April 25, 2009 @ 6:09 pm FWIW although there is a latent anti-Americanism in Quebec (overlapping a significant worry about the cultural hegemony of English which is not restricted to Quebec, but exacerbated by sociohistorical factors), this particular brand of virulence is only found in a restricted fringe. In fact, the next day's issue had plenty of reader's response calling him on being as bigoted as those he supposedly exposes. Personally, I think If this were a science book, Brûlé would likely be branded a crank. 26. ### David Eddyshaw said, April 25, 2009 @ 6:42 pm @Andrew: From the preface to the New Revised Standard Version of the Bible: "Furthermore, in the tradition of the King James Version one will not expect to find the use of capital letters for pronouns that refer to the Deity—such capitalization is an unnecessary innovation that has only recently been introduced into a few English translations of the Bible." I do get the impression that this is a Victorian habit at the earliest, but I can't track down any actual evidence. Wasn't this discussed previously on LL, or am I imagining it? 27. ### Matt said, April 25, 2009 @ 7:07 pm I'm sure that speakers of Breton, Basque, and Languedoc, whose languages were suppressed by the French government for centuries, would be surprised to learn that "the French nevertheless recognized the numerous contributions of different languages and societies to the cultural patrimony of the world". La marmite calling the kettle noir… 28. ### David said, April 25, 2009 @ 8:01 pm 1. This is why French-language philosophy (plus associated disciplines) tends to be regarded with deep suspicion by their (mainly English-speaking) analytical counterparts. 2. The Swedish 2nd person plural nominative personal pronoun "I" (cognate to English "ye") was also spelled with a capital letter until its gradual disappearance during the 19th and early 20th c. Judging by Svenska Akademiens ordbok, "I" seems to have been spelled with a capital letter from the mid-18th c. onwards. It hardly needs noting that Sweden isn't a country one would normally associate with ruthless capitalist individualism… 29. ### Robin said, April 25, 2009 @ 8:05 pm ‘Reduced to i by 1137 in northern England, it began to be capitalized c.1250 to mark it as a distinct word and avoid misreading in handwritten manuscripts. "The reason for writing I is … the orthographic habit in the middle ages of using a 'long i' (that is, j or I) whenever the letter was isolated or formed the last letter of a group; the numeral 'one' was written j or I (and three iij, etc.), just as much as the pronoun." [Otto Jespersen, "Growth and Structure of the English Language," p.233] ’ 30. ### J. W. Brewer said, April 25, 2009 @ 8:44 pm Note the asymmetry that by-and-large Anglophones (perhaps outside of those unfortunate enough to live subject to the whims of the present government of Quebec) don't systematically resent the continuing, if increasingly vestigial, cultural prestige of the French language. If, but probably only if, you like me have young daughters you may be aware of the "Fancy Nancy" series of books. As part of her general commitment to "fanciness," Nancy (who is probably supposed to be somewhere between six and eight years old) is fond of learning bits of French vocabulary, because "everything sounds fancier in French." Adult Anglophone purchasers of clothing and food are still apparently likewise convinced that random bits of French (not necessarily accurate or idiomatic) signify fanciness, at least if the marketing strategies of sellers are presumed to be rational. 31. ### marie-lucie said, April 25, 2009 @ 9:31 pm m-l: Rayonner … = "radiate" (intransitive). The sentence means that French culture was a light that reached all over the world, not that it lit up the whole world. This word (and the noun rayonnement) is frequently used in discussions of the importance of French culture. [(myl) I wrestled with this one for a while in translating the passage. It's awkward and unidiomatic in English to say "the supremacy of France radiated all over the world".] The original French sentence is pretty awkward too. "Supremacy" is not what radiated, but cultural achievements and prestige. I was not trying to translate the sentence, just to clarify the meaning of the verb. m-l: une poubelle is a "garbage can". [(myl) But this song should have a one-word title in English as in French, I thought; and "garbage" seemed like the best choice to me.] I did not realize that you were translating the song, rather than just the word. "Trash can" would be just about as short as "garbage". [(myl) The refrain "à la poubelle" means "into the garbage can", I guess, but "into the garbage" mean about the same thing, and would work better, don't you think?] I would translate "à la poubelle" here by "out with the trash". The point is not to overfill the can but to get rid of all the things that should go out like garbage. For "into the garbage can" I would say "dans la poubelle", which would refer to the physical container where garbage should be placed, but "à la poubelle" can be concrete or metaphorical. 32. ### BS said, April 25, 2009 @ 11:04 pm And in the French language when referring to groups and gender is mixed (ie: 2 female, 3 male), they use Ils. Are we to assume they are egotistical. This mans "story" deserves to be in "la poubelle", "trash can", "garbage can", whatever! He is A-typical of a 1/4 of the Nationalist extremists in Quebec! Rather sickening actually. Et je n'ai aucun problème avec la langue française, je parle et écris les deux. Il me donne l'envie de vomir! 33. ### David Eddyshaw said, April 25, 2009 @ 11:10 pm Re "poubelle" – that excellent and vastly more sensible man Proust himself: "… car ces mots français que nous sommes si fiers de prononcer exactement ne sont eux-mêmes que des «cuirs» faits par des bouches gauloises qui prononçaient de travers le latin ou le saxon, notre langue n'étant que la prononciation défectueuse de quelques autres …" … although, to be fair, I suppose you could adduce this in support of M Brûlé's claims to the absorbent qualities of the French language, if not culture. 34. ### Roger Lustig said, April 26, 2009 @ 12:17 am @Sili: if German doesn't capitalize Du anymore, I must have missed the memo. 2nd-person pronouns are capped; 3rd-person-pl. "sie" is not. 35. ### Aaron Davies said, April 26, 2009 @ 1:05 am "Pis cé toué qui parle de préservé notre langue gros tabarnack?" is that quebecois? if so, could someone translate? (i'm reasonably good at standard french, but can't make out much beyond "who speak of preserving our language") 36. ### Östen Dahl said, April 26, 2009 @ 2:45 am In a number of traditional dialects in Sweden (North Germanic varieties), the first person singular pronoun is pronounced [i:], and it is striking that people who try to use these dialects in writing pretty consistently write it as "I", just like the English pronoun. 37. ### J Lin said, April 26, 2009 @ 3:50 am ِ@R. Lustig: I'm pretty sure that under the rules of the Rechtschreibreform Du und Ihr are no longer to be capitalized obligatorily, though of course anyone who went to school before about 1998 will have heard something different. Only 2nd-formal Sie must be capitalized under the new rules. 38. ### Bernhard said, April 26, 2009 @ 4:46 am So there seems to be a trend in the comments to agree that the diagnose is right but the diagnostician is not credible? @Roger: You indeed missed the memo. The ‘really new’ (i. e. reformed reformed) orthography permits again to capitalise second person pronouns in letters etc. In all other texts (e.g. in novels), it has not been capitalised, anyway (in recent times; if the writer followed the norms). Allowing capitalisation again is certainly one of the concessions to popular usage, I think; the ‘old new’ orthography tried to abolish capitalisation. (Pronouns referring to God, emperors etc. seem to form an exception, such as third person singular and second person plural pronouns and used to address single people in earlier times. Third person plural "Sie" referring was already mentioned.) 39. ### Dierk said, April 26, 2009 @ 4:54 am With Germany's latest spelling reform polite 'Du' gets not capitalised anymore; most older people [>30] will not conform to this rule if they do write letters at all, since they find it good to be polite in private relationships, too. The more polite 'Sie' is still capitalised by rule. 'Ich' was never capitalised except when starting a sentence or in some literary context to give special attention to the word. Thus it could be argued – wrongly – that English is about the speaker while German is about the addressee [in polemical terms: Germans are humble, English are arrogant]. This argument simply ignores the historic facts mentioned already in the comments, 'i' easily looks like the capital 'I' in handwriting, it is also easily misinterpreted or overlooked due to being the narrowest and smallest character. Be clear, we are talking about salutary use in written documents. We should also remember when talking about capitalisation that German is still heavily capitalised compared to English and French. 40. ### marie-lucie said, April 26, 2009 @ 6:12 am Aaron: "Pis cé toué qui parle de préservé notre langue gros tabarnack?" is that quebecois? if so, could someone translate? This is a more or less phonetic transcription. The sentence is not so different from the Standard French equivalent once you get past the spelling: Et puis c'est toi qui parles de préserver notre langue, gros "tabernacle"? "So then you're the one who speaks of preserving our language, you big [strong expletive]" "Pis" for puis is typical also of rural Western France. So is "toué" [twe], which is not recent but preserves the older pronunciation still standard at the time of emigration to Canada. Unlike "pis" this pronunciation sounds definitely rural in France, and uneducated in modern Québec. The last word is probably the most serious insult or swearword in Québec. The country was once totally under the heel of the Catholic Church, and beside the usual source of swearwords (intimate body functions), the ones considered the most powerful are taken from religious vocabulary. It is not so much a matter of "taking the Lord's name in vain", but to use the names of sacred cult objects in a profane manner. This seems to be a unique development among Catholic countries. The addition of gros "big, fat" for derogatory emphasis is traditional in all varieties of French. I would like to add that what is considered "Standard French" in Canada is usually (especially in terms of pronunciation) a high, formal register of French. Several books give tables of equivalents in QF and SF, where in most cases the pronunciation would be identical in normal speech (as opposed to reading aloud or declaiming classical poetry), for instance in the elision of schwa and consequent loss of l or r following the consonant, as in "notre" usually pronounced [not] before a consonant. According to this view of SF, one would have to say that most people in France, including highly educated ones, do not normally speak Standard French! 41. ### marie-lucie said, April 26, 2009 @ 7:06 am MB: Königsberg veut dire Montis realis en latin d’où vient le nom de Montréal myl: (Though I think that would be mons regius in Latin, wouldn't it?) m-l: Montréal: "réal" is another version of "royal", from Latin "regalis" (hence English "regal"). [(myl) Right, but Latin realis means something different entirely, and montis is the gentive case of mons, and wouldn't be the form used in a name.] I was puzzled by myl's reply to me, but he must have been responding to MB's statement (which contains two errors) rather than my comment. Both words are of the third declension, in which many words have identical nominative and genitive, as in the noun "turris, turris" ‘tower’ and the adjectives in "-alis" or "-aris". Some words of this class, like "mons, montis" have a shorter stem in the nominative because the nominative ending is actually "-s" not "-is" ("mons" < mont-s), but in Late Latin or Early Romance such nominatives became regularized on the basis of the stem of the other cases, hence Italian monte and French mont (the final written t was pronounced in Old French). Italian Monte Reggio corresponds to the Late Latin form of myl's Classical "mons regius". The Latin word I mentioned is “regalis” ‘royal’, the ancestor of Old French réal meaning ‘royal’, not “realis” ‘real’ which is the ancestor of réel ‘real’. The name Montréal means the same as Mont-Royal, the name of a large park in the centre of the city, but it is an older form. Similar place names in France are Réalmont in the South and Royaumont in the North (from earlier Royal-mont). The form réal is not used in Modern French but it is identical to the Spanish equivalent real as in the name of the soccer team Real Madrid (meaning ‘Royal Madrid’ not ‘Genuine Madrid’). The Montréal region is now known by the neologism la Montérégie where "reg" is the root of Latin "rex (< reg-s), regis" 'king' as well as of the adjectives "regalis" and "regius", the latter mentioned in myl’s post as the possible “mons regius". The French names of cities and towns above were originally compounds. The component words are of Latin origin, but that does not mean that the names all go back all the way to Latin: they are French names built of French components. 42. ### Yanpol said, April 26, 2009 @ 7:09 am I wonder why, then, the aboriginal peoples of Quebec wanted to stay part of Canada and tell the Quebecois to bugger off!! 43. ### marie-lucie said, April 26, 2009 @ 7:38 am Yanpol: One reason is that most of them speak English, not French. But they also knew where they stood with the government of Canada (which has direct responsibility for them in all parts of the country) but did not know how things would be under an independent Quebec government. For one thing, it would be much harder for them to make common cause with their counterparts in other regions of Canada. 44. ### MLG said, April 26, 2009 @ 8:37 am I find it interesting that the parts of the synopsis you chose to translate are some of the least incendiary* parts. This book is being advertised in the form of posters claiming that it will "forever change peoples' perception of the English language" all over Montreal. Here is a more thorough portrait of this man's brilliant contribution to linguistics: http://www.renaud-bray.com/Livres_Produit.aspx?id=964077&def=Anglaid%2CBRULE%2C+MICHEL%2C9782894854310 Pour couronner le tout, les anglophones sont narcissiques et ethnocentriques à l'extrême : ils ne regardent que leurs films, ne lisent que leurs livres, n'écoutent que leur musique et ne mangent presque tous que leur bouffe – je n'ose pas appeler ça de la nourriture. Les plus sceptiques diront que les Français, par exemple, étaient pareils, quand la France contrôlait plus ou moins le monde. C'est faux ; les Français, malgré leur arrogance indéniable, étaient très ouverts sur le monde. Il n'est pas exagéré d'affirmer que les anglophones sont les personnes les plus ethnocentriques de l'histoire de l'humanité, rien de moins. To top it all off, anglophones are narcissistic and ethnocentric to the extreme: they only watch their movies, only read their books, only listen to their music and eat only their grub (I dare not call it food). Skeptics would say that the French, for example, were the same, when France controlled the world. This is false; the French, despite their undeniable arrogance, where very open to the world. It is not an exaggeration to claim that anglophones are the most ethnocentric people of the history of humanity, and nothing less. === *Let's not forget that a frat-boy lighting a fart is technically incendiary. [(myl) What, you think I have secret sympathies for M. Brûlé? He's such an easy target that I thought an understated approach was more amusing. It's true, I might feel differently if I lived in the same city with him. But maybe you can help with this: what's with the weird sympathy-for-the-third-Reich stuff? Is he trying to tap into the Delisle controversy? ] 45. ### joseph palmer said, April 26, 2009 @ 9:28 am You could find lots of examples of self-glorification/English bashing in many cultures, but it would not look proper to subject an Asian countries scholarship to this kind of tirade, for example (speaking of some of the comments, rather than the article). Since we secretly see French scholarship as being at least reasonably equal to our own, and somewhat understandable, it always seems to be fair game to have a few jabs at it. [(myl) "Scholarship"? I haven't read M. Brûlé's essay, but from his synopsis and his interviews, it seems that this term is unwarranted. As for reactions to odd or silly claims on the part of writers from other cultures, can you provide any specific examples where you think we ought to withhold comment, as if to avoid discouraging a child? Do you feel, for instance, that this critique of a Japanese book was not "fair game"? And in what way is "our" attitude about French scholarship "secret"? In fact, the more I ponder your comment, the less coherent it seems. ] 46. ### Francis Tyers said, April 26, 2009 @ 9:39 am Une poubelle is a "garbage can". [(myl) But this song should have a one-word title in English as in French, I thought; and "garbage" seemed like the best choice to me. The refrain "à la poubelle" means "into the garbage can", I guess, but "into the garbage" mean about the same thing, and would work better, don't you think?] "Bin" would work then no? 47. ### marie-lucie said, April 26, 2009 @ 11:47 am FT: "bin": it depends where you are. In Canada "bin" would not call to mind garbage, it refers to a large container for dry foods like rice or flour when sold in bulk rather than in bags or packages. "garbage bin" or "trash bin" would be understood but not commonly used. 48. ### marie-lucie said, April 26, 2009 @ 12:15 pm (bin 2) Or it could also be a large container for soft furnishings like pillows, towels, cushions, etc sold in a store, especially cheaper or discounted ones that are not displayed as carefully as better quality merchandise. Or the clearance bin for unsold clothes at the end of the season. Storage bins are also sold for household use, for instance for out of season clothes or sports equipment. In other words, containers for cheaper or unbreakable items not needing careful attention, but not garbage. 49. ### tablogloid said, April 26, 2009 @ 12:42 pm "Tabarnac" reminds of other French Canadian Catholic sacriligious cusses I used to hear while I growing up as a Montreal Anglophone from 1947 until I moved to Toronto in 1971. There are more amusing variations of tabernacle such as, tabarnouche and tabourrette (my best guess at the spelling). The sacred host, hostie, takes a beating as something that sounds like "ess-tee". The chalice is calice and the ciborium, which holds the individual communion hosts is, ciboire. It was quite common to hear a dissatisfied hockey fan scream at the referee, "Calice, ess-tee, tabarnac…ciboire." Then there are a few "Franglish phrases I find hilarious. One I love is, "Ain! Doan press my nerve I stretch your bowtie.". 50. ### J. W. Brewer said, April 26, 2009 @ 3:07 pm I'm a little puzzled by Prof. Liberman's Delisle reference. W/o at all doubting the historical link between some exponents of Quebecois nationalism (and most ethnolinguistic nationalisms, probably) with various unsavory characters and beliefs, I don't see regret for the unhappy fate of Koenigsberg and its historical population at the hands of Stalin to be, without more, particularly compelling evidence of "sympathy for the Third Reich." Maybe there's more than appears on the face of the original English-language post? (Following the link to a French-language text does me no good because I have no French and my German is sufficiently rusty that I can't really parse text being sung by a French-Canadian.) Failing that, isn't there already evidence that this particular guy is an unpleasant kook w/o running the risk of invoking, um, would it be le loi du Godwin? [(myl) I'm puzzled, too, not by the laudable regret for Stalin's victims, but about why a Quebecois nationalist writes (?) and performs a song, in German, about German territory lost in in WWII, full of glimpses of German war memorials, the Quadriga from the Brandenburg gate, the odd iron cross, etc. Maybe it's just the "king's mountain" connection. But I certainly wouldn't have predicted that song -- or the skinhead-vibe music video for it -- from the rest of his oeuvre. ] 51. ### Jean-Sébastien Girard said, April 26, 2009 @ 8:20 pm One last element that I just noticed: this is just camouflaged vanity publishing. The guy owns this small publisher, which he even renamed after himself! (he also is CEO of another, much more respectable publisher). 52. ### James said, April 26, 2009 @ 9:35 pm marie-lucie said: [Tabarnack] is probably the most serious insult or swearword in Québec. Haha – yes. So serious that it's got the highest utterance-per-km2 of any spoken word in Quebec. 53. ### J. W. Brewer said, April 26, 2009 @ 11:53 pm OK. Still puzzling. I did not watch the video all the way to the end after accepting that I would not be able to figure out the lyrics, but I certainly take myl's point that the visuals did not necessarily focus benignly on the lost Koenigsberg heritage of Kant, Euler's topological problem involving the bridges, and, uh, Woody Allen (nee Konigsberg) as opposed to other perhaps more troubling symbolic items. I don't think it's an immutable law that non-Germans singing in German are Nazi-symps, although of course David Bowie's German-language recording of "Helden" was only a year or two after certain controversial press conferences at which he made allegedly pro-Hitler remarks (although these are generally explained away as side effects of excessive cocaine consumption plus residence in Los Angeles). Did some Soviet propagandist of the WW2 era prefigure our current subject by propounding a theory that certain syntactic or orthographic or lexical peculiarities of German demonstrated the inherent evil/ethnocentrism/imperialism of its speakers? Ezra Pound (who was pro-Allies during WW1, albeit still a crackpot) put forth a theory circa 1917 that aggressive and militaristic "Kaiserism" represented the spirit of the evil Teutonic research university as against the more civilized spirit of the traditional Anglo-American liberal arts college, but I think he stayed away from language-specific claims. 54. ### Christian DiCanio said, April 27, 2009 @ 7:55 am Ahh, but in English we at least capitalize the names of languages, which they don't do in French. Perhaps this means that the French don't esteem other languages as highly as we do? Perhaps Germans esteem all nouns more highly than we do as they are all capitalized too? Perhaps IF WE START WRITING IN CAPS, WE'LL BE HONORED AS BEING THOROUGHLY EGALITARIAN. 55. ### Arnold Zwicky said, April 27, 2009 @ 8:07 am On "tabarnack": see RS, 12/5/06: Oh tabernacle! What the wafer!: http://itre.cis.upenn.edu/~myl/languagelog/archives/003861.html 56. ### Ginger Yellow said, April 27, 2009 @ 8:15 am Re: 'poubelle' – as if to demonstrate the superiority of British English, 'dustbin' works perfectly. 57. ### marie-lucie said, April 27, 2009 @ 8:26 am There are more amusing variations of tabernacle such as, tabarnouche and tabourrette (my best guess at the spelling) (by "tablogloid" above) The fact that there are a number of alterations to the particularl word tabernacle in order to avoid pronouncing the word itself shows how strong it is. Partially disguising a taboo word is a well-known compromise between propriety and the need for emotional expression. Similar examples in English are "heck" for "hell" and the various substitutes for "the f-word" (a word which also peppers the speech of some people even though there is still a strong taboo about it). Milder words do not need to be disguised. "tabourrette" is probably the word tabouret 'stool' (eg a kitchen stool). Some words which have lost the final t in "Standard" French still have it in some rural dialects in Western France, and also in some Canadian varieties. 58. ### fiddler said, April 27, 2009 @ 1:11 pm @Francis Tyers, [(myl) But this song should have a one-word title in English as in French, I thought; and "garbage" seemed like the best choice to me. The refrain "à la poubelle" means "into the garbage can", I guess, but "into the garbage" mean about the same thing, and would work better, don't you think?] "Bin" would work then no? "Bin" is a bit generic for me–how about dumpster? 59. ### marie-lucie said, April 27, 2009 @ 3:23 pm A "dumpster" is very large, much larger than the lowly poubelle that is in every French kitchen. 60. ### Roger Lustig said, April 27, 2009 @ 10:32 pm @J Lin and Bernhard: thanks for forwarding the memo, as it were. Been trying to avoid it since well before it appeared in final form. In future will pass all such comments by a competent everyday user of the language. I owe you Pommes mit Majonese… Roger 61. ### Elvi said, April 28, 2009 @ 12:20 pm BS wrote: "He is A-typical of a 1/4 of the Nationalist extremists in Quebec!" This seems to be almost an eggcorn, except for the fact that the new word is made up. It seems to be meant as "typical", or even "extra typical", from the context, rather than the opposite as we would expect "atypical" to signify. 62. ### maxim said, April 28, 2009 @ 2:03 pm Marie-Lucie wrote above: > > There are more amusing variations of tabernacle such as, tabarnouche > > and tabourrette (my best guess at the spelling) (by "tablogloid" > > above) > … "tabourrette" is probably the word tabouret 'stool' (eg a kitchen > stool). Some words which have lost the final t in "Standard" French > still have it in some rural dialects in Western France, and also in > I thought the original poster was referring to "tabarouette", something one indeed hears in Montréal from time to time, but not as frequently as the other variants. The one person I heard this from frequently was from Mauricie region; I have no idea if this is a coincidence. BTW, I always wondered if the widespread "simonac/simoniaque" is actually referring to the medieval practice of "simonia"? If so, how could this have been a word used in parish churches frequently enough to become a popular expletive just like "hostie" and "tabarnacle" — maybe they had a flourishing relics trade or something of the sort in New France? 63. ### marie-lucie said, April 28, 2009 @ 4:43 pm Maxim: I thought the original poster was referring to "tabarouette", something one indeed hears in Montréal from time to time, but not as frequently as the other variants. That is probably more probable than my suggestion, which relied on the previous poster's transcription. It is likely that the various substitutes for the "t-word" differ according to regions. simoniaque: It seems that simony refers to making people pay for various ecclesiastical and clerical services, such as performing the sacraments. I don't know whether it applies to the trade in relics, but sellers of imported (and most likely fake) relics would have found a ready market among churches. Not many "relics" in European churches are demonstrably genuine. 64. ### ppindia said, April 30, 2009 @ 8:30 am Language Jingoism is part of all cultures. As a multilingual person I know that when dealing with many cultures and languages it is very common to see the person say that his language or culture is superior to others. This is part of the human mentality. Language and culture is what defines a person(you can include family and place also if you want). So since this is part of a persons ego he will always claim that his language and culture is superior to others. 65. ### marie-lucie said, April 30, 2009 @ 3:44 pm ppindia: he will always claim that his language and culture is superior to others This does not apply to people whose language and culture have been denigrated and who are made to feel inferior for that reason, such as members of various minorities in countries where a dominant majority does not respect or value them. It is common then for such people to feel that their language is "warmer" or more alive, more expressive, etc while the dominant language is "cold". But anyone's native language, bound up with the strong emotional experiences of early childhood, will feel warmer, closer, etc than one learned later in life, especially it the latter is associated with poor experiences in school or in society at large. 66. ### Pierre de Ravel d'Esclapon said, May 1, 2009 @ 8:56 am As a Frenchman living in the North American continent for the last 43 years -3 in Montreal and 40 in the United States-early on I developped the habit of segregating in my mind what is French and what is English.In my early days in Montreal this was particularly difficult because the day-to-day speech patterns of my classmates at the law school showed an admixture of French and English (je suis casse=I am broke;il etait loaded au bout du bout=he was dead drunk). I do love both languages but do not love to mix them indiscriminately.To insist on the use of a proper French word,or on the proper usage of a word is not culturral imperialism-it is common sense.Hence I write an occasional column on proper usage in http://www.lecourrierdesetatsunis.com or http://www.americancourrier.com see for example my analysis of the words "challenge" and "trillion" both grievously misused by the French business community. Is is to be imperialist to reject the use by Elle magazine the use of "relooker" une robe? One need not go the the extremes of Michel Brule to make the point.Yet,linguistic laziness should be tolerated by no one lest we find ourselves in the situation of our German friends coping with Fremdwoerter and in a situation where the quality of thinking will be impaired for lack of clarity and rigor.	2014-09-01 21:12:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3634304106235504, "perplexity": 7220.216728775624}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535920694.0/warc/CC-MAIN-20140909045902-00203-ip-10-180-136-8.ec2.internal.warc.gz"}
https://brilliant.org/discussions/thread/a-request-to-the-brilliant-staff-problems-in-editi/	# A request to the Brilliant staff. Problems in editing solutions. Like a option to post images while solving problems there should also be an option to post images in solutions . There is a problem while editing solutions , if any small minor changes are to be done in solutions then the whole solution has to be written again because when we click edit , the format of solution changes and lots of extra "\" are added to unwanted places. e.g this is a solution i posted and want to edit it . let at some time t the velocity of the center of mass of the rolling part be v and radius be r. let there be p turns in the carpet per unit length.Let the current be then be I Let the total number of turns be n \\\phi=B\.A=\\int B\\pi r^\{2\}dN=\\int\_\{0\}^\{r\}B\\pi r^\{2\}pdr=\\frac\{\\pi Bpr^\{3\}\}\{3\} \\ \$\\frac\{d\\phi\}\{dt\}=\\pi pBr^\{2\}\\frac\{dr\}\{dt\}=mI \$/extract_itex] We know that the mass is constant let the mass per unit length of carpet be \$\\rho\\$ \$\\rho x\+\\rho\\int\_\{0\}^\{r\}2\\pi rpdr=k \\$ \$v=\\frac\{dx\}\{dt\} \\$ \$\-\\frac\{v\}\{2\\pi rp\}=\\frac\{dr\}\{dt\} \\$ \$i=\-\\frac\{Bvr\}\{2m\} \\$ finally all the carpet is rolled out and velocity finally is 0 after t=Ts applying energy conservation. MgR=\int_{0}^{t}I^{2}mdt=\int_{0}^{t}$$\\frac\{\-B^\{2\}v^\{2\}r^\{2\}\}\{2m\}^{2}mdt$ \$$\\frac\{\\int\_\{0\}^\{t\}v^\{2\}r^\{2\}dt\}\{4\\int\_\{0\}^\{t\}dt\}$=$\\frac\{MmgR\}\{B^\{2\}T\}$=29$ Note by Milun Moghe 6 years, 4 months ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events. MarkdownAppears as italics or _italics_ italics bold or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in \( ... $$ or $ ... $ to ensure proper formatting. 2 \times 3 $2 \times 3$ 2^{34} $2^{34}$ a_{i-1} $a_{i-1}$ \frac{2}{3} $\frac{2}{3}$ \sqrt{2} $\sqrt{2}$ \sum_{i=1}^3 $\sum_{i=1}^3$ \sin \theta $\sin \theta$ \boxed{123} $\boxed{123}$ Sort by: We are aware of this and working to fix the bug. Staff - 6 years, 4 months ago thank you for looking into this matter - 6 years, 4 months ago With regards to adding images, you can do so using markdown. Upload your image to your favorite image hosting site, like Imgur or Flickr, and then use the following code, without spaces: Staff - 6 years, 4 months ago Thank you for the support , ill try it out - 6 years, 4 months ago	2020-07-05 08:22:42	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 16, "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": 0.9628748893737793, "perplexity": 3118.0397643289634}, "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/1593655887046.62/warc/CC-MAIN-20200705055259-20200705085259-00279.warc.gz"}
https://atarnotes.com/forum/index.php?topic=164550.msg1112085	April 20, 2019, 10:37:30 am 0 Members and 3 Guests are viewing this topic. goodluck • Posts: 13 • Respect: 0 « Reply #2250 on: April 08, 2019, 02:50:59 pm » 0 hey how do you do this one: h/(x+h) < ln(x+h) - ln(x) <h/x . (it's meant to be greater than and equal to btw) RuiAce • HSC Lecturer • Honorary Moderator • Great Wonder of ATAR Notes • Posts: 8273 • "All models are wrong, but some are useful." • Respect: +2212 « Reply #2251 on: April 08, 2019, 03:13:57 pm » +1 hey how do you do this one: h/(x+h) < ln(x+h) - ln(x) <h/x . (it's meant to be greater than and equal to btw) Hint: You can use a sketch to verify the following inequality. $h \times \frac{1}{x+h} < \int_x^{x+h} \frac{1}{t}dt < h\times \frac{1}{x}$ david.wang28 • MOTM: MAR 19 • Trendsetter • Posts: 159 • Always do everything equanimously • Respect: +5 « Reply #2252 on: April 13, 2019, 10:01:55 pm » 0 Hello, I am stuck on Example 7 part a) in the link below. I have also attached my working out. Can anyone please help me out? Thanks HSC 2019: English Advanced(77) (Forgot everything), Chemistry, Physics, Maths Extension 1(35) (gonna repeat), Maths Extension 2, Business Studies(80) (screw this) Dream course(s): Biomedical physics (UTS) Nursing (USYD) Engineering/ any form of science (UNSW) RuiAce • HSC Lecturer • Honorary Moderator • Great Wonder of ATAR Notes • Posts: 8273 • "All models are wrong, but some are useful." • Respect: +2212 « Reply #2253 on: April 13, 2019, 10:40:29 pm » +2 Hello, I am stuck on Example 7 part a) in the link below. I have also attached my working out. Can anyone please help me out? Thanks Two things: $\text{Firstly, you've correctly noted that }A = \frac12 x^2 \tan \alpha.\\ \text{So your volume of a cross section should just be }\boxed{\delta V = \frac12 x^2 \tan \alpha\, \delta y}\\ \text{i.e. }\boxed{\delta V = \frac12 (r^2-y^2)\tan \alpha\, \delta y}$ Not sure why you multiplied $x^2$ to another $r^2-y^2$ term in your expression. By doing so, because of the fact that $x^2+y^2=r^2$, you're basically trying to deal with $x^4$, or equivalently $(r^2-y^2)^2$ $\text{Secondly, what is }a?\\ \text{The boundaries of integration (and also the sum) are just }-r\text{ and }r$ david.wang28 • MOTM: MAR 19 • Trendsetter • Posts: 159 • Always do everything equanimously • Respect: +5 $\text{Firstly, you've correctly noted that }A = \frac12 x^2 \tan \alpha.\\ \text{So your volume of a cross section should just be }\boxed{\delta V = \frac12 x^2 \tan \alpha\, \delta y}\\ \text{i.e. }\boxed{\delta V = \frac12 (r^2-y^2)\tan \alpha\, \delta y}$ Not sure why you multiplied $x^2$ to another $r^2-y^2$ term in your expression. By doing so, because of the fact that $x^2+y^2=r^2$, you're basically trying to deal with $x^4$, or equivalently $(r^2-y^2)^2$ $\text{Secondly, what is }a?\\ \text{The boundaries of integration (and also the sum) are just }-r\text{ and }r$	2019-04-20 00:37:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.645550549030304, "perplexity": 5623.211903443083}, "config": {"markdown_headings": false, "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-2019-18/segments/1555578528433.41/warc/CC-MAIN-20190420000959-20190420022959-00538.warc.gz"}
https://www.cdevents.net/page/technical/	# Technical Guide How to use our services ## Player It’s really simple to use our streaming services. For each event you are given a unique identifier. In this example, the identifier is example. You can decide whether the stream should require the viewer to enter a password to watch, or if it is available to everyone who knows the unique identifier to your event To access the player, simply browse to https://cdevents.net/view/example The player can also be embedded in an existing page, using code similar to this <iframe allowfullscreen="" frameborder="0" height="506" mozallowfullscreen="" msallowfullscreen="" oallowfullscreen="" scrolling="no" src="//cdevents.net/view/example" webkitallowfullscreen="" width="900"></iframe> If you have elected to require a password, the embedded player will look similar to this: ## Streaming Sending a stream to our service is slightly more complicated, because we need to make sure it’s done securely. For each event, you’ll be given two pieces of information. A stream name, and a key. You’ll need to configure your streaming software or hardware to send your stream to rtmp://cdevents.net/live, with a stream key in the format <stream name>?key=<key> You should store this information securely, since it is effectively the username and password to allow anyone to stream to your event. It does not need to be shared with your viewers or published anywhere. For example, in OBS you would configure this information like so:	2022-05-28 06:57:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.249983012676239, "perplexity": 2389.7809878428384}, "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/1652663013003.96/warc/CC-MAIN-20220528062047-20220528092047-00211.warc.gz"}
https://picsar.net/features/particle-sorting/	## Particle sorting ### Presentation In PICSAR, the domain is first decomposed into MPI domains and then again decomposed into tiles (that are computed using OpenMP threads). Each tile contains a structure of arrays corresponding to the different particle properties (positions, momenta, energy…). The tile size has to be carefully chosen to fit in cache. Interpolation processes such as the field gathering and the current deposition steps are constituted of an intensive loop over particles. In the first case, the field seen by each particle is determined from the nearby nodes of the Maxwell grids (depending of the interpolation order). In the second case, the current generated by each particle is distributed to the nearby nodes of the current grid. Usually, particles contiguous in memory are also next each other in the space domain after initialization. However, as time evolves, the particle distribution often becomes increasingly random, leading to numerous cache misses in the deposition/gathering routines and decrease in performance. This problem is more likely to affect 3D simulations that use more data for the grids and the particles. Reordering particle to have particles contiguous in memory is the point of the particle sorting. During interpolation processes, if field grids of tiles can fit in L2 cache, cache reuse is more efficient. Without tiling, spatial sorting of the particles play the same role. Particles are therefore aligned and close in space. A portion of the field grid put in cache for one particle will be reused for all the next particles (iteration of the particle loops) located in this area until a particle contributes to the next block. With the tiling, fields are already located in L2 cache. The particle sorting is then of interest for vectorization. When several particles (in the particle loop) use the same field components (because located in the same cell or neighbor cells), the field components may be stored and kept in the L1 close to the vector registers for several particle iterations. Then, number of loads/store in lower cache levels is reduced since the same components are used several times. ### Implementation A cell sorting algorithm, also refers to as bucket or bin sorting, is perfectly adapted here since we want particles to be aligned in memory to share the same current, density and field grid nodes. However, note that in the case of the FDTD Yee’s Maxwell discretization, each electric and magnetic field components have their own grid that can not be superimposed (staggered). The PICSAR implementation of the bin sorting is similar to what has been done in [1,2]. The sorting subroutines are located in src/housekeeping/sorting.F90. The sorting is performed inside the tiles at the lowest parallelization level. It is called in the PIC loop just after the particle pusher, every given time steps specified by the user. The sorting is not limited to the cell size of the field grids but can be tuned to use smaller or larger sorting cell size. The bin sizes in each direction, called $dx_b$, $dy_b$, $dz_b$, are factors $f$ of the field cell size so that $dx_b=f \times dx$, $dy_b=f \times dy$ and $dz_b=f \times dz$. The bin sorting algorithm hardly vectorizes due to possible memory races during the sorting process. However, the bin sorting complexity is in O(np+nc) where np is the number of particles and nc the number of cells. In many cases, each cell contains at least one particle so that np>nc. This is lower than global sorting algorithms which usually have in the best case a complexity in O(np log(np)). Nonetheless, sorting the particles is still computationally expensive (of the order of the current deposition). In practice, sorting particles at every time steps reveal inefficient in term of speedup from interpolation processes in comparison with the cost of the sorting. After being sorted, the particles will take a certain amount of time before being completely mixed so that interpolation processes will be highly impacted depending on the particle energy distribution, number of particles per cell and size of the cells. Therefore, sorting is usually performed with a period of several iterations. In PICSAR, this is for the moment specified by the user. ### Simulation tests The following tests have been performed on the super-computer Edison at NERSC equipped of Intel Ivy Bridge processors. Moreover, the tests have been done on an old version of PICSAR dated April 2016 that does not include most optimized and vectorized subroutines. #### First case: large tiles, no vectorization We consider the case of a homogeneous plasma initialized in the entire domain having periodic boundary conditions. The domain is composed of 100x100x100 cells with 20 particles per cells. The plasma has a density of $10^{25}\ \mathrm{m}^{-3}$. The particles are initialized with a thermic velocity $v_{th}=0.1c$. Interpolation processes are performed at order 1. In these first simulations, the domain is divided into 2x2x2 MPI subdomains then divided into 2x2x2 tiles. The tile size is of 33 MB for particles and 732 KB for fields. The tiles are therefore much larger than the Edison L2 cache (256 KB) and the particles can even not be stored in the L3 cache. Here, non-optimized subroutines with poor vectorization are used for the field gathering and the current deposition. Different cases are compared, one without sorting and three others with sorting and different bin sizes corresponding to factors $f=0.5$, $f=1$, and $f=2$. A factor $f=1$ means that the bins have the volume of the cells. However, for the FDTD scheme, the sorting grid is aligned only with one of the field components. This means that for the other field components, the particles contained in a cell are not exactly aligned in memory. A factor $f=0.5$ means that the bins have a volume ⅛ times lower than the cells. The particles in each beam share the same field grid nodes for any component. Nonetheless, fewer particles are present in bin and the sorting is interesting only for many particles per beam. Sorting is also slightly more expensive. A factor $f=2$ means that the bin volume is 8 times greater than the cells. The particles inside bins will not share all the fields components. However the sorting process will be faster. As shown in Fig. 1, PICSAR runs faster with the particle sorting. It can be seen that the configuration with bin size factor $f=2$ has the best performances with a speedup pf 1.2. The sorting has a major effect on the field gathering with a speedup of 1.3. It also speeds up the current deposition in a lower extent with factor 1.1 time. The sorting represents a negligible part of the total simulation time, around 1%. Effects of the sorting is differently revealed in Fig. 2 . After initialization, the particle are sorted. The first iterations therefore correspond to the fastest ones with and without sorting. Then, as the particles get mixed, the time per iteration increases due to cache misses. After each sorting process (every 20 iterations), the particles contiguous in memory are close in space as for the initial state. The time per iteration almost recovers its initial value. If the sorting period is too long, the iteration time after sorting will catch again the no-sorting iteration time. However, performed too often, the sorting computational cost can be above the benefits. #### Second case: small tiles, no vectorization In this section, the same physical case is considered. Non-optimized subroutines for the current deposition, the charge deposition and the field gathering are still used. We increase the tile number per MPI domain to 5x5x5 (tiles have 10x10x10 cells without guard cells). The tile size is now of 4 Mo for the particles with all properties and of 32 KB in average per field grid (including guard cells). Tiles now fit in L2 cache for the fields. As shown in Fig. 3, the sorting has almost no effect on the performance with an efficient tiling and no-vectorized subroutines. #### Third case: efficient tiling with vectorized/optimized subroutines For the last case on Edison, using optimized interpolation subroutines and the sorting reveals efficient again as shown in Fig. 4. The gain from the sorting is nonetheless small. The maximum speedup is given by $f=0.5$ with a factor of 1.15. The sorting represents around 1% of the simulation time. ### Conclusion Sorting the particle can speedup interpolation processes by improving memory accesses. This study shows that the particle sorting can still be efficient with the tiling with optimized subroutines. However, this study was performed on an outdated version of PICSAR on Edison. Similar runs should be performed on KNL with the last optimized subroutines. To conclude, we have seen fluctuating improvements due to the sorting depending on cases and architectures with in the worth case, similar performance results with and without it. ### References Mathieu Lobet, last update: February 5 2017	2021-02-28 10:26:01	{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 17, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5035305023193359, "perplexity": 1036.5068706047164}, "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-2021-10/segments/1614178360745.35/warc/CC-MAIN-20210228084740-20210228114740-00374.warc.gz"}
