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https://mathfashion.wordpress.com/2016/08/21/imo-2011-q3/comment-page-1/	IMO # IMO 2011 Q3 $f: \mathbb {R} \to \mathbb {R}$ satisfies $f(x+y) \leq yf(x)+f(f(x))$ for all real numbers $x$ and $y$. Show that $f(x)=0$ for all $x \leq 0$. First, set $y=0$ gives us inequality $f(f(x)) \geq f(x)$ for all $x\in \mathbb R$. Let the inequality be denoted by $P(x,y)$. $P(x, f(y)-x)$ gives us $f(f(y)) \leq (f(y)-x)f(x) + f(f (x))$ for all $x,\,y \in \mathbb R$. Similarly, we have $f(f(x)) \leq (f(x)-y)f(y) + f(f(y))$, by switching $x$ and $y$. Adding the two inequalities up, we get $2f(x)f(y) \geq xf(x) + yf(y),\, \forall x,\,y \in \mathbb R$. Let $y=2f(x)$ in this inequality, we see that $xf(x) \leq 0 \, \forall x \in \mathbb R$. If $x<0$, we see that from $xf(x) \geq 0$, we get either $f(x)=0$ or $f(x) >0$. We now try to rule out the latter case, by observing that replace $x$ in the inequality $xf(x) \leq 0$ with $f(x)$ and we get $f(x)f(f(x)) \leq 0$. This demands $f(f(x))$ to be non-positive. Now, since $f(x)$ is positive, $f(f(x))$ is non-positive, we have $f(f(x)) < f(x)$ for that certain $x<0$, which contradicts our initial inequality $f(f(x)) \geq f(x)$. Thus this cannot happen and $f(x)=0,\, \forall x<0$. It remains to show $f(0)=0$. Now, we re-use the inequality $f(f(x)) \geq f(x)$ by letting $x$ be any negative number, which gives off $f(x)=0$ and thus $f(0) \geq 0$. $P(0,y)$ gives $f(y) \leq yf(0) + f(f(0))$. Assume now that $f(0) > 0$, and let $y \to - \infty$, which requires $yf(0)+ f(f(0)) \to - \infty$ since $f(0)$ is positive, and thus we see for $y<0,\, \|y\|$ large enough, $f(y) <0$, which is impossible as now $y$ is negative and $f(y)=0$. Contradiction. Thus $f(0)>0$ has been wrong which combined with $f(0) \geq 0$ gives $f(0)=0$. Q.E.D. Comment: The functional inequality does not lead to simple non-trivial solution, and we thus hope to exploit the “badly-behavedness” of the condition by reducing it to something simple yet non-trivial. (In this problem’s case, the inequaity $xf(x) \leq 0$.) Since the condition is an inequality, it is not easy to derive an equality out of it. In this solution we established equality from both sides, to thus force the equality to be true. The motivation for the substitution $y \to f(y)-x$ is to yield a $f(f(y))$ term on one side, being symmetric to the $f(f(x))$ term on the other side. Since on the other side $y$ only appears once and appears outside $f$, we can afford to substitute a relatively complicated form of $y$. Once we reach $2f(x)f(y) \geq xf(x) + yf(y)$, substituting $y \to 2f(x)$ is simply giving up the strength of that inequality while preserving its non-triviality. It should be noted that substituting $x=y$ leads to weaker inequality $f(x)^2 \geq xf(x)$. The difficulty of the problem lies in the difficulty in dealing with the iterated application of $f$, given that it is not easy to find a non-trivial solution to this equation–under the pressure of IMO and the anxiety of the windmill problem. (I personally do not know of any elementary non-trivial solution to the inequality, which nonetheless makes it a great problem.) Also, functional inequality is much less common than functional equations. I reckon that if it appeared on the IMO as Q2 instead with the windmill problem being Q3, it would be better done–people would likely spend more time on it. And it proves itself not as challenging as the windmill problem after all! This solution is probably very close to the official solution. The problem is proposed by Igor Voronovich, Belarus, and appeared in IMO 2011 as Q3.	2017-10-22 20:42:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 64, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9605365991592407, "perplexity": 192.68087864018568}, "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/1508187825464.60/warc/CC-MAIN-20171022203758-20171022223758-00510.warc.gz"}
https://www.doubtnut.com/question-answer/suppose-the-point-with-coordinates-12-5-is-on-the-terminal-side-of-angle-theta-find-the-values-of-th-22092	Home > English > Class 11 > Maths > Chapter > Trigonometric Functions > Suppose the point with coordin... # Suppose the point with coordinates (-12 ,5) is on the terminal side of angle theta . Find the values of the six trigonometric functions of thetadot Khareedo DN Pro and dekho sari videos bina kisi ad ki rukaavat ke! Updated On: 27-06-2022 Text Solution Solution : We can draw the point (-12,5) at the XY-axis.<br> Please refer to see the graph.<br> It will become a right angle triangle with base = 12 and perpendicular = 5.<br> :. Hypotenuse = 12^2+5^2 = 13<br> As it is in second quadrant, only sin theta and cosec theta will be positive.<br> :.sin theta = 5/13<br> cos theta = -12/13<br> tan theta = -5/12<br> cot theta = -12/5<br> cosec theta = 13/5<br> sec theta = -13/12<br>	2023-02-09 02:08:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5655348300933838, "perplexity": 7385.7343460378015}, "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-2023-06/segments/1674764501066.53/warc/CC-MAIN-20230209014102-20230209044102-00584.warc.gz"}