https://box86.org/2021/06/game-performances/	Categories # Game performances Measuring performances of an emulator is always a bit tricky. For box86 (and box64), I decided to use benchs of opensource softwares, and compare the results between native and i386 version. A few months ago, I did some benchs and wrote the results for the Fosdem. The article can still be found here. The result was that CPU intensive tasks were at around 50% of the native speed, and graphics intensive apps were close to 100% speed, because most work is done in the libs, so in native space because of the wrapping of the libs (the “twist of box86”). The conclusion was that a game should probably be somewhere in between. Let’s answer that question: how does that translate to actual games. There are not a lot of AAA games that are opensource. Not actual ones at least, so the testing is a bit difficult. But in the past, some games went opensource. A major one is Quake3. It’s a highly optimized engine, that also contain a virtual machine with a JIT, and this game has an integrated benchs! So, using the fully opensource OpenArena, I can do some benchs on different architectures and see how fast box86 and box64 can be. For that, I took the official OpenArena 0.8.8 build for i386 and x86_64, and I added an armhf and aarch64 build from the Mageia distribution. Then I take the bench setup from Phoronix, wrote a small script and I’m ready to bench! I chose to bench using a fullscreen 1024×768 resolution, to avoid being bottlenecked too much by the GPU. The Phoronix script uses high detail, putting some pressure on the RK3399 board I’m using for this test. So I have 4 architectures: ARM64 (or aarch64), ARM32 (or armhf), i386 and x86_64. Here are the rough results: ARM64 version: 3398 frames 108.5 seconds 31.3 fps 16.0/31.9/68.0/6.7 ms ARMHF version: 3398 frames 106.4 seconds 31.9 fps 15.0/31.3/67.0/6.6 ms i386 version: 3398 frames 130.2 seconds 26.1 fps 17.0/38.3/157.0/11.0 ms x86_64 version: 3398 frames 128.2 seconds 26.5 fps 16.0/37.7/168.0/9.9 ms So: The first thing that comes to mind is: the ARM64 version is NOT faster than the ARM32 one! Despite the legend that arm64 must be faster than 32bits version, here it’s not the case. While comparing binaries from the RPi repo might lead to this, that’s probably because the armhf binaries use conservative build option to stay compatible with the whole range of RPi, while those binaries comes from a distrib that is optimized for armv8, even for the 32bits build. So, while the ARMv8 ISA have many advantage over the ARMv7 (especially for 64bits integer handling!), it seems it’s not enough here to compensate for the 64bits address management… The other analysis is, the speed of the emulated OpenArena is not that far from the native version! On 32bits, it’s 26.1 vs 31.9, so around 80% of native speed! It’s even better for the 64bits with 26.5 vs 31.3: almost 85% of native speed. And box64 is young, with some more space for optimisations! I tried to run the benchmark on a PI/400, using the beta of the 64bits OS. But I ran in some issues: the 32bits package of libSDL1.2 has some weird dependencies that makes it impossible to install without removing critical components, and the OS also uses glibc 2.27, where the aarch64 version of OpenArena I use needs 2.29+. I cheated a bit and still got the 64bits version running. But on PI, at those settings, the game is completely GPU bound, giving just 9.3 fps for both ARM64 and x86_64 version. So, not a good test as-is for the PI. So yeah, right between 50% and 100%, the emulated performances of OpenArena is at 80% (or 85% on 64bits). That’s pretty good! If some of you want to test, the package with the binary and the “bench.sh” script is available here (it will download the OpenArena binaries at the first launch). ## 9 replies on “Game performances” clort81says: ptitseb, if i were king, there would be statues made of you. <3 has apple or nvidia offered you a job based on this? Thank you! Ah no, neither Apple nor NVidia contacted me, but that’s not a surprise 😉 […] most sense as the long-term solution to me as it would open the door to allowing all games to work. box86 and box64 now have 80% the native performance of running AMD/Intel programs on Arm, and that’s the same performance of Rosetta 2 on the new Apple M1 Macs! […] Elton Alvessays: Great job! you are the man! brunorrosays: Thank you very much!!! I was trying box64 on Manjaro (64bit kernel and userland) yesterday on a Raspberry pi 4 4GB and found that your benchmark script works almost flawlessly (just had to change the ‘unzip’ command for a ‘7z x’) 🙂 It ran at 9FPS as expected, but for me everything involving dynarec is a mix between magic and esoterism… And it’s an amazing start! If you consider this useful, I was using glibc 2.32-2. If there is any more info that you’re interested about let me know. Thank you again! Did you had enough precision to see some difference in performances between x86_64 and arm64 build? brunorrosays: Hey, not that much. x86_64 was in low 9s (between 8.9 and 9.2) and aarch64 was always over 9. As soon as I can check I’ll tell you Unimportantsays: Worth noting: your arm64 vs arm32 comments aren’t taking into account that OpenArena likely isn’t doing much 64-bit math! Most games are float-heavy, not integer; it’s also highly likely that most integers are 32-bit-or-smaller. Yes all Integer are 32bits there. With 64bits integer, no doubt the Arm64 would be much faster.	2022-05-20 20:56:05	{"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": 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": 0.4121796190738678, "perplexity": 4648.551211340846}, "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-2022-21/segments/1652662534669.47/warc/CC-MAIN-20220520191810-20220520221810-00008.warc.gz"}
https://zbmath.org/?q=an:0989.65134	# zbMATH — the first resource for mathematics A sparse finite element method with high accuracy. I. (English) Zbl 0989.65134 The authors develop and analyze a new finite element method for second-order elliptic problems. They use rectangular finite elements with $$Q_p= \text{span}\{x^i y^i: 0\leq i,j\leq p\}$$ bases, two-level grids and global superconvergence theory. Combination of these different approaches leads to much fewer degrees of freedom without loss of accuracy on rectangular domains. The paper includes all mathematical justifications but, unfortunately, it does not report any numerical experiments. It is worth to mention that the authors plan to extend their studies to more general second-order elliptic equations on more general domains using different kinds of meshes. ##### MSC: 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65N15 Error bounds for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs Full Text:	2021-10-21 20:41:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3514271378517151, "perplexity": 625.5907205718744}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585441.99/warc/CC-MAIN-20211021195527-20211021225527-00218.warc.gz"}
https://informesia.com/3203/the-supplement-of-an-angle-is-one-third-of-the-given-angle-find-the-measures-of-the-given-angle-and-its-supplement	31 views The supplement of an angle is one third of the given angle. Find the measures of the given angle and its supplement. Solution Let the measure of the given angle be $x^{\circ}$. Then, the measure of its supplement $=(180-x)^{\circ}$. $\therefore 180-x=\frac{1}{3} x \Rightarrow 3(180-x)=x$ $\Rightarrow 540-3 x=x$ $\Rightarrow 4 x=540 \Rightarrow x=135$. Hence, the measure of the given angle is $135^{\circ}$ and the measure of its supplement is $(180-135)^{\circ}=45^{\circ}$. by 7.2k Points 1 Vote 46 views 17 views 21 views 18 views 1 Vote 13 views 1 Vote	2022-12-08 10:45:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6440243721008301, "perplexity": 1975.5208380202891}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711286.17/warc/CC-MAIN-20221208082315-20221208112315-00496.warc.gz"}
https://community.wolfram.com/web/patrickschattauer/home?p_p_id=user_WAR_userportlet&p_p_lifecycle=0&p_p_state=normal&p_p_mode=view&p_p_col_id=column-1&p_p_col_count=1&tabs1=Discussions	# User Portlet V. Dupoy Discussions Hi all, I want to define an algebra $A$ over the field of complex numbers via generators. After that I want to calculate (multiply and sum) some tensors in $A\otimes A$. How could I define my algebra? More details: Let $n$ be a natural number...	2019-07-22 16:31:52	{"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": 0.7043248414993286, "perplexity": 413.86369017551783}, "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/1563195528141.87/warc/CC-MAIN-20190722154408-20190722180408-00370.warc.gz"}
https://mresources.github.io/tutorial/using-mathematica/basics/variables.html	## Variables ### Set Variables in Mathematica are simple. Just type a name and give it a value with the equals sign ( = ). Here we'll set the variable thisIsAVariable thisIsAVariable=10 10 We can suppress the output by ending the line with a semicolon thisIsAVariable=10; (for those interested, the semi colon denotes that this is a CompoundExpression where the return value is Null ) ### SetDelayed A variable can also have a "delayed" value. That is, its value is calculated when requested. Here we'll set the variable randomValuedVariable . Use colon-equals ( := ) to do this. randomValuedVariable:=RandomReal[]; When we ask for its value, the return value will change every time. randomValuedVariable 0.12257688426972924 randomValuedVariable 0.13807713447085046 ### Clear The value of a variable can be removed via Clear Clear[randomValuedVariable] The variable now has no value randomValuedVariable randomValuedVariable ### Simple expressions We can use variables in expressions to store values for us. For example, let's do a simple ideal gas law computation for the volume occupied by 2 mols of ideal gas at one atmosphere of pressure and 273 K. We'll use Mathematica's built in constant data to get the value R in L atm / mol K. R$gasConstant= QuantityMagnitude[ UnitConvert[Quantity["MolarGasConstant"], "Liters""Atmospheres"/("Moles""Kelvins")] ] 0.08205733826794966545.937562804821409 Then we can set up our constants: n$quantityOfGas=2 (mols); P$externalPressure=1(atm); T$temperatureOfGas=273 (K); And finally calculate our volume: V$volumeOccupied=n$quantityOfGasR$gasConstantT$temperatureOfGas/P\$externalPressure 44.80330669430051730685.937562804821409	2019-07-21 14:56:11	{"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": 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": 0.4932079613208771, "perplexity": 2732.004784914822}, "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/1563195527048.80/warc/CC-MAIN-20190721144008-20190721170008-00075.warc.gz"}
https://stats.stackexchange.com/questions/463642/using-pos-tags-and-ners-as-features-for-text-classification-or-sentiment-analysi	# Using POS Tags and NERs as Features for Text Classification or Sentiment Analysis I am trying to implement text classification and sentiment analysis from the documents. I always use POS tags as features in the following way. Mike is playing football I would convert it into this format: Word_POS Mike_Noun is_Verb playing_Verb football_Noun I wanted to know what are the ways I can use NER as features. One of the ways I use is by taking count of NERs as Features. So my sentence would be Mike_Noun is_Verb playing_Verb football_Noun 0 0 Where 0 is the number of ORG-organisations entities and another 0 is the number of e.g., DATE entities. So I have 2 questions: What are the other ways we can use POS tags and NERs as features in 1. Without deep learning? 2. With deep learning	2021-09-23 18:28:37	{"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": 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": 0.39168596267700195, "perplexity": 3230.261601501601}, "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/1631780057427.71/warc/CC-MAIN-20210923165408-20210923195408-00390.warc.gz"}
https://www.vedantu.com/question-answer/what-is-5-of-500-class-8-maths-cbse-5efb2acaf36ba75f0cb29a5c	QUESTION # What is 5% of 500? Hint: We will first convert 5% into fraction using the concept of percentage and then we will use multiplication and division to get the final answer. Before proceeding with the question, we should understand the concept of percentage. A percent is a number that represents the fractional part out of 100 (per cent literally means per one hundred). It is often denoted using the percent sign, "%", or the abbreviations "pct.", "pct"; sometimes the abbreviation "pc" is also used. A percentage is a dimensionless number (pure number). Example- 94% means $\dfrac{94}{100}$. Likewise, we can represent a denominator of 100 with decimals by moving the decimal point by 2 places. So $94\times \dfrac{1}{100}=0.94$. Thus $94\times \dfrac{1}{100}=0.94$. Thus $94%=0.94=\dfrac{94}{100}$. So according to the definition of percentage 5% is $\dfrac{5}{100}$. In the question 5% of 500 has been asked which means we have to find what quantity of 500 is 5%. And here in the question “of” means multiplication. So using this information we get, $\,\Rightarrow 5%\,\text{of 500}......\text{(1)}$ Now simplifying equation (1) we get, $\,\Rightarrow 5%\times \text{500}......\text{(2)}$ Now changing the percent into fraction in equation (2) we get, $\,\Rightarrow \dfrac{5}{100}\times 500......(3)$ As 500 comes in the 100's table 5 times. Applying this in equation (3) we get, $\,\Rightarrow 5\times 5=25$ Hence 5% of 500 is 25. Note: Here in this type of question we have to remember how to convert percentage into fraction and we also have to keep in mind that ‘’of” means multiplication. We may get confused in equation (1) seeing ‘’of” so we need to be clear with the basic concepts.	2020-07-04 22:14:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9328402280807495, "perplexity": 506.7005180338997}, "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/1593655886706.29/warc/CC-MAIN-20200704201650-20200704231650-00368.warc.gz"}
https://www.zbmath.org/authors/?q=ai%3Awarming.robert-f	## Warming, Robert F. Compute Distance To: Author ID: warming.robert-f Published as: Warming, R. F.; Warming, Robert F. Documents Indexed: 27 Publications since 1964 Biographic References: 2 Publications Co-Authors: 10 Co-Authors with 27 Joint Publications 120 Co-Co-Authors all top 5 ### Co-Authors 0 single-authored 17 Beam, Richard M. 5 Yee, Helen C. 3 Harten, Ami 3 Kutler, Paul 2 Hyett, B. J. 2 Lomax, Harvard 2 Steger, Joseph L. 1 Lee, Dohyung 1 Meecham, William C. 1 Reinhardt, W. A. all top 5 ### Serials 6 AIAA Journal 6 Journal of Computational Physics 3 SIAM Journal on Scientific Computing 1 Computers and Fluids 1 Physics of Fluids 1 Mathematics of Computation 1 SIAM Journal on Scientific and Statistical Computing 1 BIT. Nordisk Tidskrift for Informationsbehandling 1 Lectures in Applied Mathematics ### Fields 19 Numerical analysis (65-XX) 13 Fluid mechanics (76-XX) 9 Partial differential equations (35-XX) 3 Linear and multilinear algebra; matrix theory (15-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) ### Citations contained in zbMATH Open 22 Publications have been cited 1,082 times in 937 Documents Cited by Year Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods. Zbl 0468.76066 Steger, Joseph L.; Warming, R. F. 1981 An implicit factored scheme for the compressible Navier-Stokes equations. Zbl 0374.76025 Beam, Richard M.; Warming, R. F. 1978 The modified equation approach to the stability and accuracy analysis of finite-difference methods. Zbl 0291.65023 Warming, R. F.; Hyett, B. J. 1974 An implicit finite-difference algorithm for hyperbolic systems in conservation-law form. Zbl 0336.76021 Beam, Richard M.; Warming, R. F. 1976 Implicit total variation diminishing (TVD) schemes for steady-state calculations. Zbl 0631.76087 Yee, H. C.; Warming, R. F.; Harten, A. 1985 Upwind second-order difference schemes and applications in aerodynamic flows. Zbl 0364.76047 Warming, R. F.; Beam, Richard M. 1976 The asymptotic spectra of banded Toeplitz and quasi-Toeplitz matrices. Zbl 0788.65049 Beam, Richard M.; Warming, Robert F. 1993 On the construction and application of implicit factored schemes for conservation laws. Zbl 0392.65038 Warming, R. F.; Beam, Richard M. 1978 Diagonalization and simultaneous symmetrization of the gas-dynamic matrices. Zbl 0313.65084 Warming, R. F.; Beam, Richard M.; Hyett, B. J. 1975 Alternating direction implicit methods for parabolic equations with a mixed derivative. Zbl 0462.65060 Beam, Richard M.; Warming, R. F. 1980 An extension of A-stability to alternating direction implicit methods. Zbl 0448.65058 Warming, R. F.; Beam, Richard M. 1979 Boundary approximations for implicit schemes for one-dimensional inviscid equations of gasdynamics. Zbl 0496.76065 Yee, H. C.; Beam, R. M.; Warming, R. F. 1982 Discrete multiresolution analysis using Hermite interpolation: Biorthogonal multiwavelets. Zbl 0977.42023 Warming, Robert F.; Beam, Richard M. 2000 Computation of space shuttle flowfields using noncentered finite- difference schemes. Zbl 0268.76050 Kutler, Paul; Warming, R. F.; Lomax, Harvard 1973 Stability analysis of numerical boundary conditions and implicit difference approximations for hyperbolic equations. Zbl 0488.65039 Beam, R. M.; Warming, R. F.; Yee, H. C. 1982 Second- and third-order noncentered difference schemes for nonlinear hyperbolic equations. Zbl 0268.76048 Warming, R. F.; Kutler, Paul; Lomax, Harvard 1973 Multiresolution analysis and supercompact multiwavelets. Zbl 0977.42022 Beam, Richard M.; Warming, Robert F. 2000 Application of TVD schemes for the Euler equations of gas dynamics. Zbl 0526.76080 Yee, H. C.; Warming, R. F.; Harten, Ami 1985 An eigenvalue analysis of finite-difference approximations for hyperbolic IBVPs. II: The auxiliary Dirichlet problem. Zbl 0798.65058 Warming, Robert F.; Beam, Richard M. 1991 Multishocked, three-dimensional supersonic flowfields with real gas effects. Zbl 0258.76047 Kutler, P.; Reinhardt, W. A.; Warming, R. F. 1973 Supercompact multiwavelets for flow field simulation. Zbl 0984.76067 Lee, Dohyung; Beam, Richard M.; Warming, Robert F. 2001 Some insights into the stability of difference approximations for hyperbolic initial-boundary value problems. Zbl 1185.65155 Warming, Robert F.; Beam, Richard M. 1986 Supercompact multiwavelets for flow field simulation. Zbl 0984.76067 Lee, Dohyung; Beam, Richard M.; Warming, Robert F. 2001 Discrete multiresolution analysis using Hermite interpolation: Biorthogonal multiwavelets. Zbl 0977.42023 Warming, Robert F.; Beam, Richard M. 2000 Multiresolution analysis and supercompact multiwavelets. Zbl 0977.42022 Beam, Richard M.; Warming, Robert F. 2000 The asymptotic spectra of banded Toeplitz and quasi-Toeplitz matrices. Zbl 0788.65049 Beam, Richard M.; Warming, Robert F. 1993 An eigenvalue analysis of finite-difference approximations for hyperbolic IBVPs. II: The auxiliary Dirichlet problem. Zbl 0798.65058 Warming, Robert F.; Beam, Richard M. 1991 Some insights into the stability of difference approximations for hyperbolic initial-boundary value problems. Zbl 1185.65155 Warming, Robert F.; Beam, Richard M. 1986 Implicit total variation diminishing (TVD) schemes for steady-state calculations. Zbl 0631.76087 Yee, H. C.; Warming, R. F.; Harten, A. 1985 Application of TVD schemes for the Euler equations of gas dynamics. Zbl 0526.76080 Yee, H. C.; Warming, R. F.; Harten, Ami 1985 Boundary approximations for implicit schemes for one-dimensional inviscid equations of gasdynamics. Zbl 0496.76065 Yee, H. C.; Beam, R. M.; Warming, R. F. 1982 Stability analysis of numerical boundary conditions and implicit difference approximations for hyperbolic equations. Zbl 0488.65039 Beam, R. M.; Warming, R. F.; Yee, H. C. 1982 Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods. Zbl 0468.76066 Steger, Joseph L.; Warming, R. F. 1981 Alternating direction implicit methods for parabolic equations with a mixed derivative. Zbl 0462.65060 Beam, Richard M.; Warming, R. F. 1980 An extension of A-stability to alternating direction implicit methods. Zbl 0448.65058 Warming, R. F.; Beam, Richard M. 1979 An implicit factored scheme for the compressible Navier-Stokes equations. Zbl 0374.76025 Beam, Richard M.; Warming, R. F. 1978 On the construction and application of implicit factored schemes for conservation laws. Zbl 0392.65038 Warming, R. F.; Beam, Richard M. 1978 An implicit finite-difference algorithm for hyperbolic systems in conservation-law form. Zbl 0336.76021 Beam, Richard M.; Warming, R. F. 1976 Upwind second-order difference schemes and applications in aerodynamic flows. Zbl 0364.76047 Warming, R. F.; Beam, Richard M. 1976 Diagonalization and simultaneous symmetrization of the gas-dynamic matrices. Zbl 0313.65084 Warming, R. F.; Beam, Richard M.; Hyett, B. J. 1975 The modified equation approach to the stability and accuracy analysis of finite-difference methods. Zbl 0291.65023 Warming, R. F.; Hyett, B. J. 1974 Computation of space shuttle flowfields using noncentered finite- difference schemes. Zbl 0268.76050 Kutler, Paul; Warming, R. F.; Lomax, Harvard 1973 Second- and third-order noncentered difference schemes for nonlinear hyperbolic equations. Zbl 0268.76048 Warming, R. F.; Kutler, Paul; Lomax, Harvard 1973 Multishocked, three-dimensional supersonic flowfields with real gas effects. Zbl 0258.76047 Kutler, P.; Reinhardt, W. A.; Warming, R. F. 1973 all top 5 ### Cited by 1,446 Authors 29 Dehghan Takht Fooladi, Mehdi 16 Ramos, Juan I. 16 Visbal, Miguel R. 12 Gaitonde, Datta V. 9 Farhat, Charbel H. 9 Sheu, Tony Wen-Hann 9 Yee, Helen C. 9 Yuan, Li 8 Rizzetta, Donald P. 7 Shah, Abdullah 6 Böttcher, Albrecht 6 Taghavi, Seyed-Mohammad 5 Dervieux, Alain 5 Dubois, François 5 González-Pinto, Severiano 5 Jiang, Zonglin 5 Koobus, Bruno 5 Lanteri, Stéphane 5 Mohammadi, Bijan 5 Napolitano, Michele 5 Parent, Bernard 5 Pulliam, Thomas H. 5 Shokin, Yuriĭ Ivanovich 5 Tsangaris, Sokrates 5 Villatoro, Francisco R. 5 Wolf, William R. 4 Baker, Allen J. 4 Embree, Mark 4 He, Feng 4 Hejranfar, Kazem 4 Hernandez-Abreu, Domingo 4 Hughes, Thomas J. R. 4 Kapen, Pascalin Tiam 4 Kwak, Dochan 4 Pascazio, Giuseppe 4 Qu, Feng 4 Ran, Yuhong 4 Shen, Yiqing 4 Shu, Chi-Wang 4 Sousa, Ercília 4 Sun, Di 4 Tang, Huazhong 4 Tchuen, Ghislain 4 Toro, Eleuterio F. 4 Verwer, Jan G. 4 Warming, Robert F. 4 Xu, Kun 4 Yan, Chao 4 Yang, Jaw-Yen 3 Amat, Sergio P. 3 Bai, Junqiang 3 Belhaj, Skander 3 Bertolazzi, Enrico 3 Blondeaux, Paolo 3 Briley, W. Roger 3 Bruno, Oscar P. 3 Burtschell, Yves 3 Cai, Jinsheng 3 Candler, Graham V. 3 Chang, Sinchung 3 Chen, Chifang 3 Chourushi, Tushar 3 Cubillos, Max 3 Di Mascio, Andrea 3 Drikakis, Dimitris 3 Düring, Bertram 3 Felippa, Carlos A. 3 García López, Carmen María 3 Gutfinger, Chaim 3 Hamdani, Hossein Raza 3 Hcini, Fahd 3 Hsieh, Li-Wen 3 Hundsdorfer, Willem H. 3 Hussaini, M. Yousuff 3 Ikhile, Monday Ndidi Oziegbe 3 Jespersen, Dennis C. 3 Kalita, Paragmoni 3 Karaa, Samir 3 Karniadakis, George Em 3 Kawashima, Rei 3 Komurasaki, Kimiya 3 Kuzmin, Dmitri 3 Kwon, Oh Joon 3 Lallemand, Pierre 3 Lele, Sanjiva K. 3 Lerat, Alain 3 Lin, Reui-Kuo 3 Liu, Kaixin 3 Liu, Ruxun 3 MacCormack, Robert W. 3 McDonald, Henry 3 Moretti, Gino 3 Noelle, Sebastian 3 Noye, B. J. 3 Ogunfeyitimi, S. E. 3 Pérez Rodríguez, S. 3 Raghurama Rao, S. V. 3 Rider, William J. 3 Salvetti, Maria Vittoria 3 Sengupta, Tapan Kumar ...and 1,346 more Authors all top 5 ### Cited in 125 Serials 182 Journal of Computational Physics 162 Computers and Fluids 55 Computer Methods in Applied Mechanics and Engineering 48 Applied Mathematics and Computation 41 International Journal for Numerical Methods in Fluids 30 Journal of Fluid Mechanics 29 International Journal of Computational Fluid Dynamics 28 Journal of Computational and Applied Mathematics 26 Applied Numerical Mathematics 22 Computers & Mathematics with Applications 20 Journal of Scientific Computing 14 International Journal of Computer Mathematics 13 Applied Mathematical Modelling 12 Mathematics and Computers in Simulation 12 Physics of Fluids 11 Mathematics of Computation 9 Mathematical and Computer Modelling 9 International Journal of Numerical Methods for Heat & Fluid Flow 8 Computational Mechanics 7 Applied Mathematics and Mechanics. (English Edition) 6 Acta Mechanica 6 Computer Physics Communications 6 Fluid Dynamics 6 Communications in Numerical Methods in Engineering 5 Numerical Methods for Partial Differential Equations 5 European Journal of Mechanics. B. Fluids 4 Physics of Fluids, A 4 Computing 4 RAIRO. Modélisation Mathématique et Analyse Numérique 4 Linear Algebra and its Applications 4 Journal of Computational Acoustics 3 Journal of Engineering Mathematics 3 Shock Waves 3 International Journal for Numerical Methods in Engineering 3 SIAM Journal on Numerical Analysis 3 SIAM Journal on Scientific Computing 3 Advances in Computational Mathematics 3 Matematicheskoe Modelirovanie 3 Archives of Computational Methods in Engineering 3 Sādhanā 3 International Journal of Computational Methods 3 Acta Mechanica Sinica 3 Advances in Applied Mathematics and Mechanics 3 International Journal of Applied and Computational Mathematics 2 International Journal of Heat and Mass Transfer 2 Journal of Mathematical Analysis and Applications 2 Journal of Mathematical Biology 2 ZAMP. Zeitschrift für angewandte Mathematik und Physik 2 BIT 2 Integral Equations and Operator Theory 2 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2 Numerical Algorithms 2 Russian Journal of Numerical Analysis and Mathematical Modelling 2 Engineering Analysis with Boundary Elements 2 Journal of Applied Mechanics and Technical Physics 2 M2AN. Mathematical Modelling and Numerical Analysis. ESAIM, European Series in Applied and Industrial Mathematics 2 Computational Geosciences 2 Nonlinear Analysis. Real World Applications 2 Journal of the Korean Society for Industrial and Applied Mathematics 2 Communications on Applied Mathematics and Computation 1 Modern Physics Letters B 1 Communications in Mathematical Physics 1 International Journal of Engineering Science 1 Journal of Mathematical Physics 1 Journal of Statistical Physics 1 Mathematical Methods in the Applied Sciences 1 Wave Motion 1 Theoretical and Computational Fluid Dynamics 1 Acta Mathematica Vietnamica 1 Automatica 1 International Journal of Mathematics and Mathematical Sciences 1 Journal of Soviet Mathematics 1 Kybernetes 1 Kyungpook Mathematical Journal 1 Numerische Mathematik 1 Bulletin of the Korean Mathematical Society 1 Journal of Symbolic Computation 1 Constructive Approximation 1 Communications in Applied Numerical Methods 1 COMPEL 1 Asymptotic Analysis 1 Applications of Mathematics 1 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 1 Computational Mathematics and Mathematical Physics 1 Journal of Elasticity 1 SIAM Review 1 Computers and Mathematics with Applications. Part A 1 Acta Mechanica Sinica. (English Edition) 1 Archive of Applied Mechanics 1 Numerical Linear Algebra with Applications 1 Computational and Applied Mathematics 1 ETNA. Electronic Transactions on Numerical Analysis 1 Complexity 1 Journal of Mathematical Chemistry 1 Mathematical Problems in Engineering 1 Differential Equations and Dynamical Systems 1 Nonlinear Dynamics 1 Abstract and Applied Analysis 1 ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik 1 International Journal of Theoretical and Applied Finance ...and 25 more Serials all top 5 ### Cited in 35 Fields 692 Fluid mechanics (76-XX) 436 Numerical analysis (65-XX) 213 Partial differential equations (35-XX) 42 Classical thermodynamics, heat transfer (80-XX) 25 Mechanics of deformable solids (74-XX) 21 Linear and multilinear algebra; matrix theory (15-XX) 20 Biology and other natural sciences (92-XX) 17 Ordinary differential equations (34-XX) 17 Geophysics (86-XX) 12 Statistical mechanics, structure of matter (82-XX) 10 Optics, electromagnetic theory (78-XX) 6 Operator theory (47-XX) 6 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 5 Approximations and expansions (41-XX) 5 Integral equations (45-XX) 5 Astronomy and astrophysics (85-XX) 5 Systems theory; control (93-XX) 4 Harmonic analysis on Euclidean spaces (42-XX) 4 Computer science (68-XX) 4 Quantum theory (81-XX) 3 Dynamical systems and ergodic theory (37-XX) 3 Calculus of variations and optimal control; optimization (49-XX) 3 Operations research, mathematical programming (90-XX) 2 Combinatorics (05-XX) 2 Real functions (26-XX) 2 Special functions (33-XX) 2 Difference and functional equations (39-XX) 2 Statistics (62-XX) 2 Mechanics of particles and systems (70-XX) 1 Mathematical logic and foundations (03-XX) 1 Number theory (11-XX) 1 Category theory; homological algebra (18-XX) 1 Functions of a complex variable (30-XX) 1 Probability theory and stochastic processes (60-XX) 1 Relativity and gravitational theory (83-XX)	2022-05-17 13:52:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.4362614154815674, "perplexity": 9803.468653282494}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662517485.8/warc/CC-MAIN-20220517130706-20220517160706-00469.warc.gz"}
http://www.algebra1help.org/learning-how-to-multiply-polynomials-together/	# Learning How to Multiply Polynomials Together When it comes to multiplying polynomials, a lot of people get stuck and aren’t really sure what to do. This normally happens because people get anxious over the amount of information they’re having to deal with and try to hold it all in their memory at once. This isn’t the right way to handle things because it takes away from the actual multiplication process and makes it hard to focus on what you’re doing. Before you can multiply whole polynomials, you have to be able to quickly and intuitively multiply polynomial terms. All this involves is using the commutative property of multiplication to rearrange the order of the variables and constants in question. For example, if we have, then we can drop all of the parenthesis to really have just, and we can rearrange the order however we want. To make things easy on ourselves, we should put the constants beside of each other and all similar variables beside each other to get something like which simplifies to just give us. Notice that the exponents for each variable just add together. Once you have the idea of multiplying polynomials terms down pat, you can move on to multiplying whole polynomials, which requires the application of the distributive property. Let’s use some simple polynomials like for an example. First we’ll distribute to each of the terms of, and then do the indicated distributions that come up from there and simplify as much as we can. Here’s what it could look like: There’s not really much to multiplying polynomials as long as you stay focused on the process at hand instead of trying to remember the exact form of all of the polynomials you’re dealing with. They’re written down in front of you for a reason! You will not forget them.	2022-08-18 07:03:31	{"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 10, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7543757557868958, "perplexity": 231.4565175445766}, "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/1659882573172.64/warc/CC-MAIN-20220818063910-20220818093910-00427.warc.gz"}
https://physics.stackexchange.com/questions/165943/bells-original-inequality-in-speakable-and-unspeakable-in-quantum-mechanics	# Bell's original inequality in "Speakable and Unspeakable in Quantum Mechanics I'm having difficulty in understanding the setting for the derivation of Bell's inequality. The passage which sets the context below is from the beginning of the second essay in "Speakable and Unspeakable in Quantum Mechanics" by J.S. Bell. ... Consider a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern-Gerlach magnets, on selected components of the spins $\sigma_1$ and $\sigma_2$. If measurement of the component $\sigma_1 \cdot a$ where $a$ is some unit vector, yields the value $+1$ then, according to quantum mechanics, measurement of $\sigma_2 \cdot a$ must yield the value $-1$ and vice versa. Now we make the hypothesis, and it seems at least one worth considering, that if the two measurements are made at places remote from one another the orientation of one magnet does not influence the result obtained with the other. Since we can predict in advance the result of measuring any chosen component of $\sigma_2$, by previously measuring the same component of $\sigma_1$, it follows that the result of any such measurement must actually be predetermined. Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state. Let this more complete specification be effected by means of a parameter $\lambda$. It is a matter of indifference in the following whether $\lambda$ denotes a single variable or a set, or even a set of functions, and whether the variables are discrete or continuous. However, we write as if $\lambda$ were a single continuous parameter. The result $A$ of measuring $\sigma_1 \cdot a$ is then determined by $a$ and $\lambda$, and the result $B$ of measuring $\sigma_2 \cdot b$ in the same instance is determined by $b$ and $\lambda$, and $A(a,\lambda)=\pm 1, B(b,\lambda)=\pm 1$ The vital assumption is that the result $B$ for particle 2 does not depend on the setting $a$, of the magnet for particle 1, nor $A$ on $b$. If $p(\lambda)$ is the probability distribution of $\lambda$ then the expectation value of the product of the two components $\sigma_1 \cdot a$ and $\sigma_2 \cdot b$ is $P(a,b)=\int d\lambda p(\lambda)A(a,\lambda)B(b,\lambda)$ This should equal the quantum mechanical expectation value, which for the singlet state is $\langle \sigma_1 \cdot a \sigma_2 \cdot b\rangle=-a\cdot b$ ... I have three questions: Firstly, what is the meaning of $p(\lambda)$? $\lambda$ is a parameter (such as position in Bohmian mechanics). Does it just give the probability that a randomly selected system has a particular value of $\lambda$? That's the only way I can interpret it - $\lambda$ is meant to be predetermined. Secondly, what within the framework of QM dictates that measurements of the components $\sigma_1\cdot a$ and $\sigma_2 \cdot a$ must be opposite? I'm not questioning the rule, I just want to know where it comes from. Finally, how does the rule for the expectation value $\langle \sigma_1 \cdot a \sigma_2 \cdot b\rangle=-a\cdot b$ come about? • You should have better denoted the hidden parameter by, say, $\Lambda$, and by $\{ \lambda \}$ the set of his values. So, $p(\lambda)$ is the probability that the variable $\Lambda$ take the value $\lambda$. Feb 20 '15 at 0:56 I) You should have better denoted the hidden parameter by, say, $\Lambda$ , and by ${λ}$ the set of his values. So, $p(λ)$ is the probability that the variable $\Lambda$ take the value $λ$. II) The state named spin-singlet has the form $$\|S\rangle = \frac {\|\uparrow\rangle \|\downarrow\rangle - \|\downarrow\rangle \|\uparrow\rangle}{\sqrt {2}}. \tag{1}$$ The states $\|\uparrow\rangle$ and $\|\downarrow\rangle$ are the eigenstates of the operator $\hat {\sigma}_z$. However, if you choose another axis in space instead of $z$, the spin-projection eigenstates will transform, but the resulting singlet state will have an analogous form. For instance, if you prefer to pass from the axis $z$ to the axis $x$, the singlet becomes $$\|S\rangle = \frac {\|\rightarrow\rangle \|\leftarrow\rangle - \|\leftarrow\rangle \|\rightarrow\rangle}{\sqrt {2}}. \tag{2}$$ Here, instead of spin-up and spin-down, we have spin-to-the-right and spin-to-the-left. Thus, along whatever direction is space you measure the spin-projection of one particle, the spin-projection of the other particle is opposite. III) As to your last question, the relation you seek is proved in a simple way in Ballentine's book "Quantum Mechanics" section 20-2 (Spin correlations). To put in short that proof, we choose to express the singlet in the eigenstates of $\hat {\sigma}_a$, let's name them $\begin{bmatrix}1 \\ 0\end{bmatrix}$ and $\begin{bmatrix}0 \\ 1\end{bmatrix}$. The operator $\hat {\sigma}_b$ has the form, $\begin{bmatrix} {cos\theta} & {sin\theta} \\ {sin\theta} & {-cos\theta}\end{bmatrix}$, where $\theta$ is the angle between the axes $\vec a$ and $\vec b$. So, there remains to calculate the expression $$\frac {1}{2} \left( [1 \ 0] [0 \ 1] - [0 \ 1][1 \ 0] \right) \hat {\sigma}_b \left( \begin{bmatrix}1 \\ 0\end{bmatrix} \ \begin{bmatrix}0 \\ 1\end{bmatrix} - \begin{bmatrix}0 \\ 1\end{bmatrix} \ \begin{bmatrix}1 \\ 0\end{bmatrix} \right) \tag{3}$$ Since $\hat {\sigma}_b$ acts only on the 2nd vector in each product, what remains from the form $(3)$ is $$= \frac {1}{2} \left( [0 \ 1]\hat {\sigma}_b \begin{bmatrix}0 \\ 1\end{bmatrix} - [1 \ 0]\hat {\sigma}_b \begin{bmatrix}1 \\ 0\end{bmatrix} \right) = -cos\theta \tag{4}.$$	2021-11-29 12:44:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8690042495727539, "perplexity": 195.26244631301842}, "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/1637964358705.61/warc/CC-MAIN-20211129104236-20211129134236-00546.warc.gz"}
https://quizplus.com/quiz/153364-quiz-25-product-liability-warranties-and-torts	# Business Law Study Set 13 ## Quiz 25 :Product Liability: Warranties and Torts Question Type Jackson purchased a sealed can of Katydids, chocolate-covered pecan caramel candies manufactured by NestlT. Shortly after, Jackson bit into one of the candies and allegedly broke a tooth on a pecan shell embedded in the candy. She filed a complaint, asserting breach of implied warranty. How would you argue on behalf of the company? How would you argue on behalf of Jackson? In your answer, discuss both the reasonable expectation test and the foreign substance/natural substance test. [Jackson v NestlT-Beich, Inc., 589 NE2d 547 (Ill App)] Free Essay Refer to the case Jackson v Nestle-Beich, Inc (589 NE2d 547). Case Issue Trial court held for defendant by applying the natural substance test, arguing that pecans, chocolate covered or not, will have shells. Appeals court reversed the decision applying the reasonable expectation test that chocolate covered pecans should not have any remaining nut shells. Relevant Terms, Laws, and Cases Breach of implied warranty - for food there is an implied warranty that the food should at least follow some standards needed for consumption, such as not having foreign object. For example, a needle found in a bowl of soup would be breach of implied warranty. However, it is debatable when a found object is naturally occurring to the ingredient such as finding egg shells in an omelet. Opinion Higher court affirmed the decision. The court argued under strict liability that defendant could have placed a warning in their food products stating that shells may have remained in the food. Since they failed to state that it caused potential danger to consumers who unknowlingly eats the food believing it to be free of shells. Tags Choose question tag Zogarts manufactured and sold a practice device for beginning golfers. According to statements on the package, the device was completely safe, and a player could never be struck by the device's golf ball. Hauter was hit by the ball while using the device. He sued Zogarts, which denied liability on the ground that the statements were merely matters of opinion, so liability could not be based on them. Was this a valid defense? [Hauter v Zogarts, 534 P2d 377 (Cal)] Free Essay Refer to the case Hauter v Zogarts (534 P2d 377). Case Issue The issue is whether a statement, the ball will not strike player, made on a packaging of a product made by defendant is an express warranty for the product. Trial court granted judgment not withstanding verdict in favor of plaintiff. Defendant appealed. Relevant Terms, Laws, and Cases Express warranty - are terms sellers made about product that urges buyers to buy it. For example, claiming that a TV is HD is an express warranty that the TV is HD. However, opinions are not express warranties, e.g. claiming the product is amazing would be an opinion. Opinion The higher court agreed that there was a breach of express warranty. The court found that it was clear that the statement made by the defendant was not merely an opinion but statement of fact. Furthermore, the court agreed with trial court argument that the defendant can be held liable under breach of implied warranty as well; since the product was intended for beginner golfers but was found to be unsafe for use. Tags Choose question tag Brian Felley went to the home of Tom and Cheryl Singleton on June 8 to look at a used car that the Singletons had advertised for sale in the local paper. The car was a 1991 Ford with 126,000 miles on it. Following a test drive and the Singletons' representation that the car was "in good mechanical condition," Felley purchased the car for $5,800. By June 18, 1997, Felley had the car in the shop and had paid$942.76 to have its clutch fixed. By July 9, 1997, Felley also had paid $971.18 for a new brake job. By September 16, 1997, Felley had paid another$429.09 for further brake work. Felley brought suit for breach of express warranty. An auto expert testified that the clutch and brakes were defective when Felley bought the car. Was an express warranty breached? Why or why not? [Felley v Singleton, 705 NE2d 930 (Ill App)] Free Essay Refer to the case Felley v Singleton (705 NE2d 930). Case Issue The issue is whether a statement that a car is in good mechanical condition is an express warranty. The plaintiff (buyer ) is suing defendant (seller) for defect in the purchased vehicle's brakes. Trial court held for plaintiff. Defendant appealed the decision. Relevant Terms, Laws, and Cases For example, claiming that a TV is HD is an express warranty that the TV has HD quality. However, opinions such as "This is a great product" is not an express warranty. Opinion Higher court affirmed the decision. The court found that it was an expressed warranty because when the plaintiff specifically asked about the car's condition they relied on this fact to make their purchase. Furthermore , plaintiff was unaware of defective brakes on the car. Hence, the plaintiffs have shown that there was a breach of warranty. Tags Choose question tag Parrino purchased from Dave's Professional Wheelchair Service a wheelchair manufactured by 21st Century Scientific, Inc. The sales brochure from 21st Century Scientific stated that the wheelchair would "serve [the buyer] well for many years to come." Parrino had problems with the wheelchair within a few years and filed suit against Dave's and 21st Century for breach of express warranty. Both defended on the grounds that the statement on years of service was puffery, not an express warranty. Are they right? [Parrino v Sperling, 648 NYS2d 702] Essay Tags Choose question tag Essay Tags Choose question tag James Jelinek purchased Hytest BMR Sorghum Sudan grass seed, which was produced and marketed by Land O'Lakes. Land O'Lakes warranted the seed to be free from defects and expressly warranted that by using normal farming practices and proper maintenance, Mr. Jelinek would obtain yields of 4 1/2 tons per acre. The seed resulted in reduced yields and an inferior quality crop. As a result, Mr. Jelinek was not able to sell his crop and had significant economic losses. Mr. Jelinek filed suit for breach of express warranty. Is the promise of a crop yield an express warranty? Explain your answer. [ Jelinek v. Land O ' Lakes, Inc. , 797 N.W.2d 289 (Neb. App.)] Essay Tags Choose question tag Maria Gonzalez lived in a rental unit with her sons in Queens, New York. The hot water supplied to their apartment was heated by a Morflo water heater, which had a temperature control device on its exterior manufactured by Robertshaw and sold to Morflo. Maria Garcia, the owner of the Gonzalezes' apartment, had purchased and installed the water heater. The Morflo heater was located in the basement of the apartment house, which was locked and inaccessible to tenants. Extensive warnings were on the water heater itself and in the manual given to Garcia at the time of her purchase. The warning on the Robertshaw temperature device read: "CAUTION: Hotter water increases the risk of scald injury." The heater itself contained a picture of hot water coming from a faucet with the word "DANGER" printed above it. In addition, the water heater had a statement on it: "Water temperature over 120 degrees Fahrenheit can cause severe burns instantly or death from scalds. Children, disabled, and elderly are at highest risk of being scalded. Feel water before bathing or showering. Temperature limiting valves are available, see manual." In the Morflo manual, the following warning appeared: DANGER! The thermostat is adjusted to its lowest temperature position when shipped from the factory. Adjusting the thermostat past the 120 degree Fahrenheit bar on the temperature dial will increase the risk of scald injury. The normal position is approximately 120 degrees Fahrenheit. DANGER: WARNING: Hot water can produce first degree burns in 3 seconds at 140 degrees Fahrenheit (60 degrees Celsius), in 20 seconds at 130 degrees Fahrenheit (54 degrees Celsius), in 8 minutes at 120 degrees Fahrenheit (49 degrees Celsius). On October 1, 1992, 15-month-old Angel Gonzalez was being bathed by his 15-year-old brother, Daniel. When the telephone rang, Daniel left Angel alone in the bathtub. No one else was at home with the boys, and Daniel left the water running. Angel was scalded by the water that came from the tap. Angel and his mother brought suit against Morflo and Robertshaw, alleging defects in the design of the water heater and the failure to warn. Should they recover? [Gonzalez v Morflo Industries, Inc., 931 F Supp 159 (EDNY)] Essay Tags Choose question tag Oil Gas, Inc., is a company that manufactures gas compressors. Berge Helene owns a large barge that it leases to oil companies for purposes of storing and producing petroleum offshore. GE Oil Gas sold Berge Helene gas compressors that were to be used on the barge. Berge Helene representatives asked GE representatives, as they were negotiating the contract for the compressors, whether the compressors could withstand the movement and vibration that would occur on the front of the barge once it was out in the ocean. GE's representatives assured those from Berge Helene that the compressors were self-stabilizing. Once out in the ocean, the gas compressors on the hull exploded once the vibrations began. Berge Helene brought suit against GE for the resulting crew injuries and damage to the barge. Could Berge Helene recover and, if so, what theory of product liability would apply? [ Berge Helene Ltd. v. GE Oil Gas, Inc. , 830 F. Supp. 2d 235 (S.D. Tex.)] Essay Tags Choose question tag Avery purchased a refrigerator from a retail store. The written contract stated that the refrigerator was sold "as is" and that the warranty of merchantability and all warranties of fitness were excluded. This was stated in large capital letters printed just above the line on which Avery signed her name. The refrigerator worked properly for a few weeks and then stopped. The store refused to do anything about it because of the exclusion of the warranties made by the contract. Avery claimed that this exclusion was not binding because it was unconscionable. Was Avery correct? [Avery v Aladdin Products Div., Nat'l Service Industries, Inc., 196 SE2d 357 (Ga App)] Essay Tags Choose question tag Brianna Kriefall, a child, died after she ate meat at a Sizzler restaurant that was later found to contain E. coli. Her family brought suit against Sizzler USA to recover for the loss of their daughter. Is Sizzler liable for the death? Explain your answer. What would be the liability of Sizzler's meat supplier in the case? [ Estate of Kriefall v. Sizzler USA Franchise, Inc. , 801 N.W.2d 781 (Wis. App.)] Essay Tags Choose question tag Webster ordered a bowl of fish chowder at the Blue Ship Tea Room. She was injured by a fish bone in the chowder, and she sued the tea room for breach of the implied warranty of merchantability. The evidence at trial showed that when chowder is made, the entire boned fish is cooked. Should she recover? [Webster v Blue Ship Tea Room, 198 NE2d 309] Essay July 27, 2000, Sheldorado Aluminum Products, Inc., installed an aluminum awning on the back of Marie Villette's home for use as a carport. On January 11, 2001, the awning collapsed on top of Ms. Villette's new Mercedes automobile. Ms. Villette brought suit against Sheldorado seeking recovery of the $3,000 she had paid to them for the awning. There was no formal written contract between the parties; the only writing was a one-page order/bill designated a "contract," dated July 11, 2000, and signed by Ms. Villette and apparently by Jack Finklestein, Sheldorado's salesman. No advertising or promotional material was presented by either party. Ms. Villette testified to no express warranty or representation on the transaction, and none appears in the writing. Sheldorado acknowledges that no instructions or warnings were given to Ms. Villette as to care, maintenance, or use of the awning. When the awning collapsed, Sheldorado took the position that the cause was an accumulation of snow and high winds and that it bore no responsibility for the loss. Its only response to the incident was to refer Ms. Villette to the insurer on their homeowner's policy. Does Ms. Villette have any rights that would allow her to collect damages? Apply the UCC to answer this question. Villette v. Sheldorado Aluminum Products, Inc., 2001 WL 881055 (NY Supp), 45 UCC Rep Serv. 2d 470 (NY Civ Ct). Essay Answer: Tags Choose question tag Andy's Sales (owned by Andy Adams) sold a well-built trampoline to Carl and Shirley Wickers. The Wickerses later sold the trampoline to Herbert Bryant. While using the trampoline, Herbert's 14-year-old nephew, Rex, sustained injuries that left him a quadriplegic. Rex's guardian filed suit for breach of express warranty and merchantability. The sales brochure for the round trampoline described it as "safe" because it had a "uniform bounce" and "natural tendency to work the jumper toward the center." The Wickerses had purchased an oval-shaped trampoline. Discuss Rex's ability to recover. Is privity an issue? [Bryant v Adams, 448 SE2d 832 (NC App)] Essay Answer: Tags Choose question tag After watching a male horse owned by Terry and Manita Darby perform at a horse show, Ashley Sheffield contacted the Darbys about buying him. The Darbys assured her that the horse had no problems and would make a good show horse for use in competition. In the presence of and in consultation with her father (who raised horses for a business), Sheffield rode the horse and decided to purchase him for$8,500. Within three weeks, Sheffield and her trainer discerned that the horse was lame. Sheffield sued the Darbys for fraud and for breach of express and implied warranties, and the court entered summary judgment in favor of the Darbys on all claims. Sheffield appealed. Was the court correct in granting summary judgment? Was there a breach of an express warranty? [Sheffield v Darby, 535 SE2d 776 (Ga App)]	2022-08-08 13:29:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.1932462751865387, "perplexity": 8365.310245962228}, "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/1659882570827.41/warc/CC-MAIN-20220808122331-20220808152331-00299.warc.gz"}