https://slave2.omega.jstor.org/stable/j.ctt7s1h6	# The Calculus Lifesaver: All the Tools You Need to Excel at Calculus Edition: SCH - School edition Pages: 752 https://www.jstor.org/stable/j.ctt7s1h6 1. Front Matter (pp. i-vi) (pp. vii-xvii) 3. WELCOME! (pp. xviii-xxii) 4. ACKNOWLEDGMENTS (pp. xxiii-xxiv) 5. CHAPTER 1 Functions, Graphs, and Lines (pp. 1-24) Trying to do calculus without using functions would be one of the most pointless things you could do. If calculus had an ingredients list, functions would be first on it, and by some margin too. So, the first two chapters of this book are designed to jog your memory about the main features of functions. This chapter contains a review of the following topics: functions: their domain, codomain, and range, and the vertical line test; inverse functions and the horizontal line test; composition of functions; odd and even functions; graphs of linear functions and polynomials in general, as well as... 6. CHAPTER 2 Review of Trigonometry (pp. 25-40) To do calculus, you really need to know trigonometry. Truth be told, we won’t see much trig at first, but when it comes, it doesn’t let up. So we might as well do a thorough review of the most important aspects of trig: angles in radians and the basics of the trig functions; trig functions on the real line (not just angles between 0° and 90°); graphs of trig functions; and trig identities. Time to refresh your memory. . . . The first thing I want to remind you about is the notion of radians. Instead of saying that there... 7. CHAPTER 3 Introduction to Limits (pp. 41-56) Calculus wouldn’t exist without the concept of limits. This means that we are going to spend a lot of time looking at them. It turns out that it’s pretty tricky to define a limit properly, but you can get an intuitive understanding of limits even without going into the gory details. This will be enough to tackle differentiation and integration. So, this chapter contains only the intuitive version; check out Appendix A for the formal version. All in all, here’s what we’ll look at in this chapter: an intuitive idea of what a limit is; left-hand, right-hand, and two-sided limits,... 8. CHAPTER 4 How to Solve Limit Problems Involving Polynomials (pp. 57-74) In the previous chapter, we looked at limits from a mostly conceptual viewpoint. Now it’s time to see some of the techniques used to evaluate limits. For the moment, we’ll concentrate on limits involving polynomials; later on we’ll see how to deal with trig functions, exponentials, and logarithms. As we’ll see in the next chapter, differentiation involves taking limits of ratios, so most of our focus will be on this type of limit. When you’re taking the limit of a ratio of two polynomials, it’s really important to notice where the limit is being taken. In particular, the techniques for... 9. CHAPTER 5 Continuity and Differentiability (pp. 75-98) In general, there’s only one special thing about the graph of a function: it just has to obey the vertical line test. That’s not particularly exclusive. The graph could be all over the place—a little bit here, a vertical asymptote there, or any number of individual disconnected points wherever the hell they feel like being. So now we’re going to see what happens if we’re a little more exclusive: we want to look at two types ofsmoothness. First, continuity: intuitively, this means that the graph now has to be drawn in one piece, without taking the pen off... 10. CHAPTER 6 How to Solve Differentiation Problems (pp. 99-126) Now we’ll see how to apply some of the theory from the previous chapter to solve problems involving differentiation. Finding derivatives from the formula is possible but cumbersome, so we’ll look at a few rules that make life a lot easier. All in all, here’s what we’ll tackle in this chapter: finding derivatives using the definition; using the product, quotient, and chain rules; finding equations of tangent lines; velocity and acceleration; finding limits which are derivatives in disguise; how to differentiate piecewise-defined functions; and using the graph of a function to draw the graph of its derivative. Let’s say we... 11. CHAPTER 7 Trig Limits and Derivatives (pp. 127-148) So far, most of our limits and derivatives have involved only polynomials or poly-type functions. Now let’s expand our horizons by looking at trig functions. In particular, we’ll focus on the following topics: the behavior of trig functions at small, large, and other argument values; derivatives of trig functions; and simple harmonic motion. Consider the following two limits: $\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin (5x)}{x}\quad \quad \text{and}\quad \quad \underset{x\to \infty }{\mathop{\lim }}\,\frac{\sin (5x)}{x}.$ They look almost the same. The only difference is that the first limit is taken asx→ 0 while the second is taken asx→ ∞. What a difference, though! As we’ll soon see, the answers and the techniques... 12. CHAPTER 8 Implicit Differentiation and Related Rates (pp. 149-166) Let’s take a break from trying to work out how to differentiate everything in sight. It’s time to look at implicit differentiation, which is a nice generalization of regular differentiation. We’ll then see how to use this technique to solve word problems involving changing quantities. Knowing how fast one quantity is changing allows us to find how fast a different, but related, quantity is changing too. Anyway, the summary for this chapter is the same as the title: implicit differentiation; and related rates. Consider the following two derivatives: $\frac{d}{dx}\left( {{x}^{2}} \right)\quad \quad \text{and}\quad \frac{d}{dx}\left( {{y}^{2}} \right)$ The first is just 2x, as we’ve seen. So isn’t the... 13. CHAPTER 9 Exponentials and Logarithms (pp. 167-200) Here’s a big old chapter on exponentials and logarithms. After we review the properties of these functions, we need to do some calculus with them. It turns out that there’s a special base, the numbere, that works out particularly nicely. In particular, doing calculus withexand loge(x) is a little easier than dealing with 2xand log3(x), for example. So we need to spend some time looking ate. There are other things we want to look at as well; all in all, the plan is to check out the following topics: review of the basics of exponentials... 