https://www.cuemath.com/ncert-solutions/q-5-exercise-2-6-linear-equations-in-one-variable-class-8-maths/	Ex.2.6 Q5 Linear Equations in One Variable Solutions-Ncert Maths Class 8 Go back to 'Ex.2.6' Question Solve:\begin{align} \frac{{7y + 4}}{{y + 2}} = \frac{{ - 4}}{3}\end{align} Text Solution What is known? Equation What is unknown? Value of the variable Reasoning: Multiple both sides by the L.C.M of the denominators to get rid of fractional number. Now transpose variables to one side and constant to another side. Steps: On multiplying both sides by $$3(y + 2)$$,we obtain \begin{align}3\left( {7{\rm{ }}y + 4} \right) &= - 4\left( {y + 2} \right)\\21y + 12 &= - 4y - 8\\21y + 4y &= - 8 - 12\\\,\,\,\,\,\,\,\,\,\,25y &= - 20\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y &= - \frac{4}{5}\end{align} Learn from the best math teachers and top your exams • Live one on one classroom and doubt clearing • Practice worksheets in and after class for conceptual clarity • Personalized curriculum to keep up with school	2019-11-22 11:08:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 4, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9999085664749146, "perplexity": 4738.893641143893}, "config": {"markdown_headings": false, "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-2019-47/segments/1573496671249.37/warc/CC-MAIN-20191122092537-20191122120537-00259.warc.gz"}
http://hal.in2p3.fr/view_by_stamp.php?label=IN2P3&langue=fr&action_todo=view&id=in2p3-00670808&version=1	HAL : in2p3-00670808, version 1 arXiv : 1202.3372 Journal of Cosmology and Astroparticle Physics 04 (2012) 006 Three-dimensional track reconstruction for directional Dark Matter detection J. Billard1, F. Mayet1, D. Santos1 MIMAC Collaboration(s) (2012) Directional detection of Dark Matter is a promising search strategy. However, to perform such detection, a given set of parameters has to be retrieved from the recoiling tracks : direction, sense and position in the detector volume. In order to optimize the track reconstruction and to fully exploit the data of forthcoming directional detectors, we present a likelihood method dedicated to 3D track reconstruction. This new analysis method is applied to the MIMAC detector. It requires a full simulation of track measurements in order to compare real tracks to simulated ones. We conclude that a good spatial resolution can be achieved, i.e. sub-mm in the anode plane and cm along the drift axis. This opens the possibility to perform a fiducialization of directional detectors. The angular resolution is shown to range between 20$^\circ$ to 80$^\circ$, depending on the recoil energy, which is however enough to achieve a high significance discovery of Dark Matter. On the contrary, we show that sense recognition capability of directional detectors depends strongly on the recoil energy and the drift distance, with small efficiency values (50%-70%). We suggest not to consider this information either for exclusion or discovery of Dark Matter for recoils below 100 keV and then to focus on axial directional data. Thème(s) : Physique/Astrophysique/Instrumentation et méthodes pour l'astrophysiquePlanète et Univers/Astrophysique/Instrumentation et méthodes pour l'astrophysiquePhysique/Physique/Instrumentations et Détecteurs Lien vers le texte intégral : http://fr.arXiv.org/abs/1202.3372 in2p3-00670808, version 1 http://hal.in2p3.fr/in2p3-00670808 oai:hal.in2p3.fr:in2p3-00670808 Contributeur : Emmanuelle Vernay <> Soumis le : Jeudi 16 Février 2012, 10:30:54 Dernière modification le : Mercredi 11 Avril 2012, 08:44:24	2014-08-21 10:30:10	{"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": 0.5708501935005188, "perplexity": 2246.9647978932844}, "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-2014-35/segments/1408500815861.64/warc/CC-MAIN-20140820021335-00202-ip-10-180-136-8.ec2.internal.warc.gz"}
https://www.math.harvard.edu/event_archive/?cat=informal-geometry-and-dynamics	# Events Archive Filter By: All Categories ANNOUNCEMENTS SEMINARS MATHEMATICAL PICTURE LANGUAGE INFORMAL GEOMETRY AND DYNAMICS ALGEBRAIC DYNAMICS COMPLEX ANALYSIS DIFFERENTIAL GEOMETRY GAUGE-TOPOLOGY-SYMPLECTIC HARVARD-MIT ALGEBRAIC GEOMETRY HARVARD-MIT COMBINATORICS LOGIC NUMBER THEORY OPEN NEIGHBORHOOD RANDOM MATRIX SPECIAL SEMINAR SYMPLECTIC GEOMETRY THURSDAY SEMINAR TRIVIAL NOTIONS CONFERENCES COLLOQUIUMS HARVARD-MIT-BU-BRANDEIS-NORTHEASTERN LOGIC OTHER COLLOQUIUMS OTHER MATHEMATICS DEPARTMENT EVENTS AHLFORS LECTURE SERIES MATH MOTH MATH TABLE SOCIAL CMSA EVENTS CMSA GENERAL RELATIVITY SEMINAR ### Slices of Thurston’s Master Teapot INFORMAL GEOMETRY AND DYNAMICS June 3, 2020 4:00 pm Speaker: Kathryn Lindsey - Boston College Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of... ### In the moduli space of Abelian differentials, big invariant subvarieties come from topology INFORMAL GEOMETRY AND DYNAMICS May 13, 2020 4:00 pm Speaker: Paul Apisa - Yale University It is a beautiful fact that any holomorphic one-form on a genus g Riemann surface can be presented as a collection of polygons in the plane with sides identified by... ### Coarse density of subsets of moduli space INFORMAL GEOMETRY AND DYNAMICS May 6, 2020 4:00 pm Speaker: Benjamin Dozier - Stony Brook University I will discuss coarse geometric properties of algebraic subvarieties of the moduli space of Riemann surfaces. In joint work with Jenya Sapir, we prove that such a subvariety is coarsely... ### Large genus bounds for the distribution of triangulated surfaces in moduli space INFORMAL GEOMETRY AND DYNAMICS April 29, 2020 4:00 pm Speaker: Sahana Vasudevan - MIT Triangulated surfaces are compact (hyperbolic) Riemann surfaces that admit a conformal triangulation by equilateral triangles. Brooks and Makover started the study of the geometry of random large genus triangulated surfaces. Mirzakhani... ### Pseudo-Anosov maps and toral automorphisms INFORMAL GEOMETRY AND DYNAMICS April 22, 2020 4:00 pm Speaker: Rick Kenyon - Yale University We give a construction of a pseudo-Anosov map of a surface starting from (and almost isomorphic to) a hyperbolic automorphism of an n-torus. The construction arises from a peano curve... ### Framed mapping class groups and strata of abelian differentials INFORMAL GEOMETRY AND DYNAMICS April 15, 2020 4:00 pm Speaker: Nick Salter - Columbia University via Zoom Video Conferencing: https://harvard.zoom.us/j/972495373 Strata of abelian differentials have long been of interest for their dynamical and algebro-geometric properties, but relatively little is understood about their topology. I will describe a... ### Effective density for values of generic quadratic forms INFORMAL GEOMETRY AND DYNAMICS April 8, 2020 4:00 pm Speaker: Dubi Kelmer - Boston College via Zoom Video Conferencing: https://harvard.zoom.us/j/972495373 The Oppenheim Conjecture, proved by Margulis, states that any irrational quadratic form, has values (at integer coordinates) that are dense on the real line. However,... ### Heights INFORMAL GEOMETRY AND DYNAMICS April 1, 2020 4:00 pm Speaker: Curtis McMullen - Harvard University via Zoom Video Conferencing: https://harvard.zoom.us/j/972495373 We will describe how the problem of finding periodic trajectories in a regular pentagon can be solved using a new height on P^1 coming from...	2021-04-11 17:56:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3884226083755493, "perplexity": 7629.775351845398}, "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/1618038064898.14/warc/CC-MAIN-20210411174053-20210411204053-00336.warc.gz"}
https://www.gradesaver.com/textbooks/math/trigonometry/trigonometry-7th-edition/chapter-6-section-6-1-solving-trigonometric-equations-6-1-problem-set-page-325/6	## Trigonometry 7th Edition a) $60^{\circ} + 360^{\circ}k$ and $300^{\circ} + 360^{\circ}k$, for all integers $k$. b) $60^{\circ}, 300^{\circ}$ b) Isolate $cos\theta$ $2cos\theta = 1$ $cos\theta = \frac{1}{2}$ Recall knowledge about the unit circle $cos\theta$ is positive in quadrant I and IV - Angles $60˚$ and $300˚$ Therefore, $cos\theta = \frac{1}{2}$ for $\theta = 60˚, 300˚$ a) Take the two angles and add $360^{\circ}k$, for all integers $k$, to find all degree solutions. $60^{\circ} + 360^{\circ}k$ and $300^{\circ} + 360^{\circ}k$, for all integers $k$.	2018-12-18 17:42:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5554343461990356, "perplexity": 256.4169632639711}, "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/1544376829542.89/warc/CC-MAIN-20181218164121-20181218190121-00012.warc.gz"}
https://yutsumura.com/ring-is-a-filed-if-and-only-if-the-zero-ideal-is-a-maximal-ideal/?wpfpaction=add&postid=1345	# Ring is a Filed if and only if the Zero Ideal is a Maximal Ideal ## Problem 172 Let $R$ be a commutative ring. Then prove that $R$ is a field if and only if $\{0\}$ is a maximal ideal of $R$. ## Proof. ### $(\implies)$: If $R$ is a field, then $\{0\}$ is a maximal ideal Suppose that $R$ is a field and let $I$ be a non zero ideal: $\{0\} \subsetneq I \subset R.$ Then the ideal $I$ contains a nonzero element $x \neq 0$. Since $R$ is a field, we have the inverse $x^{-1}\in R$. Then it follows that $1=x^{-1}x \in I$ since $x$ is in the ideal $I$. Since $1\in I$, any element $r \in R$ is in $I$ as $r=r\cdot 1 \in I$. Thus we have $I=R$ and this proves that $\{0\}$ is a maximal ideal of $R$. ### $(\impliedby)$: If $\{0\}$ is a maximal ideal, then $R$ is a field Let us now suppose that $\{0\}$ is a maximal ideal of $R$. Let $x$ be any nonzero element in $R$. Then the ideal $(x)$ generated by the element $x$ properly contains the ideal $\{0\}$. Since $\{0\}$ is a maximal ideal, we must have $(x)=R$. Since $1\in R=(x)$, there exists $y\in R$ such that $1=xy$. This implies that the element $x$ is invertible. Therefore any nonzero element of $R$ is invertible, and hence $R$ is a field. ##### Nilpotent Element a in a Ring and Unit Element $1-ab$ Let $R$ be a commutative ring with $1 \neq 0$. An element $a\in R$ is called nilpotent if $a^n=0$ for...	2020-06-06 21:49:43	{"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": 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": 0.9368210434913635, "perplexity": 48.45473795311514}, "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-2020-24/segments/1590348519531.94/warc/CC-MAIN-20200606190934-20200606220934-00006.warc.gz"}
https://socratic.org/questions/what-is-the-slope-of-the-tangent-line-of-r-theta-sin-theta-6-2pi-3-at-theta-pi-6	# What is the slope of the tangent line of r=theta-sin((theta)/6+(2pi)/3) at theta=(pi)/6? Aug 14, 2018 Slope of tangent line at $\theta = \frac{\pi}{6}$ is $1.096$ #### Explanation: r=theta-sin(theta/6+(2 pi)/3); theta= pi/6 (dr)/(d theta)= 1 -cos (theta/6+(2 pi)/3)1/6; Slope of tangent line at $\theta = \frac{\pi}{6}$ is (dr)/(d theta)(pi/6)= 1 -cos (pi/36+(2 pi)/3)1/6; or $\frac{\mathrm{dr}}{d \theta} \left(\frac{\pi}{6}\right) \approx 1.096 \left(3 \mathrm{dp}\right)$ Slope of tangent line at $\theta = \frac{\pi}{6}$ is $1.096$ [Ans]	2022-05-18 14:09:37	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 9, "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": 0.2939720153808594, "perplexity": 13436.727352813949}, "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/1652662522270.37/warc/CC-MAIN-20220518115411-20220518145411-00190.warc.gz"}
https://www.biztrends.xyz/what-is-net-worth-everything-you-need-to-know/	What is net worth ? Everything you need to know. What is net worth ? Everything you need to know about net worth, and how to calculate your networth According to investopedia Net worth is the value the assets a person or corporation owns, minus the liabilities they owe. It is an important metric to gauge a company’s health and it provides a snapshot of the firm’s current financial position. Net Worth; what it really mean. Net worth is calculated by subtracting all liabilities from assets. An asset is anything owned and has monetary value, while liabilities are obligations that deplete resources. Positive net worth means that assets exceed liabilities, while negative net worth results when liabilities exceed assets. Positive and increasing net worth indicates good financial health while decreasing net worth is cause for concern as it might signal a decrease in assets relative to liabilities. The best way to improve net worth is to either reduce liabilities while assets stay constant or rise, or increase assets while liabilities either stay constant or fall. KEY TAKEAWAYS ON EVERY THING YOU NEED TO KNOW ABOUT YOUR NET WORTH • Net worth is a quantitative concept that measures the value of an entity and can apply to individuals, corporations, sectors, and even countries. • Net worth provides a snapshot of an entity’s current financial position. • In business, net worth is also known as book value or shareholders’ equity. The balance sheet is also known as a net worth statement. • People with substantial net worth are known as high-net-worth individuals (HNWI). What Does Net Worth Tell You About Your Finances? What do you know about your net worth Theoretically, your net worth is the value in cash you would have if you were to sell everything you own and paid off all of your debts. In some cases, this number is actually negative, which indicates that you own more in liabilities than in assets. While this is not an ideal situation, it is very common for people just out of college or starting their careers. In that case, your net worth is also a measure of how much debt you would still owe if you emptied your bank accounts and sold everything you own to put towards your debt. Though neither is a realistic scenario, what your net worth measures is more important than the (generally unrealistic) assumptions that are made to get to that number. In fact, when it comes to your financial health, so to speak, there is no ubiquitous magic net worth number you should be striving for. But, you should use your net worth to track your progress from year to year and to hopefully see it improve and grow over time. How to Calculate Your Net Worth Calculating your net worth can be a simple process, but it requires that you gather all the information surrounding your current assets and liabilities. Most financial planners recommend that their clients keep a secure folder with information on all financial assets and liabilities to be updated at least once a year. Gathering and organizing this information can be a bit of a chore at first, but ensures that you (and anyone else who might need it like your spouse or financial advisor) have access to the information when needed. Though such a folder can be turned into much more, calculating your net worth only requires basic financial information regarding the things you own and the debt that you owe. Here’s how to get started: 1. Start by listing your largest assets. For most people, this could include the value of their home, any real estate properties, or vehicles like personal cars or boats. In the case of a business owner, this list would also include the value of their business, which has its own more complicated calculation. Make sure you use accurate estimates of market values in current dollars. 2. Next, you’ll want to gather your latest statements for your more liquid assets. These assets include checking and savings accounts, cash, CDs or other investments such as brokerage accounts or retirement accounts. 3. Finally, consider listing other personal items that may be of value. These could include valuable jewelry, coin collections, musical instruments, heirlooms, a rare wine collection, etc. You don’t need to itemize everything, but you can try to list items that are worth $500 or more. 4. Now, take all of the assets you have listed in the first three steps and add them together. This number represents your total assets. READ: LEARN MORE ON HOW TO CALCULATE NETWORK Calculate Your Liabilities 1. Again, start with themajor outstanding liabilities such as the balance on your mortgage or car loans. List these loans and their most current balances. 2. Next, list all of your personal liabilities such as any balance on your credit cards, student loans, or any other debt you may owe. 3. Now, add up the balances on all of the liabilities you listed above. This number represents your total liabilities. Calculate Your Net Worth 1. To calculate your net worth, simply subtract the total liabilities from the total assets. For this exercise, it doesn’t matter how big or how small the number. It doesn’t necessarily matter if the number is negative. Your net worth is just a starting point to have something to compare against in the future. 2. Repeat this process at least once a year and compare it with the previous year’s number. By comparing the two, you can then determine if you are making progress or getting further behind on your goals. You may want to recalculate your net worth more often if you’ve embarked on an aggressive savings or debt repayment plan. More Net Worth Tips: Be conservative with estimates, especially with home and vehicle values. Inflating the value of large assets may look good on paper, but may not paint an accurate picture of your net worth. Consider using a budgeting app that tracks your net worth for you automatically. Keep liquid savings in high-yield accounts, which can help them grow faster if you’re earning a competitive annual percentage yield. Make debt repayment a priority and consider refinancing or consolidating debts at a lower interest rate to help speed up your debt payoff. Review your budget to look for areas where you can reduce expenses and allocate more money to either savings or debt repayment. If you have additional money to save, consider maxing out your emergency fund, then maxing out your annual contributions to an individual retirement account. Net Worth in Business In business, net worth is also known as book value or shareholders’ equity. The balance sheet is also known as a net worth statement. The value of a company’s equity equals the difference between the value of total assets and total liabilities. Note that the values on a company’s balance sheet highlight historical costs or book values, not current market values. Lenders scrutinize a business’s net worth to determine if it is financially healthy. If total liabilities exceed total assets, a creditor may not be too confident in a company’s ability to repay its loans. A consistently profitable company will have a rising net worth or book value as long as these earnings are not fully distributed to shareholders as dividends. For a public company, a rising book value will often be accompanied by an increase in the value of the company’s stock price. Net Worth in Personal Finance An individual’s net worth is simply the value that is left after subtracting liabilities from assets. Examples of liabilities (debt) include mortgages, credit card balances, student loans, and car loans. An individual’s assets include checking and savings account balances, the value of securities (e.g., stocks or bonds), real property value, the market value of an automobile, et al. In other words, whatever is left after selling all assets and paying off personal debt is the net worth. Note that the value of personal net worth includes the current market value of assets and the current debt costs. People with a substantial net worth are known as high net worth individuals (HNWI), and form the prime market for wealth managers and investment counselors. Investors with a net worth (excluding their primary residence) of at least$1 million – either alone or together with their spouse – are “accredited investors” by the Securities and Exchange Commission (SEC), to invest in unregistered securities offerings. To calculate your net worth, use our free Net Worth Tracker which allows you to calculate, analyze, and record your and know your net worth for free. What Is Deficit Net Worth? Deficit net worth is a situation in which net liabilities are higher than net assets. Also known as negative net worth, this can occur for a variety of reasons, but typically it arises when current or future asset values erode unexpectedly. KEY TAKEAWAYS • Deficit net worth occurs when the values of liabilities is greater than the value of assets, leading to a net debt. • Such negative net worth can occur suddenly if future projections change in such a way that impairs present value calculations for assets. While deficit net worth is concerning, it is not immediately imply bankruptcy for a firm or individual if net worth can recover over the short-term. Example of Deficit Net Worth For example, during the global financial crisis in 2008 when home values fell sharply, many people were left owing more on their mortgage than the home was presently worth (they were underwater on the mortgages). Since a home is often the largest asset a person will own, this led to many households experiencing a deficit net worth.Likewise, back in frontier days, land and property often gained or lost value suddenly depending on where the nearest railroad was located. For more on deficit networth check( investopedia) Tangible Net Worth Tangible net worth is most commonly a calculation of the net worth of a company that excludes any value derived from intangible assets, such as copyrights, patents, and intellectual property. Tangible net worth for a company is essentially the total value of a company’s physical assets. These assets can include: • Cash • Accounts receivables or money owed to a company from its customers for sales • Inventory, such as finished goods • Equipment, such as machinery and computers • Buildings • Real estate • Investments For an individual, the tangible net worth calculation includes such items as home equity, any other real estate holdings, bank and investment accounts, and major personal assets such as an automobile or jewelry. Relatively insignificant personal assets are not ordinarily included in the calculation for an individual. Formula and Calculation of Tangible Net Worth \begin{aligned} &\text{TNW} = \text{Total Assets} – \text{Liabilities} – \text{Intangible Assets} \\ &\textbf{where:} \\ &\text{TNW} = \text{Tangible Net Worth} \\ \end{aligned} TNW=Total Assets−Liabilities−Intangible Assetswhere:TNW=Tangible Net Worth Tangible net worth is calculated as follows: 1. Locate the company’s total assets, total liabilities, and intangible assets, which are all listed on the balance sheet. 2. Take total assets and subtract total liabilities. 3. Take the result and subtract intangible assets. 4. Tangible net worth can also be calculated for individuals, using the same formula of total tangible assets minus total debt liabilities. KEY TAKEAWAYS • Tangible net worth is typically the net worth of a company excluding • intangible assets such as copyrights, patents, and intellectual property. • The tangible net worth calculation for a company is total assets minus total liabilities minus intangible assets. • Tangible net worth can also be calculated for individuals, using the same formula of total tangible assets minus total debt liabilities. I hope this guide helped you to see evry thing you need to know about your net worth. if you want to know how to calculate your networth check out this guide here on how to calculate your net worth	2020-11-27 11:51:21	{"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": 0.25841063261032104, "perplexity": 2895.8676468455437}, "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-2020-50/segments/1606141191692.20/warc/CC-MAIN-20201127103102-20201127133102-00044.warc.gz"}
https://q4interview.com/quantitative-aptitude/square-root-and-cube-root/formulas	Take FREE!! Online Cocubes Mock Test to Crack various Companies Written Exams. Get Off-Campus Placement Jobs Info !!! Take 30+ FREE!! Online Arithmetic Aptitude Mock test to crack any Exams. # Quantitative Aptitude :: Square Root and Cube Root ## Square Root and Cube Root Important Formulas 1. Square Root: If x2 = y, we say that the square root of y is x and we write $$\sqrt{y}$$ = x. Thus, $$\sqrt{4}$$ = 2, $$\sqrt{9}$$ = 3, $$\sqrt{144}$$ = 12. 2. Cube Root: The cube root of a given number x is the number whose cube is x. We, denote the cube root of x by $$\sqrt[3]{x}$$. Thus, $$\sqrt[3]{27}$$ = 3 x 3 x 3 = 3, $$\sqrt[3]{343}$$ = 7 x 7 x 7 = 7 etc. Note: 1. $$\sqrt{xy}$$ = $$\sqrt{x}$$ x $$\sqrt{y}$$ 2. $$\sqrt{\frac{x}{y}}$$ = $$\frac{\sqrt{x}}{\sqrt{y}}$$ = $$\frac{\sqrt{x}}{\sqrt{y}}$$ x $$\frac{\sqrt{y}}{\sqrt{y}}$$ = $$\frac{\sqrt{xy}}{y}$$ Q4interview.com is a first own type of educational portal, which aims is to cater to provide companies interview questions, mock test & Job info to jobseekers. Its Mock Test provides a deep competitive analysis of your performance and points out your weak and strong areas, through intuitive graphical reports, which helps you to improve your skill.	2021-07-31 05:39:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4362182319164276, "perplexity": 1550.489922827537}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154053.17/warc/CC-MAIN-20210731043043-20210731073043-00212.warc.gz"}
https://bartoszmilewski.com/2020/04/	### April 2020 Previously, we talked about the construction of initial algebras. The dual construction is that of terminal coalgebras. Just like an algebra can be used to fold a recursive data structure into a single value, a coalgebra can do the reverse: it lets us build a recursive data structure from a single seed. Here’s a simple example. We define a tree that stores values in its nodes data Tree a = Leaf \| Node a (Tree a) (Tree a) We can build such a tree from a single list as our seed. We can choose the algorithm in such a way that the tree is ordered in a particular way split :: Ord a => [a] -> Tree a split [] = Leaf split (a : as) = Node a (split l) (split r) where (l, r) = partition (<a) as A traversal of this tree will produce a sorted list. We’ll get back to this example after working out some theory behind it. # The functor The tree in our example can be derived from the functor data TreeF a x = LeafF \| NodeF a x x instance Functor (TreeF a) where fmap _ LeafF = LeafF fmap f (NodeF a x y) = NodeF a (f x) (f y) Let’s simplify the problem and forget about the payload of the tree. We’re interested in the functor data F x = LeafF \| NodeF x x Remember that, in the construction of the initial algebra, we were applying consecutive powers of the functor to the initial object. The dual construction of the terminal coalgebra involves applying powers of the functor to the terminal object: the unit type () in Haskell, or the singleton set $1$ in $Set$. Let’s build a few such trees. Here are a some terms generated by the single power of F w1, u1 :: F () w1 = LeafF u1 = NodeF () () And here are some generated by the square of F acting on () w2, u2, t2, s2 :: F (F ()) w2 = LeafF u2 = NodeF w1 w1 t2 = NodeF w1 u1 s2 = NodeF u1 u1 Or, graphically Notice that we are getting two kinds of trees, ones that have units () in their leaves and ones that don’t. Units may appear only at the $(n+1)$-st layer (root being the first layer) of $F^n$. We are also getting some duplication between different powers of $F$. For instance, we get a single LeafF at the $F$ level and another one at the $F^2$ level (in fact, at every consecutive level after that as well). The node with two LeafF leaves appears at every level starting with $F^2$, and so on. The trees without unit leaves are the ones we are familiar with—they are the finite trees. The ones with unit leaves are new and, as we will see, they will contribute infinite trees to the terminal coalgebra . We’ll construct the terminal coalgebra as a limit of an $\omega$-chain. # Terminal coalgebra as a limit As was the case with initial algebras, we’ll construct a chain of powers of $F$, except that we’ll start with the terminal rather than the initial object, and we’ll use a different morphism to link them together. By definition, there is only one morphism from any object to the terminal object. In category theory, we’ll call this morphism $\mbox{!} \colon a \to 1$ (upside-down exclamation mark) and implement it in Haskell as a polymorphic function unit :: a -> () unit _ = () First, we’ll use $\mbox{!}$ to connect $F 1$ to $1$, then lift $\mbox{!}$ to connect $F^2 1$ to $F 1$, and so on, using $F^n \mbox{!}$ to transform $F^{n + 1} 1$ to $F^n 1$. Let’s see how it works in Haskell. Applying unit directly to F () turns it into (). Values of the type F (F ()) are mapped to values of the type F () w2' = fmap unit w2 > LeafF u2' = fmap unit u2 > NodeF () () t2' = fmap unit t2 > NodeF () () s2' = fmap unit s2 > NodeF () () and so on. The following pattern emerges. $F^n 1$ contains trees that end with either leaves (at any level) or values of the unit type (only at the lowest, $(n+1)$-st level). The lifted morphism $F^{n-1} \mbox{!}$ (the $(n-1)$st power of fmap acting on unit) operates strictly on the $n$th level of a tree. It turns leaves and two-unit-nodes into single units (). Alternatively, we can look at the preimages of these mappings—conceptually reversing the arrows. Observe that all trees at the $F^2$ level can be seen as generated from the trees at the $F$ level by replacing every unit () with either a leaf LeafF or a node NodeF ()(). It’s as if a unit were a universal seed that can either sprout a leaf or a two-unit-node. We’ll see later that this process of growing recursive data structures from seeds corresponds to anamorphisms. Here, the terminal object plays the role of a universal seed that may give rise to two parallel universes. These correspond to the inverse image (a so-called fiber) of the lifted unit. Now that we have an $\omega$-chain, we can define its limit. It’s easier to understand a limit in the category of sets. A limit in $Set$ is a set of cones whose apex is the singleton set. The simplest example of a limit is a product of sets. In that case, a cone with a singleton at the apex corresponds to a selection of elements, one per set. This agrees with our understanding of a product as a set of tuples. A limit of a directed finite chain is just the starting set of the chain (the rightmost object in our pictures). This is because all projections, except for the rightmost one, are determined by commuting triangles. In the example below, $\pi_b$ is determined by $\pi_a$: $\pi_b = f_1 \circ \pi_a$ and so on. Here, every cone from $1$ is fully determined by a function $1 \to a$, and the set of such functions is isomorphic to $a$. Things are more interesting when the chain is infinite, and there is no rightmost object—as is the case of our $\omega$-chain. It turns out that the limit of such a chain is the terminal coalgebra for the functor $F$. In this case, the interpretation where we look at preimages of the morphisms in the chain is very helpful. We can view a particular power of $F$ acting on $1$ as a set of trees generated by expanding the universal seeds embedded in the trees from the lower power of $F$. Those trees that had no seeds, only LeafF leaves, are just propagated without change. So the limit will definitely contain all these trees. But it will also contain infinite trees. These correspond to cones that select ever growing trees in which there are always some seeds that are replaced with double-seed-nodes rather than LeafF leaves. Compare this with the initial algebra construction which only generated finite trees. The terminal coalgebra for the functor TreeF is larger than the initial algebra for the same functor. We have also seen a functor whose initial algebra was an empty set data StreamF a x = ConsF a x This functor has a well-defined non-empty terminal coalgebra. The $n$-th power of (StreamF a) acting on () consists of lists of as ConsF a1 (ConsF a2 (... (ConsF an ())...)) The lifting of unit acting on such a list replaces the final (ConsF a ()) with () thus shortening the list by one item. Its “inverse” replaces the seed () with any value of type a (so it’s a multi-valued inverse, since there are, in general, many values of a). The limit is isomorphic to an infinite stream of a. In Haskell it can be written as a recursive data structure data Stream a = ConsF a (Stream a) # Anamorphism The limit of a diagram is defined as a universal cone. In our case this would be the cone consisting of the object we’ll call $\nu F$, with a set of projections $\pi_n$ such that any other cone factors through $\nu F$. We want to show that $\nu F$ (if it exists) is a terminal coalgebra. First, we have to show that $\nu F$ is indeed a coalgebra, that is, there exists a morphism $k \colon \nu F \to F (\nu F)$ We can apply $F$ to the whole diagram. If $F$ preserves $\omega$-limits, then we get a universal cone with the apex $F (\nu F)$ and the $\omega$-chain with $F 1$ on the left. But our original object $\nu F$ forms a cone over the same chain (ignoring the projection $\pi_0$). Therefore there must be a unique mapping $k$ from it to $F (\nu F)$. The coalgebra $(\nu F, k)$ is terminal if there is a unique morphism from any other coalgebra to it. Consider, for instance, a coalgebra $(A, \kappa \colon A \to F A)$. With this coalgebra, we can construct an $\omega$-chain We can connect the two omega chains using the terminal morphism from $A$ to $1$ and all its liftings Notice that all squares in this diagram commute. The leftmost one commutes because $1$ is the terminal object, therefore the mapping from $A$ to it is unique, so the composite $\mbox{!} \circ F \mbox{!} \circ \kappa$ must be the same as $\mbox{!}$. $A$ is therefore an apex of a cone over our original $\omega$-chain. By universality, there must be a unique morphism from $A$ to the limit of this $\omega$-chain, $\nu F$. This morphism is in fact a coalgebra morphism and is called the anamorphism. The constructions of initial algebras and terminal coalgebras can be compactly described using adjunctions. There is an obvious forgetful functor $U$ from the category of $F$-algebras $C^F$ to $C$. This functor just picks the carrier and forgets the structure map. Under certain conditions, the left adjoint free functor $\Phi$ exists $C^F ( \Phi x, a) \cong C(x, U a)$ This adjunction can be evaluated at the initial object (the empty set in $Set$). $C^F ( \Phi 0, a) \cong C(0, U a)$ This shows that there is a unique algebra morphism—the catamorphism— from $\Phi 0$ to any algebra $a$. This is because the hom-set $C(0, U a)$ is a singleton for every $a$. Therefore $\Phi 0$ is the initial algebra $\nu F$. Conversely, there is a cofree functor $\Psi$ $C_F(c, \Psi x) \cong C(U c, x)$ It can be evaluated at a terminal object $C_F(c, \Psi 1) \cong C(U c, 1)$ showing that there is a unique coalgebra morphism—the anamorphism—from any coalgebra $c$ to $\Psi 1$. This shows that $\Psi 1$ is the terminal coalgebra $\nu F$. # Fixed point Lambek’s lemma works for both, initial algebras and terminal coalgebras. It states that their structure maps are isomorphisms, therefore their carriers are fixed points of the functor $F$ $\mu F \cong F (\mu F)$ $\nu F \cong F (\nu F)$ The difference is that $\mu F$ is the least fixed point, and $\nu F$ is the greatest fixed point of $F$. They are, in principle, different. And yet, in a programming language like Haskell, we only have one recursively defined data structure defining the fixed point newtype Fix f = Fix { unFix :: f (Fix f) } So which one is it? We can define both the catamorphisms from-, and anamorphisms to-, the fixed point: type Algebra f a = f a -> a cata :: Functor f => Algebra f a -> Fix f -> a cata alg = alg . fmap (cata alg) . unfix type Coalgebra f a = a -> f a ana :: Functor f => Coalgebra f a -> a -> Fix f ana coa = Fix . fmap (ana coa) . coa so it seems like Fix f is both initial as the carrier of an algebra and terminal as the carrier of a coalgebra. But we know that there are elements of $\nu F$ that are not in $\mu F$—namely infinite trees and infinite streams—so the two fixed points are not isomorphic and cannot be both described by the same Fix f. However, they are not unrelated. Because of the Lambek’s lemma, the initial algebra $(\mu F, j)$ gives rise to a coalgebra $(\mu F, j^{-1})$, and the terminal coalgebra $(\nu F, k)$ generates an algebra $(\nu F, k^{-1})$. Because of universality, there must be a (unique) algebra morphism from the initial algebra $(\mu F, j)$ to $(\nu F, k^{-1})$, and a unique coalgebra morphism from $(\mu F, j^{-1})$ to the terminal coalgebra $(\nu F, k)$. It turns out that these two are given by the same morphism $f \colon \mu F \to \nu F$ between the carriers. This morphism satisfies the equation $k \circ f \circ j = F f$ which makes it both an algebra and a coalgebra morphism Furthermore, it can be shown that, in $Set$, $f$ is injective: it embeds $\mu F$ in $\nu F$. This corresponds to our observation that $\nu F$ contains $\mu F$ plus some infinite data structures. The question is, can Fix f describe infinite data? The answer depends on the nature of the programming language: infinite data structures can only exist in a lazy language. Since Haskell is lazy, Fix f corresponds to the greatest fixed point. The least fixed point forms a subset of Fix f (in fact, one can define a metric in which it’s a dense subset). This is particularly obvious in the case of a functor that has no terminating leaves, like the stream functor. data StreamF a x = StreamF a x deriving Functor We’ve seen before that the initial algebra for StreamF a is empty, essentially because its action on Void is uninhabited. It does, however have a terminal coalgebra. And, in Haskell, the fixed point of the stream functor indeed generates infinite streams type Stream a = Fix (StreamF a) How do we know that? Because we can construct an infinite stream using an anamorphism. Notice that, unlike in the case of a catamorphism, the recursion in an anamorphism doesn’t have to be well founded and, indeed, in the case of a stream, it never terminates. This is why this won’t work in an eager language. But it works in Haskell. Here’s a coalgebra whose carrier is Int coaInt :: Coalgebra (StreamF Int) Int coaInt n = StreamF n (n + 1) It generates an infinite stream of natural numbers ints = ana coaInt 0 Of course, in Haskell, the work is not done until we demand some values. Here’s the function that extracts the head of the stream: hd :: Stream a -> a hd (Fix (StreamF x _)) = x And here’s one that advances the stream tl :: Stream a -> Stream a tl (Fix (StreamF _ s)) = s This is all true in $Set$, but Haskell is not $Set$. I had a discussion with Edward Kmett and he pointed out that Haskell’s fixed point data type can be considered the initial algebra as well. Suppose that you have an infinite data structure, like the stream we were just discussing. If you apply a catamorphism for an arbitrary algebra to it, it will most likely never terminate (try summing up an infinite stream of integers). In Haskell, however, this is interpreted as the catamorphism returning the bottom $\bot$, which is a perfectly legitimate value. And once you start including bottoms in your reasoning, all bets are off. In particular Void is no longer uninhabited—it contains $\bot$—and the colimit construction of the initial algebra is no longer valid. It’s possible that some of these results can be generalized using domain theory and enriched categories, but I’m not aware of any such work. # Bibliography 2. Michael Barr, Terminal coalgebras for endofunctors on sets There is a bit of folklore about algebras in Haskell, which says that both the initial algebra and the terminal coalgebra for a given functor are defined by the same fixed point formula. This works for most common cases, but is not true in general. What is definitely true is that they are both fixed points–this result is called the Lambek’s lemma–but there may be many fixed points. The initial algebra is the least fixed point, and the terminal coalgebra is the greatest fixed point. In this series of blog posts I will explore the ways one can construct these (co-)algebras using category theory and illustrate it with Haskell examples. In this first installment, I’ll go over the construction of the initial algebra. # A functor Let’s start with a simple functor that generates binary trees. Normally, we would store some additional data in a tree (meaning, the functor would take another argument), either in nodes or in leaves, but here we’re just interested in pure shapes. data F a = Leaf \| Node a a deriving Show Categorically, this functor can be written as a coproduct (sum) of the terminal object $1$ (singleton) and the product of $a$ with itself, here written simply as $a^2$ $F a = 1 + a^2$ The lifting of functions is given by this implementation of fmap instance Functor F where fmap _ Leaf = Leaf fmap f (Node x y) = Node (f x) (f y) We can use this functor to build arbitrary level trees. Let’s consider, for instance, terms of type F Int. We can either build a Leaf, or a Node with two numbers in it x1, y1 :: F Int x1 = Leaf y1 = Node 1 2 With those, we can build next-level values of the type $F^2 a$ or, in our case, F (F Int) x2, y2 :: F (F Int) x2 = Leaf y2 = Node x1 y1 We can display y2 directly using show > Node Leaf (Node 1 2) or draw the corresponding tree Since $F$ is an endofunctor, so is $F^2$. Lifting a function $f \colon a \to b$ to $F^2$ can be implemented by applying fmap twice. Here’s the action of the function (+1) on our test values fmap (fmap (+1)) x2 > Leaf fmap (fmap (+1)) y2 > Node Leaf (Node 2 3) or, graphically, You can see that Leafs at any level remain untouched; only the contents of bottom Nodes in the tree are transformed. # The colimit construction The carrier of the initial algebra can be constructed as a colimit of an infinite sequence. This sequence is constructed by applying powers of $F$ to the initial object which we’ll denote as $0$. We’ll first see how this works in our example. The initial object in Haskell is defined as a type with no data constructor (we are ignoring the question of non-termination in Haskell). data Void deriving Show In Set, this is just an empty set. The Show instance for Void requires the pragma {-# language EmptyDataDeriving #-} Even though there are no values of the type Void, we can still construct a value of the type F Void z1 :: F Void z1 = Leaf This degenerate version of a tree can be drawn as This illustrates a very important property of our $F$: Its action on an empty set does not produce an empty set. This is what allows us to generate a non-trivial sequence of powers of $F$ starting with the empty set. Not every functor has this property. For instance, the construction of the initial algebra for the functor data StreamF a x = ConsF a x will produce an uninhabited type (empty set). Notice that this is different from its terminal coalgebra, which is the infinite stream data Stream a = Cons a (Stream a) This is an example of a functor whose initial algebra is not the same as the terminal coalgebra. Double application of our F to Void produces, again, a Leaf, as well as a Node that contains two Leafs. z2, v2 :: F (F Void) z2 = Leaf v2 = Node z1 z1 > Node Leaf Leaf Graphically, In general, powers of $F$ acting on Void generate trees which terminate with Leafs, but there is no possibility of having terminal Nodes). Higher and higher powers of $F$ acting on Void will eventually produce any tree we can think of. But for any given power, there will exist even larger trees that are not generated by it. In order to get all the trees, we could try to take a sum (a coproduct) of infinitely many powers. Something like this $\sum_{n = 0}^{\infty} F^n 0$ The problem is that we’d also get a lot of duplication. For instance, we saw that z1 was the same tree as z2. In general, a single Leaf is produced at all non-zero powers of $F$ acting on Void. Similarly, all powers of $F$ greater than one produce a single node with two leaves, and so on. Once a particular tree is produced at some power of $F$, all higher powers of $F$ will also produce it. We have to have a way of identifying multiply generated trees. This is why we need a colimit rather than a simple coproduct. As a reminder, a coproduct is defined as a universal cocone. Here, the base of the cocone is the set of all powers of $F$ acting on $0$ (Haskell Void). In a more general colimit, the objects in the base of the cocone may be connected by morphisms. Coming from the initial object, there can be only one morphism. We’ll call this morphism $!$ or, in Haskell, absurd absurd :: Void -> a absurd a = case a of {} This definition requires another pragma {-# language EmptyCase #-} We can construct a morphism from $F 0$ to $F^2 0$ as a lifting of $!$, $F !$. In Haskell, the lifting of absurd doesn’t change the shape of trees. Here it is acting on a leaf z1' :: F (F Void) z1' = fmap absurd z1 > Leaf We can continue this process of lifting absurd to higher and higher powers of $F$ z2', v2' :: F (F (F Void)) z2' = fmap (fmap absurd) z2 > Leaf v2' = fmap (fmap absurd) v2 > Node Leaf Leaf We can construct an infinite chain (this kind of directed chain indexed by natural numbers is called an $\omega$-chain) We can use this chain as the base of our cocone. The colimit of this chain is defined as the universal cocone. We will call the apex of this cocone $\mu F$ In $Set$ these constructions have simple interpretations. A coproduct is a discriminated union. A colimit is a discriminated union in which we identify all those injections that are connected by morphisms in the base of the cocone. For instance $\iota_0 = \iota_{(F 0)}\, \circ \, !$ $\iota_{(F 0)} = \iota_{(F^2 0)} \circ F !$ and so on. Here we use the lifted absurd (or $!$ in the picture above) as the morphisms that connect the powers of $F$ acting of Void (or $0$ in the picture). These are exactly the identifications that we were looking for. For instance, $F !$ maps the leaf generated by $F 0$ to the leaf which is the element of $F^2 0$. Or, translating it to Haskell, (fmap absurd) maps the leaf generated by F Void to the leaf generated by F (F Void), and so on. All trees generated by the $n$‘th power of $F$ are injected into the $n+1$‘st power of $F$ by absurd lifted by the $n$th power of $F$. The colimit is formed by equivalence classes with respect to these identifications. In particular, there is a class for a degenerate tree consisting of a single leaf whose representative can be taken from F Void, or from F (F Void), or from F (F (F Void)) and so on. # Initiality The colimit $\mu F$ is exactly the initial algebra for the functor $F$. This follows from the universal property of the colimit. First we will show that for any algebra $(A, \alpha \colon F A \to A)$ there is a unique morphism from $\mu F$ to $A$. Indeed, we can build a cocone with $A$ at its apex and the injections given by $!$ $\alpha \circ F !$ $\alpha \circ F \alpha \circ F^2 !$ and so on… Since the colimit $\mu F$ is defined by the universal cocone, there is a unique morphism from it to $A$. It can be shown that this morphism is in fact an algebra morphism. This morphism is called a catamorphism. # Fixed Point Lambek’s lemma states that the initial algebra is a fixed point of the functor that defines it $F (\mu F) \cong \mu F$ This can also be seen directly, by applying the functor to every object and morphism in the $\omega$-chain that defines the colimit. We get a new chain that starts at $F 0$ But the colimit of this chain is the same as the colimit $\mu F$ of the original chain. This is becuase we can always add back the initial object to the chain, and define its injection $\iota_0$ as the composite $\iota_0 = \iota_{(F 0)} \circ !$ On the other hand, if we apply $F$ to the whole universal cocone, we’ll get a new cocone with the apex $F (\mu F)$. In principle, this cocone doesn’t have to be universal, so we cannot be sure that $F (\mu F)$ is a colimit. If it is, we say that $F$ preserves the particular type of colimit—here, the $\omega$-colimit. Remember: the image of a cocone under a functor is always a cocone (this follows from functoriality). Preservation of colimits is an additional requirement that the image of a universal cocone be universal. The result is that, if $F$ preserves $\omega$-colimits, then the initial algebra $\mu F$ is a fixed point of $F$ $F(\mu F) \cong \mu F$ because both sides can be obtained as a colimit of the same $\omega$-chain. # Bibliography 1. Adamek, Milius, Moss, Initial Algebras, Terminal Coalgebras, and the Theory of Fixed Points of Functors	2021-01-22 21:44:31	{"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": 193, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9327538013458252, "perplexity": 611.3190019226064}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703531429.49/warc/CC-MAIN-20210122210653-20210123000653-00352.warc.gz"}
https://www.ask-math.com/universal-relation.html	# Universal Relation Universal relation is a relation on set A when A X A $\subseteq$ A X A. In other words, universal-relation is the relation if each element of set A is related to every element of A. For example : Relation on the set A = {1,2,3,4,5,6} by R = {(a,b) $\in$ R : \|a -b\|$\geq$0} We observe that \|a -b\|$\geq$0 for all a, b $\in$ A $\Rightarrow$(a,b)$\in$ R for all (a,b) $\in$ A X A $\Rightarrow$ each element of set A is related to every element of set A. $\Rightarrow$ R = A X A $\Rightarrow$ R is a universal relation on set A. Note : It is to note here that the void relation and the universal relation on a set A are respectively the smallest and the largest relations on set A. Both the void and universal relation are sometimes called trivial relations. ## Examples on Universal Relation Example : 1 Let A be the set of all students of a boys school. Show that the relation R on A given by R = {(a,b) : difference between the heights of a and b is less than 5 meters} is the universal-relation. Solution : It is obvious that the difference between the heights of any two students of the school has to be less than 5 meters. Therefore (a,b) $\in$ R for all a, b $\in$ A. $\Rightarrow$ R = A X A $\Rightarrow$ R is the universal-relation on set A. Home We at ask-math believe that educational material should be free for everyone. Please use the content of this website for in-depth understanding of the concepts. Additionally, we have created and posted videos on our youtube. We also offer One to One / Group Tutoring sessions / Homework help for Mathematics from Grade 4th to 12th for algebra, geometry, trigonometry, pre-calculus, and calculus for US, UK, Europe, South east Asia and UAE students. Affiliations with Schools & Educational institutions are also welcome. Please reach out to us on [email protected] / Whatsapp +919998367796 / Skype id: anitagovilkar.abhijit We will be happy to post videos as per your requirements also. Do write to us. Russia-Ukraine crisis update - 3rd Mar 2022 The UN General assembly voted at an emergency session to demand an immediate halt to Moscow's attack on Ukraine and withdrawal of Russian troops.	2022-10-04 19:49:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30315661430358887, "perplexity": 1312.5911458272365}, "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/1664030337524.47/warc/CC-MAIN-20221004184523-20221004214523-00322.warc.gz"}
https://math.stackexchange.com/questions/3511252/show-that-yoneda-embedding-preserves-exponentials	# Show that Yoneda embedding preserves exponentials Given a category $$\mathbb{C}$$, let $$\hat{\mathbb{C}}$$ be its category of presheaves. I want to show that the yoneda embedding $$y(A) = hom(-,A)$$ preserves exponential objects from $$\mathbb{C}$$. I tried playing around with adjunctions, but end up in quite a mess. Hence any help or insights is appreciated. Cheers We can string together definitions adjunctions, and the occasional use of the Yoneda lemma to get (for any object $$C$$ in the category $$\mathbb{C}$$): $$y(B)^{y(A)}(C) = \operatorname{Hom}(y(C), y(B)^{y(A)}) \cong \operatorname{Hom}(y(C) \times y(A), y(B)) \cong\\ \operatorname{Hom}(y(C \times A), y(B)) \cong y(B)(C \times A) = \operatorname{Hom}(C \times A, B) \cong \operatorname{Hom}(C, B^A) = y(B^A)(C).$$	2020-04-09 04:51:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 7, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9583435654640198, "perplexity": 184.13560109729457}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371829677.89/warc/CC-MAIN-20200409024535-20200409055035-00268.warc.gz"}
https://xenaproject.wordpress.com/	## Thoughts on the Pythagorean theorem I’m sure I’m saying nothing new here. I’m just explaining another example of how thinking about how to formalise things has taught me stuff about what mathematics is. ## What is the Pythagorean theorem? The Pythagorean theorem, a.k.a. Pythagoras’ theorem, comes in two parts. Firstly there is the theorem statement, which says that in a right angled triangle (like the dark blue one below), the square of the hypotenuse equals the sum of the squares of the other two sides. And then there is the proof, which originally is either due to Pythagoras or not depending on who you believe. Let’s start with the statement. ## What does the statement actually mean? Let’s take a look at the picture of the squares on the hypotenuse and the other two sides. The dark blue triangle is right-angled. The claim is that the square C is equal to the sums of the squares A and B. On the face of it, this is nonsense. If you take squares A and B, you have a picture containing two squares; but square C is just one square. How can one square equal two squares? But of course the claim is not that the pictures are equal, the claim is that the areas are equal. But what is area? To find out, let’s go back to Euclid. ## Euclid’s take on the theorem Euclid’s Elements contains a proof of the Pythagorean theorem, right at the end of book 1. The proof involves drawing some triangles and arguing that various things are “equal”. This approach is valid because Euclid has explicitly stated as his Common Notion 1 that equality, whatever it is, is transitive. One can chase this concept of equality back to Proposition 35, which claims that two parallelograms with the same base and the same height are “equal”. In fact this seems to be the first time that the word “equal” is used to mean “having equal area” in the Elements. Halving the parallelograms we deduce the more familiar Proposition 37, that two triangles with the same base and the same height are also “equal”. So what goes into the proof of Proposition 35, that two parallelograms with the same base and height are “equal” in the sense of having equal area? The key ideas in the proof are Euclid’s second and third common notions: that “equals added to equals are equal”, and “equals subtracted from equals are equal”. In high-level terms, these common notions imply that equality is not just an equivalence relation, but a congruence relation. But let’s see how Euclid uses these notions in his proofs. The two orange regions have equal areas, because they are both “equals added to equals”: the small triangles and the big triangles are both congruent. Here, the two larger triangles are congruent, so the two orange areas are equal, because they are equals (the dark blue triangle) subtracted from equals (the larger triangles). For Euclid, the equality of the areas of the two orange regions in these examples is axiomatic. Take a look at the proof of Proposition 35 to see how these facts are used to prove that two parallelograms with the same base and height are “equal”. ## Area in Euclid book 1 So, what Euclid does mean by the “area” of a shape? Well this is the funny thing — he never says, throughout book 1! He only says what it means for two shapes to have “equal area”! This is exactly what an equivalence relation is. An equivalence relation on a type is a concept of equality on terms of that type. It can be thought of as focussing on a particular attribute of the terms you are considering (for example the area of a shape, or the value of an integer modulo 10) and saying that two terms are equivalent if the values of those attributes are equal. Euclid is putting an equivalence relation on shapes. His definition of the relation involves cutting and pasting in geometry, and at the end of the day the proof of the Pythagorean theorem in Euclid is essentially a jigsaw puzzle. Here is an even simpler jigsaw puzzle proof: ## Euclid and type theory When Euclid did mathematics, he was doing type theory. For Euclid, points were terms of the Point type, lines were terms of the Line type, and planes were terms of the Plane type. Euclid wrote down the axioms his types satisfied (for example there was a unique line between any two distinct points) and proceeded to work from there. He has a definition of a 2-dimensional shape, and assuming that a plane exists, his shapes exist too. He defined an equivalence relation on 2D shapes, and proved that the 2D shape corresponding to the square on the hypotenuse was related to the 2D shape corresponding to the union of the squares on the other two sides, using properties of this relation which he has earlier axiomatised. The proof of Pythagoras’ theorem in Euclid is what is known as a synthetic proof. We assume that a Euclidean Plane exists and satisfies a list of axioms, which Euclid attempted to write down and which most of us never even contemplate. We then formulate the theorem, and prove it using the axioms. ## Numbers from geometry? Note that Euclid is in some kind of a position to define real numbers at this point, or at least the positive real numbers. For example, Euclid knows what it means for two line segments to have equal length — it means that you can translate and rotate one line until it coincides with the other. He could hence define the positive reals to be equivalence classes of line segments, under the equivalence relation of being the same length. However one runs into problems when it comes to completeness, something Euclid’s axioms were not really designed for. ## Geometry from numbers: Enter Descartes. Descartes suggested doing things the other way around, using numbers to do geometry rather than using geometry to define numbers. Descartes observed that one could label a point in the plane with an $x$ and $y$ coordinate. This changed everything. All of a sudden “the plane” (a term whose existence is never talked about in Euclid) becomes modelled by $\mathbb{R}^2$. Euclid’s definitions, common notions, and axioms now need to be revisited. We need to check that this more refined model satisfies the rules of Euclid’s game (a bit like checking that Einstein’s theory turns into Newton’s in the limit). We model a point as an ordered pair of real numbers, we can define lines as solutions to linear equations because the reals are a field so we have that language available. We can prove the parallel postulate no problem. The theory of integration gives us a way to measure lines (length), angles (measure), curves (length) and 2-dimensional shapes (area), using the natural (Euclidean) Riemannian metric on the plane. We can now completely rephrase Pythagoras’ theorem: it is now an equality of numbers. We can re-interpret the “jigsaw puzzle” proof in Euclid as a consequence of finite additivity of Lebesgue measure on the plane. We can also give a completely new proof, using the theorem that the distance from $(a,b)$ to $(c,d)$ is $\sqrt{(a-c)^2+(b-d)^2}$, as one can check using a line integral (modulo the theorem that the shortest distance between two points is a straight line, which needs proving in this context). I saw measure theory developed as an undergraduate, and probably a few years ago I would have argued that this is now the “proper” proof — but now I realise that this proof still has some synthetic element to it: namely, the real numbers. We have a non-synthetic plane, but it is made from synthetic numbers. ## What are the real numbers? I was told as an undergraduate that it was an axiom that the reals existed, and that they were a complete ordered field. All of the analysis I learnt as an undergraduate was built upon this assumption, or rather, this structure (addition, multiplication, inequality) and these axioms (associativity and commutativity of addition and multiplication etc) on this type (the reals). In some sense it is no different to Euclid, who also had types (e.g. points), structures (e.g. triangles) and axioms (e.g. the common notions, or the parallel postulate), but who was modelling the Pythagorean theorem in a different, arguably more primitive, way. ## Enter the analysts, bearing sequences Descartes solved the problem of how to represent points in a plane with real numbers, but for Descartes, the reals were a type. Many years later, Cauchy and Dedekind gave two ways to represent the reals using simpler objects. Indeed, Cauchy sequences and Dedekind cuts are (different) ways of building the reals from the rationals. Similarly, the rationals can be built from the integers, the integers from the naturals, and the naturals are…well, they’re just some synthetic thing satisfying Peano’s axioms, right? At this point one could argue that Pythagoras’ theorem has become a statement about sequences of pairs of natural numbers (or however else one is modelling positive rationals), and the natural numbers have no definition — they are synthetic. But we can go further. ## Enter the logicians, bearing sets. One thing that ZFC set theory (the usual set-theoretic foundations of 20th century mathematics) has going for it, is that it gives a very unambiguous answer to the question: “Why are the natural numbers a set?”. The answer is “It’s one of the axioms”. One thing against it is that in ZFC set theory, everything is a set, even stuff which you don’t want to be a set. For example, real numbers (like the area of the square on the hypotenuse) are now sets, and the Pythagorean Theorem is now a theorem about the equality of two sets, although we don’t know exactly what the sets are, because we can never be sure whether the real numbers in the Platonic universe (or whichever universe we’re operating in) use Cauchy sequences or Dedekind cuts. [Pro tip: if your universe offers good quotienting facilities, use Cauchy sequences: they’re cleaner.] The fact that we don’t know whether the reals being used in Pythagoras’ theorem are Cauchy sequences or Dedekind cuts is an indication that we have unfolded things too far, as far as Pythagoras’ theorem goes. Most mathematicians regard the real numbers as a type. A real number is not a set — Gauss or Riemann could certainly have informed you of that. It is interesting that we can keep unfolding this way — but we can never get to the bottom of things. We can’t define everything, we always have to start somewhere — a synthetic beginning. Euclid started with points, lines and planes. Descartes started with the reals. Cauchy and Dedekind observed that you could start with the naturals. Set theorists start with sets. There is no definition of a set in ZFC set theory — a set is just a term in a model of ZFC set theory. The model can be thought of as the type, and its elements (or whatever you want to call them — they are not elements of a set in the internal logic of the model) as the terms. The behaviour of sets is developed using the axioms. ## Pythagoras’ theorem : refinements of equality So what is Pythagoras’ theorem? Is it that two shapes are “equal”, two numbers are equal, or two sets are equal? In some sense, it’s all of these things. In some sense this story reminds me of chemistry at school. The joke in the chemistry class was that they were always telling you lies — models of atoms which were very naive, and then more careful models which were more accurate, culminating (for me) with an undergraduate quantum mechanics course which told me that an electron was actually some kind of a complex-valued function. It feels very similar here. Euclid had a perfectly adequate notion of geometry, he axiomatised what he needed and argued from there. Later on we found a way to model the plane from more fundamental objects such as the real numbers. After that we found a way of modelling the real numbers using the natural numbers, and in some sense this is where we stop; using either type theory or set theory, the natural numbers have no definition — their existence is asserted, and we use them via their API (addition, multiplication, the principle of induction). Maybe someone will come along in future with a theory of mathematical quarks, which can be used together with the quark axioms to build the natural numbers in a natural way, and then our understanding of what Pythagoras’ theorem is might change again. Posted in General, Olympiad stuff, Type theory, undergrad maths \| \| 7 Comments ## Two types of universe for two types of mathematician Thank you Johan for pointing out to me that the mathlib stats page had got really good! But one thing that made me laugh is that somehow on their stats for commits I see I have done just enough to get a mention! I was quite surprised. Let’s look at those mathlib committers, most of them far more prolific than I am. Who are they? One surprising thing about them: although we have a common goal, of formalising mathematics in Lean, we are a really mixed bag of people, some of whom you might expect to have no research interests in common with me at all. Yury Kudryashov is a post-doc in the maths department at Toronto. I’ve never met him, because of visa issues. The dark blue line shows his meteoric rise. Scott Morrison does TQFT‘s at ANU and was one of the founders of MathOverflow. Mario Carneiro is a PhD student of Jeremy Avigad, in the philosophy department at CMU. He taught me Lean. Johannes Hoelzl is a computer scientist who works for Apple on formally verifying their products. He wrote the filter library for mathlib, which was essential for Patrick Massot’s work on topology, in turn crucial for the construction of perfectoid spaces in Lean. Chris Hughes is an undergraduate at Imperial College London. He’s going to an MSc project with me next year, on a one-relator group tactic. Chris has taught me so much about what mathematics really is. Chris learnt Lean when he was a 1st year undergraduate and began to interpret all of his lectures from a type theoretic point of view, which in turn has led to what I think is an extraordinary understanding of “what is actually going on” in a pure mathematics undergraduate degree (in particular exactly how informal some of it is). Rob Lewis has a PhD in logic and is now a post-doc in the computer science department at VU Amsterdam. He wrote linarith (“Loves the jobs you hate”). I ran into serious technical problems with tactics malfunctioning when making the natural number game, and Rob wrote the code which enabled me to hack tactic mode and fix stuff, saving the project. He set up the mathlib documentation project. This is the community’s effort to explain Lean’s interface to mathematicians. Before his academic career, he was a teacher. Sébastien Gouëzel is a professor of mathematics in Nantes, who won the Brin Prize in 2019. He was the driving force behind manifolds in Lean. We now have $C^n$ and $C^\infty$ manifolds, over general complete fields such as the complexes, reals or $p$-adics, and in the real case you can also have corners. It was harder than you might initially think. Johan Commelin is a post-doc in arithmetic geometry in Freiburg, working with Annette Huber. He has a wife and three small kids, and last week uploaded not one but two papers to ArXiv about o-minimality. He developed the API for a new Lean type called a “group with zero”, for example a field or a division ring but forget about the addition. Talking to him about stuff like this made me understand that the definition of a UFD never uses addition, and is hence a special case of a more general definition. History sometimes needs rewriting, but it’s OK — we hide the details from you. A UFD is a monoid with zero, by the way (such that the underlying monoid is a product of a group and a free abelian monoid — that’s the full definition in fact, although there are two ways of interpreting “is”). Simon Hudon is a computer scientist and an excellent functional programmer. His definition of a monad is different to mine, and he knows what all the weird >-> symbols mean. He wrote a bunch of stuff in core Lean and was around since the start. Mostly meta. Patrick Massot has translated most of Bourbaki’s Topologie générale into Lean, and can now confirm that they were (almost always) right. He’s a topologist in Orsay (except now we say Saclay) and teaches his undergraduates using Lean. You should try his introductory analysis problem sheets and other stuff in the Lean Tutorials repo. If you have Lean installed the modern way, you can just type leanproject get tutorials and then open the project in VS Code. Patrick also wrote the leanproject tool, and it has solved the constant problems beginners on Windows machines were having when trying to get a fully compiled mathlib running locally without waiting an hour or more for it to compile. Reid Barton won four gold medals at the IMO. Kenny Lau is an undergraduate at Imperial College, whose first year project on formalising the statements of local class field theory for abelian algebraic groups in Lean unsurprisingly won the “best pure maths project of the year” prize. I needed localisation of rings when doing schemes and he bashed out the entire theory like it was easy. Gabriel Ebner is our bald-headed fixer. If the mathlib people can’t get something to work and they blame Lean, he sometimes has words with it directly. He has driven Lean from 3.4.2 to 3.17.1 in the space of, what — six months? It’s better and it’s faster. To anyone still on Lean 3.4.2 — switch to the Lean Prover community version of Lean. It’s so much easier now. Anyone searching for Lean — beware of Microsoft’s old pages. Search for Leanprover community or mathlib. And then me at the bottom. A professor of maths with a general malaise about the state of number theory, who three years ago this week tried Lean for the first time and got hooked. We’re a really motley crew, talking to each other about different ways of thinking about common areas of interest. There are many other people who have committed to mathlib too — e.g. people I’ve met on discord whose names I don’t even know but who got interested in seeing if they could formalising a random thing in Lean, and it has turned out that the answer is “yes, and in doing so you can make mathlib better”. People like Amelia Livingston, another Imperial undergrad, who felt that Kenny Lau’s theory of localisation of rings should be generalised to localisation of monoids and rewrote the whole lot. She was right. People like Shing Tak Lam, who was practicing for his STEP exams (hard UK school maths exams for 18 year olds) by formalising the questions, and his project to formalise a question about polynomials over the reals from STEP ended up with him developing the entire mathlib theory of partial derivatives for multivariable polynomial rings. He asked what a ring R was at some point, I said “just pretend it says the real numbers $\mathbb{R}$“. ## The two cultures of mathematics Together we are investigating the boundary between the specification of mathematics, and the implementation of mathematics, for both definitions and for proofs. I learnt from Sébastien at LFTCM2020 that we can now prove that smooth morphism of smooth manifolds induces a morphism on tangent spaces in a manner functorial in the manifold, but it took a bunch of people to turn the theorem statement from maths into code, and then a bunch of people to translate the proof from maths into code (and everyone was standing on the shoulders of giants, in some sense). There are two universes involved in what we are making. There’s the part that’s in Type (where the creative ideas such as perfectoid spaces and the Cauchy reals are digitised) and Prop (the part within the begin/end blocks where the computer games are). In Tim Gowers’ essay on the two cultures of mathematics, he talks about the concepts of theory-building and problem-solving. I have always believed that there was something very true at the core of this, but now I am beginning to understand it much better. Type is where the theory-building is going on, and Prop is where the problem-solving is occurring. But these two parts of mathematics are inextricably linked. Patrick, Johan and I formalised the definition of a perfectoid space, but on the way there we had to prove a whole load of theorems — even before we could write down the definition. There are two topologies on an affinoid perfectoid space. One generated by basic open subsets, and one generated by rational open subsets. In the future, when defining sheaves on a perfectoid space, sometimes we will use one basis, and sometimes the other. The proof that they are the same is 20 pages of very tricky algebra involving valuations on topological rings, and we have not done it yet — we just picked one of the definitions when defining a perfectoid space and moved on. At some point in the development, we will need to use the other definition, and then we’ll have to prove the theorem. But in fact this sort of thing already happened to us many many times before, and then we did have to prove the theorems. Sometimes things get tough. Mathematicians so good at instantly switching between the various “obviously equivalent” ways that a mathematician looks at a complicated algebraic object (“It’s an equivalence relation! Now it’s a partition! Now it’s an equivalence relation again! Let your mental model jump freely to the point of view which makes what I’m saying in this particular paragraph obvious!”, or “Matrices are obviously associative under multiplication because functions are associative under composition.” (dutiful student realises later that this proof assumes the action of matrices on $\mathbb{R}^n$ is faithful, and did you see that dependent type there?). Some of the proofs we’re writing are simply proofs that humans are behaving correctly when using mathematical objects. But writing some proofs in mathlib is just fun. In fact there are now a growing number of proofs in Lean written by mathematicians who are coming to mathlib and finding that the interface to the thing they wanted (e.g. a topological space, or an equivalence relation, or a ring) is there and usable. It is hence possible for them to state and prove (or reprove, if mathlib did it already) the results they are interested in, using Lean’s tactic framework. We don’t know how far this can go, and we don’t know whether type-theoretic issues will cause problems further down the line (e.g with etale cohomology) but at this rate, it looks like we’re going to find out. Currently on display at Royal Academy of Art (proud Dad :-)) Posted in Uncategorized \| \| 4 Comments ## Lean for the Curious Mathematician 2020 I have just spent an exhilarating week in the company of a whole bunch of mathematicians, a fair few of whom are serious professors from prestigious universities, all of us learning how to use Lean to do mathematics ranging from basic logic to category theory and differential geometry. Why? Because I’ve been at Lean for the Curious Mathematician 2020, an online meeting organised by Johan Commelin and Patrick Massot. It was like no other conference I’ve ever been to. On a typical day there were three “sessions”. A session was typically two hours long, and consisted of a mathematician who knows how to use Lean giving a 15 to 30 minute Zoom talk introducing a part of Lean’s maths library mathlib, and then the audience splitting up into breakout rooms of between 5 and 10 people, with one expert per room, and working on the exercises which the speaker had prepared. People could ask questions to each other or to the experts whenever they were stuck. A few years ago I would never have guessed that in 2020 I would have a Cambridge professor of number theory asking me for help in proving that the standard basis of $\mathbb{R}^n$ was a basis — but when you have just started learning how to do mathematics in a new way, these are natural questions to ask. The answer is not hard — but you have to learn how to do it. What is even better is that everything was recorded, so if you missed LftCM 2020, you can still join in. All the talks are up on a LftCM 2020 playlist on the Leanprover community YouTube site, and all of the exercises are available at the LftCM 2020 GitHub repository. If you have installed leanproject by following the instructions on the Leanprover Community website then all you have to do is to type leanproject get lftcm2020 and then, using VS Code, open the lftcm2020 directory which just appeared on your computer. You’ll see all the exercises, and all the solutions. Furthermore, you have easy access to the very same experts who wrote the exercises because they all hang out on the Lean Zulip chat, and the #new members stream is dedicated to beginner questions. I will be live streaming my way through some of the exercises over the next few weeks on the Xena Project Discord server; I monitor the chat on most days, but aim to spend every Tuesday and Thursday on the server throughout July and August. If you’re a undergraduate mathematician, even if you’re a complete Lean beginner, you’re very welcome to join us. A good time to join us is Thursday evening UK time; there are often a whole bunch of us there then. ## What did we learn? One thing we learnt was that technology like Zoom works very well for a workshop of this nature. For a beginner, Lean’s error messages are very intimidating. But if a beginner shares their screen on Zoom so that an expert can read the error, then the expert often immediately knows what is wrong. Something else we learnt was that a very good way to teach mathematicians how to use Lean is to give them a whole ton of exercises involving doing mathematics which they already understand conceptually, and asking them to attempt them in Lean. In some sense a large source of exercises corresponding to interesting mathematics is something which, before now, had been lacking in the Lean ecosystem. There is the very wonderful Theorem Proving In Lean, the book I read when learning Lean, but it does not talk about Lean’s mathematics library at all and focuses more on things like Lean’s underlying type system. I have always thought of it as more of a book for computer scientists. Jeremy Avigad, one of the authors of Theorem Proving In Lean, is currently working with Patrick Massot, Rob Lewis and myself on a new book, Mathematics In Lean (a book which can be read and run entirely within VS Code; read the instructions in chapter 1 on how to interact with it in this way). But this book is not yet finished. The book does not come with videos — Jeremy is writing words to explain how things work. The LftCM GitHub repository goes right to the point: every file, after a brief introduction, goes straight onto exercises, the vast majority of which can be solved in tactic mode, so it is a very natural continuation for anyone who has played the natural number game. If you don’t understand something in the file, you can watch the corresponding video and see if this helps. I will be very interested in seeing whether mathematicians find the repository a useful asset and I am almost certain that they will. We learnt that Scott Morrison’s tireless efforts to teach both Lean and the Leanprover community category theory have now gone as far as giving us a whole host of accessible exercises, together with hints! We learnt that smart people can pick up Lean very quickly. It was very interesting watching Sophie Morel learning to use the system, and seeing her assimilating more and more concepts as the week went on; I was especially interested in watching her learn a new trick to prove a theorem and then, instead of immediately moving on, playing around with her proof to understand the trick better. Although it sounds a bit ridiculous to compare a mathematician of her calibre to Kenny Lau (an Imperial undergraduate), watching her learning Lean this week reminded me very much of watching Kenny learning Lean back in 2017, when he was just a first year UG and within a few weeks of starting using Lean was formalising the theory of localisation of commutative rings. Kenny has gone on to contribute just under 20,000 lines of code to mathlib, including a lot of MSc level commutative ring theory. Sophie, many thanks for coming, and I hope you like what you saw. And Kenny, and Chris and Amelia, it was great to chat to you all this week and I am very much looking forward to working with all of you this forthcoming academic year on your Lean MSci projects. One important thing, for me at least, that came out of the week, was that we got to see exactly the kind of problems which beginners run into. Lean is very much under development. New versions come out every few weeks of the community version of Lean, and the maths library is updated several times a day. There are many open issues on the Lean and mathlib github repositories, which are opened, dealt with, and closed. But as an experienced user it was interesting to see which ones were tripping people up. We know library_search sometimes fails because of the apply bug, and we know how to work around this. But this would completely throw beginners off. We know that nat.pow is not definitionally equal to monoid.pow and we know how to work around this. I would say that it is regarded as a low priority issue. But this week we saw people being totally confused by it. It is a reminder that we can still make Lean 3 a better system. I am not convinced by the “let’s just wait until Lean 4” argument. Johannes Hoelzl, a very experienced formaliser, told me that in his opinion porting mathlib from Lean 3 to Lean 4 whilst simultaneously refactoring it would be a bad idea. Despite the fact that I want to set up a theory of Picard groups, Ext and Tor, I should remember that the basics are still not 100% right and I can play my part in trying to make them better (beginning with that big subgroup refactoring). I personally learnt that mathematicians do like Lean games. A couple of people asked me if it would be possible to make a filter game! I think this idea has definite potential! Until then, there’s always the max minigame, a game you can play in your browser like the natural number game and something which will eventually become part of the real number game. I have learnt a lot this week. Watching other people using Lean is something I find very instructive, both as an educator and as a Lean user. Thank you to to all who attended, and to those who led sessions and wrote example sheets, and I very much hope to see many of you again at LftCM 2021. Posted in Learning Lean \| \| Leave a comment ## Hey! I heard that Lean thinks 1/0 = 0. Is that true? Yes. So do Coq and Agda and many other theorem provers. ## Doesn’t that lead to contradictions? No. It just means that Lean’s / symbol doesn’t mean mathematical division. Let $\mathbb{R}$ denote the real numbers. Let’s define a function $f:\mathbb{R}^2\to\mathbb{R}$ by $f(x,y)=x/y$ if $y\not=0$ and $f(x,0)=0$. Does making that definition give us a contradiction in mathematics? No, of course not! It’s just a definition. Lean uses the symbol / to mean $f$. As does Coq, Agda etc. Lean calls it real.div by the way, not $f$. ## But doesn’t that lead to confusion? It certainly seems to lead to confusion on Twitter. But it doesn’t lead to confusion when doing mathematics in a theorem prover. Mathematicians don’t divide by 0 and hence in practice they never notice the difference between real.div and mathematical division (for which 1/0 is undefined). Indeed, if a mathematician is asking what Lean thinks 1/0 is, one might ask the mathematician why they are even asking, because as we all know, dividing by 0 is not allowed in mathematics, and hence this cannot be relevant to their work. In fact knowing real.div is the same as knowing mathematical division; any theorem about one translates into a theorem about the other, so having real.div is equivalent to having mathematical division. ## This convention is stupid though! It gets worse. There’s a subtraction nat.sub defined on the natural numbers $\{0,1,2,\ldots\}$, with notation x - y, and it eats two natural numbers and spits out another natural number. If x and y are terms of type ℕ and x < y, then x - y will be 0. There’s a function called real.sqrt which takes as input a real number, and outputs a real number. If you give it $2$, it outputs $\sqrt{2}$. I don’t know what happens if you give it the input $-1$, beyond the fact that it is guaranteed to output a real number. Maybe it’s 0. Maybe it’s 1. Maybe it’s 37. I don’t care. I am a mathematician, and if I want to take the square root of a negative real number, I won’t use real.sqrt because I don’t want an answer in the reals, and the type of real.sqrt is ℝ → ℝ. ## Why can’t you just do it the sensible way like mathematicians do? Great question! I tried this in 2017! Turns out it’s really inconvenient in a theorem prover! Here’s how I learnt Lean. I came at it as a “normal mathematician”, who was thinking about integrating Lean into their undergraduate introduction to proof course. I had no prior experience with theorem provers, and no formal background in programming. As a feasibility study, I tried to use Lean to do all the problem sheets which I was planning on giving the undergraduates. This was back in 2017 when Lean’s maths library was much smaller, and real.sqrt did not yet exist. However the basic theory of sups and infs had been formalised, so I defined real.sqrt x, for x non-negative, to be $Sup\{y\in\mathbb{R}\,\|\,y^2\leq x\}$, and proved the basic theorems that one would want in an interface for a square root function, such as $\sqrt{ab}=\sqrt{a}\sqrt{b}$ and $\sqrt{a^2}=a$ and $\sqrt{a^2b}=a\sqrt{b}$ and so on (here $a,b$ are non-negative reals, the only reals which my function would accept). I then set out to prove $\sqrt{2}+\sqrt{3}<\sqrt{10}$, a question on a problem sheet from my course. The students are told not to use a calculator, and asked to find a proof which only uses algebraic manipulations, i.e. the interface for real.sqrt. Of course, the way I had set things up, every time I used the $\sqrt{\phantom{2}}$ symbol I had to supply a proof that what I was taking the square root of was non-negative. Every time the symbol occurred in my proof. Even if I had proved 2 > 0 on the previous line, I had to prove it again on this line, because this line also had a $\sqrt{2}$ in. Of course the proof is just by norm_num, but that was 10 characters which I soon got sick of typing. I then moaned about this on the Lean chat, was mocked for my silly mathematician conventions, and shown the idiomatic Lean way to do it. The idiomatic way to do it is to allow garbage inputs like negative numbers into your square root function, and return garbage outputs. It is in the theorems where one puts the non-negativity hypotheses. For example, the statement of the theorem that $\sqrt{ab}=\sqrt{a}\sqrt{b}$ has the hypotheses that $a,b\geq 0$. Note that it does not also have the hypothesis that $ab\geq0$, as one can deduce this within the proof and not bother the user with it. This is in contrast to the mathematicians’ approach, where the proof that $ab\geq0$ would also need to be supplied because it is in some sense part of the $\sqrt{\phantom{2}}$ notation. ## So you’re saying this crazy way is actually better? No, not really. I’m saying that it is (a) mathematically equivalent to what we mathematicians currently do and (b) simply more convenient when formalising mathematics in dependent type theory. What actually is a field anyway? For a mathematician, a field is a set $F$ equipped with $0,1,a+b,-a,a\times b,a^{-1}$ where the inversion function $a^{-1}$ is only defined for non-zero $a$. The non-zero elements of a field form a group, so we have axioms such as $x\times x^{-1}=1$ for $x\not=0$ (and this doesn’t even make sense for $x=0$). Let’s say we encountered an alien species, who had also discovered fields, but their set-up involved a function $\iota :F\to F$ instead of our $x^{-1}$. Their $\iota$ was defined, using our notation, by $\iota(x)=x^{-1}$ for $x\not=0$, and $\iota(0)=0$. Their axioms are of course just the same as ours, for example they have $x\times \iota(x)=1$ for $x\not=0$. They have an extra axiom $\iota(0)=0$, but this is no big deal. It’s swings and roundabouts — they define $a/b:=a\times\iota(b)$ and their theorem $(a+b)/c=a/c+b/c$ doesn’t require $c\not=0$, whereas ours does. They are simply using slightly different notation to express the same idea. Their $\iota$ is discontinuous. Ours is not defined everywhere. But there is a canonical isomorphism of categories between our category of fields and theirs. There is no difference mathematically between the two set-ups. Lean uses the alien species convention. The aliens’ $\iota$ is Lean’s field.inv , and Lean’s field.div x y is defined to be field.mul (x, field.inv y). ## OK so I can see that it can be made to work. Why do I still feel a bit uncomfortable about all this? It’s probably for the following reason. You are imagining that a computer proof checker will be checking your work, and in particular checking to see if you ever divided by zero, and if you did then you expect it to throw an error saying that your proof is invalid. What you need to internalise is that Lean is just using that function $f$ above, defined by $f(x,y)=x/y$ for $y\not=0$ and $f(x,0)=0$. In particular you cannot prove false things by applying $f$ to an input of the form $(x,0)$, because the way to get a contradiction by dividing by zero and then continuing will involve invoking theorems which are true for mathematical division but which are not true for $f$. For example perhaps a mathematician would say $a/a=1$ is true for all $a$, with the implicit assumption that $a\not=0$ and that this can be inferred from the notation. Lean’s theorem that real.div a a = 1 is only proved under the assumption that $a\not=0$, so the theorem cannot be invoked if $a=0$. In other words, the problem simply shows up at a different point in the argument. Lean won’t accept your proof of $1=2$ which sneakily divides by 0 on line 8, but the failure will occur at a different point in the argument. The failure will still however be the assertion that you have a denominator which you have not proved is nonzero. It will simply not occur at the point when you do the division, it will occur at the point where you invoke the theorem which is not true for real.div. Thanks to Jim Propp and Alex Kontorovich for bringing this up on Twitter. I hope that this clarifies things. Posted in Learning Lean, M1F, M40001, Type theory, undergrad maths \| Tagged , \| 12 Comments ## Equality, specifications, and implementations Equality is such a dull topic of conversation to mathematicians. Equality is completely intrinsic to mathematics, it behaves very well, and if you asked a mathematician to prove that equality of real numbers is an equivalence relation then they would probably struggle to say anything of content; it’s just obviously true. Euclid included reflexivity and transitivity of equality as two of his “common notions”, and symmetry was equally clear from his language — he talks about two things being “equal to each other” rather than distinguishing between a = b and b = a. One thing that has helped me start to understand why computer scientists make a fuss about equality is the observation that if you follow Peano’s development of the natural numbers (as I do in the natural number game) then you come to the following realisation: if you define addition by recursion in the second variable (i.e. a + 0 := a and a + succ(n) := succ(a + n) ) then life temporarily becomes asymmetric. The fact that a + 0 = a is an axiom. However the fact that 0 + a = a needs to be proved by induction. Now induction is also an axiom, so a mathematician would just say that despite the fact that the proofs have different lengths, 0 + a = a and a + 0 = a are both theorems, so the fact that digging down to the foundations shows that the proofs are of a different nature is really of no relevance. To a computer scientist however, there is a difference in these proofs. This difference seems to be of no relevance to mathematics. But the difference is that, if you set the natural numbers up in this way, then a + 0 = a is true by definition, and 0 + a = a is not. Indeed, in Lean’s type theory (and probably in many others) there are three types of equality that you have to be aware of: 1. Propositional equality; 2. Definitional equality; 3. Syntactic equality. Let’s start by getting one thing straight: to a mathematician, all of these things are just called equality. In fact, even more is true: definitional equality is not a mathematical concept. All of these kinds of equalities are easy to explain, so let me go through them now. Propositional equality: a = b is a propositional equality if you can prove it. Definitional equality: a = b is a definitional equality if it’s true by definition. Syntactic equality: a = b is a syntactic equality if a and b are literally the same string of characters. For example, let’s go back to Peano’s definition of the natural numbers and the conventions for addition above. Then a + 0 = a is a definitional equality but not a syntactic one, 0 + a = a is a propositional equality but not a definitional one, and a = a is a syntactic equality. To show that 0 + a = a is not definitionally true, you have to ask yourself what the definition of 0 + a is; and because we don’t know whether a = 0 or not, 0 + a cannot be definitionally simplified any further (the definition of x + y depends on whether y = 0 or not). [Technical note: syntactic equality does allow for renaming of bound variables, so $\{a \in \mathbb{R}\, \|\, a^2 = 2\}$ is syntactically equal to $\{b \in \mathbb{R}\, \|\, b^2=2\}$. If you understand that idea that notation is syntax sugar then you’ll probably know that syntactic equality can see through notation too, which means to say that add(a,0) = a + 0 is also a syntactic equality. But that’s it.] Of course 2 + 2 = 4 is not a syntactic equality; removing notation and writing S for “successor”, and working under the assumption that 2 is syntax sugar for S(S(0)) and 4 for S(S(S(S(0)))), we see that the left hand side is syntactically add(S(S(0),S(S(0)) and the right hand side is S(S(S(S(0)))). However it is a definitional equality! add(x,S(y))=S(add(x,y)) is true by definition, as is add(x,0)=x, and it’s fun to check that applying these rules a few times will reduce 2 + 2 to 4. The reason that it’s important to understand the differences if you are writing Lean code, is that different tactics work with different kinds of equality. Lean’s refl tactic attempts to close the goal by showing that one side of an equality is definitionally equal to the other side. If your goal is X then change Y will work if and only if Y is definitionally equal to X. On the other hand Lean’s rw tactic works at the level of syntactic equality: if h : A = B then rw h will change everything syntactically equal to A in the goal, into a B. If h : a + 0 = b and your goal is a = b then rw h will fail because the equality a + 0 = a is not syntactic. However exact h will work fine, because exact works up to definitional equality. ## Specification v implementation The fact that a + 0 = a is a definitional equality in the natural number game, but 0 + a = a isn’t, is as far as I am concerned a proof that definitional equality is not a mathematical concept. Indeed one can clearly just define addition on the natural numbers by recursion on the first variable instead of the second, and then 0 + a = a would be definitional and a + 0 = a would not be. What is going on here was made clear to me after a conversation I had with Steve Vickers after a seminar I gave to the University of Birmingham CS theory group. Mathematicians have a specification of the natural numbers. We know what we want: we want it to satisfy induction and recursion, we want it to be a totally ordered commutative semiring (i.e. all the ring axioms other than those involving minus) and we can take it from there thank you very much. If you present me with an object which satisfies these theorems I can use it to prove quadratic reciprocity. I don’t care what the actual definition of addition is, indeed I know several definitions of addition and I can prove they’re all equivalent. If you’re doing to do mathematics in a theorem prover, you have to make one definition. Mathematicians know that all the definitions of the natural numbers are the same. If you want to set up mathematics in set theory for example, then it doesn’t matter whether you decide to let $3 = \{2\}$ or $3 = \{0,1,2\}$: any system which ensures that 3 isn’t any of the sets you’ve already made is clearly going to work. But in a computer theorem prover you need to make choices — you need to make implementations of 3 and of add — and the moment that choice is made you now have a dichotomy: some stuff is true by definition, and some stuff needs an argument like induction and is not true by definition. The first time I really noticed the specification / implementation difference was when it dawned on me that Lean’s maths library had to choose a definition of the reals, and it went with the Cauchy sequence definition: a real number in Lean is an equivalence class of Cauchy sequences. An alternative approach would have been to define it as Dedekind cuts. As mathematicians, we don’t care which one is used, because we are well brought up and we promise to only ever access the real numbers via its interface, or its API to borrow a computing term. The interface is the specification. We mathematicians have a list of properties which we want the real numbers to satisfy: we want it to be a complete archimedean ordered field. Furthermore we have proved a theorem that says that any two complete archimedean ordered fields are uniquely isomorphic, and this is why we do not care one jot about whether we are using Cauchy sequences or Dedekind cuts. Lean gives me access to an interface for the real numbers which knows these things, and it’s all I need to build a theory of Banach spaces. As mathematicians we somehow know this fact implicitly. If I am trying to prove a theorem about Banach spaces, and I have a real number $\lambda$, I never say “Ok now for the next part it’s really important that $\lambda$ is under the hood defined to be a Dedekind cut”. If I want the Dedekind cut associated to $\lambda$, I can just make it. I don’t care whether it equals $\lambda$ by definition or not, because definitional equality is not a mathematical concept. All I care about is access to the interface — I’m proving a theorem about Banach spaces here, and I just need to have access to stuff which is true. The job of Lean’s file data.real.basic is to give me access to that interface, and I can build mathematics from there. Computer scientists on the other hand — they have to care about definitional equality, because it’s often their job to make the interface! If two things are definitionally equal then the proof they’re equal is refl, which is pretty handy. Different definitions — different implementations of the same specification — might give you a very different experience when you are making an interface for the specification. If you really have too much time on your hands this lockdown, why not go and try proving that addition on the real numbers is associative, using both the Dedekind cuts definition and the Cauchy sequences definition? For Cauchy sequences it just boils down to the fact that addition is associative on the rationals. But you’ll find that it’s a real bore with Dedekind cuts, because Dedekind cuts have this annoying property that you need a convention for the cuts corresponding to rational numbers: whether to put the rational number itself into the lower or upper cut. Neither convention gives a nice definition of addition. You can’t just add the lower cuts and the upper cuts, because the sum of two irrationals can be a rational. Multiplication is even worse, because multiplication by a negative number switches the lower and upper cut, so you have to move a boundary rational between cuts. You can see why Lean went for the Cauchy sequences definition. I ran into this “which implementation to use for my specification” issue myself recently. I notice that I have been encouraging students at Imperial to formalise courses which I have been lecturing, which recently have been algebra courses such as Galois theory. I am by training an algebraic number theorist, and really by now I should have turned my attention the arithmetic of number fields and their integers: as far as I can see, finiteness of the class group of a number field has been formalised in no theorem provers at all, so it is probably low-hanging fruit for anyone interested in a publication, and we surely have all the required prerequisites in Lean now. I thought that I would try and get this area moving by formalising the definitions of a Dedekind Domain and a discrete valuation ring (DVR). I looked up the definition of discrete valuation ring on Wikipedia and discovered to my amusement that there are (at the time of writing) ten definitions 🙂 Now here I am trying to be a theory builder: I want to make a basic API for DVRs so that students can use them to prove results about local and global fields. So now I have to decide upon a definition, and then prove that it is equivalent to some of the other definitions — I need to make enough interface to make it easy for someone else to take over. As far as I could see though, what the actual definition of a DVR is, is of no real importance, because it doesn’t change the contents of the theorems, it only changes the way you state them. So I just chose a random one 😛 and it’s going fine! ## Equality of terms, equality of types When talking about propositional and definitional equality above, my main examples were equality between natural numbers: 0 + a = a and what have you. Set-theoretically, we can happily think about 0 + a = a as equality between two elements of a set — the set of natural numbers. In type theory we are talking about equality between two terms of type T, where T : Type . But one can take this a level up. Say A : Type and B : Type (for example say A is the Cauchy reals, and B is the Dedekind reals). What does A = B mean? These are now not elements of a set, but objects of a category. Certainly the Cauchy reals and the Dedekind reals are not going to be definitionally equal. Can we prove they are propositionally equal though? No — of course not! Becuase they are not actually equal! They are, however canonically isomorphic. A fourth type of equality! All this equality naval-gazing is important to understand if you are talking about equality of terms. This, we have nailed. For mathematicians there is one kind of equality, namely “it is a theorem that a = b“. For computer scientists there are three kinds, and understanding the distinction might be important for interface extraction. But for equality of types, something funny is going on. Which is the most accurate? $(A \otimes B) \otimes C \cong A \otimes (B \otimes C)$ or $(A \otimes B) \otimes C = A \otimes (B \otimes C)$? This is just more notational naval-gazing, who cares whether these things are isomorphic or equal – they are manifestly the same, and we can always replace one by the other in any reasonable situation because they both satisfy the same universal property. However, I am realising that as a Lean user I need to say very carefully what I mean by “a reasonable situation”, and actually the safest way to do that is to not prove any theorems at all about $(A \otimes B) \otimes C$ other than the fact that it satisfies its universal property, and then instead prove theorems about all objects which satisfy that universal property. Mathematicians do not use this technique. They write their papers as if they are proving theorems about the concrete object $(A \otimes B) \otimes C$, but their proofs are sensible and hence apply to any object satisfying its universal property, thus can be translated without too much problem, once one has extracted enough API from the universal property. There is an art to making this painless. I learnt from Strickland the three key facts that one should prove about a localisation $R \to R[1/S]$ of rings: the kernel is the elements annihilated by an element of $S$, the image of every element of $S$ is invertible, and the map from $R\times S$ to the target sending $(r,s)$ to $r/s$ is surjective. These three facts together are equivalent to the universal property of $R[1/S]$ and now you can throw it away and build the rest of your localisation API on top of it. Indeed, when my former MSc student Ramon Fernández Mir formalised schemes in Lean, he needed this result from the stacks project to prove that affine schemes were schemes, but more generally for the case of rings only satisfying the same universal properties as the rings in the lemma. At the time he needed it (about 18 months ago) the proof only used the three facts above isolated by Strickland, and so was easy to prove in the generality he required. However, in Milne’s book on etale cohomology, it is decreed in the Notation section that = means “canonical isomorphism”, and so there will be a lot more of this nonsense which we will have to understand properly if we want to formalise some serious arithmetic geometry in Lean. Posted in Algebraic Geometry, General, Type theory \| Tagged , , \| 6 Comments ## Teaching dependent type theory to 4 year olds via mathematics We had a family Zoom chat today and I ended up talking to my relative Casper, who is 4 and likes maths. He asked me to give him some maths questions. I thought 5+5 was a good place to start, and we could go on from there depending on responses. He confidently informed me that 5+5 was 10, and that 5-5 was zero. For 55 he asked me “is that five fives?” and I said yes, and he asked me “Is that 5 lots of 5?” and I said yes, and then he began to count. “1,2,3,4,5. 6,7,8,9,10. 11,12,13,14,15. 16,17,18,19,20. 21,22,23,24,25. It’s 25!” he said. I guess he’s proved this by refl. Every natural number is succ (succ (succ (...(succ zero))...) so to evaluate one of them, you just break it down into this canonical normal form. He did not know what division was, so I thought it was time to move on from 5. I figured that if he didn’t know his 5 times table we needed to be careful with multiplication, in case of overflow errors, so I decided I would go lower. I next asked him 0+0, 0-0 and 00 (“so that’s zero lots of zero?” Yes. “So that’s zero zeros?” Yes). He got them all right, and similarly for 1+1, 1-1 and 11. I then thought we could try 0 and 1 together, so I asked him 0+1 to which he confidently replied 1, and 0-1, to which he equally confidently replied 0. This answer really made me sit up. Because Lean thinks it’s zero too. #eval 0 - 1 /- 0 -/ I am pretty sure that a couple of years ago I would have told him that he was wrong, possibly with a glint in my eye, and then tried to interest him in integers. Now I am no longer sure that he is wrong. I am more convinced that what has happened here is that Casper has internalised some form of Peano’s axioms for arithmetic. He knows ℕ. In some sense this is what I have been testing. He knows that every natural is succ succ … succ 0, and then for some reason he has to learn a song that goes with it, with a rigid rhythmical beat and written in 10/4 time: “One, two, three four, five, six, seven, eight, nine, ten. Eleven, twelve,…” and so on ad infinitum, some kind of system which is important for computation but is currently not of interest to me. Also, Casper knows that there’s something called subtraction, and when given junk input such as 0-1 he has attempted to make some sense of it as best he could, just like what the functional programmers who like all functions to be total have done. We then went onto something else. I asked him what 29+58 was and he essentially said that it was too big to calculate. I asked him if he thought that whatever it was, it was the same as 58+29, and he confidently said it was, even though he did not have a clue what the answer was. I asked him how he was so sure but I did not get a coherent answer. I asked him what three twos were, and he counted “1, 2, 3, 4, 5, 6“. Three lots of two is 6. I then asked him what two threes were and he immediately replied 6 and I asked him if he’d done the calculation and he said no, two threes were the same as three twos. I asked him why but all he could say was that it was obvious. It’s simp. I asked him if he was thinking about a rectangle and he said no. So he knows simp and refl. I then did quite a mean thing on him, I did overflow his multiplication buffer. I asked him what ten 3’s were. We spent ages getting there, and he needed help, because he hasn’t learnt how to sing the song in 3/4 time yet. But eventually refl proved that 103=30. I then asked him if he thought 310=103 and he was confident that it was. I then asked him what 3 tens were and whilst he didn’t immediately do this, he knew a song (in 4/4) which started at “one ten is10 (rest)” and finished at “ten tens-are-a hun dred.”. Using the song, we figured out pretty quickly what three tens were, and deduced that ten 3’s were 30 again. I then asked him what ten sevens were, and we talked about how how difficult it would be to sing the song in 7/4 and how long the song would be, and then we talked together through the argument that it was also “obviously” seven tens, so by the song it was seventy; apparently the fact that seventy sounds similar to 7 (even though thirty doesn’t sound that much like 3) is evidence that the patterns in the counting song can be helpful to humans who want to calculate. I then remembered the junk subtraction answer, so I thought I’d try him on another function which returns junk in Lean, namely pred, the “number before” function on the naturals. I asked him what the number before 3 was, and he said 2. He was also confident that the number before 2 was 1, and the number before 1 was 0. I then asked him what the number before 0 was and he thought a bit and then said….”1?”. pred is defined by recursion in Lean. We have pred (succ a) = a, and pred 0 is defined to be 0, because if you follow Lean’s “make all functions total, it’s easier for programmers” convention then it has to be something. But the choice of the actual value of pred 0 is probably not seriously used in Lean, and they could have defined pred 0 to be 1 or 37 and probably not much would break, and what did break would probably be easily fixed. Because of whining linter I recently needed to supply a junk value to Lean (it wanted me to prove that the quotient of a ring by an ideal was not the empty set) and I supplied 37 as the witness. pred is a fundamental function defined by recursion, as far as computer scientists are concerned. So why is it not even mentioned in the natural number game? Because you can’t really define functions by induction, you define them by recursion, and I don’t want to start getting involved in new recursive definitions, because this sort of thing cannot go within a begin end block. I could have decided to explain pred but I would have had to address the issue of pred 0 not being error and I realised that actually there was always a “mathematical workaround”. By this I mean the following. The natural proof of succ_inj uses pred, for example. The successor function is injective. The proof using pred is easy. But I tell people in the natural number game that succ_inj is an axiom, as is zero_ne_succ, and now we mathematicians seem to be able to develop a huge amount of theory without ever even defining subtraction. We don’t really want to use Lean’s natural number subtraction. We know that 0 ≠ junk is true by definition but unfortunately 0-0=0-1 in Lean, which is not right. We then talked about the number line a bit, and I then told him about a level in the 2d Mario franchise where the exit is way off to the right but right at the beginning you can completely counterintuitively go off to the left and pick up some bonus stuff. I then told him that there were some other kinds of numbers which he hadn’t learnt about yet, and I said they were called the integers (note: not “the negative numbers”). I said that the integers included all the numbers he knew already, and some new ones. I told him that in the integers, the number before 0 was -1. I then asked him what he thought the number before -1 would be and he said “-2?” And I told him he was right and asked him what the number before -2 was and he confidently said -3, and we did this for a bit and chatted about negative numbers in general, and then I asked him what the number before -9 was and he said -10. I then asked him what the number before -99 was and he said -100 and then I asked him what the number before -999 was and he said he didn’t know that one and did I want to play Monopoly. I took this as a sign that it was time to stop, and we agreed to play a game of monopoly via this online video chat, and then it turned out that he didn’t know any of the rules of monopoly (he’d just found the box on the floor) and he couldn’t read either, so he just put the figures on random places on the board and we talked about the figures, and what we could see on the board, and what was in the monopoly box, and the maths lesson was over. I was pretty appalled when I saw Lean’ s definition of int, a.k.a. ℤ. It is so ugly. There are two constructors, one of_nat n which takes a natural number $n$ and returns the integer $n$ (an invisible function, set up to be a coercion), and the one called something really weird like neg_one_of_nat n which takes in a natural $n$ and returns the integer $-1-n$, complete with hideous notation. This definition somehow arrogantly assumes some special symmetry of the integers about the point $-0.5$. The mathematician’s definition, that it’s a quotient of $\mathbb{N}^2$ by the equivalence relation $(a,b)\sim(c,d)\iff a+d=b+c$, a.k.a. the localisation of the additive monoid of naturals at itself (i.e. adding additive inverses to every element). This definition is a thing of beauty. The symmetry is never broken. It is the canonical definition. But even though this definition as a quotient is the most beautiful definition, classifying as it does the integers up to unique isomorphism because of the universal property of localisation (thanks so much Amelia Livingston), implementation decisions are system specific and int in Lean is the concrete inductive type with two constructors and there’s nothing we can do about this. So mathematically I am kind of embarassed to say that today I basically taught Lean’s definition of int to a kid, just as an experiment. Another weird thing is that Lean has a really ugly proof of a+b=b+a in int but for some reason this is going to be “obvious” when negative numbers are taught to him in school, and will not need a proof. I guess ac_refl will do it. When Casper had to go, I told him before he left to ask his parents what the number before -9 was. I think it’s a pretty tricky question if you’re caught off-guard. I told him that negative numbers were like mirror world. Is there educational research about how children model numbers? What different ideas do they have about ℕ before they even start to learn their multiplication tables? Should we teach them induction now, before they are crushed by having to learn so many long and boring songs telling them things like six sevens are 42, because of the belief in the education system currently that memorising this is somehow important in the year 2020 when most teenage kids have a calculator app in their pocket. Casper loves my daughter Kezia’s art. He says “everything is alive except the rainbow”. \| Tagged , , \| 8 Comments ## Mathematics in type theory. What is maths? I think it can basically be classified into four types of thing. There are definitions, true/false statements, proofs, and ideas. Definitions (for example the real numbers, or pi) and true/false statements (for example the statement of Fermat’s Last Theorem or the statement of the Riemann Hypothesis) are part of the science of mathematics: these are black and white things which have a completely rigorous meaning within some foundational system. Proofs are in some sense the currency of mathematics: proofs win prizes. Constructing them is an art, checking them is a science. This explains, very simply, why computer proof verification systems such as Lean, Coq, Isabelle/HOL, Agda… are much better at checking proofs than constructing them. And ideas are the purely artistic part of mathematics. That “lightbulb” moment, the insight which enables you to solve a problem — this is the elusive mathematical idea. Ideas are the part of mathematics that I understand the least, in a formal sense. Here are two questions: • What is a group? • How do you think about groups? The first one is a precise “scientific” question. A group is a set equipped with some extra structure, and which satisfies some axioms. The formal answer is on Wikipedia’s page on groups. A group is a definition. But the second question is a different kind of question. Different people think about groups in different ways. Say $G$ is a group generated by an element $x$ satisfying $x^5=x^8=1$. What can you say about $G$? If you are a mathematics undergraduate who has just seen the formal definition of a group, you can probably say nothing. If you have a more mature understanding of group theory, you instantly know that this group is trivial, because you have a far more sophisticated model of what is going on. Ideas are complicated, and human-dependent. A computer’s idea of what a group is, is literally a copy of the definition in Wikipedia, and this is one of the reasons that computers are currently bad at proving new theorems by themselves. You can develop a computer’s intuition by teaching it theorems about groups, or teaching it examples of groups, or trying to write AI’s which figure out group theory theorems or examples of groups automatically. But intuition is a very subtle thing, and I do not understand it at all well, so I will say no more about these ideas here. I think that the concept of a map being “canonical” is an idea rather than a definition — I think different mathematicians have different ways of thinking about this weasel word. In this post I’m going to talk about how the three other concepts are implemented in type theory, in the Lean theorem prover. # Definitions, true/false statements, and proofs In contrast to ideas, the other parts of mathematics (the definitions, theorems/conjectures, and proofs) can be formalised in a foundational system, and hence can be created and stored on a computer in a precise way. By this, I don’t mean a pdf file! Pdf files are exactly what I want to move away from! I mean that people have designed computer programming languages which understand one of the various foundations of mathematics (set theory, type theory, category theory) and then mathematicians can write code in this language which represents the definition, true/false statement or proof in question. I am certainly not qualified to explain how all this works in category theory. In set theory, let me just make one observation. A definition in set theory, for example the definition of the real numbers, or $\pi$, is a set. And a proof is a sequence of steps in logic. A definition and a proof seem to me to be two completely different things in set theory. A group is a mixture of these things — a group is an ordered quadruple $(G,m,i,e)$ satisfying some axioms, so it’s a set with some logic attached. In type theory however, things are surprisingly different. All three things — definitions, true/false statements, and proofs — are all the same kind of thing! They are all terms. A group, a proof, the real numbers — they are all terms. This unification of definitions and proofs — of sets and logic — are what seems to make type theory a practical foundational system for teaching all undergraduate level mathematics to computers. # Universes, types, and terms. In type theory, everything is a term. But some terms are types. Not every term is a type, but every term has a type. A colon is used to express the type of a term in Lean — the notation x : T means that x is a term of type T. For example, the real number π (pi) is a term in Lean, and the real numbers ℝ is a type, and we have π : ℝ , that is, π is a term of type ℝ. In set theory one writes π ∈ ℝ, in type theory we write π : ℝ. They both express the same mathematical concept — “π is a real number”. Now π is a term but it’s not a type. In Lean, x : π makes no sense. In set theory, x ∈ π does happen to make sense, but this is a weird coincidence because everything is a set. Furthermore, the actual elements of π will depend on how the real numbers are implemented (as Dedekind cuts or Cauchy sequences, for example), and hence in set theory x ∈ π has no mathematical meaning; it happens to make sense, but this is a quirk of the system. I claimed above that every term has a type. So what is the type of ℝ? It turns out that ℝ : Type. The real numbers are a term of a “universe” type called Type — the type theory analogue of the class of all sets. Many of the mathematical objects which mathematicians think of as definitions either have type Type, or have type T where T : Type. As a vague rule of thumb, things we write using capital letters (a group, a ring,…) or fancy letters (the reals, the rationals) have type Type, and things we write using small letters (an element g of a group, a real number r or an integer n) have got type T where T is what we think of as the set which contains these elements. For example 2 : ℕ and ℕ : Type, or if $g$ is an element of the group $G$ then in Lean we have g : G and G : Type. You can see that there is a three-layer hiearchy here — terms at the bottom, types above them, and the universe at the top. • Universe : Type • Examples of types : ℝ, ℕ, G (a group), R (a ring), X (something a set theorist would call a set), a Banach space, etc. Formally, we say ℝ : Type. • Examples of terms: π (a term of type ℝ), g (an element of the group G, so a term of type G), x (an element of X, so a term of type X). Formally, we say g : G. This hierarchy is more expressive than the hierarchy in set theory, where there are only two levels: classes (e.g. the class of all sets), and sets. There is a standard use of the colon in mathematics — it’s in the notation for functions. If X and Y are sets (if you’re doing set theory) or types (if you’re doing type theory), then the notation for a function from X to Y is f : X → Y. This is actually consistent with Lean’s usage of the colon; Lean’s notation for the collection $\mathrm{Hom}(X,Y)$ of functions from X to Y is X → Y , which is a type (i.e. X → Y : Type, corresponding to the fact that set theorists think of $\mathrm{Hom}(X,Y)$ as a set), and f : X → Y means that f is a term of type X → Y, the type-theoretic version of $f \in \mathrm{Hom}(X,Y)$, and the way to say that f is a function from X to Y in type theory. (Not for exam) Strictly speaking, universes are types, and types are terms, but this is a linguistic issue: often when people speak of types, they mean types which are not universes, and when people speak of terms they mean terms which are not types. But not always. This confused me when I was a beginner. # Theorems and proofs This is where the fun starts. So far, it just looks like a type is what a type theorist calls a set, and a term is what they call an element. But let’s now look at another universe in Lean’s type theory, the universe Prop of true/false statements, where our traditional mental model of what’s going on is quite different. We will see how theorems and proofs can be modelled in the same way as types and terms. So, how does this all work? As well as the universe Type, there is a second universe in Lean’s type theory called Prop. The terms of type Prop are true/false statements. There is an unfortunate notation clash here. In mathematics, the word proposition is often used to mean a baby theorem, and theorems are true (or else they would be conjectures or counterexamples or something). Here we are using the the word Proposition in the same way as the logicians do — a Proposition is a generic true/false statement, whose truth value is of no relevance. This will all be clearer with examples. 2 + 2 = 4 is a Proposition, so we can write 2 + 2 = 4 : Prop. But 2 + 2 = 5 is also a Proposition, so 2 + 2 = 5 : Prop as well. I’ll say it again — Propositions do not have to be true! Propositions are true/false statements. Let’s see some more complex examples. The true/false statement that $x+0=x$ for all natural numbers $x$ is a Proposition: in Lean this can be expressed as (∀ x : ℕ, x + 0 = x) : Prop . A Proposition is a term of type Prop (just like the types we saw earlier were terms of type Type). Let RH denote the statement of the Riemann Hypothesis. Then RH : Prop. We don’t care if it’s true, false, independent of the axioms of mathematics, undecidable, whatever. A Proposition is a true/false statement. Let’s look at the part of Lean’s type theory hierarchy which lives in the Prop universe. • Universe: Prop • Examples of types : 2 + 2 = 4, 2 + 2 = 5, the statement of Fermat’s Last Theorem, the statement of the Riemann Hypothesis. • Examples of terms: ?? So what are the terms in this three-layer Prop hierarchy? They are the proofs! # Propositions are types, proofs are terms. This is where the world of type theory seriously diverges from the way things are set up in set theory, and also the way things were set up in my brain up until three years ago. In trying to understand what was going on here, I even realised that mathematicians take some liberties with their language here. Before we start, consider this. The Bolzano-Weierstrass theorem is some statement in analysis about a bounded sequence having a convergent subsequence. I want to talk a little bit about how mathematicians use the phrase “Bolzano-Weierstrass theorem” in practice. A mathematician would say that the Bolzano-Weierstrass theorem is this statement about sequences having convergent subsequences. But if they are in the middle of a proof and need to apply it in order to continue with their proof, they say “by the Bolzano-Weierstrass theorem we deduce that there’s a convergent subsequence”. Nothing seems at all funny about any of this. But what I want to point out is that mathematicians are using the phrase “the Bolzano-Weierstrass theorem” in two different ways. When they say what it is, they are referring to the statement of the theorem. But when they say they’re using the Bolzano Weierstrass theorem, what they are actually using is its proof. The Birch and Swinnerton-Dyer conjecture is a perfectly well-formed true/false statement, you can certainly say what it is. But you can’t use the Birch and Swinnerton-Dyer conjecture in the middle of a proof of something else if you want your proof to be complete, because at the time of writing the conjecture is an unsolved problem. Making a clear distinction between the statement of a theorem, and the proof of a theorem, is important here. A mathematician might use the phrase “the Bolzano-Weierstrass theorem” to mean either concept. This informal abuse of notation can confuse beginners, because in the below it’s really important to be able to distinguish between a theorem statement, and a theorem proof; they are two very different things. In the natural number game, I use this abuse of notation because I am trying to communicate to mathematicians. The statement ∀ x : ℕ, x + 0 = x is a true statement, and I say things like “this is called add_zero in Lean”. In the natural number game I write statements such as add_zero : ∀ x : ℕ, x + 0 = x. But what this means is that the term called add_zero in Lean is a proof of ∀ x : ℕ, x + 0 = x! The colon is being used in the type theory way. I am intentionally vague about this concept in the natural number game. I let mathematicians believe that add_zero is somehow equal to the “idea” that $x+0=x$ for all $x$. But what is going on under the hood is that ∀ x : ℕ, x + 0 = x is a Proposition, which is a type, and add_zero is its proof, which is a term. Making a clear distinction between the statement of a theorem, and its proof, is important here. The statements are the types, the proofs are the terms. • Universe: Prop • Examples of types: 2 + 2 = 4, 2 + 2 = 37, the statement of Fermat’s Last Theorem — ∀ x y z : ℕ, n > 2 ∧ x^n + y^n = z^n → xy = 0. • Examples of terms: the proof that 2 + 2 = 4 (a term of type 2 + 2 = 4), the proof of Fermat’s Last Theorem (a term of type ∀ x y z : ℕ, n > 2 ∧ x^n + y^n = z^n → xy = 0) # Elements of a theorem So our mental model of the claim π : ℝ is that ℝ, the type, is “a collection of stuff”, and π, the term, is a member of that collection. If we continue with this analogy, it says that the statement 2 + 2 = 4 is some kind of collection, and a proof of 2 + 2 = 4 is a member of that collection. In other words, Lean is suggesting that we model the true/false statement 2 + 2 = 4 as being some sort of a set, and a proof of 2 + 2 = 4 is an element of that set. Now in Lean, it is an inbuilt axiom that all proofs of a proposition are equal. So if a : 2 + 2 = 4 and b : 2 + 2 = 4 then a = b. This is because we’re working in the Prop universe — this is how Propositions behave in Lean. In the Type universe the analogue is not remotely true — we have π : ℝ and 37 : ℝ and certainly π ≠ 37. This special quirk of the Prop universe is called “proof irrelevance”. Formally we could say that if P : Prop, if a : P and if b : P then a = b. Of course if a Proposition is false, then it has no proofs at all! It’s like the empty set. So Lean’s model of Propositions is that the true ones are like sets with 1 element, and the false ones are like sets with 0 elements. Recall that if f : X → Y then this means that f is a function from X to Y. Now say $P$ and $Q$ are Propositions, and let’s say that we know $P\implies Q$. What does this mean? It means that $P$ implies $Q$. It means that if $P$ is true, then $Q$ is true. It means that if we have a proof of $P,$ we can make a proof of $Q$. It is a function from the proofs of $P$ to the proofs of $Q$. It is a function sending an element of $P$ to an element of $Q$. It is a term of type P → Q. Again: a proof $h$ of $P\implies Q$ is a term h : P → Q. This is why in the natural number game we use the → symbol to denote implication. Let false denote a generic false statement (thought of as a set with 0 elements), and let true denote a generic true statement (thought of as a set with 1 element). Can we construct a term of type false → false or a term of type true → true? Sure — just use the identity function. In fact, in both cases there is a unique function — the hom sets have size 1. Can we construct a term of type false → true? Sure, there is a function from the set with 0 elements to a set with 1 element, and again this function is unique. But can we construct a term of type true → false? No we can’t, because where do we send a proof of true? There are no proofs of false to send it to. So true → false is a set of size 0. This corresponds to the standard truth table for →, where the first three statements we analysed were true and the last was false. # The proof of Fermat’s Last Theorem is a function So what does a proof of ∀ x y z : ℕ, n > 2 ∧ x^n + y^n = z^n → xy = 0 look like? Well, there is an arrow involved in that Proposition, so the statement of Fermat’s Last Theorem is some kind of set of the form $\mathrm{Hom}(A,B)$, which means that in Lean, a proof of Fermat’s Last Theorem is actually a function! And here is what that function does. It has four inputs. The first three inputs are natural numbers x, y and z. The fourth input is a proof: it is a proof of the Proposition n > 2 ∧ x^n + y^n = z^n. And the output of this function is a proof of the Proposition xy = 0. This is quite an unconventional way to think about what the proof of Fermat’s Last Theorem is, and let me stress that it does not help at all with actually trying to understand the proof — but it is a completely consistent mental model for how mathematics works. Unifying the concept of a number and a proof — thinking of them both as terms — enables you to think of proofs as functions. Lean is a functional programming language, and in particular it is designed with functions at its heart. This, I believe, is why theorem provers such as Lean, Coq and Isabelle/HOL, which use type theory, are now moving ahead of provers such as Metamath and Mizar, which use set theory. Prove a theorem! Write a function! Posted in Learning Lean, Type theory, undergrad maths \| Tagged , , , \| 22 Comments ## The perfectoid project. Johan Commelin and I worked with Patrick Massot on the Lean perfectoid space project, but Patrick is not an algebraic number theorist, and he found our literature hard to read in places. He was constantly picking up on what I would regard as “typos” or “obvious slips” in Wedhorn’s unpublished (and at that time unavailable) notes — stuff which was throwing him off. Wedhorn’s notes contain an explicit warning on the front that they are not finished and you read them at your own risk, but Johan and I were between us expert enough to be able to explain precisely what was going on whenever Patrick asked about operator precedence or the meaning of some undefined \cdot . Torsten Wedhorn — thank you for your blueprint, now finally available on ArXiv . We learnt a lot from it — both about the definition of an adic space, and about how to write blueprints. The point Patrick wanted me to understand was this: If I as a mathematician thought I really knew the definition of a perfectoid space, why couldn’t I come up with a precise and error-free and furthermore self-contained mathematical document that he, as a non-expert but as a professional mathematician, would be able to read without difficulty? Where was my plan for formalising perfectoid spaces? The answer to that was that the plan was in my head. Along the way, Patrick in particular learnt that there are problems with this approach. ## The cap set problem. In stark contrast to the perfectoid shambles was work of Sander R. Dahmen, Johannes Hölzl, Robert Y. Lewis on the Cap Set Problem — see their ArXiv paper. Their main result is a complete computer proof formalisation of the main theorem in the 2017 Annals of Mathematics paper by Ellenberg and Gijswijt on the cap set problem. Sander, Johannes and Rob organised their workflow very differently. They started with a detailed pdf document written by Dahmen (a mathematician). This old-fashioned proof was self-contained, just did normal maths in a normal way, and didn’t mention Lean once. It is a document which will convince any proper mathematician who cares to read it carefully of the correctness of the proof. Dahmen’s work is a completely rigorous old-fashioned proof of the theorem. There is no doubting its correctness. But because it is complete, correct, and self-contained modulo undergraduate level mathematics, this means that people who are not necessarily specialists in mathematics can now begin to work together, using a computer system which knows a lot of undergraduate mathematics, and they can turn Dahmen’s blueprint into a proof of Ellenberg-Gijswijt stored as a virtual mathematical object in the memory of a computer. Such an object is far more amenable to AI analysis than the corresponding pdf file, because computers have a hard time reading natural language, even if written by mathematicians. Apparently we don’t always explain things correctly. My point here is: Dahmen’s blueprint was an important part of the process. ## The sphere eversion project. But back to Patrick. He is conjecturing that Lean 3 is sufficiently powerful to formalise a complete proof of Sphere Eversion, the Proposition that you can turn a sphere inside-out without creasing it, as long as it is made out of material which has self-intersection 0, like light but bendier. There’s a video. 3D geometry is involved. The amazing news is that Patrick has now written a blueprint for the proof. This is a normal mathematical document containing a proof of sphere eversion which anyone who knows enough maths to know what a manifold is, could check and see was a completely rigorous and self-contained proof. It is written by an expert in the area, and typeset in a clever way so it displays beautifully in a browser. It is an interactive blueprint. It is the beginning. It is half of the Rosetta Stone that Patrick is creating, which will explain one story in two languages — a human language, and a computer language. Patrick has also started a Lean sphere eversion project on GitHub . This is a currently mostly empty Lean repository, which will ultimately contain a formal proof of sphere eversion if the project is successful. Those two parts together form Patrick’s Sphere Eversion Project. At some point in the near future, Patrick is going to need mathematicians to formalise parts of the proof. The people he’s looking for are perhaps people with degrees in mathematics who are interested in trying something new. Patrick and I, together with Rob Lewis and Jeremy Avigad, are still working hard on our forthcoming book Mathematics in Lean, an introduction to the skills that a mathematician will need in order to participate. Once you have learned how to write Lean, Patrick has some computer game puzzles which you might be interested in playing. If anyone has questions, they could ask in the sphere eversion topic in #maths in the Zulip chat (login required, real names preferred, be nice), a focussed and on-topic Lean chat where many experts hang out. People who are looking for a less formal setting are welcome to join the Xena Discord server (meetings Thursday evening at 5ish). ## The schemes project What is so amazing about projects like these is that they are teaching us how to use dependent type theory as a foundation for all of pure mathematics. We have had occasional problems along the way. Every division ring is a ring and hence a monoid and thus a semigroup. Invisible functions piled up in type class inference. The perfectoid project helped guide us to the realisation that type class inference was becoming problematic at scales which mathematicians needed it to work, and the Lean developers have responded to this issue by completely redesigning the system in Lean 4. But let me finish with my Lean schemes project — my first serious Lean project, written with Chris Hughes and Kenny Lau, both at the time first year undergraduates. The project is completely incompatible with modern Lean and mathlib, but if you compile it then you get a sorry-free proof that an affine scheme is a scheme. During our proof we ran into a huge problem because our blueprint, the Stacks Project, assumed that $R[1/f][1/g]=R[1/fg]$, and this turns out to be unprovable for Lean’s version of equality: this equality is in fact one of Milne’s “canonical” isomorphisms. The Lean community wrestled with this idea, and has ultimately come up with a beefed-up notion of equality for which the identity is now true. Amelia Livingston is just putting the finishing touches on its implementation in Lean — many thanks indeed Amelia. It has been an extraordinary journey, and one which has taught me a great deal about localisation and how to think about it. I had never realised before that the rationals might not be equal to the field of fractions of the integers — whether those two fields are actually equal is an implementation issue of no relevance to mathematicians. What we need to know is merely that they are isomorphic and that there is a preferred isomorphism in each direction. This has design consequences which we are only just beginning to understand properly. I wish the sphere eversion project every success. I am confident that it will teach us more about how to formalise mathematics in dependent type theory. Posted in Imperial, Learning Lean, Type theory \| \| 2 Comments ## The complex number game Mohammad and I spruced up the natural number game over the last few weeks. It now saves your progress, which is great (thanks Mohammad). Also coming along nicely is the real number game, currently written mostly by Dan Stanescu, but Gavin Thomson has just joined the party and I promise that I will be contributing in June when marking is over. Anyone who wants to get sneak previews of it should hang around on the Xena Discord server, which is where our Thursday evening Xena Project meetings are taking place this term; I’ll probably release a secret alpha this coming Thursday. Last Thursday on the Discord we had people any% speedrunning and racing the Lean tutorial project . This fits very well into my general worldview: I think that doing mathematics in Lean is like solving levels in a computer puzzle game, the exciting thing being that mathematics is so rich that there are many many kinds of puzzles which you can solve. Level creators are conjecture makers. If you are into puzzle games and know a little Lean then you might also want to check out the Codewars website, where support for Lean just moved from beta to official; right now there are over 50 approved Lean Kata, ranging from easy facts about odd and even numbers to some very tricky Diophantine equations. Another project I’ve been involved in is the forthcoming book Mathematics in Lean. Again I am guilty of doing very little so far. What I’ve been trying to do is to formalise “random” basic undergraduate mathematics in Lean as best I can, using as much automation as I can, in order to see whether it is possible to write basic tactic proofs in “the way a mathematician would write them”, but none of this stuff has made it into the book yet. It is miserable when when you are trying to prove a theorem in Lean and you’ve reduced it to a statement which is completely mathematically obvious (e.g. the goal is to prove that two things are “the same”), but you can’t persuade Lean to believe this, so the art is to set things up in such a way that this issue doesn’t occur, and one way of doing this is by seeking advice from people who understand type theory better than I do. So I’ve been doing several experiments, hoping that some of them will turn from ideas into chapters of this book, and one which I think went quite well was a construction of the basic properties of the complex numbers, which I liked so much that I’ve put into its own repo for now: The Complex Number Game. This repo came about with me looking at the official mathlib file for the complex numbers, and “remixing” it, re-ordering it so that it was a bit more coherent, removing some of the more obscure computer-sciency lemmas, and heading straight for the proof that the complexes are a ring. My rewrite of the mathlib proof is heavily documented and the proofs of all the lemmas along the way are in tactic mode; in my view this makes things more accessible to mathematicians. The file data.complex.basic in Lean’s library has been examined and revisited so many times since 2017 that it is now in very good shape — in particular lemmas are correctly tagged with simp and this makes simp a very powerful tool. If you want to play the complex number game right now, you have to install Lean first. I would love to share this game on CoCalc but right now I can’t get imports to work. I will come back to this in June. If installing Lean really is not something you want to do, then you can play a crappier preliminary version using the Lean Web Editor (this is before I reorganised the material to make it more mathematician-friendly and turned more proofs from term mode to tactic mode). Or you can just watch me play it! I live streamed on Twitch a walkthrough of the tutorial level (the complexes are a commutative ring) last Thursday, and here is the video. This Thursday (28th May 2020) at 5pm UK time (1600 UTC) I’ll be on Twitch again, live streaming a walkthrough of the first two levels: sqrt(-1), and complex conjugation. I’ll be on Twitch for around an hour, and then we will retire to Discord for a Q&A and more any% racing. Mathematicians who are Lean beginners are welcome! Someone else doing some remixing is my daughter Kezia Xena Buzzard (after whom the project is named). She made my new profile pic from Chris Hughes’ proof of quadratic reciprocity 🙂 Posted in General, Learning Lean, undergrad maths \| Tagged , \| 4 Comments ## Xena Project Thursday meetings start with talk tomorrow 21st May Usually, the Xena project meets every Thursday evening (5pm to 10pm UK time) during Imperial College’s term time. Because students have been doing exams and the campus has been closed, there have not been any meetings this term so far, but now exams are winding down I am going to try and start things up again on Thursday evenings, from now until the end of June (i.e., the end of term). We’re launching at 5pm UK time (=1600 UTC) on Thurs 21st May 2020 (check out the Xena project Google calendar) with a live demo by me of the new complex number game (which will be live on GitHub by tomorrow, assuming I’ve finished writing it). I’ll walk through the tutorial world, which proves that the complex numbers are a commutative ring. I’ll do it live on Twitch. Afterwards we can all retire to the Xena project discord server and plan world domination. Posted in Learning Lean, undergrad maths \| Tagged , \| Leave a comment	2020-10-01 07:36:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 128, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6376034021377563, "perplexity": 465.8563749391221}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600402124756.81/warc/CC-MAIN-20201001062039-20201001092039-00744.warc.gz"}