14. CHAPTER 10 Inverse Functions and Inverse Trig Functions (pp. 201-224) In the previous chapter, we looked at exponentials and logarithms. We got a lot of mileage out of the fact thatexand ln(x) are inverses of each other. In this chapter, we’ll look at some more general properties of inverse functions, then examine inverse trig functions (and their hyperbolic cousins) in greater detail. Here’s the game plan: using the derivative to show that a function has an inverse; finding the derivative of inverse functions; inverse trig functions, one by one; and inverse hyperbolic functions. In Section 1.2 of Chapter 1, we reviewed the basics of inverse functions. I strongly... 15. CHAPTER 11 The Derivative and Graphs (pp. 225-244) We have seen how to differentiate functions from several different families: polynomials and poly-type functions, trig and inverse trig functions, exponentials and logs, and even hyperbolic functions and their inverses. Now we can use this knowledge to help us sketch graphs of functions in general. We’ll see how the derivative helps us understand the maxima and minima of functions, and how the second derivative helps us to understand the so-called concavity of functions. All in all, we have the following agenda: global and local maxima and minima (that is, extrema) of functions, and how to find them using the derivative;... 16. CHAPTER 12 Sketching Graphs (pp. 245-266) Now it’s time to look at a general method for sketching the graph ofy=f(x) for some given functionf. When we sketch a graph, we’re not looking for perfection; we just want to illustrate the main features of the graph. Indeed, we’re going to use the calculus tools we’ve developed: limits to understand the asymptotes, the first derivative to understand maxima and minima, and the second derivative to investigate the concavity. Here’s what we’ll look at: the useful technique of making a table of signs; a general method for sketching graphs; and five examples of how to... 17. CHAPTER 13 Optimization and Linearization (pp. 267-292) We’re now going to look at two practical applications of calculus: optimization and linearization. Believe it or not, these techniques are used every day by engineers, economists, and doctors, for example. Basically, optimization involves finding the best situation possible, whether that be the cheapest way to build a bridge without it falling down or something as mundane as finding the fastest driving route to a specific destination. On the other hand, linearization is a useful technique for finding approximate values of hard-to-calculate quantities. It can also be used to find approximate values of zeroes of functions; this is called Newton’s... 18. CHAPTER 14 LʹHôpitalʹs Rule and Overview of Limits (pp. 293-306) We’ve used limits to find derivatives. Now we’ll turn things upside-down and use derivatives to find limits, by way of a nice technique called l’Hôpital’s Rule. After looking at various varieties of the rule, we’ll give a summary, followed by an overview of all the methods we’ve used so far to evaluate limits. So, we’ll look at: l’Hôpital’s Rule, and four types of limits which naturally lead to using the rule; and a summary of limit techniques from earlier chapters. Most of the limits we’ve looked at are naturally in one of the following forms: $\underset{x\to a}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)},\quad \quad \underset{x\to a}{\mathop{\lim }}\,\left( f\left( x \right)-g\left( x \right) \right),\quad \quad \underset{x\to a}{\mathop{\lim }}\,f\left( x \right)g\left( x \right),\quad \quad \text{and}\quad \quad \underset{x\to a}{\mathop{\lim }}\,f{{\left( x \right)}^{g\left( x \right)}}$. Sometimes you can just... 19. CHAPTER 15 Introduction to Integration (pp. 307-324) So far as calculus is concerned, differentiation is only half the story. The other half concerns integration. This powerful tool enables us to find areas of curved regions, volumes of solids, and distances traveled by objects moving at variable speeds. In this chapter, we’ll spend some time developing the theory we need to define the definite integral. Then, in the next chapter, we’ll give the definition and see how to apply it. So here’s the plan for the preliminaries on integration: sigma notation and telescoping sums; the relationship between displacement and area; and using partitions to find areas. Consider the... 20. CHAPTER 16 Definite Integrals (pp. 325-354) Now it’s time to get some facts straight about definite integrals. First we’ll give an informal definition in terms of areas; then we’ll use our ideas about partitions from the previous chapter to tighten up the definition. After one (exhausting) example of applying the tightened-up definition, we’ll see what else we can say about definite integrals. More precisely, we’ll look at the following topics: signed areas and definite integrals; the definition of the definite integral; an example using this definition; basic properties of definite integrals; using integrals to find unsigned areas, the area between two curves, and areas between a... 21. CHAPTER 17 The Fundamental Theorems of Calculus (pp. 355-382) Here it is: the big kahuna. I’m talking about the Fundamental Theorems of Calculus, which not only provide the key for finding definite integrals without using messy Riemann sums, but also show how differentiation and integration are connected to each other. Without further ado, here’s the roadmap for the chapter: we’ll investigate functions which are based on integrals of other functions; the First Fundamental Theorem, and the basic idea of antiderivatives; the Second Fundamental Theorem; and indefinite integrals and their properties. After all this theoretical stuff, we’ll look at a lot of different examples in the following categories: problems based... 22. CHAPTER 18 Techniques of Integration, Part One (pp. 383-408) Let’s kick off the process of building up a virtual toolkit of techniques to find antiderivatives. In this chapter, we’ll look at the following three techniques: the method of substitution (otherwise known as “change of variables”); integration by parts; and using partial fractions to integrate rational functions. Then, in the next chapter, we’ll look at some more techniques involving trig functions. Using the chain rule, we can easily differentiateex2with respect toxand see that $\frac{d}{dx}\left( {{e}^{{{x}^{2}}}} \right)\ =\ 2x{{e}^{{{x}^{2}}}}$. The factor 2xis the derivative ofx2, which appears in the exponent. Now, as we saw in Section 17.4 of the... 23. CHAPTER 19 Techniques of Integration, Part Two (pp. 409-430) In this chapter, we’ll finish gathering our techniques of integration by taking an extensive look at integrals involving trig functions. Sometimes one has to use trig identities to solve these types of problems; on other occasions there are no trig functions present, so you have to introduce some by making a trig substitution. After we finish all this trigonometry, there’ll be a quick wrap-up of the techniques from this and the previous chapter so that you can keep it all together. So, this is what we’ll look at in this chapter: integrals involving trig identities; integrals involving powers of trig... 24. CHAPTER 20 Improper Integrals: Basic Concepts (pp. 431-450) This is a difficult topic, so I’m devoting two chapters to it. This chapter serves as an introduction to improper integrals. The next chapter gets into the details of how to solve problems involving improper integrals. If you are reading this chapter for the first time, you should probably take care to try to understand all the points in it. On the other hand, if you are reviewing for a test, most likely you’ll want to skim over the chapter, noting the boxed formulas and the sections marked as important, and concentrate on the next chapter. Here’s what we’ll actually... 25. CHAPTER 21 Improper Integrals: How to Solve Problems (pp. 451-476) Let’s get practical and look at a lot of examples of improper integrals. As we go along, we’ll summarize the main methods. In the previous chapter, we introduced some tests that will turn out to be really useful. To use them effectively, you have to understand how some common functions behave, especially near 0 and near ∞. By “common functions,” I mean our usual suspects: polynomials, trig functions, exponentials, and logarithms. So, here’s the game plan for this chapter: what to do when you first see an improper integral, including how to deal with multiple problem spots and functions which... 26. CHAPTER 22 Sequences and Series: Basic Concepts (pp. 477-500) Here’s the good news: infinite series are pretty similar to improper integrals. So a lot, but not all, of the relevant techniques are shared and we don’t need to reinvent the wheel. In order to define what an infinite series is, we’ll also need to look at sequences. Just as in the case of improper integrals, I’m devoting two chapters to sequences and series: this first chapter covers general principles, while the next one is more practical and contains methods for solving problems. If you’re reading this for the first time, go ahead and check out the details of this... 27. CHAPTER 23 How to Solve Series Problems (pp. 501-518) The scenario: you are given a series$\sum\nolimits_{n=1}^{\infty }{\ {{a}_{n}}}$,and you want to know whether or not it converges. If it does converge, then perhaps you’d like to know its value (that is, what it converges to). The series has to be pretty special in order to find a nice expression for its value. Of course, the series may not start atn= 1 as in the above series—it could ben= 0 or some other value ofn. This chapter is all about giving you a blueprint of how to proceed. Here’s a possible flowchart for how... 28. CHAPTER 24 Introduction to Taylor Polynomials, Taylor Series, and Power Series (pp. 519-534) We now come to the important topics of power series and Taylor polynomials and series. In this chapter, we’ll see a general overview of these topics. The following two chapters will deal with problem-solving techniques in the context of the material in this chapter. Here’s what we’ll look at first: approximations, Taylor polynomials, and a Taylor approximation theorem; how good our approximations are, and the full Taylor Theorem; the definition of power series; the definition of Taylor series and Maclaurin series; and convergence issues involving Taylor series. Here’s a nice fact: for any real numberx, we have ${{e}^{x}}\ \cong \ 1\ +\ x\ +\ \frac{{{x}^{2}}}{2}\ +\ \frac{{{x}^{3}}}{6}$. Also,... 