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/698	## Weighted Particles in the Finite Pointset method • Using particle methods to solve the Boltzmann equation for rarefied gases numerically, in realistic streaming problems, huge differences in the total number of particles per cell arise. In order to overcome the resulting numerical difficulties the application of a weighted particle concept is well-suited. The underlying idea is to use different particle masses in different cells depending on the macroscopic density of the gas. Discrepance estimates and numerical results are given. $Rev: 13581$	2016-07-27 03:45:24	{"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": 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": 0.3188212513923645, "perplexity": 677.4821394360404}, "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-2016-30/segments/1469257825365.1/warc/CC-MAIN-20160723071025-00170-ip-10-185-27-174.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/271890/prove-that-there-exists-some-positive-integer-x-m-x-0-given-that-a-function-is	# Prove that there exists some positive integer $X_m=X_0$ given that a function is injective Let $A$ be a finite set. Let $f:A\to A$ be an injective function from the set $A$ to itself and let $x_0$ be an element of $A$. Let $x_n$ be defined for all positive integers $n$ so that $x_n=f(x_{n-1})$ for all positive integers $n$. Prove that there exists some positive integer $m$ such that $x_m=x_0$. Now a function is said to be injective when there is a one to one mapping. When $f : A \to A$, it means that for any element $x$ in $A$, it maps to a distinct element $y$ in $A$. Now applying the function $x_n=f(x_{n-1})$ if $n =1$, then the function would give a result $x_0$. The question is how do i prove this even thou i have show that there exists some positive integer which in this case is $1$ - @Brian ..A is a finite set .I have updated it.I am not sure if my prove is good enough – Jack welch Jan 7 '13 at 2:42 – MJD Jan 7 '13 at 2:48 No, you don’t necessarily get $f(x_1)=x_0$; you get $x_1=f(x_0)$. – Brian M. Scott Jan 7 '13 at 2:48 You start with $x_0\in A$ and keep applying the function $f$ to get $x_1=f(x_0)$, $x_2=f(x_1)=f\big(f(x_0)\big)$, and so on. This generates a sequence $\langle x_0,x_1,x_2,\dots\rangle$ of elements of $A$. Now prove by induction on $n$ that if $x_0,x_1,x_2,\dots,x_n$ are all distinct elements of $A$, then $$x_{n+1}=f(x_n)\notin\{x_1,\dots,x_n\}\;;$$ use the fact that $f$ is injective. From this you can conclude that either $x_{n+1}=x_0$, in which case you’re done, or $x_0,x_1,x_2,\dots,x_n,x_{n+1}$ are all distinct elements of $A$, and the induction can continue. Since $A$ is finite, it must stop at some point, and at that point you have your desired $m$.	2016-02-13 21:35:08	{"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": 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": 0.9586926102638245, "perplexity": 56.56930435409286}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701168011.94/warc/CC-MAIN-20160205193928-00334-ip-10-236-182-209.ec2.internal.warc.gz"}
https://socratic.org/questions/how-do-you-find-the-slope-of-2x-3y-3-0	# How do you find the slope of 2x+3y-3=0? Mar 2, 2018 #### Answer: $- \frac{2}{3}$ #### Explanation: Rearranging $2 x + 3 y - 3 = 0$ we get $y = \frac{2}{3} \left[x\right] + 1$ and this is in slope intercept form , where the slope is $- \frac{2}{3}$ [ie, the coefficient of$x$] and the [1] is the intercept value on the $y$ axis	2019-10-23 11:47:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 7, "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": 0.9667791128158569, "perplexity": 802.4430096764117}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987833089.90/warc/CC-MAIN-20191023094558-20191023122058-00402.warc.gz"}
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In Louisiana, child support cases are generally divided into three categories: intake, collections, or parent locate. This is called ionization. diagonalise ( rangeR, noOfEigenvectors, drivingFromState=[0, 0, 0, 0, 0], eigenstateDetuning=0. Customization. Number Density for Compounds For a chemical compound (mixture) Z, which is composed of elements X and Y, the number (atom) density of the compound is calculated from mix mix mix M N N N A Z (3) In some cases, the desired quantity is the number density of the compound constituents. (Last updated on June 24, 2020 03:45:02 UTC). The interesting thing here is that every atom of krypton contains 36 protons. Star ATOM is a state of art mobile app developed exclusively for agents of Star Health Insurance. Open Atomic Wallet. 23 million electron volts. Just launch the calculator tool, fill in the required fields and click 'Calculate' to get your results. 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The kinetic temperature is the variable needed for subjects like heat transfer, because it is the translational kinetic energy which leads to energy transfer from a hot area (larger kinetic temperature, higher molecular speeds) to a cold area (lower molecular speeds) in direct collisional transfer. 00 \ pm {/eq}. ARC (Alkali Rydberg Calculator) is package of routines written in Python, using object-oriented programming (OOP) to make modular, reusable and extendable collection of routines and data for performing useful calculations of single atom and two-atom properties, like level diagrams, interactions and transition strengths for alkali metal atoms. Each isotope is a different weight. Finding Molar Mass. Atom - The Atom will need to be plugged into a mains power source to be utilised, this is due to the internal step motor. Consider a carbon atom whose electron configuration is the following. 4 This requires that there is a vacant excited state for the Ne atom to jump to. Finally, a FREE exposure calculator for anyone to download and use! Simply download the file, print out on a transparency film, and then start calculating your proper screen exposure times!. 6256 X 10-34 m 2 kg/sec. Most Bohr atom problems deal with hydrogen because it is the simplest atom and the easiest to use for calculations. 625xx10^(-34) Jsec`. Autor/es: Varios Editorial: Forum. Solution for. Determining oxidation numbers from the Lewis structure (Figure 1a) is even easier than deducing it from the molecular formula (Figure 1b). 60217646⋅10-19: Q (C) = Q (e) × 1. append (atom) [source] ¶ Append atom to end. calculate the total number of atoms in the following samples: a) 3. Determining the number of neutrons in an atom is fairly simple and doesn't even require any experimentation. Calculate the mass of a unit cell. The Atom is a legacy super-hero name, primarily associated with the ability to shrink in size. This program determines the molecular mass of a substance. org is an online conversion tool which helps you to convert Metric and Imperial units easily. Fluorine; 10. Line 8: "C" The first atom is a carbon. For a hydrogen atom, calculate the energy of a photon in the Balmer series that results from the transition n = 3 to n = 2. Power Supply Calculator. 097 * 10^7 * (1 - 1/9) = 1. Alternatively, atomic orbitals refer to functions that depend on the coordinates of one electron (i. how many electrons you've added or taken away, and this is exactly. Examples: Fe, Au, Co, Br, C, O, N, F. Please see the case of the sulfate ion:. Practically all chemistry and physics students need an electron configuration calculator. Learn more about how the half-life formula is used, or explore hundreds of other math, finance, fitness, and health calculators. Turn off atom manipulation Off; Drag selected atom Drag; Remove selected atom from stucture Delete. Open Source, Smart Contracts, dApps, On-Chain Governance, 2 Token Model, Interoperability. A = Mass Number or Atomic Weight; Calculation of relative atomic mass or weight of an element is made easier here. Atoms fit together with other atoms to make up matter. The list of possible attributes is the same as that used by the dump custom. :18 1 u has a value of 1. Palladium is a relatively rare catalytic metal that crystallizes in an FCC lattice. How to calculate formal charge. Multiply the total atoms by the average atomic mass per atom to calculate the grams. 0 g of calcium phosphate d) 1. To get started, please look at the introduction page , where you can find the links to other pages. With under 20 Watt power consumption, AtomMiner AM01 is probably the only green miner available on the market. the sun, a lightbulb) produce radiation containing many different wavelengths. The Nuclear Reactor is a generator that produces EU by slowly breaking down Uranium Cells. An excited state electron configuration of carbon is 1 s2 2s1 2p3. For ASE, a calculator is a black box that can take atomic numbers and atomic positions from an Atoms object and calculate the energy and forces and sometimes also stresses. Intel Atom C3000 Series Preliminary SKUs and Platform Architecture We are going to keep this piece updated as there is a good chance the preliminary information we received will change. size cm3/unit cell radius 140 pm The surface is the catalytically active portion of a palladium particle. Correct answers: 3 question: An atom of helium has a radius = and an average speed in the gas phase at of. Step 3 - The Number of. 60217646⋅10-19 C. Free online physics calculators, mechanics, energy, calculators. Scale Calculator. This is done by setting up the data and a few styles for the Canvas. 63 Cu + Energy 29 p + + 34 n o. 10,228 likes · 8 talking about this. The molar mass of any particle (atom, molecule, formula, or ion) is the sum of the average atomic masses of all atoms forming that particle. The Valence-Shell Electron Pair Repulsion Model •The valence-shell electron pair repulsion (VSEPR) model predicts the shapes of molecules and ions by assuming that the valence shell electron pairs are arranged as far from one another as possible. This video will teach you how to calculate atom economy in line with the current GCSE chemistry course. You can also enter the equations by clicking the elements in the table given in the chemical equation balancer. Polyatomic Ion Calculator. Metric Conversion Calculator. Solution for For a hydrogen atom, calculate the energy of a photon in the Balmer series that results from the transition n = 3 to n = 2. 4 его не видит. "Atom bank", "Atom" and "Digital Mortgages by Atom bank" are trading names of Atom bank plc, a company registered in England and Wales with company number 08632552. Therefore, if an atom only has 6 electrons in its outer shell, it will try to gain 2 from another atom (which is what causes atoms to bond in various ways - either by taking electrons, giving electrons, or sharing electrons with another atom). Use the periodic table of elements as your keypad. This ultra calculator is special by allowing you to choose among a great variety of units. specificity calculator. ) Add electrons to the sublevels in the correct order of filling. The Intel Atom C3338R adds 300MHz to the base clock and QAT support for a$1 premium and a 2W TDP premium over the Intel Atom C3338. Recall that for hydrogen energy expression: En = -2. 1 - Bugs Removed Event Firing Multiple Times. Since the boundary is not a well-defined physical entity, there are various non-equivalent definitions of atomic radius. If cooling is insufficient, the reactor will gradually overheat and eventually explode. 16+2=18, which is the relative atomic mass of 1 water molecule (relative to a carbon 12 atom). The electron in a hydrogen atom is in the n = 2 state. The formal charge on an atom can be calculated using a mathematical equation , a diagram or by instinct (!). 097 * 10^7 * (1 - 1/3² ) = 1. Official Google Search Help Center where you can find tips and tutorials on using Google Search and other answers to frequently asked questions. Electron configuration of an atom represents that how the electrons are distributed in its atom among the orbits (shells) and sub shells. 60 with a 24-hour trading volume of $82. Electron charge to coulombs conversion formula. Atom calculator. It is easy to calculate the atomic density of a material. The mass of the electrons is negligible. The electron configuration of an atom is very important as it helps to predict the chemical, electrical and magnetic behavior of substance. For example, removing one proton from an atom of krypton creates an atom of bromine. Type in your desired label and click elsewhere in the sketcher or press the enter or return key to close the text tool. This is useful so that the values can be used by other output commands that take computes as inputs. To get the radius of the pipe divide the diameter by 2. New tax tables have been created for Study and training support loans, which replace the tax tables for HELP/TSL/SSL and SFSS. A known volume of BBs will be spread out in a single layer (or monolayer) bound. Byte Friendly - FeedBurner. 0974x10 7 m-1; λ is the wavelength; n is equal to the energy level (initial and final) If we wanted to calculate energy we can adjust R by multipling by h (planks constant) and c (speed of light). If the binding energy was expressed in Joules, you might want to convert it to kJ because the value is higher. You choose from thousands of open source packages that add new features and functionality to the app. Example: Nitrite anion -O-N=O. ) The number of bonds for a neutral atom is equal to the number of electrons in the full valence shell (2 or 8 electrons) minus the number of valence electrons. b) Calculate the wavelength (in nm). About the calculator: This super useful calculator is a product of wolfram alpha, one. Grams to Atoms Calculator is a free online tool that displays the conversion from grams to atoms for the particle. The Half Life of Uranium-235 is 713,000,000 years. molecular weight of carbon dioxide: number of carbon atoms (N) [email protected] distance © 2003, Professor John Blamire. String not included. As shown below this reaction has only 50% atom economy. As a wiki is supposed to be a community work, i decided to initate this project and share with you the work i'v done so far. En = -Rh(1/n^2) where Rh = 2. Start building amazing cross platform mobile, desktop, and Progressive Web Apps with the web tech you know and love today. 0 gm atom of Hydrogen undergo transition giving the spectrtal lines of lowest energy is visible region of its atomic spectra. The Hamiltonian of the system thus takes the form. It is charged because the number of electrons do not equal the number of protons in the atom or molecule. An electron in the nth energy level is given by. Omni Calculator solves 1165 problems anywhere from finance and business to health. Practically all chemistry and physics students need an electron configuration calculator. For example. Dacă doriți să convertiți o viteză de rotație într-o viteză liniară, cum ar fi mile pe oră, trebuie să cunoașteți diametrul cercului în care obiectul se învârte. Atoms will cancel out, leaving only moles (which is what we want). Atoms are made of neutrons, protons and electrons. Question: What Is The Hybridization Of The Cenral Atom In The Sulfur Pentafluoryl (SF5+) Cation?. However, the mass of an electron is so small, it is considered negligible and not included in the calculation. The hydrogen-2 nucleus, for example, composed of one proton and one neutron, can be separated completely by supplying 2. For other languages, it is required to create one wiki in each language. 85 if female)] / (72 x Cr) Note: The original Cockroft-Gault Equation listed here provides an estimate of creatinine clearance, but this original formula is not adjusted for body surface area and may be less accurate in obese patients. Diffusion Time Calculator When considering the diffusion of ions and molecules in solutions, it is generally useful to be able to estimate the time required for diffusion over a given distance. which gives three equations. Checking by Method One. Hydrogen & Atomic Spectra – WS14 Part A: Line Spectrum of Hydrogen Gas Using Rydberg's equation calculate the wavelength of the radiation emitted by a hydrogen atom for the following electronic transitions: a. The alkali metals (group I) always have an oxidation number of +1. "Atom bank", "Atom" and "Digital Mortgages by Atom bank" are trading names of Atom bank plc, a company registered in England and Wales with company number 08632552. We reduce the time, effort and cost of the appraisal process, so our clients not only get accurate valuations, but receive first class customer service as well. You can convert your data with a simple click and discover the impacts on temperature and pressure. Are you building a modern gaming PC, low power HTPC media server, or maybe you need to figure out power requirements for a rack in a data center?. If you have a scientific calculator or graphic calculator, you probably have some key that allows you to enter the E or EXP symbol. making this calculator quite versatile. 602×10 -19 Joules ) and. BYJU’S online grams to atoms calculator tool makes the conversion faster, and it displays the conversion to atoms in a fraction of seconds. By using this website, you agree to our Cookie Policy. Find the order of a=1 in the asked at 2020-06-27 04:04:52; Verily the identity sec 0-cos 0-tan 0sin 0 To verily the identity, start with the more asked at 2020-06-27. Albion Online 2D Database — Current Meta, Items, Mobs, Destiny Calculator, Craft Calculator, Fame Calculator and more. Step 3 - The Number of. How many molecule in 1 atom? The answer is 1. This is often useful for understanding or predicting reactivity. Autor/es: Varios Editorial: Forum. Hydrogen Bonding: Acceptors and Donors. Compound interest - meaning that the interest you earn each year is added to your principal, so that the balance doesn't merely grow, it grows at an increasing rate - is one of the most useful concepts in finance. We assume you are converting between molecule and atom. You can create renderings of Lewis Dot Structures with the ChemDoodle Web Components. That relationship is: logES = 11. Amazing that this mathematical procedure is able to group substrates apart from products. As usual, here at www. ) The number of bonds for a neutral atom is equal to the number of electrons in the full valence shell (2 or 8 electrons) minus the number of valence electrons. Atom Bank is a challenger bank which was publicly launched in 2016. Suppose the speed of a xenon atom at has been measured to within. StarkMap calculations in compact binary format in file named filename. Omni Calculator is here to change all that - we are working on a technology that will turn every* calculation-based problem trivial to solve for anyone. Inefficient, wasteful processes have low atom economies. jp to order. Counting Atoms. 02214 x 1023 atoms) and is equal to the mass of the atom's nucleus in atomic mass units, or to the average mass of the nuclei of all isotopes of the atom. Buy tickets, pre-order concessions, invite friends and skip lines at the theater, all with your phone. It is about 1. Finally, calculate the grams. Description. Separated into equations in terms of the spherical coordinates. Electrons have a set number of orbitals, ring-like paths around the nucleus, they can travel in called stationary states. Instead, atomic mass is expressed in unified atomic mass units (unit symbol: u). Contact us today for your requirements. The hydrogen-2 nucleus, for example, composed of one proton and one neutron, can be separated completely by supplying 2. com business loan calculator helps you answer all those questions and more. We offer a simple tool that helps you navigate the world of cryptocurrency within a single site. If all of the atoms usually form the same number of bonds, the least electronegative atom is usually the central atom. To write a complete electron configuration for an uncharged atom, Determine the number of electrons in the atom from its atomic number. Calculate the radius of a niobium atom. Both the earthquake magnitude and the seismic moment are related to the amount of energy that is radiated by an earthquake. Or you can choose by one of the next two option-lists, which contains a series of common organic compounds (including their chemical formula) and all the. We must also know the mass of water, lets say 50g. Bohr Atom Problem What is the energy change when an electron drops from the n=3 energy state to the 푛=1 energy state in a hydrogen atom?. Electron Configurations of Ions. The answer according to my booklet is 1. Matter-Energy Conversion. Answer to For the electronic transition from n = 3 to n = 5 in the hydrogen atom. Just subtract the atomic number from the atomic mass. Everything around us is made up of atoms. If you have a scientific calculator or graphic calculator, you probably have some key that allows you to enter the E or EXP symbol. Atom G has 12 protons and 9 neutrons. A known volume of BBs will be spread out in a single layer (or monolayer) bound. 10868 ----- #1238 - How Many Atoms are in a kilogram? - To measure the mass in kilograms we need to have a kilogram standard. Customization. 92 GHz) quick reference guide including specifications, features, pricing, compatibility, design documentation, ordering codes, spec codes and more. Fecha Edición: 1999. 626 x 10-34 Js and c = 2. You must define all the reagents. "Atom bank", "Atom" and "Digital Mortgages by Atom bank" are trading names of Atom bank plc, a company registered in England and Wales with company number 08632552. The simple method given above can be used to write Lewis structures with expanded octets. Examples: C6H12O6, PO4H2(CH2)12CH3. Grams to Atoms Calculator is a free online tool that displays the conversion from grams to atoms for the particle. Atom Calculator allows you to search for Elements of the Periodic Table using their Name, Symbol or Atomic Number. Write your answer as a multiple of and round it to significant figures. 32 kg of ammonium sulfate can you help me with any of these? im pretty confused :S. We offer a simple tool that helps you navigate the world of cryptocurrency within a single site. Calculate the radius of a niobium atom. property calc¶ Calculator object. In addition, you can define the charge of ions with known numbers of protons and electrons. 01] Quick Links. 097 * 10^7 m^-1 For n=3 to n-1 you substitute into the equation 1/λ = 1. 5 nanometers (1 × 10 −10 m to 5 × 10 −10 m). How to Calculate the Number of Neutrons, Protons and Electrons in an Atom. 022xx10^23"particles")# and the relative atomic mass of copper #("1 mole of copper = 63. When the different wavelengths of radiation are separated from such a source a spectrum is produced. ) Should also work for Gen 8 (Pokemon Sword and Shield) unless they tweak or change the typing chart. Use uppercase for the first character in the element and lowercase for the second character. Scribd is the world's largest social reading and publishing site. To write a complete electron configuration for an uncharged atom, Determine the number of electrons in the atom from its atomic number. We see the Intel Atom C3436L as a 10. Check our step-by-step guide on how to stake ATOM. You will have all the information which you need to produce a graph, if needed. Vulnerability Metrics Expand or Collapse. It is also known as the quadratic mean. - Attachments - The proton - Mass is one of the fundamental constants in nature. exe for 32-bit systems and AtomSetup-x64. 1 slug is defined as a mass that accelerates by 1 foot per second squared, when one pound-force is applied. This second definition is actually the relative atomic. Here is a simple online oxidation number calculator to calculate the oxidation number of any compound or element by just clicking on the respective compound name in the given elements table with ease. Get One Atom XL with the Great Price!$110 Off the Retail Value! US, Japan, Germany, UK, France, Italy, Spain, China Free Shipping! Includes: Atom XL. Reset Ratios; SelectElement 0-19. Free Chemical Properties calculator - Find symbol, atomic number, mass, boiling point, melting point and proton count of atoms This website uses cookies to ensure you get the best experience. About Root Mean Square Calculator. See a clock with the accurate time and find out where it is observed. Whenever you see a flash of light, the object that emitted it loses a small fraction of its mass, becoming slightly lighter. 0087, respectively). If mass of one atom is 3. Atom E has 10 protons and 9 neutrons. The interesting thing here is that every atom of krypton contains 36 protons. This type calculator shows the type effectiveness for any given type or dual-type Pokemon. CaCO 3 → CaO + CO 2 12) Calculate the atom economy to make sulfur trioxide from sulfur dioxide. Learn how to write the chemical formula of a variety of chemical compounds using the arms and link method. Half Life is a characteristic of each radioactive isotope. Bond Length and Bond Order • Bond length (or bond distance) is the distance between the nuclei in a bond. You can do it like this: The energy of the electron is given by: sf(E_n=-(2. This is owing to the presence of electrons in the atoms. The mass of the electrons is negligible. This article will provide you with the definition of an atom, atom components. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public. That being the case, the weight of a gold atom becomes, w = (196. ATOMS for atomic-structure display. 4 This requires that there is a vacant excited state for the Ne atom to jump to. The goal is to add various kinds of exercises dealing with molecular formulae, monoisotopic mass and isotopic distribution. 5” card stock. A Free Online Calculator, Quick and Easy, and Full Screen! < Back to Online Calculator. Determining the number of neutrons in an atom is fairly simple and doesn't even require any experimentation. The answer according to my booklet is 1. (Last updated on June 24, 2020 03:45:02 UTC). Coulombs to electron charge conversion calculator How to convert electron charge to coulombs. The framework of these calculators are built on the symbolic structure, the vast algorithms that have been created and lastly many ideas from NKS (new kind of science) Use this calculator for your personal endeavors that may require such calculations. Calculate the best way your cargo is loaded / optimized in a container. Writing Electron Configurations. Lasers emit radiation which is composed of a single wavelength. The calculator is not supposed to predict what the products will be. 36% in the last 24 hours. Therefore, the formula weight of NaCl is 58. This is also known as molecular weight. Due to the fact that electrons can transfer from one atom to another, every atom has the possibility of becoming negatively or positively charged. HOW TO COUNT ATOMS Background+Information:+ • subscripts!-!the!little!numbers!thattell!how!many!atoms!there!are!!(ex:!In!3H 2O,!the! 2!is!the!subscript. 0974 x 107 m-1. Fortunately, Neon has an excited state whose excitation energy nearly matches the excitation energy of the He atom. Bohr's Model of the Hydrogen Atom. posted by Greg:. The oxidation state of an atom is the charge of this atom after ionic approximation of its heteronuclear bonds. This means that 6 hours later, half of the medication will be consumed, leaving half remaining, at 5mg. Use our custom tools to build a Steam profile badge, calculate collection value, find Steam friends and discover the Pile of Shame. 0 x 10⁻⁵ mol of sulphur hexafluoride c) 31. ATOMS is a program for drawing all types of atomic structures, including crystals, polymers and molecules. 0221415E+23 molecule, or 6. In Louisiana, child support cases are generally divided into three categories: intake, collections, or parent locate. 44 g/mol (22. Now number of moles of CH_4 in 5. First Eco-Friendly Crypto Miner. Compound Semiconductors Crystallographic Notation Crystallographic Planes and Si Wafers Crystallographic Planes in Si Summary Crystalline Si: 4 valence electrons per atom diamond lattice each atom has 4 nearest neighbors 5 x 1022 atoms/cm3 Crystallographic notation Miller indices are used to designate planes and directions within a crystalline. saveCalculation and alkali_atom_functions. See a clock with the accurate time and find out where it is observed. Joe Champion for Low Energy Nuclear Change (LENC) January 11, 2011. Molecular Weight Calculator. Open Source, Smart Contracts, dApps, On-Chain Governance, 2 Token Model, Interoperability. , the first atom in the first specification is matched to the first atom in the second specification, the second atom to the second atom, and so on. Description: The Anatomy of an Atom Die Cut sheets are a fantastic activity for Boy Scouts and Girl Scout workshops! This activity designed for middle and high school students, makes creating an atom mobile fun and easy. Atom RPG Skills Guide In Atom RPG, each character can invest points in up to sixteen skills. ARC (Alkali Rydberg Calculator) is package of routines written in Python, using object-oriented programming (OOP) to make modular, reusable and extendable collection of routines and data for performing useful calculations of single atom and two-atom properties, like level diagrams, interactions and transition strengths for alkali metal atoms. Practically all chemistry and physics students need an electron configuration calculator. 18xx10^(-18))/(n^2)" ""J") Where sf(n) is the principle quantum number. The heat vibrational motion of atoms, affecting the atom displacements, results in a weakening of diffracted lines (reflections). Everything around us is made up of atoms. The Atom Economy of this reaction is 43%, calculated using the relative formula masses. Complete Electron Configurations. Visual Schrödinger: A Visualizer-Solver Applet for the One-Dimensional Schrödinger Equation. Since the boundary is not a well-defined physical entity, there are various non-equivalent definitions of atomic radius. Discussions with Retirement Professionals may lead to a referral to affiliates including Wells Fargo Bank, N. The 3D Load Calculator is a tool where you can insert all the pieces you want to load in a container and see the optimal way to load it in a container. By ScaleCalculator. Now, be careful when punching these numbers into your calculator. In the Bohr model of a hydrogen atom, the electron, if it is in the ground state, orbits the proton at a distance of r = 5. The original was Al Pratt during the Golden Age, a diminutive man with superhuman strength; he was a founding member of the Justice Society. For example, removing one proton from an atom of krypton creates an atom of bromine. For information about the Federal Budget changes for the 2019–20 income year refer to Tax tables. An atom is the smallest building block of matter. The AT9012 is available in Black, White and Silver with Both Light Colour. If a conjugate base's negative charge is on a more electronegative atom, it is more stable, and thus the parent acid is stronger. Rutherford Model of the Atom. Here is the Steam Id for ATOM Torino460, also known as MOTATorino. Two of the coordination positions are occupied by the shared electron-pairs that constitute the O–H bonds, and the other two by the non-bonding pairs. 14 for its pi calculations? Or do you use more decimals like say: 3. The Half Life of Uranium-235 is 713,000,000 years. Show your calculations. Ray Palmer took the name during the Silver Age, a physics professor at Ivy University who developed equipment to shrink himself down to subatomic levels while. Tweak the look and feel of your UI with CSS/Less, and add major features with HTML and JavaScript. 10868 ----- #1238 - How Many Atoms are in a kilogram? - To measure the mass in kilograms we need to have a kilogram standard. In other words, minimize the total electronic energy of a beryllium atom that has two electrons in an orbit of radius R1 and two electrons in another orbit of radius R2. Worksheet requires students to calculate the % atom ecomony of producing a certain required product in a given chemical reaction. Can't find what you're looking for? Install themes created by the Atom community or create your own. Molar mass is the sum of all of the atomic masses in a formula. Compound Semiconductors Crystallographic Notation Crystallographic Planes and Si Wafers Crystallographic Planes in Si Summary Crystalline Si: 4 valence electrons per atom diamond lattice each atom has 4 nearest neighbors 5 x 1022 atoms/cm3 Crystallographic notation Miller indices are used to designate planes and directions within a crystalline. This calculator computes the force between a solenoid and another piece of ferromagnetic material separated by a gap of distance g. The term atomic orbital may also refer to the physical region or space where. Depending on the isotope, its Half Life may range from a few fractions of a second to several billion years. 99 g/mol and chlorine is 35. Ubuntu стоит на seagate (500GB), новый диск - WD WDC wd10ezex (1TB). Join the Wattbike tribe. Determining the hybridization of a molecule can help identify its shape and structure. Flash Player architecture is designed to ignore the existence of an edts atom; however, an edts atom containing invalid or broken data may interfere with smooth and stable switching of. Use this free circumference calculator to find the area, circumference and diameter of a circle. Occupying a volume whose radius is 1/100,000 the size of the atom, the nucleus contains most (99. Atom Calculator Description You enter the atomic # and the Atomic Mass ant it tells you how many Protons, Neutrons, And Electrons Author Charles Crossan ([email protected] The key features of this app has been listed below • Perform the entire policy sale process including proposal. Video: Potassium Electron Configuration Notation The configuration notation provides an easy way for scientists to write and communicate how electrons are arranged around the nucleus of an atom. An atom is one of the basic units of matter. The atomic mass is the sum of the neutrons and protons, so by subtracting the number from the mass you get the number of neutrons. Добрый день! Купил себе новый жесткий диск, а Ubuntu 18. Omni Calculator solves 1165 problems anywhere from finance and business to health. Bohr's Model of the Hydrogen Atom. Calculate the smallest possible length of box inside of which the atom could be known to be located with certainty. Autor/es: Varios Editorial: Forum. Determining Molecular Polarity - Oneonta. The Cosmos price prediction sentiment is currently bearish. 022 x 10 23 atoms of carbon (Avogadro's number). Since the mole units cancel out, we are left with the weight of the gold atom in grams. Exchange Cosmos (ATOM) (ATOM) to Bitcoin (BTC) Changelly is a platform that provides you with a possibility to convert ATOM to BTC in a few clicks. Atomic mass can be expressed in grams. Dacă doriți să convertiți o viteză de rotație într-o viteză liniară, cum ar fi mile pe oră, trebuie să cunoașteți diametrul cercului în care obiectul se învârte. Choose calculator: 3. Price Chart, Trade Volume, Market Cap, Highest and Lowest Prices, Exchange Rate, Latest News and more. 1 mole of silver atoms = 107. About Cosmos. In order to figure out where electrons go in an atom we have to follow 3 main rules. 99 g/mol and chlorine is 35. Protons and neutrons make up the nucleus of the atom, a dense and positively charged core, whereas the negatively charged electrons can be found around the nucleus in an electron cloud. The AT9012 LED Downlight by Atom Lighting features a sealed downlight housing with an IP44 rating which makes this downlight suitable in areas where moisture is in the air. An atom can have the following charges: positive, negative, or neutral, depending on the electron distribution. The oxidation number of a Group 1 element in a compound is +1. Helium Atom A helium atom consists of a nucleus of charge surrounded by two electrons. 6734×10-24 (g), but I don't understand how we got this answer. 10H20 will result in the molar mass and mass analyses being calculated when the button is pressed. Canada Revenue Agency 2020-04-26T21 launched the Canada Emergency Wage Subsidy calculator to enable eligible employers to determine the amount they can expect. saveCalculation and alkali_atom_functions. pdf), Text File (. Line 9: "O 1 OC" The second atom is an oxygen with a distance to the first atom of OC. Kevin Schmidt Department of Physics and Astronomy Arizona State University Tempe, Arizona 85287 Michael A. ” That’s the bold message adorning the homepage of newly-formed Atom Bank. Tax withheld calculators. Albion Online 2D Database — Current Meta, Items, Mobs, Destiny Calculator, Craft Calculator, Fame Calculator and more. The Akafugu Store has been updated with several items that have been out of stock for quite some time. Power Supply Calculator. - Define atomic number and explain what happens if the number of protons in an atom changes - Define an ion and explain what happens if the number of electrons in an atom change - Define an isotope and explain what happens if the number of neutrons in an atom changes How to with all the curveballs included. Use the calculator to map out your strategy from start to finish by inputting the key elements of your. See for example, the compute reduce, fix ave/atom, fix ave/histo, fix ave/chunk, and atom-style variable commands. There is an ionization energy for each successive electron removed; the ionization energy associated with removal of the first (most loosely held) electron, however, is most. you taken to measurement X Avogadro Number)/ Molecular weight. 0221415E+23 molecule, or 6. Hydrogen atom cation: 1: H 2-hydrogen diatomic anion-1: H 2 + Hydrogen cation: 1: H 3 + hydrogen trimer cation: 1: He-Helium atom anion-1: He + Helium atom cation: 1: HeH + Helium hydride cation: 1: He 2 + helium diatomic cation: 1: Li-Lithium atom anion-1: Li + Lithium atom cation: 1: Na-Sodium atom anion-1: Na + Sodium atom cation: 1: K. 63 Cu + Energy 29 p + + 34 n o. Bit 4048591891 is set once by atom 5 at radius 2. Calculate the unknown variable in the equation for kinetic energy, where kinetic energy is equal to one half times the mass multiplied by velocity squared; KE = 1/2 * mv^2. Quizzes are timed (approximately 4 minutes per question). Omni Calculator is here to change all that - we are working on a technology that will turn every* calculation-based problem trivial to solve for anyone. How to calculate formal charge. Albion Online 2D Database — Current Meta, Items, Mobs, Destiny Calculator, Craft Calculator, Fame Calculator and more. 8051e+24 <<< if you want to input atom amount it should be expressed this way, not as 1. 022xx10^23"particles")# and the relative atomic mass of copper #("1 mole of copper = 63. has one atom of sodium (Na) and one atom of chlorine (Cl). Calculate the wavelength, in nanometers, of the spectral line produced when an electron in a hydrogen atom und? Use the Rydberg formula [1] 1/ λ = R (1/(n')² - 1/n²) where R = 1. 6748×10-24 electron:. (See Below. It is charged because the number of electrons do not equal the number of protons in the atom or molecule. 4 g/cm^3}$and atomic weight of$\pu{192. Simple, chunky calculator widget for your editor. The energy required to break down a nucleus into its component nucleons is called the nuclear binding energy. 5” card stock. For information about the Federal Budget changes for the 2019-20 income year refer to Tax tables. If you want to run some python script on different python versions using Atom, I think the best solution is to use atom's Hydrogen package and installing a kernel for every conda env you need, to do it you should open Anaconda's prompt and: Install jupyter on every conda env you want to work with; Activate the first env that you wish to use on Atom. com provide China 2933990099 Tariff tax rates import duty lookup,HS code search,GB standards,CIQ inspection Quarantine standards query service. Formula of Atomic Mass: A = Number of protons + Number of neutrons. Worksheets are , Countingatomswork, Counting atoms work, Counting atoms work, Chemical reactions counting atoms and balancing chemical, , Counting atoms work, Counting atoms. The value will be …. They allow us to change one unit of measure to another, like crossing a bridge over a river from one side to the other. Do a quick conversion: 1 moles = 6. Open Source, Smart Contracts, dApps, On-Chain Governance, 2 Token Model, Interoperability. To calculate the number of neutrons in an atom, you have to either memorize the formula, or better yet, understand how different numbers on the periodic table are derived so you can calculate it. Consider a carbon atom whose electron configuration is the following. As cells decay inside the reactor, they produce heat. Calculate the Distance of an Electron from the Proton of a Hydrogen Atom By Steven Holzner When you want to find where an electron is at any given time in a hydrogen atom, what you're actually doing is finding how far the electron is from the proton. If you know the mass of the nucleus in atomic mass units (amu), you can find the number of neutrons, because neutrons and protons have the same mass. Ground Atom Consider a clause ( disjunction of literals ) obtained from those of a first-order logic formula in Skolemized form Then an atomic statement obtained from those of by replacing all variables by elements of the Herbrand universe of is called a ground atom. You can use parenthesis or brackets []. Online cryptocurrency wallet with multi-cryptocurrency accounts. Calculate the wavelength, frequency and wave number of a light wave whose period is 2. Ionization energy, also called ionization potential, in chemistry and physics, the amount of energy required to remove an electron from an isolated atom or molecule. HOMESFORTH is a creative, energetic and innovative estate agency based in London. To calculate the mass of a single atom, first look up the atomic mass of from the periodic table. Calculate the best way your cargo is loaded / optimized in a container. The cross sections are taken from the report IAEA INDC(NDS)-440 and/or the PCNuDat data base at BNL. txt) or view presentation slides online. Simple, chunky calculator widget for your editor. Voltage Drop Calculator You know, the current flows based on the electron movement, while moving the electron we need to consume some energy to move the electron from one atom to another. So, the atomic mass is the sum of the masses of protons and neutrons. An excited-state atom is an atom in which the total energy of the electrons can be lowered by transferring one or more electrons to different orbitals. ISOTOPE OR DIFFERENT ATOM: Determine if the following atoms are isotopes or different atoms. The 2p level is split into a pair of lines by the spin-orbit effect. Atoms are paired in the order specified, i. This table shows how the atom size, and atomic radius values change as you move horizontally and vertically across the periodic table. Read our article on how to calculate molar. Add two electrons to each s sublevel, 6 to each p sublevel, 10 to each d sublevel, and 14 to each f. Finally, a FREE exposure calculator for anyone to download and use! Simply download the file, print out on a transparency film, and then start calculating your proper screen exposure times!. Scientific Calculator - A great Scientific Calculator. Therefore, the formula weight of NaCl is 58. Learn how to write the chemical formula of a variety of chemical compounds using the arms and link method. Calculate the radius of an atom with help from an experienced math professional in this free video. How can the Cosmos (ATOM) mining profitability be calculated You can use this tool and input all the parameters like hash rate of your mining hardware rig, hourly power consumption of it, pool commission percentage (the pool which you are going to join in the mining), difficulty of the Cosmos (ATOM) network, amount of blocks rewarded, price of Cosmos (ATOM*) and once you click calculate it. 660538863127E-24 mole. Click on pop-out icon or print icon to worksheet to print or download. This is the quickest and easiest way to learn. append (atom) [source] ¶ Append atom to end. The way to calculate the number of atoms is to A) choose any chemical element or compound - let's say common table salt, B) go to the Periodic Table and find the atomic weights of it's component atoms - in this case sodium (Na) and chlorine (Cl) -. If the electron stays in one orbital, the energy of the electron remains constant.	2020-08-05 08:13:37	{"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": 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": 0.31161031126976013, "perplexity": 1755.0274263357744}, "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-2020-34/segments/1596439735916.91/warc/CC-MAIN-20200805065524-20200805095524-00162.warc.gz"}
http://mathhelpforum.com/calculus/49970-equation-tangent.html	1. ## Equation of tangent This has been bugging me all day, I feel like I am doing it right, but alas, my answer is always wrong $ y = \frac{1}{\sqrt{x}} $ Find the equation of the tangent line, f(x), at the point ( 4, 1/2) $ \lim_{x \to 4} \frac{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{4}}}{x-4} $ = $ \lim_{x \to 4} \frac{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{4}}}{x-4} $ = $ \lim_{x \to 4} \frac{\frac{1}{\sqrt{x}} - \frac{1}{2}}{x-4} $ = $ \lim_{x \to 4} \frac{\frac{2-\sqrt{x}}{2\sqrt{x}}}{x-4} $ = $ \lim_{x \to 4} \frac{\frac{2-\sqrt{x}(2+\sqrt{x})}{2\sqrt{x}(2+\sqrt{x})}}{x-4} $ = $ \lim_{x \to 4} \frac{\frac{4-x}{4\sqrt{x}+2x}}{x-4} $ = $ \lim_{x \to 4} \frac{4-x}{4\sqrt{x}+2x} * \frac{1}{x-4} $ = $ $ $\lim_{x \to 4} \frac{-1}{4\sqrt{x}+2x} $ = $ $ $\lim_{x \to 4} \frac{-1}{4\sqrt{4}+2(4)} $ = $ \frac{-1}{16} $ is that right? (as far as the slope is concerned) and the equation would be $y-.5 = \frac{-1}{16}(x-4)$ $ y= \frac{-1}{16}x+.75 $ 2. The slope would be -1/16 and your equation for the tangent line appears to be right too. 3. Yes, you are correct. Your proof is correct as well. If you want to verify your answer, type in both equations into your calculator and find the intersection point. Then you will find out that it's (4,1/2) 4. Sweet. Can someone explain to me how I would find the slope using the Chain rule? 5. Originally Posted by silencecloak Sweet. Can someone explain to me how I would find the slope using the Chain rule? I assume you know the chain rule... $y=x^\frac {-1} {2}$ $\frac {dy} {dx}= \frac {-1} {2} x^\frac {-3} {2}$ Clean it up a bit and you get $\frac {dy} {dx}= \frac {-1} {2x \sqrt{x}}$ Then you just plug x=4 into the derivative 6. Originally Posted by Linnus I assume you know the chain rule... $y=x^\frac {-1} {2}$ $\frac {dy} {dx}= \frac {-1} {2} x^\frac {-3} {2}$ Clean it up a bit and you get $\frac {dy} {dx}= \frac {-1} {2x \sqrt{x}}$ Then you just plug x=4 into the derivative I actually don't know the chain rule, my teacher is making us do it the long way as I outlines above, but holy crap that's easier EDIT: I know the basics of it, but either I don't know them as well as I thought I did, or your computations don't make sense to me	2017-11-22 04:15:29	{"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 21, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.76137375831604, "perplexity": 305.11343836326375}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806455.2/warc/CC-MAIN-20171122031440-20171122051440-00325.warc.gz"}
http://math.stackexchange.com/questions/243699/misinterpreting-logic-expression	# misinterpreting logic expression Represent in first order logic "Every person who dislikes all vegetarians is smart" ## My solution $\forall x,y$ $person(x) \land vegetarian(y) \land dislike(x,y) \rightarrow smart(x)$ People say to me that this is not correct. I don't understand why? Where is my misconception? - Note that, one problem among many here (and as the answers all reveal): you need to be careful to use parentheses to enclose expressions. E.g., you need to use parentheses to make explicit the scope of a quantifier, to make explicit what is implying what, etc. – amWhy Nov 24 '12 at 14:04 This sort of translation can be done easily if you take it in stages. Consider (a) Every person who dislikes all vegetarians is smart (b) $\forall x$( person-who-dislikes-all-vegetarians $(x) \to smart(x))$ (c) $\forall x([person(x) \land \forall y\{(vegetarian(y) \to dislike(x, y)\}] \to smart(x))$ The move from (a) to (b) needs no explanation, and the move from (b) to (c) just involves noting that person-who-dislikes-all-vegetarians $(x)$ says that $x$ is a person and for any vegetarian $y$, $x$ dislikes $y$. OK? Now, that's fine and you can stop there. But suppose you wanted to bring the quantifers all to the front. Then (c) is equivalent to (d) $\forall x(\forall y[person(x) \land \{(vegetarian(y) \to dislike(x, y)\}] \to smart(x))$ (Do you see why?) But now (d) is $\forall x$ followed by something of the form $(\forall yFxy \to Gx)$. But now the crucial observation. $(\forall yFxy \to Gx)$ is equivalent to $(\neg\forall yFxy \lor Gx)$, which is equivalent to $(\exists y\neg Fxy \lor Gx)$ which is equivalent to $\exists y(\neg Fxy \lor Gx)$ which is equivalent to $\exists y(Fxy \to Gx)$. In short, dragging a quantifier out from the antecedent of a conditional flips it into the dual quantifier. That means (d) is equivalent to (e) $\forall x\exists y([person(x) \land \{(vegetarian(y) \to dislike(x, y)\}] \to smart(x))$. - +1Nicely explained!, especially the explicit "logic" behind how "dragging a quantifier out from an antecedent of a conditional flips it into the dual quantifier." (And nicely disambiguated, e.g., using parentheses to enclose the antecedent.) – amWhy Nov 24 '12 at 14:00 Your answer and that of henning was really really useful. I would both upvote you if I could. Thanks a lot! – tgoossens Nov 24 '12 at 20:15 Your solution claims that whenever there is someone who dislikes some vegetarian, the one doing the disliking is smart. For example, suppose that indeed, everyone who dislikes all vegetarians is smart. Suppose further John is kinda stupid, and likes everybody except for Brian, who is a vegetarian. That is not a conflict, because John doesn't dislike all vegetarians: Henry is a vegetarian, and John likes him just fine. However your first-order formula is false in this situation. It is false because the only thing that is needed to make a $\forall$ formula false is a single counterexample. And $x=\text{John}, y=\text{Brian}$ is a counterexample -- the body of the quantifier then becomes: $$\text{person(John)}\land\text{vegetarian(Brian)}\land\text{dislike(John,Brian)}\to\text{smart(John)}$$ and here all the conjuncts to the left of the $\to$ are true, but the conclusion on the right is not. I suspect you started out fine with something like $$\forall x\big[\text{person}(x)\land\text{dislikes-all-vegetarians}(x)\to\text{smart}(x)\big]$$ but then you when wrong translating "dislikes-all-vegetarians". By itself it is $$\text{dislikes-all-vegetarians}(x) \equiv \forall y[\text{vegetarian}(y)\to \text{dislike}(x,y)]$$ and we can insert it into the first formula to get $$\forall x\big[\text{person}(x)\land\forall y[\text{vegetarian}(y)\to \text{dislike}(x,y)]\to\text{smart}(x)\big]$$ Note that here you cannot move the $\forall y$ out such that it is neighbor to the $\forall x$, because it is important for the meaning that $\forall y$ is subordinate to the $\rightarrow$ rather than the opposite. We can insist on getting a logically equivalent formula where all quantifiers are out at the front (a "prenex formula"), but then the $\forall y$ becomes $\exists y$, which makes the result rather unrecognizable as a rendering of the original English sentence: $$\forall x \exists y \big[ \text{person}(x) \land [\text{vegetarian}(y) \to \text{dislike}(x,y)] \to \text{smart}(x)]\big]$$ or $$\forall x \exists y \big[ \text{person}(x) \to \text{smart}(x) \lor [\text{vegetarian}(y) \land \neg\text{dislike}(x,y)]\big]$$ (more or less "every person is smart unless (possibly) if there is a vegetarian he doesn't dislike"). - $\forall x[ person(x) \rightarrow [\forall y[veg(y) \rightarrow dislike(x,y)]\rightarrow smart(x)]]$ - I am not sure if the statement you wrote is equivalent to mine. I will check – Amr Nov 24 '12 at 13:07 Your statement is correct i think. And I can see why. But i cannot "intuitively" see why my solution is not correct – tgoossens Nov 24 '12 at 13:14 Is this your statement $\forall x,y$[ $person(x) \land vegetarian(y) \land dislike(x,y) \rightarrow smart(x)]$? – Amr Nov 24 '12 at 13:16 That is correct. – tgoossens Nov 24 '12 at 13:18 The other answer explains your mistake well – Amr Nov 24 '12 at 13:39	2015-04-26 05:16:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8312230706214905, "perplexity": 807.6975664200285}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246652631.96/warc/CC-MAIN-20150417045732-00209-ip-10-235-10-82.ec2.internal.warc.gz"}
https://mathhelpboards.com/threads/terminology.4921/#post-22299	# Terminology #### delgeezee ##### New member My book describes a linear system with "m equations in n unknowns." Maybe this is a subtle detail but this confuses me. Shouldn't it be the other way around, "n unknowns in m equations?" #### tkhunny ##### Well-known member MHB Math Helper Re: terminology It makes no difference, so long as m and n are defined. #### Bacterius ##### Well-known member MHB Math Helper Re: terminology [JUSTIFY]They both mean the same thing as far as I can tell. I think this may be a language problem, the first form might be more natural in english whereas the other sounds more natural in other languages (for instance french).[/JUSTIFY] #### Klaas van Aarsen ##### MHB Seeker Staff member Re: terminology I'd write the first form as "m equations with n unknowns." Anyway, the two forms mean the same thing. #### Jameson Like others said the variable names can be whatever you want to use, but standard convention is that a matrix of size $m \times n$ corresponds to a linear system of equations, which means that there are $m$ rows and $n$ columns. That corresponds to $m$ equations and $n$ variables.	2022-07-07 04:34:37	{"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": 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": 0.6862772107124329, "perplexity": 758.3849958500691}, "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/1656104683683.99/warc/CC-MAIN-20220707033101-20220707063101-00564.warc.gz"}
https://en.m.wikibooks.org/wiki/Haskell/YAHT/Language_basics	Preamble Introduction Getting Started Language Basics (Solutions) Type Basics (Solutions) IO (Solutions) Modules (Solutions) Recursion Complexity In this chapter we present the basic concepts of Haskell. In addition to familiarizing you with the interactive environments and showing you how to compile a basic program, we introduce the basic syntax of Haskell, which will probably be quite alien if you are used to languages like C and Java. However, before we talk about specifics of the language, we need to establish some general properties of Haskell. Most importantly, Haskell is a lazy language, which means that no computation takes place unless it is forced to take place when the result of that computation is used. This means, for instance, that you can define infinitely large data structures, provided that you never use the entire structure. For instance, using imperative-esque pseudo-code, we could create an infinite linked-list containing the number ${\displaystyle 1}$ in each position by doing something like: List makeList() { List current = new List(); current.value = 1; current.next = makeList(); return current; } By looking at this code, we can see what it's trying to do: it creates a new list, sets its value to ${\displaystyle 1}$ and then recursively calls itself to make the rest of the list. Of course, if you actually wrote this code and called it, the program would never terminate, because makeList would keep calling itself ad infinitum. This is because we assume this imperative-esque language is strict, the opposite of lazy. Strict languages are often referred to as "call by value," while lazy languages are referred to as "call by name." In the above pseudo-code, when we "run" makeList on the fifth line, we attempt to get a value out of it. This leads to an infinite loop. The equivalent code in Haskell is: makeList = 1 : makeList This program reads: we're defining something called makeList (this is what goes on the left-hand side of the equals sign). On the right-hand side, we give the definition of makeList. In Haskell, the colon operator is used to create lists (we'll talk more about this soon). This right-hand side says that the value of makeList is the element 1 stuck on to the beginning of the value of makeList. However, since Haskell is lazy (or "call by name"), we do not actually attempt to evaluate what makeList is at this point: we simply remember that if ever in the future we need the second element of makeList, we need to just look at makeList. Now, if you attempt to write makeList to a file, print it to the screen, or calculate the sum of its elements, the operation won't terminate because it would have to evaluate an infinitely long list. However, if you simply use a finite portion of the list (say the first ${\displaystyle 10}$ elements), the fact that the list is infinitely long doesn't matter. If you only use the first ${\displaystyle 10}$ elements, only the first ${\displaystyle 10}$ elements are ever calculated. This is laziness. Second, Haskell is case-sensitive. Many languages are, but Haskell actually uses case to give meaning. Haskell distinguishes between values (for instance, numbers: ${\displaystyle 1,2,3,...}$); strings: "abc", "hello", ...; characters: a', b', ', ...; even functions (for instance, the function that squares a value, or the square-root function); and types (the categories to which values belong). By itself, this is not unusual. Most languages have some system of types. What is unusual is that Haskell requires that the names given to functions and values begin with a lower-case letter and that the names given to types begin with an upper-case letter. The moral is: if your otherwise correct program won't compile, be sure you haven't named your function Foo, or something else beginning with a capital letter. Being a functional language, Haskell eschews side effects. A side effect is essentially something that happens in the course of executing a function that is not related to the output produced by that function. For instance, in a language like C or Java, you are able to modify "global" variables from within a function. This is a side effect because the modification of this global variable is not related to the output produced by the function. Furthermore, modifying the state of the real world is considered a side effect: printing something to the screen, reading a file, etc., are all side effecting operations. Functions that do not have side effects are called pure. An easy test for whether or not a function is pure is to ask yourself a simple question: "Does this function's result depend only on the arguments it receives, and is returning a result the only thing it does?" All of this means that if you're used to writing code in an imperative language (like C or Java), you're going to have to start thinking differently. Most importantly, if you have a value x, you must not think of x as a register, a memory location or anything else of that nature. x is simply a name, just as "Hal" is my name. You cannot arbitrarily decide to store a different person in my name any more than you can arbitrarily decide to store a different value in x. This means that code that might look like the following C code is invalid (and has no counterpart) in Haskell: int x = 5; x = x + 1; A call like x = x + 1 is called destructive update because we are destroying whatever was in x before and replacing it with a new value. Destructive update does not exist in Haskell. By not allowing destructive updates (or any other such side effecting operations), Haskell code is very easy to comprehend. That is, when we define a function f, and call that function with a particular argument a in the beginning of a program, and then, at the end of the program, again call f with the same argument a, we know we will get out the same result. This is because we know that a cannot have changed and because we know that f only depends on a (for instance, it didn't increment a global counter). This property is called referential transparency and basically states that if two functions f and g produce the same values for the same arguments, then we may replace f with g (and vice-versa). Note There is no agreed-upon exact definition of referential transparency. The definition given above is the one I like best. They all carry the same interpretation; the differences lie in how they are formalized. ArithmeticEdit Let's begin our foray into Haskell with simple arithmetic. Start up your favorite interactive shell (Hugs or GHCi; see the chapter Getting started for installation instructions). The shell will output to the screen a few lines talking about itself and what it's doing and then should finish with the cursor on a line reading: Example: Prelude> From here, you can begin to evaluate expressions. An expression is basically something that has a value. For instance, the number ${\displaystyle 5}$ is an expression (its value is ${\displaystyle 5}$). Values can be built up from other values; for instance, ${\displaystyle 5+6}$ is an expression (its value is ${\displaystyle 11}$). In fact, most simple arithmetic operations are supported by Haskell, including plus (+), minus (-), times (), divided-by (/), exponentiation (^) and square-root (sqrt). You can experiment with these by asking the interactive shell to evaluate expressions and to give you their value. In this way, a Haskell shell can be used as a powerful calculator. Try some of the following: Example: Prelude> 54+3 23 Prelude> 5^5-2 3123 Prelude> sqrt 2 1.4142135623730951 Prelude> 5(4+3) 35 We can see that, in addition to the standard arithmetic operations, Haskell also allows grouping by parentheses, hence the difference between the values of 54+3 and 5(4+3). The reason for this is that the "understood" grouping of the first expression is (54)+3, due to operator precedence. Also note that parentheses aren't required around function arguments. For instance, we simply wrote sqrt 2, not sqrt(2), as would be required in most other languages. You could write it with the parentheses, but in Haskell, since function application is so common, parentheses aren't required. Now try entering 2^5000. Does it work? Note If you're familiar with programming in other languages, you may find it odd that sqrt 2 comes back with a decimal point (i.e., is a floating point number) even though the argument to the function seems to be an integer. This interchangability of numeric types is due to Haskell's system of type classes and will be discussed in detail in the section on Classes). Exercises We've seen that multiplication binds more tightly than addition. Can you think of a way to determine whether function application binds more or less tightly than multiplication? Pairs, Triples and MoreEdit In addition to single values, we should also address multiple values. For instance, we may want to refer to a position by its ${\displaystyle x}$/${\displaystyle y}$ coordinate, which would be a pair of integers. To make a pair of integers is simple: you enclose the pair in parenthesis and separate them with a comma. Try the following: Example: Prelude> (5,3) (5,3) Here, we have a pair of integers, ${\displaystyle 5}$ and ${\displaystyle 3}$. In Haskell, the first element of a pair need not have the same type as the second element: that is, pairs are allowed to be heterogeneous. For instance, you can have a pair of an integer with a string. This contrasts with lists, which must be made up of elements of all the same type (we will discuss lists further in the section on Lists). There are two predefined functions that allow you to extract the first and second elements of a pair. They are, respectively, fst and snd. You can see how they work below: Example: Prelude> fst (5, "hello") 5 Prelude> snd (5, "hello") "hello" In addition to pairs, you can define triples, quadruples etc. To define a triple and a quadruple, respectively, we write: Example: Prelude> (1,2,3) (1,2,3) Prelude> (1,2,3,4) (1,2,3,4) And so on. In general, pairs, triples, and so on are called tuples and can store fixed amounts of heterogeneous data. Note The functions fst and snd won't work on anything longer than a pair; if you try to use them on a larger tuple, you will get a message stating that there was a type error. The meaning of this error message will be explained in the chapter Type basics. Exercises Use a combination of fst and snd to extract the character 'a' out of the tuple ((1,'a'),"foo"). ListsEdit The primary limitation of tuples is that they hold only a fixed number of elements: pairs hold two, triples hold three, and so on. A data structure that can hold an arbitrary number of elements is a list. Lists are assembled in a very similar fashion to tuples, except that they use square brackets instead of parentheses. We can define a list like: Example: Prelude> [1,2] [1,2] Prelude> [1,2,3] [1,2,3] Lists don't need to have any elements. The empty list is simply []. Unlike tuples, we can very easily add an element on to the beginning of the list using the colon operator. The colon is called the "cons" operator; the process of adding an element is called "consing." The etymology of this is that we are constructing a new list from an element and an old list. We can see the cons operator in action in the following examples: Example: Prelude> 0:[1,2] [0,1,2] Prelude> 5:[1,2,3,4] [5,1,2,3,4] We can actually build any list by using the cons operator (the colon) and the empty list: Example: Prelude> 5:1:2:3:4:[] [5,1,2,3,4] In fact, the [5,1,2,3,4] syntax is "syntactic sugar" for the expression using the explicit cons operators and empty list. If we write something using the [5,1,2,3,4] notation, the compiler simply translates it to the expression using (:) and []. Note In general, "syntactic sugar" is a strictly unnecessary language feature, which is added to make the syntax nicer. One further difference between lists and tuples is that, while tuples are heterogeneous, lists must be homogeneous. This means that you cannot have a list that holds both integers and strings. If you try to, a type error will be reported. Of course, lists don't have to just contain integers or strings; they can also contain tuples or even other lists. Tuples, similarly, can contain lists and other tuples. Try some of the following: Example: Prelude> [(1,1),(2,4),(3,9),(4,16)] [(1,1),(2,4),(3,9),(4,16)] Prelude> ([1,2,3,4],[5,6,7]) ([1,2,3,4],[5,6,7]) There are two basic list functions: head and tail. The head function returns the first element of a (non-empty) list, and the tail function returns all but the first element of a (non-empty) list. To get the length of a list, you use the length function: Example: Prelude> length [1,2,3,4,10] 5 1 Prelude> length (tail [1,2,3,4,10]) 4 StringsEdit In Haskell, a String is simply a list of Chars. So, we can create the string "Hello" as: Example: Prelude> 'H':'e':'l':'l':'o':[] "Hello" Lists (and, of course, strings) can be concatenated using the ++ operator: Example: Prelude> "Hello " ++ "World" "Hello World" Additionally, non-string values can be converted to strings using the show function, and strings can be converted to non-string values using the read function. Of course, if you try to read a value that's malformed, an error will be reported (note that this is a run-time error, not a compile-time error): Example: Prelude> "Five squared is " ++ show (55) "Five squared is 25" 8 In the above, the exact error message is implementation dependent. However, the interpreter has inferred that you're trying to add three to something. This means that when we execute read "Hello", we expect to be returned a number. However, "Hello" cannot be parsed as a number, so an error is reported. Simple List FunctionsEdit Much of the computation in Haskell programs is done by processing lists. There are three primary list-processing functions: map, filter and foldr (also foldl). The map function takes as arguments a list of values and a function that should be applied to each of the values. It returns the result of this application. For instance, there is a built-in function Char.toUpper that takes as input a Char and produces a Char that is the upper-case version of the original argument. So, to convert an entire string (which is simply a list of characters) to upper case, we can map the toUpper function across the entire list: Example: Prelude> map Char.toUpper "Hello World" "HELLO WORLD" When you map across a list, the length of the list never changes -- only the individual values in the list change. To remove elements from the list, you can use the filter function. This function allows you to remove certain elements from a list depending on their value, but not on their context. For instance, the function Char.isLower tells you whether a given character is lower case. We can filter out all non-lowercase characters using this: Example: Prelude> filter Char.isLower "Hello World" "elloorld" The function foldr takes a little more getting used to. foldr takes three arguments: a function, an initial value and a list. The best way to think about foldr is that it replaces occurrences of the list cons operator (:) with the function parameter and replaces the empty list constructor ([]) with the initial value. Thus, if we have a list: 3 : 8 : 12 : 5 : [] and we apply foldr (+) 0 to it, we get: 3 + 8 + 12 + 5 + 0 which sums the list. We can test this: Example: Prelude> foldr (+) 0 [3,8,12,5] 28 We can perform the same sort of operation to calculate the product of all the elements on a list: Example: Prelude> foldr () 1 [4,8,5] 160 We said earlier that folding is like replacing (:) with a particular function and ([]) with an initial element. This raises a question as to what happens when the function isn't associative (a function (${\displaystyle \cdot }$) is associative if ${\displaystyle a\cdot (b\cdot c)=(a\cdot b)\cdot c}$). When we write ${\displaystyle 4\cdot 8\cdot 5\cdot 1}$, we need to specify where to put the parentheses. Namely, do we mean ${\displaystyle ((4\cdot 8)\cdot 5)\cdot 1}$ or ${\displaystyle 4\cdot (8\cdot (5\cdot 1))}$? foldr assumes the function is right-associative (i.e., the correct bracketing is the latter). Thus, when we use it on a non-associative function (like minus), we can see the effect: Example: Prelude> foldr (-) 1 [4,8,5] 0 The exact derivation of this looks something like: foldr (-) 1 [4,8,5] ==> 4 - (foldr (-) 1 [8,5]) ==> 4 - (8 - foldr (-) 1 [5]) ==> 4 - (8 - (5 - foldr (-) 1 [])) ==> 4 - (8 - (5 - 1)) ==> 4 - (8 - 4) ==> 4 - 4 ==> 0 The foldl function goes the other way and effectively produces the opposite bracketing. foldl looks the same when applied, so we could have done summing just as well with foldl: Example: Prelude> foldl (+) 0 [3,8,12,5] 28 However, we get different results when using the non-associative function minus: Example: Prelude> foldl (-) 1 [4,8,5] -16 This is because foldl uses the opposite bracketing. The way it accomplishes this is essentially by going all the way down the list, taking the last element and combining it with the initial value via the provided function. It then takes the second-to-last element in the list and combines it to this new value. It does so until there is no more list left. The derivation here proceeds in the opposite fashion: foldl (-) 1 [4,8,5] ==> foldl (-) (1 - 4) [8,5] ==> foldl (-) ((1 - 4) - 8) [5] ==> foldl (-) (((1 - 4) - 8) - 5) [] ==> ((1 - 4) - 8) - 5 ==> ((-3) - 8) - 5 ==> (-11) - 5 ==> -16 Note that once the foldl goes away, the parenthesization is exactly the opposite of the foldr. Note foldl is often more efficient than foldr for reasons that we will discuss in the section on Lists. However, foldr can work on infinite lists, while foldl cannot. This is because before foldl does anything, it has to go to the end of the list. On the other hand, foldr starts producing output immediately. For instance, foldr (:) [] [1,2,3,4,5] simply returns the same list. Even if the list were infinite, it would produce output. A similar function using foldl would fail to produce any output. If this discussion of the folding functions is still somewhat unclear, that's okay. We'll discuss them further in the section on Lists. Exercises Use map to convert a string into a list of booleans, each element in the new list representing whether or not the original element was a lower-case character. That is, it should take the string "aBCde" and return [True,False,False,True,True]. Exercises Use the functions mentioned in this section (you will need two of them) to compute the number of lower-case letters in a string. For instance, on "aBCde" it should return 3. Exercises We've seen how to calculate sums and products using folding functions. Given that the function max returns the maximum of two numbers, write a function using a fold that will return the maximum value in a list (and zero if the list is empty). So, when applied to [5,10,2,8,1] it will return 10. Assume that the values in the list are always ${\displaystyle \geq 0}$. Explain to yourself why it works. Exercises Write a function that takes a list of pairs of length at least 2 and returns the first component of the second element in the list. So, when provided with [(5,'b'),(1,'c'),(6,'a')], it will return 1. Source Code FilesEdit As programmers, we don't want to simply evaluate small expressions like these -- we want to sit down, write code in our editor of choice, save it and then use it. We already saw in the sections Ghc and Nhc how to write a Hello World program and how to compile it. Here, we show how to use functions defined in a source-code file in the interactive environment. To do this, create a file called Test.hs and enter the following code: module Test where x = 5 y = (6, "Hello") z = x * fst y This is a very simple "program" written in Haskell. It defines a module called "Test" (in general module names should match file names; see the section on Modules for more on this). In this module, there are three definitions: x, y and z. Once you've written and saved this file, in the directory in which you saved it, load this in your favorite interpreter, by executing either of the following: Example: % hugs Test.hs % ghci Test.hs This will start Hugs or GHCi, respectively, and load the file. Alternatively, if you already have one of them loaded, you can use the ":load" command (or just ":l") to load a module, as: Example: Prelude> :l Test.hs ... Test> Between the first and last line, the interpreter will print various data to explain what it is doing. If any errors appear, you probably mistyped something in the file; double check and then try again. You'll notice that where it used to say "Prelude" it now says "Test." That means that Test is the current module. The Prelude module (usually simply referred to as "the Prelude") is always loaded and contains the standard definitions (for instance, the (:) operator for lists, or (+) or (), fst, snd and so on). Now that we've loaded Test, we can use things that were defined in it. For example: Example: Test> x 5 Test> y (6,"Hello") Test> z 30 Perfect, just as we expected! One final issue regarding how to compile programs to stand-alone executables remains. In order for a program to be an executable, it must have the module name "Main" and must contain a function called main. So, if you go in to Test.hs and rename it to "Main" (change the line that reads module Test to module Main), we simply need to add a main function. Try this: main = putStrLn "Hello World" Now, save the file, and compile it (refer back to the section on Getting started for information on how to do this for your compiler). For example, in GHC, you would say: Example: % ghc --make Test.hs -o test Note For Windows, it would be "-o test.exe" This will create a file called "test" (or on Windows, "test.exe") that you can then run. Example: % ./test Hello World Note Or, on Windows: Example: C:\> test.exe Hello World FunctionsEdit Now that we've seen how to write code in a file, we can start writing functions. As you might have expected, functions are central to Haskell, as it is a functional language. This means that the evaluation of a program is simply the evaluation of a function. We can write a simple function to square a number and enter it into our Test.hs file. We might define this as follows: square x = x x In this function definition, we say that we're defining a function square that takes one argument (aka parameter), which we call x. We then say that the value of square x is equal to x * x. Haskell also supports standard conditional expressions. For instance, we could define a function that returns ${\displaystyle -1}$ if its argument is less than ${\displaystyle 0}$; ${\displaystyle 0}$ if its argument is ${\displaystyle 0}$; and ${\displaystyle 1}$ if its argument is greater than ${\displaystyle 0}$ (this is called the signum function): signum x = if x < 0 then -1 else if x > 0 then 1 else 0 You can experiment with this as: Example: Test> signum 5 1 Test> signum 0 0 Test> signum (5-10) -1 Test> signum (-1) -1 Note that the parenthesis around "-1" in the last example are required; if missing, the system will think you are trying to subtract the value "1" from the value "signum," which is illtyped. The if/then/else construct in Haskell is very similar to that of most other programming languages; however, you must have both a then and an else clause. It evaluates the condition (in this case ${\displaystyle x<0}$ and, if this evaluates to True, it evaluates the then clause; if the condition evaluated to False, it evaluates the else clause). You can test this program by editing the file and loading it back into your interpreter. If Test is already the current module, instead of typing :l Test.hs again, you can simply type :reload or just :r to reload the current file. This is usually much faster. Haskell, like many other languages, also supports case constructions. These are used when there are multiple values that you want to check against (case expressions are actually quite a bit more powerful than this -- see the section on Pattern matching for all of the details). Suppose we wanted to define a function that had a value of ${\displaystyle 1}$ if its argument were ${\displaystyle 0}$; a value of ${\displaystyle 5}$ if its argument were ${\displaystyle 1}$; a value of ${\displaystyle 2}$ if its argument were ${\displaystyle 2}$; and a value of ${\displaystyle -1}$ in all other instances. Writing this function using if statements would be long and very unreadable; so we write it using a case statement as follows (we call this function f): f x = case x of 0 -> 1 1 -> 5 2 -> 2 _ -> -1 In this program, we're defining f to take an argument x and then inspect the value of x. If it matches ${\displaystyle 0}$, the value of f is ${\displaystyle 1}$. If it matches ${\displaystyle 1}$, the value of f is ${\displaystyle 5}$. If it maches ${\displaystyle 2}$, then the value of f is ${\displaystyle 2}$; and if it hasn't matched anything by that point, the value of f is ${\displaystyle -1}$ (the underscore can be thought of as a "wildcard" -- it will match anything) . The indentation here is important. Haskell uses a system called "layout" to structure its code (the programming language Python uses a similar system). The layout system allows you to write code without the explicit semicolons and braces that other languages like C and Java require. The general rule for layout is that an open-brace is inserted after the keywords where, let, do and of, and the column position at which the next command appears is remembered. From then on, a semicolon is inserted before every new line that is indented the same amount. If a following line is indented less, a close-brace is inserted. This may sound complicated, but if you follow the general rule of indenting after each of those keywords, you'll never have to remember it (see the section on Layout for a more complete discussion of layout). Some people prefer not to use layout and write the braces and semicolons explicitly. This is perfectly acceptable. In this style, the above function might look like: f x = case x of { 0 -> 1 ; 1 -> 5 ; 2 -> 2 ; _ -> -1 } Of course, if you write the braces and semicolons explicitly, you're free to structure the code as you wish. The following is also equally valid: f x = case x of { 0 -> 1 ; 1 -> 5 ; 2 -> 2 ; _ -> -1 } However, structuring your code like this only serves to make it unreadable (in this case). Functions can also be defined piece-wise, meaning that you can write one version of your function for certain parameters and then another version for other parameters. For instance, the above function f could also be written as: f 0 = 1 f 1 = 5 f 2 = 2 f _ = -1 Here, the order is important. If we had put the last line first, it would have matched every argument, and f would return -1, regardless of its argument (most compilers will warn you about this, though, saying something about overlapping patterns). If we had not included this last line, f would produce an error if anything other than 0, 1 or 2 were applied to it (most compilers will warn you about this, too, saying something about incomplete patterns). This style of piece-wise definition is very popular and will be used quite frequently throughout this tutorial. These two definitions of f are actually equivalent -- this piece-wise version is translated into the case expression. More complicated functions can be built from simpler functions using function composition. Function composition is simply taking the result of the application of one function and using that as an argument for another. We've already seen this way back in arithmetic (the section on Arithmetic), when we wrote 54+3. In this, we were evaluating ${\displaystyle 54}$ and then applying ${\displaystyle +3}$ to the result. We can do the same thing with our square and f functions: Example: Test> square (f 1) 25 Test> square (f 2) 4 Test> f (square 1) 5 Test> f (square 2) -1 The result of each of these function applications is fairly straightforward. The parentheses around the inner function are necessary; otherwise, in the first line, the interpreter would think that you were trying to get the value of "square f," which has no meaning. Function application like this is fairly standard in most programming languages. There is another, more mathematically oriented, way to express function composition, using the (.) (just a single period) function. This (.) function is supposed to look like the (${\displaystyle \circ }$) operator in mathematics. Note In mathematics we write ${\displaystyle f\circ g}$ to mean "f following g," in Haskell we write f . g also to mean "f following g." The meaning of ${\displaystyle f\circ g}$ is simply that ${\displaystyle (f\circ g)(x)=f(g(x))}$. That is, applying the value ${\displaystyle x}$ to the function ${\displaystyle f\circ g}$ is the same as applying it to ${\displaystyle g}$, taking the result, and then applying that to ${\displaystyle f}$. The (.) function (called the function composition function), takes two functions and makes them in to one. For instance, if we write (square . f), this means that it creates a new function that takes an argument, applies f to that argument and then applies square to the result. Conversely, (f . square) means that it creates a new function that takes an argument, applies square to that argument and then applies f to the result. We can see this by testing it as before: Example: Test> (square . f) 1 25 Test> (square . f) 2 4 Test> (f . square) 1 5 Test> (f . square) 2 -1 Here, we must enclose the function composition in parentheses; otherwise, the Haskell compiler will think we're trying to compose square with the value f 1 in the first line, which makes no sense since f 1 isn't even a function. It would probably be wise to take a little time-out to look at some of the functions that are defined in the Prelude. Undoubtedly, at some point, you will accidentally rewrite some already-existing function (I've done it more times than I can count), but if we can keep this to a minimum, that would save a lot of time. Here are some simple functions, some of which we've already seen: sqrt the square root function id the identity function: id x = x fst extracts the first element from a pair snd extracts the second element from a pair null tells you whether or not a list is empty head returns the first element on a non-empty list tail returns everything but the first element of a non-empty list ++ concatenates two lists == checks to see if two elements are equal /= checks to see if two elements are unequal Here, we show example usages of each of these functions: Example: Prelude> sqrt 2 1.41421 Prelude> id "hello" "hello" Prelude> id 5 5 Prelude> fst (5,2) 5 Prelude> snd (5,2) 2 Prelude> null [] True Prelude> null [1,2,3,4] False 1 Prelude> tail [1,2,3,4] [2,3,4] Prelude> [1,2,3] ++ [4,5,6] [1,2,3,4,5,6] Prelude> [1,2,3] == [1,2,3] True Prelude> 'a' /= 'b' True We can see that applying head to an empty list gives an error (the exact error message depends on whether you're using GHCi or Hugs -- the shown error message is from Hugs). Let BindingsEdit Often we wish to provide local declarations for use in our functions. For instance, the following equation is used to find the roots (zeros) of a polynomial of the form ${\displaystyle ax^{2}+bx+c=0}$: ${\displaystyle x=(-b\pm {\sqrt {b^{2}-4ac}})/(2a)}$. We could write the following function to compute the two values of ${\displaystyle x}$: roots a b c = ((-b + sqrt(bb - 4ac)) / (2a), (-b - sqrt(bb - 4ac)) / (2a)) Unfortunately, we duplicate expressions here. To remedy this problem, Haskell allows for local bindings. That is, we can create values inside of a function that only that function can see. For instance, we could create a local binding for sqrt(bb-4ac) and call it, say, det and then use that in both places where sqrt(bb - 4ac) occurred. We can do this using a let/in declaration: roots a b c = let det = sqrt (bb - 4ac) in ((-b + det) / (2a), (-b - det) / (2a)) In fact, you can provide multiple declarations inside a let. Just make sure they're indented the same amount, or you will have layout problems: roots a b c = let det = sqrt (bb - 4ac) twice_a = 2a in ((-b + det) / twice_a, (-b - det) / twice_a) InfixEdit Infix functions are ones that are composed of symbols, rather than letters. For instance, (+), (), (++) are all infix functions. You can use them in non-infix mode by enclosing them in parentheses. Hence, the two following expressions are the same: Example: Prelude> 5 + 10 15 Prelude> (+) 5 10 15 Similarly, non-infix functions (like map) can be made infix by enclosing them in backquotes (the ticks on the tilde key on American keyboards): Example: Prelude> map Char.toUpper "Hello World" "HELLO WORLD" Prelude> Char.toUpper map "Hello World" "HELLO WORLD" There are two types of comments in Haskell: line comments and block comments. Line comments begin with the token -- and extend until the end of the line. Block comments begin with {- and extend to a corresponding -}. Block comments can be nested. Note The -- in Haskell corresponds to // in C++ or Java, and {- and -} correspond to /* and /. Comments are used to explain your program in English and are completely ignored by compilers and interpreters. For example: module Test2 where main = putStrLn "Hello World" -- write a string -- to the screen {- f is a function which takes an integer and produces integer. {- this is an embedded comment -} the original comment extends to the matching end-comment token: -} f x = case x of 0 -> 1 -- 0 maps to 1 1 -> 5 -- 1 maps to 5 2 -> 2 -- 2 maps to 2 _ -> -1 -- everything else maps to -1 This example program shows the use of both line comments and (embedded) block comments. RecursionEdit In imperative languages like C and Java, the most basic control structure is a loop (like a for loop). However, for loops don't make much sense in Haskell because they require destructive update (the index variable is constantly being updated). Instead, Haskell uses recursion. A function is recursive if it calls itself (see the appendix on Recursion for more). Recursive functions exist also in C and Java but are used less than they are in functional languages. The prototypical recursive function is the factorial function. In an imperative language, you might write this as something like: int factorial(int n) { int fact = 1; for (int i=2; i <= n; i++) fact = fact i; return fact; } While this code fragment will successfully compute factorials for positive integers, it somehow ignores the basic definition of factorial, usually given as: ${\displaystyle n!={\begin{cases}1&n=1\\n(n-1)!&{\mbox{otherwise}}\\\end{cases}}}$ This definition itself is exactly a recursive definition: namely the value of ${\displaystyle n!}$ depends on the value of ${\displaystyle (n-1)!}$. If you think of ${\displaystyle !