29. CHAPTER 25 How to Solve Estimation Problems (pp. 535-550) In the previous chapter, we showed how Taylor polynomials can be used to estimate (or approximate, if you prefer) certain quantities. We also saw that the remainder term could be used to get an idea of how good the approximation actually is. In this chapter, we’ll develop these techniques and look an number of examples. So, here’s the plan for the chapter: a review of the most important facts about Taylor polynomials and series; how to find Taylor polynomials and series; estimation problems; and a different method for analyzing the error. Here are the most important facts about Taylor polynomials... 30. CHAPTER 26 Taylor and Power Series: How to Solve Problems (pp. 551-574) In this chapter, we’ll look at how to solve four different classes of problems involving Taylor series, Taylor polynomials and power series: how to find where power series converge or diverge; how to manipulate Taylor series to get other Taylor series or Taylor polynomials; using Taylor series or Taylor polynomials to find derivatives; and using Maclaurin series to find limits. Let’s say we have a power series aboutx=a: $\sum\limits_{n=0}^{\infty }{{{a}_{n}}{{\left( x\ -\ a \right)}^{n}}}$. As we saw in the case of geometric series, a power series might converge for somexand diverge for otherx. The question that we want to... 31. CHAPTER 27 Parametric Equations and Polar Coordinates (pp. 575-594) So far, we’ve sketched the graphs of many equations of the formy=f(x) with respect to Cartesian coordinates. Now we’re going to look at things in a different way: first, we’ll look at what happens when the coordinatesxandyare not directly related, but are instead related by a common parameter; and then we’ll see what happens when we replace the whole darn coordinate system with something entirely different. Of course, we have to do some calculus too. So here’s the program for this chapter: parametric equations, graphs and finding tangents; converting from polar coordinates to... 32. CHAPTER 28 Complex Numbers (pp. 595-616) Why should some quadratics have all the fun? The quadraticx2− 1 gets the privilege of having two roots (1 and −1), but poor oldx2+1 doesn’t have any, since its discriminant is negative. To even things up a little, let’s introduce the concept of complex numbers. Using complex numbers, any quadratic has two roots.*(You have to count the double rootaof (xa)2as two roots.) Anyway, here’s what we’re going to be doing with complex numbers: the complex plane, and Cartesian and polar forms... 33. CHAPTER 29 Volumes, Arc Lengths, and Surface Areas (pp. 617-644) We have used definite integrals to find areas. Now we’re going to use them to find volumes, lengths of curves, and surface areas. For volumes and surface areas, we’ll pay special attention to solids which are formed by revolving a region in the plane about some axis which lies in the plane; such solids are calledsolids of revolution. In the case of volumes, we’ll also look at some more general solids. Here, then, is the game plan for this chapter: finding volumes of solids of revolution using the disc and shell methods; finding volumes of more general solids; finding... 34. CHAPTER 30 Differential Equations (pp. 645-668) A differential equation is an equation involving derivatives. These things are really useful for describing how quantities change in the real world. For example, if you want to understand how fast a population grows, or even how quickly you can pay off a student loan, a differential equation can help model the situation and give you a decent answer. In this final chapter, we’ll see how to solve certain types of differential equations. In particular, here’s what we’ll look at: an introduction to differential equations; separable first-order differential equations; first-order linear differential equations; first- and second-order constant-coefficient differential equations; and... 35. APPENDIX A Limits and Proofs (pp. 669-702) 36. APPENDIX B Estimating Integrals (pp. 703-716) 37. LIST OF SYMBOLS (pp. 717-718) 38. INDEX (pp. 719-728)	2021-06-22 02:33:10	{"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": 0.8435215353965759, "perplexity": 674.019142529771}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488504969.64/warc/CC-MAIN-20210622002655-20210622032655-00157.warc.gz"}
https://www.physicsforums.com/threads/interpret-current-density-in-plasma-as-material-property.888560/	# Interpret current density in plasma as material property? Tags: 1. Oct 10, 2016 ### lampCable 1. The problem statement, all variables and given/known data An electromagnetic wave propagates through a gas of N free electrons per unit volume. Neglecting damping, show that the index of refraction is given by $$n^2 = 1 - \frac{\omega_P^2}{\omega^2},$$ where the plasma frequency $$\omega_P = \sqrt{\frac{Ne^2}{\epsilon_0m_e}}.\quad(1)$$ We assume that the incident wave is a plane wave, and that on each electron $F = qE$. 2. Relevant equations Maxwell's fourth equation in vacuum where there exist charges and current: $$\nabla\times\textbf{B} = \mu_0\textbf{J} + \mu_0\epsilon_0\frac{\partial\textbf{E}}{\partial t}.$$ 3. The attempt at a solution Integrating Newtons second law and neglecting any source velocity we get $$\textbf{v} = \frac{iq}{m\omega}\textbf{E}_0e^{i(kz-\omega t)}.$$ The current density, then, is $$\textbf{J} = Nq\textbf{v} = -\frac{Nq^2}{m\omega^2}\frac{\partial\textbf{E}}{\partial t}.\quad(2)$$ Using now Maxwell's fourth equation in vacuum together with (1) and (2) we get $$\nabla\times\textbf{B} = \mu_0\epsilon_0\bigg(1-\frac{\omega_P^2}{\omega^2}\bigg)\frac{\partial\textbf{E}}{\partial t}.