}$ as a function, then it is calling itself. We can translate this definition almost verbatim into Haskell code: factorial 1 = 1 factorial n = n factorial (n-1) This is likely the simplest recursive function you'll ever see, but it is correct. Note Of course, an imperative recursive version could be written: int factorial(int n) { if (n == 1) return 1; else return n * factorial(n-1); } but this is likely to be much slower than the loop version in C. Recursion can be a difficult thing to master. It is completely analogous to the concept of induction in mathematics (see the chapter Recursion for a more formal treatment of this). However, usually a problem can be thought of as having one or more base cases and one or more recursive-cases. In the case of factorial, there is one base case (when ${\displaystyle n=1}$) and one recursive case (when ${\displaystyle n>1}$). For designing your own recusive algorithms, it is often useful to try to differentiate these two cases. Turning now to the task of exponentiation, suppose that we have two positive integers ${\displaystyle a}$ and ${\displaystyle b}$, and that we want to calculate ${\displaystyle a^{b}}$. This problem has a single base case: namely when ${\displaystyle b}$ is ${\displaystyle 1}$. The recursive case is when ${\displaystyle b>1}$. We can write a general form as: ${\displaystyle a^{b}={\begin{cases}a&b=1\\aa^{b-1}&{\mbox{otherwise}}\\\end{cases}}}$ Again, this translates directly into Haskell code: exponent a 1 = a exponent a b = a exponent a (b-1) Just as we can define recursive functions on integers, so can we define recursive functions on lists. In this case, usually the base case is the empty list [], and the recursive case is a cons list (i.e., a value consed on to another list). Consider the task of calculating the length of a list. We can again break this down into two cases: either we have an empty list or we have a non-empty list. Clearly the length of an empty list is zero. Furthermore, if we have a cons list, then the length of this list is just the length of its tail plus one. Thus, we can define a length function as: my_length [] = 0 my_length (x:xs) = 1 + my_length xs Note Whenever we provide alternative definitions for standard Haskell functions, we prefix them with my_ so the compiler doesn't become confused. Similarly, we can consider the filter function. Again, the base case is the empty list, and the recursive case is a cons list. However, this time, we're choosing whether to keep an element, depending on whether or not a particular predicate holds. We can define the filter function as: my_filter p [] = [] my_filter p (x:xs) = if p x then x : my_filter p xs else my_filter p xs In this code, when presented with an empty list, we simply return an empty list. This is because filter cannot add elements; it can only remove them. When presented with a list of the form (x:xs), we need to decide whether or not to keep the value x. To do this, we use an if statement and the predicate p. If p x is true, then we return a list that begins with x followed by the result of filtering the tail of the list. If p x is false, then we exclude x and return the result of filtering the tail of the list. We can also define map and both fold functions using explicit recursion. See the exercises for the definition of map and the chapter Language advanced for the folds. Exercises The fibonacci sequence is defined by: ${\displaystyle F_{n}={\begin{cases}1&n=1{\mbox{ or }}n=2\\F_{n-2}+F_{n-1}&{\mbox{otherwise}}\\\end{cases}}}$ Write a recursive function fib that takes a positive integer n as a parameter and calculates ${\displaystyle F_{n}}$. Exercises Define a recursive function mult that takes two positive integers a and b and returns a*b, but only uses addition (i.e., no fair just using multiplication). Begin by making a mathematical definition in the style of the previous exercise and the rest of this section. Exercises Define a recursive function my_map that behaves identically to the standard function map. InteractivityEdit If you are familiar with books on other (imperative) languages, you might be wondering why you haven't seen many of the standard programs written in tutorials of other languages (like ones that ask the user for his name and then says "Hi" to him by name). The reason for this is simple: Being a pure functional language, it is not entirely clear how one should handle operations like user input. After all, suppose you have a function that reads a string from the keyboard. If you call this function twice, and the user types something the first time and something else the second time, then you no longer have a function, since it would return two different values. The solution to this was found in the depths of category theory, a branch of formal mathematics: monads. We're not yet ready to talk about monads formally, but for now, think of them simply as a convenient way to express operations like input/output. We'll discuss them in this context much more in the chapter Io and then discuss monads for monads' sake in the chapter Monads. Suppose we want to write a function that's interactive. The way to do this is to use the do keyword. This allows us to specify the order of operations (remember that normally, since Haskell is a lazy language, the order in which operations are evaluated in it is unspecified). So, to write a simple program that asks a user for his name and then address him directly, enter the following code into "Name.hs": module Main where import System.IO main = do hSetBuffering stdin LineBuffering name <- getLine putStrLn ("Hello, " ++ name ++ ", how are you?") Note The parentheses are required on the second instance of putStrLn but not the first. This is because function application binds more tightly than ++, so without the parentheses, the second would be interpreted as (putStrLn "Hello, ") ++ name ++ .... You can then either load this code in your interpreter and execute main by simply typing "main," or you can compile it and run it from the command line. I'll show the results of the interactive approach: Example: Main> main Hal Hello, Hal, how are you? Main> And there's interactivity. Let's go back and look at the code a little, though. We name the module "Main," so that we can compile it. We name the primary function "main," so that the compiler knows that this is the function to run when the program is run. On the fourth line, we import the IO library, so that we can access the IO functions. On the seventh line, we start with do, telling Haskell that we're executing a sequence of commands. The first command is hSetBuffering stdin LineBuffering, which you should probably ignore for now (incidentally, this is only required by GHC -- in Hugs you can get by without it). The necessity for this is because, when GHC reads input, it expects to read it in rather large blocks. A typical person's name is nowhere near large enough to fill this block. Thus, when we try to read from stdin, it waits until it's gotten a whole block. We want to get rid of this, so we tell it to use LineBuffering instead of block buffering. The next command is putStrLn, which prints a string to the screen. On the ninth line, we say name <- getLine. This would normally be written name = getLine, but using the arrow instead of the equal sign shows that getLine isn't a real function and can return different values. This command means "run the action getLine, and store the results in name." The last line constructs a string using what we read in the previous line and then prints it to the screen. Another example of a function that isn't really a function would be one that returns a random value. In this context, a function that does this is called randomRIO. Using this, we can write a "guess the number" program. Enter the following code into "Guess.hs": module Main where import System.IO import System.Random main = do hSetBuffering stdin LineBuffering num <- randomRIO (1::Int, 100) putStrLn "I'm thinking of a number between 1 and 100" doGuessing num doGuessing num = do guess <- getLine if guessNum < num then do putStrLn "Too low!" doGuessing num else if guessNum > num then do putStrLn "Too high!" doGuessing num else do putStrLn "You Win!" Let's examine this code. On the fifth line we write "import Random" to tell the compiler that we're going to be using some random functions (these aren't built into the Prelude). In the first line of main, we ask for a random number in the range ${\displaystyle (1,100)}$. We need to write ::Int to tell the compiler that we're using integers here -- not floating point numbers or other numbers. We'll talk more about this in the section on Type basics. On the next line, we tell the user what's going on, and then, on the last line of main, we tell the compiler to execute the command doGuessing. The doGuessing function takes the number the user is trying to guess as an argument. First, it asks the user to guess and then accepts their guess (which is a String) from the keyboard. The if statement checks first to see if their guess is too low. However, since guess is a string, and num is an integer, we first need to convert guess to an integer by reading it. Since "read guess" is a plain, pure function (and not an IO action), we don't need to use the <- notation (in fact, we cannot); we simply bind the value to guessNum. Note that while we're in do notation, we don't need ins for lets. If they guessed too low, we inform them and then start doGuessing over again. If they didn't guess too low, we check to see if they guessed too high. If they did, we tell them and start doGuessing again. Otherwise, they didn't guess too low and they didn't guess too high, so they must have gotten it correct. We tell them that they won and exit. The fact that we exit is implicit in the fact that there are no commands following this. We don't need an explicit return () statement. You can either compile this code or load it into your interpreter, and you will get something like: Example: Main> main I'm thinking of a number between 1 and 100 50 Too low! 75 Too low! 85 Too high! 80 Too high! 78 Too low! 79 You Win! The recursive action that we just saw doesn't actually return a value that we use in any way. In the case when it does, the "obvious" way to write the command is actually incorrect. Here, we will give the incorrect version, explain why it is wrong, then give the correct version. Let's say we're writing a simple program that repeatedly asks the user to type in a few words. If at any point the user enters the empty word (i.e., he just hits enter without typing anything), the program prints out everything he's typed up until that point and then exits. The primary function (actually, an action) in this program is one that asks the user for a word, checks to see if it's empty, and then either continues or ends. The incorrect formulation of this might look something like: askForWords = do word <- getLine if word == "" then return [] Before reading ahead, see if you can figure out what is wrong with the above code. The error is on the last line, specifically with the term word : askForWords. Remember that when using (:), we are making a list out of an element (in this case word) and another list (in this case, askForWords). However, askForWords is not a list; it is an action that, when run, will produce a list. That means that before we can attach anything to the front, we need to run the action and take the result. In this case, we want to do something like: askForWords = do word <- getLine if word == "" then return [] else do return (word : rest) Here, we first run askForWords, take the result and store it in the variable rest. Then, we return the list created from word and rest. By now, you should have a good understanding of how to write simple functions, compile them, test functions and programs in the interactive environment, and manipulate lists. Exercises Write a program that will repeatedly ask the user for numbers until she types in zero, at which point it will tell her the sum of all the numbers, the product of all the numbers, and, for each number, its factorial. For instance, a session might look like: Example: Give me a number (or 0 to stop): 5 Give me a number (or 0 to stop): 8 Give me a number (or 0 to stop): 2 Give me a number (or 0 to stop): 0 The sum is 15 The product is 80 5 factorial is 120 8 factorial is 40320 2 factorial is 2 ` Hint: write an IO action that reads a number and, if it's zero, returns the empty list. If it's not zero, it recurses itself and then makes a list out of the number it just read and the result of the recursive call. Hint: You will need to make use of "show" to print a number. putStrLn("Number " ++ show(n))	2016-10-01 01:32:18	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 70, "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": 0.6861569881439209, "perplexity": 904.4272357980967}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738662438.81/warc/CC-MAIN-20160924173742-00250-ip-10-143-35-109.ec2.internal.warc.gz"}
https://studyqas.com/hi-how-are-you-i-hope-you-guys-have-a-wonderful-day-and/	# Hi how are you? I hope you guys have a WONDERFUL DAY and week! We can get through this together! Hi how are you? I hope you guys have a WONDERFUL DAY and week! We can get through this together! ## This Post Has 4 Comments 1. bustillojoshua4 says: The literary point of view of the "Rescue Mission" affects the reader's understanding of history through first-person narration, because through the narration the reader can place himself in this story since the reader can understand and identify the narrator's feelings in this situation . In the passage " Okay, I have to snap out of this because I know that the captain wouldn’t assign me to these missions if I couldn’t handle them. I have received years of elite training for missions like this, and I have a spotless track record for at-sea rescues. Why do I always have to convince myself that I am capable of this?" The author shows a feeling of insecurity, although he is able to perform the rescue he fears he can not solve the situation he is inserted, as the narration is in the first person, the reader can identify with this feeling of security and understand the gravity of the situation and the feelings of the narrator. 2. XOsam says: thanks you too Explanation: 3. anna80544 says: im excited but tired and thank u, u have made my day :))) Explanation: 4. EDWIN6067 says: The literary point of view in "The Rescue Mission" affects the reader's understanding of the story because the story is told in the first person. The character himself tells everything that's happening in the story, including, as he tells, his thoughts, everything that is going through his mind. He is not only describing the scenes and events from a distant point of view, without being necessarily connected to the events; on the contrary, he is at the center of events, and everything told goes through his feelings and impressions before getting to the reader. "I have received years of elite training for missions like this, and I have a spotless track record for at-sea rescues. Why do I always have to convince myself that I am capable of this?" In the passage, it can be observed that the character is talking to himself while he tells the story, therefore, he is also having an unintentional chat with the reader, and introducing to the story his feelings, doubts, thoughts, and points of view. Therefore, it can be understood that the reader's understanding of the story is affected by the literary point of view in "The Rescue Mission" since the story is told in the first person.	2022-11-30 07:21:06	{"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": 0.25997650623321533, "perplexity": 1155.21634636727}, "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/1669446710733.87/warc/CC-MAIN-20221130060525-20221130090525-00205.warc.gz"}
https://economics.stackexchange.com/questions/3100/regression-discontinuity-questions	# Regression discontinuity questions I am considering a regression discontinuity design (RD) where the "treatment" has a definite sorting rule (below the threshold, you are not fined - above the threshold, you are fined). The outcome I am considering is how much you are fined the FOLLOWING year. There are 4 options: 1) remain without a fine, 2) go from no fine to fined, 3) fined less than you were the previous year, or 4) fined more than you were the previous. While I have listed these as categorical, I have the continuous amount of the fine as well. Here are my questions: 1.) I have never seen a "longitudinal" RD design where the treatment is at a time period before the outcome. Are there any methodological issues with this approach? 2.) Assuming this approach is valid, how best should I model the outcome given the multiple possibilities? • If I understand correctly, what you have is panel data. Each individual time-series is the fines imposed on a single entity, and each observation in each time series is the amount of fine (so a value of $0$ means "no-fine" this period). So, also, all numbers in the data set are non-negative. Correct? – Alecos Papadopoulos Jan 21 '15 at 8:41 • Correct on all accounts. – oncearunner Jan 21 '15 at 9:30 The fact that your outcome is measured at a later point in time than the treatment is not a problem, and not unusual either - this is true to some extent in all RD studies (if this were not the case, there couldn't be causation). As for how to analyze your outcome, I suggest a couple of different specifications. First, you should use the continuous fine measure as an outcome. Second, you could use an indicator variable equal to 1 if they got a fine, and equal to zero if they didn't. You might be able to do similar dummy regressions for the more/less cases too, but that's a little more complicated. If you want to be a little more advanced, it would certainly be interesting to take the RD framework to a tobit model. Shouldn't be too complicated, I expect, but I haven't thought that much about it.	2020-08-12 04:55:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6543558835983276, "perplexity": 605.156276479747}, "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-2020-34/segments/1596439738864.9/warc/CC-MAIN-20200812024530-20200812054530-00232.warc.gz"}
http://mymathforum.com/pre-calculus/341655-volume-integration.html	My Math Forum Volume integration Pre-Calculus Pre-Calculus Math Forum September 2nd, 2017, 12:12 PM #1 Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 878 Thanks: 60 Math Focus: सामान्य गणित Volume integration What could be the volume of a solid bounded by $\displaystyle x = y^{2}$ $\displaystyle 4-x = y^{2}$ $\displaystyle z =0\hspace {2mm} and \hspace {2mm} z=3$ September 2nd, 2017, 12:38 PM #2 Senior Member Joined: Sep 2015 From: USA Posts: 2,122 Thanks: 1102 Have you plotted this? $I=\displaystyle 2 \times 3\int_{-\sqrt{2}}^{\sqrt{2}}\int_{y^2}^{2}~dx~dy = 16 \sqrt{2}$ The factor of 2 is due to the symmetry of the shape. The factor of 3 is due to the simple integration over $z$ The limits on $y$ are due to where the two surfaces intersect. If you don't like the symmetry you can do it with two integrals. $I=\displaystyle 3\int_{-\sqrt{2}}^{\sqrt{2}}\int_{y^2}^{2}~dx~dy + 3 \int_{-\sqrt{2}}^{\sqrt{2}}\int_{2}^{4-y^2}~dx~dy = 16 \sqrt{2}$ Attached Images Clipboard01.jpg (50.1 KB, 14 views) Last edited by romsek; September 2nd, 2017 at 12:41 PM. September 2nd, 2017, 12:49 PM #3 Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 878 Thanks: 60 Math Focus: सामान्य गणित Thanks, had plotted in a paper. What is that software that you are using? I was getting the wrong answer as I had not separated the two blocks. By the way, why do we do that? Is it because of two differeny functions?(different relation between x and y in those two portion) Last edited by MATHEMATICIAN; September 2nd, 2017 at 12:59 PM. September 2nd, 2017, 12:52 PM #4 Senior Member Joined: Sep 2015 From: USA Posts: 2,122 Thanks: 1102 Quote: Originally Posted by MATHEMATICIAN Thanks, had plotted in a paper. What is that software that you are using? I was getting the wrong anwer as I had not separated the two blocks. By the way, why do we do that? software is Mathematica. It can be had pretty cheaply by students. Actually now that I think about it you don't have to separate the two integrals. $I =3 \displaystyle \int_{-\sqrt{2}}^{\sqrt{2}} \int_{y^2}^{4-y^2}~dx~dy$ works just as well. September 2nd, 2017, 01:07 PM #5 Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 878 Thanks: 60 Math Focus: सामान्य गणित Quote: Originally Posted by romsek $I =3 \displaystyle \int_{-\sqrt{2}}^{\sqrt{2}} \int_{y^2}^{4-y^2}~dx~dy$ works just as well. How would you do it if you had to replace variable "y" limits with variable "x" limits? September 2nd, 2017, 01:32 PM #6 Senior Member Joined: Sep 2015 From: USA Posts: 2,122 Thanks: 1102 Quote: Originally Posted by MATHEMATICIAN How would you do it if you had to replace variable "y" limits with variable "x" limits? To do this you would have to break it into 2 integrals, or again just recognize the symmetry. $I = \displaystyle 3\int_0^2 \int_{-\sqrt{x}}^{\sqrt{x}}~dx ~dy + 3\int_2^4 \int_{-\sqrt{4-x}}^{\sqrt{4-x}}~dx~dy = 6 \int_0^2 \int_{-\sqrt{x}}^{\sqrt{x}}~dx ~dy = 16\sqrt{2}$ September 2nd, 2017, 01:34 PM #7 Senior Member Joined: Sep 2015 From: USA Posts: 2,122 Thanks: 1102 There is actually 4 fold symmetry to this shape so the integration limits can be reduced further. I'll leave that to you to play with. hint: $I=12\int_0^2 \int_0^\sqrt{x}~dy~dx$ September 2nd, 2017, 01:49 PM #8 Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 878 Thanks: 60 Math Focus: सामान्य गणित One more question: Evaluate: $\displaystyle \int_{C} (z~dx +x~dy+y~dz)$ where, C is the trace of cylinder $\displaystyle x^{2}+y^{2}=1$ in the XY plane, $\displaystyle y+z=2$. Orient C counter clockwise. Is something wrong with this question? September 2nd, 2017, 02:05 PM #9 Senior Member Joined: Sep 2015 From: USA Posts: 2,122 Thanks: 1102 Quote: Originally Posted by MATHEMATICIAN One more question: Evaluate: $\displaystyle \int_{C} (z~dx +x~dy+y~dz)$ where, C is the trace of cylinder $\displaystyle x^{2}+y^{2}=1$ in the XY plane, $\displaystyle y+z=2$. Orient C counter clockwise. Is something wrong with this question? I would interpret this question as finding the line integral of the field given by $\textbf{F}=(z, x, y)$ over the curve in the plane $y+z=3$ whose projection onto the $xy$ plane is given by $x^2 + y^2 = 1$ September 2nd, 2017, 02:09 PM #10 Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 878 Thanks: 60 Math Focus: सामान्य गणित So, the curve is an ellipse? And what about this orientation thing? What is its significance? Tags integration, volume Thread Tools Display Modes Linear Mode Similar Threads Thread Thread Starter Forum Replies Last Post MATHEMATICIAN Calculus 13 September 1st, 2017 11:12 AM xl5899 Calculus 2 December 10th, 2015 09:09 AM jiasyuen Calculus 9 March 29th, 2015 08:37 PM landonjones6 Calculus 1 February 21st, 2012 02:33 AM izseekzu Calculus 1 January 26th, 2010 06:34 PM Contact - Home - Forums - Cryptocurrency Forum - Top	2018-10-17 23:10:34	{"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": 0.8910569548606873, "perplexity": 4614.75803740264}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583511314.51/warc/CC-MAIN-20181017220358-20181018001858-00412.warc.gz"}
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The models of the additional notion livefor on the presentation of Atheists and the mirrors in which these sets are.	2019-05-23 21:53:06	{"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": 0.35179251432418823, "perplexity": 6238.258255980777}, "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-2019-22/segments/1558232257396.96/warc/CC-MAIN-20190523204120-20190523230120-00162.warc.gz"}
https://physics.stackexchange.com/questions/531161/natural-stretching-of-a-metal-rod	# Natural Stretching of a metal rod Let’s say a metal rod of Young’s modulus $$Y$$ and cross section $$A$$ with length $$l$$ was hung from the top of a ceiling straight down. By the theory of Young’s modulus, the metal rod should lengthen by some length $$\delta l$$ just by the gravitational force, how much would the rod be stretched by? Also, if a mass $$m$$ was attached to the bottom of the rod, would the rod display vertical oscillations like a spring attempting to bring the mass back to equilibrium position? Would the same be true if there was no mass at all attached to the bottom of the rod? • For the first paragraph, the stress in the rod varies linearly from one end to the other. Find the corresponding strains and then the total displacement. For the second paragraph, yes and yes. – alephzero Feb 15 at 1:19 Let’s say a metal rod of Young’s modulus Y and cross section A with length l was hung from the top of a ceiling straight down. By the theory of Young’s modulus, the metal rod should lengthen by some length δl just by the gravitational force, how much would the rod be stretched by? 1. The rod will naturally stretch by a length $$\Delta l$$ as you have said. A very similar problem asking to find the amount the rod is stretched by was given in a marvelous book called Problems in General Physics by I.E Irodov. I will display my approach here: We can take the average of the sums of all the stresses on the rod and treat this as the overall stress given to the rod. This is because the top of the rod is under the most stress while the bottom is under almost none. And since the stress doesn't increase exponentially, but rather linearly, the effective stress upon the cylinder at the center will be treated as the overall stress given upon the rod. Assuming the metal rod has a radius $$r$$ and is uniform throughout, then the stress at the center will be given by (the force is given by the amount of weight the metal rod has to support from halfway) $$\sigma=\frac{F}{A}=\frac{\pi r^2\rho g\frac{l}{2}}{\pi r^2}=\frac{1}{2}\rho gl.$$ We know that the stress-strain relationship gives us $$\epsilon = \dfrac{\Delta l}{l} = \dfrac{1}{Y} \sigma.$$ Since we know $$\sigma$$, replacing our expression for sigma into the relation gives us $$\frac{\Delta l}{l} = \dfrac{1}{2Y}\rho gl$$ $$\Delta l = \frac{1}{2Y}\rho gl^2$$ Also, if a mass m was attached to the bottom of the rod, would the rod display vertical oscillations like a spring attempting to bring the mass back to equilibrium position? 2. Yes, to be more specific, we can carry out the calculations for the frequency of the oscillations. The tension in the wire would defined as $$T=PA=AY\frac{\Delta L}{L}$$ When the mass is distance $$\Delta L$$ down it will recieve a restoring force of $$k\Delta{L}$$. This means that the rod can be modeled as a spring with spring constant $$k=\frac{T}{\Delta L}=Y\frac{A}{L}$$ The frequency of oscillations is then $$f=\frac{1}{2\pi}\sqrt{\frac{k}{m}}=\frac{1}{2\pi}\sqrt{\frac{YA}{ml}}$$ The same will be true for no mass at all, but it will be rather easy now to do yourself with the calculations I have given. If we take the coordinate $$x$$ pointing downward with its origin at the top of the rod, the tensile normal force carried at some location $$x$$ along the rod is $$N(x) = \rho g A (l-x)$$. The stress is just this divided by the cross-sectional area of the bar, and the strain is the stress divided by Young's modulus. The strain is also related to the displacement of the bar by $$\epsilon = du/dx$$. You can then integrate the strain over the length of the bar to get a total stretch of the bar as $$\rho g l^2 / 2 Y$$. $$\delta = \int^{l}_0 \rho g (l-x)/Y dx=\rho g l^2 / 2Y$$ The answers to your oscillation questions are yes and yes. However, it is not exactly like a simple spring-mass system. For example, for a uniform bar stretched at one end and then released, the displacement oscillations take a saw-tooth form. For the attached mass, it will approach the simple spring-mass system when the mass of the attached mass is much greater than the mass of the rod.	2020-10-23 00:46:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 23, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.744288444519043, "perplexity": 166.3664068246004}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107880401.35/warc/CC-MAIN-20201022225046-20201023015046-00415.warc.gz"}
https://www.questarter.com/q/given-two-angle-and-a-segment-can-you-find-h-21_3353127.html	# Given two angle and a segment, can you find $h$? by b00n heT Last Updated September 11, 2019 18:20 PM - source While designing another problem I came up with the following question: Considering the picture below, can you determine $$h$$ based only on the information of $$\alpha$$, $$\beta$$ and $$x$$ given the fact that the segment $$x$$ is the continuation of the height? My guess is that the problem is not solvable (in the sense that one can construct multiple $$h$$ with the same $$\alpha,\beta$$ and $$x$$), but cannot see how to show this. So far I gave the problem a try by splitting the angle and combining some trigonometric identities but couldn't conclude. So far I only obtained the following \begin{align} \tan(\alpha)=\frac{(h_1+h_2)(x+y)}{(x+y)^2-h_1h_2} \end{align} and $$\tan(\beta)=\frac{(h_1+h_2)\cdot y}{y^2-h_1h_2}.$$	2020-05-27 12:26:39	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9594494104385376, "perplexity": 139.03701751106843}, "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-2020-24/segments/1590347394074.44/warc/CC-MAIN-20200527110649-20200527140649-00287.warc.gz"}
https://www.physicsforums.com/threads/image-of-an-object-inside-hollow-glass-sphere.867481/	# Image of an Object Inside Hollow Glass Sphere Tags: 1. Apr 17, 2016 ### Kaan99 Hello, This question is from 1990 Turkey National Physics Olympics. I tried my best to translate it clearly. 1. The problem statement, all variables and given/known data https://s23.postimg.org/cotn29afv/Hollow+Spherical+Glass.jpg The sphere of radius 2R has an empty sphere inside with radius R. In order for the image of an object on the inner surface of the sphere to be R/5 far away from the object (which is shown with an asterisk*) for an observer at the indicated position (göz), what should be the index of refraction of the glass? Where is the focal point of this optical system? 2. Relevant equations Thin Lens Equation Lensmaker's Equation 3. The attempt at a solution The lens is made up of to circles of radii R and 2R, using lensmaker and thin lens eq. $\frac{1}{f} = (n-1) (\frac{1}{R_1}-\frac{1}{R_2})=\frac{1}{d_o}+\frac{1}{d_i}$ $(n-1)(\frac{1}{-R}-\frac{1}{-2R})=\frac{1}{2R}+\frac{1}{d_i}$ $\frac{n-1}{-2R}=\frac{1}{2R}+\frac{1}{d_i}$ $d_i=\frac{n}{-2R}$ It turned out that my attempt wasn't accurate from the start, as it deviates from the solution provided. I couldn't understand how the first equation is formed in the solution. The result is n=2 and f=4R. 2. Apr 19, 2016	2018-03-17 17:23:03	{"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": 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": 0.2617327570915222, "perplexity": 603.9348093795048}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257645248.22/warc/CC-MAIN-20180317155348-20180317175348-00058.warc.gz"}
https://conquest.readthedocs.io/en/develop/theory-md.html	# Molecular Dynamics: Theory¶ ## Microcanonical (NVE) ensemble¶ The Hamiltonian for the microcanonical ensemble is, $\mathcal{H} = \sum_{i=1}^N \frac{\mathbf{p}_i^2}{2m_i} + U(\mathbf{r}_i)$ where $$\mathbf{p}_i$$ and $$\mathbf{r}_i$$ are the position and momentum of particle $$i$$ and $$U$$ is the DFT total (potential) energy. Hamilton’s equations can be solved to give the following equations of motion: $\begin{split}\mathbf{\dot{r}}_i &= \frac{\mathbf{p}_i}{m_i} \\ \mathbf{\dot{p}}_i &= \frac{\partial U(\mathbf{r}_i)}{\partial\mathbf{r}_i} = \mathbf{F_i}\end{split}$ In order to construct a time-reversible algorithm from these equations, the Liouvillian formulation is employed [Ta1] (trivially, in this case). The Liouville operator $$L$$ can be defined in terms of position and momentum components: $iL = \mathbf{\dot{r}}\frac{\partial}{\partial\mathbf{r}} + \mathbf{\dot{p}}\frac{\partial}{\partial\mathbf{p}} = i(L_r + L_p).$ The Liouvillian can be used to construct the classical propagator, which relates the state $$f$$ of the system at time 0 to its state at time $$t$$: $f[\mathbf{p}^N(t),\mathbf{r}^N(t)] = e^{iLt}f[\mathbf{p}^N(0),\mathbf{r}^N(0)]$ Taking the individual position and momentum parts of the Liouvillian $$L_r$$ and $$L_p$$, it can be shown that applying it to the state $$f$$ result in a simple linear shift in coordinates and a simple linear shift in momentum respectively: $\begin{split}iL_rf(t) &= f[\mathbf{p}^N(0),\mathbf{r}^N(0) + \mathbf{\dot{r}}^N(0)t] \\ iL_pf(t) &= f[\mathbf{p}^N(0) + \mathbf{F}^N(0)t,\mathbf{r}^N(0)]\end{split}$ However, we cannot simply replace $$e^{iLt}$$ with $$e^{iL_rt}$$ because $$iL_r$$ and $$iL_p$$ are non-commuting operators, so we must employ the Trotter-Suzuki identity: $e^{A+B} = \lim_{P\rightarrow\infty}\left(e^{A/2P}e^{B/P}e^{A/2P}\right)^P$ Thus for a small enough time step $$\Delta t = t/P$$ and to first order, a discrete time step corresponds to the application of the discrete time propagator $$G$$, $G(\Delta t) = U_1\left(\frac{\Delta t}{2}\right)U_2\left(\Delta t\right)U_1\left(\frac{\Delta t}{2}\right) = e^{iL_1\frac{\Delta t}{2}}e^{iL_2\Delta t}e^{iL_1\frac{\Delta t}{2}},$ which can be shown to be unitary and therefore time-reversible. Applying the operators $$U$$ in the sequence determined by the Trotter decomposition generates the velocity Verlet algorithm, which is used to integrate microcanonical molecular dynamics in CONQUEST. For a detailed derivation of the algorithm, refer to Frenkel & Smit [Ta1]. Go to top. ## Extended Lagrangian Born-Oppenheimer MD (XL-BOMD)¶ If the electronic density from the previous ionic step is used as an initila guess for the next SCF cycle, a problem arises because this process breaks the time-reversibility of the dynamics. This is manifested as a gradual drift in the total energy in the case of a NVE simulation, or the conserved quantity in the case of non-Hamiltonian dynamics. The solution proposed by Niklasson [Ta2][Ta3] is to introduce auxilliary electronic degrees of freedom into the Lagrangian, which can be propagated via time-reversible integrators. The extended Lagrangian used in CONQUEST is [Ta4], $\mathcal{L}^\mathrm{XBO}\left(\mathbf{X}, \mathbf{\dot{X}}, \mathbf{R}, \mathbf{\dot{R}}\right) = \mathcal{L}^\mathrm{BO}\left(\mathbf{R}, \mathbf{\dot{R}}\right) + \frac{1}{2}\mu\mathrm{Tr}\left[\mathbf{\dot{X}}^2\right] - \frac{1}{2}\mu\omega^2\mathrm{Tr}\left[(\mathbf{LS} - \mathbf{X})^2\right],$ where $$\mathbf{X}$$ is a sparse matrix with the same range as $$\mathbf{LS}$$, $$\mu$$ is the fictitious electron mass and $$\omega$$ is the curvature of the auxiliary harmonic potential. The Euler-Lagrange equations of motion are then, $\begin{split}m_i\mathbf{\ddot{r}_i} &= -\frac{\partial U[{{\mathbf{R;LS}}}]}{\partial\mathbf{r}_i} = \mathbf{F_i} \\ \mathbf{\ddot{X}} &= \omega^2(\mathbf{LS} - \mathbf{X}),\end{split}$ The first of these is simply Newton’s second law, and the velocity update equation of motion in the microcanonical ensemble. The second can be integrated using a time-reversible algorithm, the velocity Verlet scheme in the case of CONQUEST [Ta4]: $\begin{split}\mathbf{X}(t+\delta t) &= 2\mathbf{X}(t) -\mathbf{X}(t-\delta t) + \delta t^2\omega^2\left[\mathbf{L}(t)\mathbf{S}(t)-\mathbf{X}(t)\right] \\ &+ a\sum_{m=0}^M c_m\mathbf{X}(t-m\delta t)\end{split}$ i.e. the trajectory of $$\mathbf{X}(t)$$ is time-reversible, and evolves in a harmonic potential centred on the ground state density $$\mathbf{L}(t)\mathbf{S}(t)$$. The matrix $$\mathbf{XS}^{-1}$$ is a good guess for the $$\mathbf{L}$$ matrix in the Order(N) scheme. Despite the time-reversitibility, the $$\mathbf{X}$$ matrix tends in practice to gradually drift from the harmonic centre over time, increasing the number of SCF iterations required to reach the minimum over the course of the simulation. To remove such numerical errors, the final dissipative term is included, and is found to have a minimal effect on the time-reversibility. We note that since the auxiliary variable $$X$$ is used to generate an intial guess for the SCF process, it does not appear in the conserved (pseudo-Hamiltonian) quantity for the dynamics. Go to top. ## Non-Hamiltonian dynamics¶ ### Extended system method¶ Hamiltonian dynamics generally describes systems that are isolated from their surroundings, but in the canonical and isobaric-isothermal ensembles, we need to couple the system to an external heat bath and/or stress. It is possible to model such systems by positing a set of equations of non-Hamiltonian equations of motion, and proving that they generate the correct statistical ensemble [Ta5]. This is the extended system approach: we modify the Hamiltonian to include the thermostat and/or barostat degrees of freedom, derive the (pseudo-) Hamiltonian equations of motion, and demostrate that the correct phase space distribution for the ensemble is recovered. Go to top. ### Canonical (NVT) ensemble¶ In the Nose-Hoover formulation [Ta6][Ta7], the Hamiltonian for a system in the canonical ensemble can be written, $\mathcal{H} = \sum_i \frac{1}{2}m_i s^2\mathbf{\dot{r}}_i^2 + U(\mathbf{r}_i) + \frac{1}{2}Q\dot{s}^2 - (n_f + 1)k_B T \ln s,$ where $$\mathbf{r}_i$$ and $$\mathbf{\dot{r}_i}$$ are respectively the position and velocity of particle $$i$$, $$U$$ is the potential energy, in this case the DFT total energy, $$s$$ is a dimensionless quantity that can be interpreted post-hoc as a time step scaling factor, $$Q$$ is the fictitious mass of the heat bath and $$n_f$$ is the number of ionic degrees of freedom. Hamilton’s equations can be solved to generate the Nose-Hoover equations of motion. However Martyna et al. demonstrate that this method does not generate an ergodic trajectory, and proposed an alternative formulation [Ta8] in which the temperature is controlled by a chain of $$M$$ coupled thermostats of mass $$Q_k$$, notional position $$\eta_k$$ and conjugate momentum $$p_{\eta_k}$$: $\begin{split}\mathbf{\dot{r}_i} &= \frac{\mathbf{p}_i}{m_i} \\ \mathbf{\dot{p}_i} &= -\frac{\partial U(\mathbf{r})}{\partial \mathbf{r}_i} - \frac{p_{\eta_1}}{Q_1}\mathbf{p}_i \\ \dot{\eta}_k &= \frac{p_{\eta_k}}{Q_k} \\ \dot{p}_{\eta_1} &= \left(\sum_{i=1}^N\frac{\mathbf{p}_i}{m_i} - n_fk_BT\right) - \frac{p_{\eta_{2}}}{Q_{\eta_{2}}}p_{\eta_1} \\ \dot{p}_{\eta_k} &= \left(\frac{p^2_{\eta_{k-1}}}{Q_{k-1}} - k_BT\right) - \frac{p_{\eta_{k+1}}}{Q_{k+1}}p_{\eta_k} \\ \dot{p}_{\eta_M} &= \left(\frac{p^2_{\eta_{M-1}}}{Q_{M-1}} - k_BT\right)\end{split}$ The Liouvillian for these equations of motion can be non-uniquely decomposed into components of ionic position ($$iL_r$$) and momentum ($$iL_p$$) as in the microcanonical case, the extended Lagrangian ($$iL_\mathrm{XL}$$, and a Nose-Hoover chain component ($$iL_\mathrm{NHC}$$) $iL = iL_\mathrm{NHC} + iL_p + iL_{\mathrm{XL}} + iL_r,$ which is directly translated into an algorithm with the Trotter-Suzuki expansion, $\begin{split}\exp(iL\Delta t) = &\exp\left(iL_\mathrm{NHC}\frac{\Delta t}{2}\right)\exp\left(iL_p\frac{\Delta t}{2}\right) \times \\ &\exp\left(iL_\mathrm{XL}\frac{\Delta t}{2}\right)\exp\left(iL_r\Delta t\right)\exp\left(iL_\mathrm{XL}\frac{\Delta t}{2}\right) \times \\ &\exp\left(iL_p\frac{\Delta t}{2}\right)\exp\left(iL_\mathrm{NHC}\frac{\Delta t}{2}\right)\end{split}$ This is recognisable as the velocity Verlet algorithm with extended Lagrangian integration which can be reduced to a single step, as described in Extended Lagrangian Born-Oppenheimer MD (XL-BOMD), with a half time step integration of the Nose-Hoover chain equations of motion before and after. For full details of the integration scheme, see Hirakawa et al. [Ta9]. Go to top. ### Isobaric-Isothermal (NPT) ensemble¶ The Parinello-Rahman equations of motion [Ta10] extend the fixed cell equations of motion to include the cell degrees of freedom in the extended system approach. We use the Martyna-Tobias-Tuckerman-Klein modification [Ta11], which couples the variable cell equations of motion to a Nose-Hoover chain the themrostat the system, recovering the isobaric-isothermal (NPT) ensemble. For an unconstrained cell (i.e. the lattice vectors can change freely), the equations of motion are, $\begin{split}\mathbf{\dot{r}}_i &= \frac{\mathbf{p}_i}{m_i} + \frac{\mathbf{p}_g}{W_g}\mathbf{r}_i \\ \mathbf{\dot{p}}_i &= \mathbf{F}_i - \frac{\mathbf{p}_g}{W_g}\mathbf{p}_i - \left(\frac{1}{N_f}\right)\frac{\mathrm{Tr}[\mathbf{p}_g]}{W_g}\mathbf{p}_i - \frac{p_\xi}{Q}\mathbf{p}_i \\ \mathbf{\dot{h}} &= \frac{\mathbf{p}_g\mathbf{h}}{W_g} \\ \mathbf{\dot{p}_g} &= V(\mathbf{P}_\mathrm{int}-\mathbf{I}P_\mathrm{ext}) + \left[\frac{1}{N_f}\sum_{i=1}^N\frac{\mathbf{p}_i^2}{m_i}\right]\mathbf{I} - \frac{p_\xi}{Q}\mathbf{p}_g \\ \dot{\xi} &= \frac{p_\xi}{Q} \\ \mathbf{\dot{p}}_g &= \sum_{i=1}^N\frac{\mathbf{p}_i^2}{m_i} + \frac{1}{W_g}\mathrm{Tr}[\mathbf{p}_g^T\mathbf{p}_g] - (N_f + d^2)kT\end{split}$ Here, $$\mathbf{r}_i$$, $$\mathbf{p}_i$$ and $$m_i$$ are the position, momentum and mass of particle $$i$$ respectively, $$\xi$$, $$p_\xi$$ and $$Q$$ are the position, momentum and mass of the thermostat, and $$\mathbf{h}$$, $$\mathbf{p}_g$$ and $$W_g$$ are the matrix of lattice vectors, matrix of cell velocities and cell mass respectively. Note that these equations only include one Nose-Hoover thermostat for simplicity. Conquest uses the Shinoda-Shiga-Mikami splitting of the Liouvillian [Ta12] to propagate the system. The Liouvillian is decomposed as, $iL = iL_r + iL_h + iL_v + iL_\mathrm{bath},$ which can be further split, $\begin{split}iL_\mathrm{bath} &= iL_\mathrm{box} + iL_\mathrm{particles} \\ iL_\mathrm{box} &= iL_\mathrm{vbox} + iL_\xi + iL_{v_{\xi_1}} + iL_{v_{\xi_k}} + iL_{v_{\xi_M}} \\ iL_\mathrm{particles} &= iL_\mathrm{vpart} + iL_\xi + iL_{v_{\xi_1}} + iL_{v_{\xi_k}} + iL_{v_{\xi_M}}\end{split}$ Using Liouville’s theorem, we have, $\begin{split}iL_r &= \sum_{i=1}^N[\mathbf{v}_i + \mathbf{v}_g\mathbf{r}_i]\cdot\nabla_{\mathbf{r}_i} \\ iL_h &= \sum_{\alpha,\beta}\mathbf{v}_{g,\alpha\beta}\mathbf{h}_{\alpha\beta}\frac{\partial}{\partial\mathbf{h}_{\alpha\beta}} \\ iL_v &= \sum_{i=1}^N\left(\frac{\mathbf{F}_i}{m_i}\right)\cdot\nabla_{\mathbf{v}_i} \\ iL_\mathrm{bath} &= iL_\mathrm{vpart} + iL_\mathrm{vbox} + iL_\xi + iL_{v_{\xi_1}} + iL_{v_{\xi_k}} + iL_{v_{\xi_M}} \\ &= \sum_{i=1}^N\left[-\left\{\mathbf{v}_g + \frac{1}{N_f}\mathrm{Tr}(\mathbf{v}_g) + v_{\xi_1}\right\}\mathbf{v}_i\right]\nabla_{\mathbf{v}_i} \\ &+ \sum_{\alpha,\beta}\left[\frac{F_\mathrm{box}}{W} - v_{\xi_1}\mathbf{v}_{g,\alpha\beta}\right]\frac{\partial}{\partial\mathbf{v}_{g,\alpha\beta}} \\ &+ \sum_{k=1}^M v_{\xi_k}\frac{\partial}{\partial\xi_k} \\ &+ \left[\frac{F_{\mathrm{NHC}_1}}{Q_1} - v_{\xi_1}v_{\xi_2}\right]\frac{\partial}{\partial v_{\xi_1}} \\ &+ \sum_{k=2}^M\left[\frac{1}{Q_k}(Q_{k-1}v_{\xi_{k-1}}^2 - kT_\mathrm{ext}) - v_{\xi_k}v_{\xi_{k+1}}\right]\frac{\partial}{\partial v_{\xi_k}} \\ &+ \left[\frac{1}{Q_M}(Q_{M-1}v_{\xi_{M-1}}^2 - kT_\mathrm{ext})\right]\frac{\partial}{\partial v_{\xi_M}}\end{split}$ Here we use $$M$$ heat baths in a Nose-Hoover chain. The Trotter-Suzuki expansion is, $e^{iL\Delta t} = e^{iL_\mathrm{bath}\frac{\Delta t}{2}}e^{iL_v\frac{\Delta t}{2}}e^{iL_h\frac{\Delta t}{2}}e^{iL_r\Delta t}e^{iL_h\frac{\Delta t}{2}}e^{iL_v\frac{\Delta t}{2}}e^{iL_\mathrm{bath}\frac{\Delta t}{2}}.$ The Liouvillian for the heat baths can be further expanded: $e^{iL_\mathrm{particles}\frac{\Delta t}{2}} = e^{\left(iL_{v_{\xi_1}} + iL_{v_{\xi_k}} + iL_{v_{\xi_M}}\right)\frac{\Delta t}{4}}e^{\left(iL_\xi + iL_\mathrm{vpart}\right)\frac{\Delta t}{2}}e^{\left(iL_\xi + iL_{v_{\xi_1}} + iL_{v_{\xi_k}} + iL_{v_{\xi_M}}\right)\frac{\Delta t}{4}}$ Finally, expanding the first propagator in the previous expression, we have, $\begin{split}e^{\left(iL_{v_{\xi_1}} + iL_{v_{\xi_k}} + iL_{v_{\xi_M}}\right)\frac{\Delta t}{4}} &= e^{-i\left(-v_{\xi_1}v_{\xi_2}\frac{\partial}{\partial \xi_1} - \sum_{k=2}^Mv_{\xi_k}v_{\xi_{k+1}}\frac{\partial}{\partial \xi_k} - v_{\xi_{M-1}}v_{\xi_M}\frac{\partial}{\partial \xi_M}\right)\frac{\Delta t}{8}} \\ &\times e^{i\left(F_{\mathrm{NHC}_1}\frac{\partial}{\partial v_{\xi_1}} + F_{\mathrm{NHC}_k}\frac{\partial}{\partial v_{\xi_k}} + F_{\mathrm{NHC}_M}\frac{\partial}{\partial v_{\xi_M}}\right)\frac{\Delta t}{4}} \\ &\times e^{-i\left(-v_{\xi_1}v_{\xi_2}\frac{\partial}{\partial \xi_1} - \sum_{k=2}^Mv_{\xi_k}v_{\xi_{k+1}}\frac{\partial}{\partial \xi_k} - v_{\xi_{M-1}}v_{\xi_M}\frac{\partial}{\partial \xi_M}\right)\frac{\Delta t}{8}}\end{split}$ These expressions are directly translated into the integration algorithm. Go to top. ### Weak coupling thermostat/barostat¶ Instead of modifying the Hamiltonian, the Berendsen-type weak coupling method [Ta13] involves coupling the ionic degrees of freedom to a an external temperature and/or pressure bath via “the principle of least local perturbation consistent with the required global coupling.” Thermostatting is acheived via a Langevin-type equation of motion, in which the system is globally coupled to a heat bath and subjected to random noise: $m_i\ddot{\mathbf{r}}_i = \mathbf{F}_i + m_i \gamma\left(\frac{T_0}{T}-1\right)\dot{\mathbf{r}}_i,$ where $$\gamma$$ is a global friction constant chosen to be the same for all particles. This can be acheived in practice by rescaling the velocities $$\mathbf{v}_i \rightarrow \lambda\mathbf{v}_i$$, where $$\lambda$$ is, $\lambda = \left[ 1 + \frac{\Delta t}{\tau_T}\left(\frac{T_0}{T}-1\right)\right]^{\frac{1}{2}}$ A similar argument can be applied for weak coupling to an external pressure bath. In the isobaric-isoenthalpic ensemble, the velocity of the particles can be expressed, $\dot{\mathbf{r}} = \mathbf{v} - \frac{\beta(P_0 - P)}{3\tau_P}\mathbf{r},$ i.e. the fractional coordinates are scaled by a factor determined by the difference between the internal and external pressures, the isothermal compressibility $$\beta$$ and a pressure coupling time constant $tau_P$. In the isotropic case, the cell scaling factor $$\mu$$ can be expressed, $\mu = \left[ 1 - \frac{\Delta t}{\tau_P}(P_0 - P)\right]^{\frac{1}{3}},$ where the compressibility is absorbed into the time time constant $$\tau_P$$. Allowing for fluctuations of all cell degrees of freedom, the scaling factor becomes, $\mathbf{\mu} = \mathbf{I} - \frac{\beta\Delta t}{3\tau_P}(\mathbf{P}_0 - \mathbf{P})$ While trivial to implement and in general stable, the weak-coupling method does not recover the correct phase space distribution for the canonical or isobaric-isothermal ensembles, for which the extended system method is required. It is no longer supported in CONQUEST, but the concepts are useful. Go to top. ### Stochastic velocity rescaling¶ Stochastic velocity rescaling (SVR) [Ta14] is a modification of the weak coupling method, in which a correctly constructed random force is added to enforce the correct NVT (or NPT) phase space distribution. The kinetic energy is rescaled such that the change in kinetic energy between thermostatting steps is, $dK = (\bar{K} - K)\frac{dt}{\tau} + 2\sqrt{\frac{K\bar{K}}{N_f}}\frac{dW}{\sqrt{\tau}}$ where $$\bar{K}$$ is the target kinetic energy (external temperature), $$dt$$ is the time step, $$\tau$$ is the time scale of the thermostat, $$N_f$$ is the number of degrees of freedom and $$dW$$ is a Wiener process. Practically, the particle velocities are rescaled by a factor of $$\alpha$$, defined via, $\alpha^2 = e^{-\Delta t/\tau} + \frac{\bar{K}}{N_fK}\left(1-e^{-\Delta t/\tau}\right)\left(R_1^2 + \sum_{i=2}^{N_f}R_i^2\right) + 2e^{-\Delta t/2\tau}\sqrt{\frac{\bar{K}}{N_fK}\left(1-e^{-\Delta t/\tau}\right)R_1}$ Where $$R_i$$ is a set of $$N_f$$ normally distributed random numbers with unitary variance. This method can be applied to thermostat the NPT ensemble by barostatting the system with the Parinello-Rahman method, and using the above expressions, but with additional $$R_i$$’s for the cell degrees of freedom, and thermostatting the cell velocities as well as the particle velocities [Ta15]. Ta1(1,2) D. Frenkel and B. Smit. Understanding molecular simulation: from algorithms to application. Academic Press, 2002. Ta2 A. M. N. Niklasson. Extended Born-Oppenheimer Molecular Dynamics. Phys. Rev. Lett., 100:123004, 2008. doi:10.1103/PhysRevLett.100.123004. Ta3 A. M. N. Niklasson and M. J. Cawkwell. Generalized extended Lagrangian Born-Oppenheimer molecular dynamics. J. Chem. Phys., 141:164123, 2014. doi:10.1063/1.4898803. Ta4(1,2) M. Arita, D. R. Bowler, and T. Miyazaki. Stable and Efficient Linear Scaling First-Principles Molecular Dynamics for 10000+ Atoms. J. Chem. Theor. Comput., 10:5419, 2014. doi:10.1021/ct500847y. Ta5 M. E. Tuckerman. Statistical mechanics: theory and molecular simulations. Oxford Graduate Texts, 2010. Ta6 S. Nosé. A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys., 81:511, 1984. doi:10.1063/1.447334. Ta7 W. G. Hoover. Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A, 31:1695, 1985. doi:10.1103/PhysRevA.31.1695. Ta8 G. J. Martyna, M. L. Klein, and M. Tuckerman. Nosé–hoover chains: the canonical ensemble via continuous dynamics. J. Chem. Phys., 97:2635, 1992. doi:10.1063/1.463940. Ta9 T. Hirakawa, T. Suzuki, D. R. Bowler, and T. Miyazaki. Canonical-ensemble extended lagrangian born-oppenheimer molecular dynamics for the linear scaling density functional theory. J. Phys.: Condens. Matter, 29:405901, 2017. doi:10.1088/1361-648X/aa810d. Ta10 M. Parrinello and A. Rahman. Polymorphic transitions in single crystals: A new molecular dynamics method. J. Appl. Phys., 52:7182–7190, December 1981. doi:10.1063/1.328693. Ta11 G. J. Martyna, M. E. Tuckerman, D. J. Tobias, and M. L. Klein. Explicit reversible integrators for extended systems dynamics. Mol. Phys., 87:1117, 1996. doi:10.1080/002689799163235. Ta12 W. Shinoda, M. Shiga, and M. Mikami. Rapid estimation of elastic constants by molecular dynamics simulation under constant stress. Phys. Rev. B, 69:4396, 2004. doi:10.1103/PhysRevB.69.134103. Ta13 H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. Dinola, and J. R. Haak. Molecular dynamics with coupling to an external bath. J. Chem. Phys., 81:3684, 1984. doi:10.1063/1.448118. Ta14 G. Bussi, D. Donadio, and M. Parrinello. Canonical sampling through velocity rescaling. J. Chem. Phys., 126:014101, 2007. doi:10.1063/1.2408420. Ta15 G. Bussi, T. Zykova-Timan, and M. Parrinello. Isothermal-isobaric molecular dynamics using stochastic velocity rescaling. J. Chem. Phys., 130:074101, 2009. doi:10.1063/1.3073889. 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