$$ Question: Is it legit to here define $$\epsilon_r\mu_r = 1 - \frac{\omega_p^2}{\omega^2}$$ for the purpose of calculating the refractive index? My argument for this is as follows. Since $$n = \sqrt{\epsilon_r\mu_r}$$ it doesn't matter really (for the purpose of determining the refractive index) how $\epsilon_r$ and $\mu_r$ are chosen per se, so long as their product satisfy the above definition. In this sense it is then possible to convert a case where we have a region with current density, to one where we simply have a material with $\epsilon_r\epsilon_r\neq1$ instead. But since we are free to choose $\epsilon_r$ and $\mu_r$, the analogy could not go further to where the two are used separately. 2. Oct 10, 2016 Your derivation looks quite legitimate. The next step in deriving the E-M wave (at least the $B$ part) is to take the curl of both sides of your equation. With a vector identity $\nabla \times \nabla \times B=\nabla (\nabla \cdot B)-\nabla^2 B$, and using Faraday's law on the other side, you have the wave equation for $B$ and the wave velocity is determined precisely by what you have already presented. editing... Alternatively you could begin with Faraday's law $\nabla \times E=-dB/dt$ and again take the curl of both sides of the equation and proceed in a similar fashion to get the wave equation for the electric field $E$.	2017-10-22 20:27:32	{"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": 2, "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": 0.83647620677948, "perplexity": 236.43778574787228}, "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-2017-43/segments/1508187825436.78/warc/CC-MAIN-20171022184824-20171022204824-00782.warc.gz"}
https://www.varsitytutors.com/isee_upper_level_quantitative-help/how-to-find-the-length-of-an-edge	# ISEE Upper Level Quantitative : How to find the length of an edge ## Example Questions ### Example Question #1 : How To Find The Length Of An Edge A cube has sidelength one and one-half feet; a rectangular prism of equal volume has length 27 inches and height 9 inches. Give the width of the prism in inches. Explanation: One and one half feet is equal to eighteen inches, so the volume of the cube, in cubic inches, is the cube of this, or cubic inches. The volume of a rectangular prism is Since its volume is the same as that of the cube, and its length and height are 27 and 9 inches, respectively, we can rewrite this as The width is 24 inches. ### Example Question #1 : How To Find The Length Of An Edge A cube has sidelength one and one-half feet; a rectangular prism of equal surface area has length 27 inches and height 9 inches. Give the width of the prism in inches. Explanation: One and one half feet is equal to eighteen inches, so the surface area of the cube, in square inches, is six times the square of this, or square inches. The surface area of a rectangular prism is determined by the formula . So, with substitutiton, we can find the width: inches ### Example Question #1 : Solid Geometry A rectangular prism has volume one cubic foot; its length and width are, respectively, 9 inches and inches. Which of the following represents the height of the prism in inches?	2018-10-18 02:38:40	{"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": 0.8411359786987305, "perplexity": 1085.3342010650283}, "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/1539583511642.47/warc/CC-MAIN-20181018022028-20181018043528-00290.warc.gz"}
https://math.stackexchange.com/questions/1875649/volumes-of-revolution-cosine-function	# Volumes of Revolution- cosine function How do you find the volume of the solid of revolution formed by revolving this function around the $x$-axis: $y=2.33 \cos\left(\frac{25\pi}{119}\left(x-2.47\right)\right)$ between the limits $x=2.47$ and $x=4.85$? Full working would be greatly appreciated! • Are you familiar with any standard methods for this? You want to use what is called "disc method" (en.wikipedia.org/wiki/Solid_of_revolution#Disc_method). – Carser Jul 30 '16 at 3:52 • Yes I am familiar with that method. That is what I am trying to use but I am just unsure with how to expand the brackets when you have to square the entire function: (2.33cos(25π119(x−2.47)))^2. Could you please show me how to expand the bracket? Does the 2.33 become squared and the cos become squared or does something happen to inside the cos bracket as well – Emma Jul 30 '16 at 4:11 The volume is given by $$V = \pi \int_{2.47}^{4.85} \left( 2.33 \cos \left( \frac{25 \pi}{119}(x-2.47) \right) \right)^2 \ dx$$ $$= \pi (2.33)^2 \int_{2.47}^{4.85} \cos^2 \left(\frac{25 \pi}{119} (x-2.47) \right) \ dx$$ Letting $u=(x-2.47)$, then $du= dx$ and we can substitute to get $$= \pi (2.33)^2 \int_{2.47}^{4.85} \cos^2 \left(\frac{25 \pi}{119}u\right) \ du$$ and if we can use $$\int \cos^2(a x) dx = \frac{2ax+\sin(2ax)}{4a}$$ then we have $$= \pi (2.33)^2 \left[ \frac{2\frac{25 \pi}{119}u+\sin(2\frac{25 \pi}{119}u)}{4 \frac{25 \pi}{119}} \right]_{2.47}^{4.85}$$ which I'll happily leave for you to simplify.	2020-01-17 13:27:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8770497441291809, "perplexity": 193.36507246572404}, "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-2020-05/segments/1579250589560.16/warc/CC-MAIN-20200117123339-20200117151339-00088.warc.gz"}
http://tex.stackexchange.com/questions/4684/longrightarrow-doesnt-look-good-in-12pt?answertab=active	# \Longrightarrow doesn't look good in 12pt I'm using the standard CM fonts in 12pt (and amsmath if this is relevant). My problem is that \Longrightarrow gives something like this output, which I find a bit annoying. I know that it's a composed symbol, but in 10pt and in 11pt it looks perfect, so why doesn't it also work in 12pt? By the way, I produced the image by typing \Huge\Longrightarrow on mathurl.com, so it's not an artefact of my TeX system (and it tells us that mathurl uses 12pt ...). Moreover, it's not just something on the screen, I also see it on a printout (where I don't know which LaTeX distribution was used for typesetting). I think I've first seen this when I was still using LaTeX 2.09 a long time ago! - See programming.itags.org/tex/124077 for a way to hack it. Basically you should re-define the \Longrightarrow command from scratch and build it from a rescaled cmr10 = and the \Rightarrow wihch is already scaled from cmsys10. – Willie Wong Oct 29 '10 at 15:37 @Willie: Hmm, interesting idea, and looks better than the solution that I just posted, but the TeX file provided there by Ulrike doesn't compile for me: It tells me ! LaTeX Error: Encoding scheme OT1 ' unknown. – Hendrik Vogt Oct 29 '10 at 15:55 I think they changed OT1 to ot1 not too long ago. Nice trick btw. – Taco Hoekwater Oct 29 '10 at 16:01 There is a to-be-released package on Github that implements this fix, see github.com/phst/longarrows (the documentation is outdated, but the package might already work). – Philipp Oct 29 '10 at 16:03 @Taco: No, that's not the reason. The reason was me being stupid: In Ulrike's post at Willie's link there's a linebreak where there shouldn't be any. And of course OT1 with a space at the end is not known. – Hendrik Vogt Oct 29 '10 at 16:26 (Posting as CW since I don't like to take credit for my good Google Fu) A hack was provided in http://programming.itags.org/tex/124077/ where you re-implement the construction of \Longrightarrow by using instead of the cmr12 version of the = sign, the cmr10 version suitably rescaled. So this way the \Longrightarrow will be just like the 10-pt version but bigger. The following is the example provided by Ulrike Fischer in that thread. \documentclass[12pt]{article} \DeclareFontFamily{OT1}{cmrx}{} \DeclareFontShape{OT1}{cmrx}{m}{n}{<->cmr10}{} \begin{document}\pagestyle{empty} $\Longrightarrow$ %Redefine \Longrightarrow as the following to get the 'fixed' version $\mathrel{% \mbox{\fontfamily{cmrx}\fontencoding{OT1}\selectfont=}}% \joinrel\Rightarrow$ \end{document} The disadvantage of the above solution is that it's dangerous to use it as a patch for existing files: The fixed \Longrightarrow is slightly longer than the original one, so this might affect linebreaks. Here's a version that doesn't have this drawback: \documentclass[12pt]{article} \DeclareFontFamily{OT1}{cmrx}{} \DeclareFontShape{OT1}{cmrx}{m}{n}{<->cmr10}{} \let\saveLongrightarrow\Longrightarrow \makeatletter \renewcommand*{\Longrightarrow}{% \mathrel{\rlap{\fontfamily{cmrx}\fontencoding{OT1}\selectfont=}% \hphantom{\saveLongrightarrow}% \llap{$\m@th\Rightarrow$}}} \makeatother \begin{document} $a\saveLongrightarrow b$ $a\Longrightarrow b$ \end{document} - I think I've taken some credit for Google Fu on two answers at least. But this way I might feel free to edit your answer later on. – Hendrik Vogt Oct 29 '10 at 17:35 There is a preliminary package on Github that implements the fix suggested in the comments, see http://github.com/phst/longarrows (the documentation is outdated, but the package might already work). - OK, I managed to produce some hack that works. The idea is to use \textcolor{white} from the xcolor package, similarly as in my not so great approach to lowering \widetilde. Here's the code: \makeatletter \newcommand{\myRelbar}{% {\Rightarrow}% \llap{\textcolor{white}{\rule[-0.2ex]{1.1ex}{2ex}}}% \kern-1.5ex} \let\saveLongrightarrow\Longrightarrow \renewcommand{\Longrightarrow}{% \mathrel{\rlap{$\m@th\myRelbar\myRelbar$}% \phantom{{\saveLongrightarrow}}% \llap{$\m@th\Rightarrow$}}} \makeatother The problem is the the LaTeX \Relbar is just =, and that there in 12pt the to horizontal bars are too close to each other. So I construct \myRelbar by taking \Rightarrow and chopping off the arrow head with a white box. This is not quite long enough, so I take two of those. I've tried a second version that uses \scalebox from the graphicx package. I strongly discourage the use of this version; it's just an attempt based on my short discussion with Taco Hoekwater, and it shows that Taco was right: Scaling makes it a bit better, but it's still not perfect. \Relbar\joinrel\mathrel{\raisebox{0.024ex}{\scalebox{0.956}{${\Rightarrow}$}}} - Another faux solution: use tikz to draw the arrow for you: \newcommand{\tikzLongrightarrow}{% \mathbin{\tikz{\draw[arrows={-latex},line width=1.2pt,double=white] (0,0) -- (3em,0);}} } \let\oldLongrightarrow=\Longrightarrow \def\Longrightarrow{\tikzLongrightarrow} One disadvantage is that tikz is a lot to include just for one symbol. Not a problem if you're already including tikz. Another disadvantage is that it will take more work to make it look like the regular \Longrightarrow. - Hmm, no, this just looks too different from the regular arrow, and I actually have no idea how to make it look as it should. And as you already wrote, TikZ is really a lot to include. But still a fun solution. – Hendrik Vogt Oct 29 '10 at 14:26 The problem is that while there exists a cmr12 font (which is used by pdflatex), there is no cmsy12, so cmsy10 is scaled to 12pt to make it match. Unfortunately, there are subtle differences between the 10pt and 12pt versions of the computer modern fonts (slightly different metafont parameter settings), and the result of the mismatch is the effect your are witnessing. Side note: when you zoom in a lot, you will see that the actual difference is quite small, but the effect is worsened by the different antialiasing\|hinting used for the two separate fonts. The only 'solution' I can come up with is to patch the ot1cmr.fd file such that it will use cmr10 instead of cmr12, but perhaps a LaTeX expert will know a better approach. - Thanks for the explanations, Taco. I did zoom in a lot and saw that the actual difference is not too big, but in the printout in my hand it looks even worse than in the picure I included in the question. But the printout also looks as if bitmap fonts were used. – Hendrik Vogt Oct 29 '10 at 13:53 @Taco: So \Longrightarrow takes = from cmr12 and a scaled \Rightarrow from cmsy10, correct? Can't I affect this scaling? – Hendrik Vogt Oct 29 '10 at 13:57 Yes, you understand what LateX does correctly, but it is not so much the scaling itself that is causing the trouble but the fact that cmr10 is slightly different from cmr12. – Taco Hoekwater Oct 29 '10 at 14:26 @Taco: If the \Rightarrow was scaled just a little bit less, then the result ought to look a lot better, or am I mistaken? – Hendrik Vogt Oct 29 '10 at 14:28 Not per se. It is quite likely that the two metafont parameters for the stroke width and the gap width are not changed by the exact same ratio between cmr10 and cmr12. After all, that is what optical scaling is all about: it is what makes cmr5 at 17pt noticeably different from cmr17`. – Taco Hoekwater Oct 29 '10 at 14:30	2014-03-09 12:46:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9384807348251343, "perplexity": 1458.8276590727546}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999678381/warc/CC-MAIN-20140305060758-00012-ip-10-183-142-35.ec2.internal.warc.gz"}
http://mathoverflow.net/revisions/23110/list	As for your first question, it seems hard to characterize all such curves. There two types of families I have thought of. Of course you can multiply curves that satisfy the property, so let us only talk about minimal curves with respect to the property. • As you mention, the real line. • Genus 0 or 1 curves with a rational point of infinite order and the required property (mentioned in comments) that $y$ appears only with even powers. This uses the simple fact that conjugate numbers in $\mathbb{Q}(\sqrt{-1})$ have minimal polynomial over the rationals. If we want to extend this to higher genus, there is the following question: Given a rational class in the jacobian of a curve, does it have infinitely many multiples such that $a(x)$, in its representation as a reduced divisor $[(a(x), b(x,y)]$, has only real roots? If so, then the property holds for all curves with a rational point on the jacobian of infinite order. • The unit circle fits into the above. But as you mention, there is another way, which we can generalize: curves of the form $\|\|f(z)\|\|=1, z=x+iy$. The question is now: how many points in the union of all CM fields does this equation have? If I had to guess, infinitely many, so I think these probably satisfy the property. • 2 deleted 2 characters in body Yes. There exist many such curves: For any rational monic polynomial without multiple roots if $x+iy$ is a root, then $x-iy$ is a root as well. So if your curve $C$ satisfies $y\not =0,\ (x,y)\in C \rightarrow (x,-y)\not\in C$ then it cannot contain the rootset of any polynomial with complex roots. To finish, make the curve have finitely many real roots. For example: $$C: x-y^3=0$$x-y=0$$1 Yes. There exist many such curves: For any rational monic polynomial without multiple roots if x+iy is a root, then x-iy is a root as well. So if your curve C satisfies y\not =0,\ (x,y)\in C \rightarrow (x,-y)\not\in C then it cannot contain the rootset of any polynomial with complex roots. To finish, make the curve have finitely many real roots. For example:$$C: x-y^3=0	2013-05-25 10:12:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7992794513702393, "perplexity": 124.41156457145094}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368705926946/warc/CC-MAIN-20130516120526-00058-ip-10-60-113-184.ec2.internal.warc.gz"}
http://math.stackexchange.com/tags/brownian-motion/hot	# Tag Info ## Hot answers tagged brownian-motion 6 From the stochastic integral analogue of integration by part, which is a corollary of Ito's Lemma, $$d(W_t\ln t) = \frac{W_t}{t}dt+(\ln t)dW_t,$$ so $$I:=\int_0^t \frac{W_s}{s}ds=W_t\ln t-\int_0^t (\ln s)dW_s=\int_0^t \ln\frac{t}{s}dW_s$$ as $W_t\ln t\to 0$ in probability, as $t\to 0^+$. Since the random variable $I$ is a linear combination of independent ... 3 I think both of you and your book are right! The point here is $d(S_t)d(e^{-rt})=0$. The reason is $dt\cdot dW=0$ and $dt\cdot dt=0$. 3 Edit: This answer shows the identity $$\mathbb{E} \left( \left\| \sum_{i=0}^{n-1} \int_{t_i^n}^{t_{i+1}^n} (W(t_i^n)-W(t)) \, dt \right\|^2 \right) = \mathbb{E} \left( \sum_{i=0}^{n-1} \left[\int_{t_i^n}^{t_{i+1}^n} (W(t_i^n)-W(t)) \, dt\right]^2 \right). \tag{1}$$ Fix $i<j$. Then, by the tower property, \begin{align} ... 2 The Wikipedia article you cite provides everything you need to evaluate the analytical solution of the Ornstein–Uhlenbeck process. However, for a beginner, I agree that it may not be very clear. 1. Simulating SDEs You should first be familiar with how to simulate this process using the Euler–Maruyama method. The stochastic differential equation (SDE) ... 2 We should focus on simulating \int_0^t e^{\theta (s)}\, \mathrm{d}W_s. If f(s)=e^{\theta (s)} is continuously differentiable, you could use the fact that\sum_{i=1}^{[tn]}f(s_i^)\Big(W(s_i)-W(s_{i-1})\Big)\to\int_0^tf(s)dW(s)$$in quadratic mean, for s_i^* \in [s_{i-1},s_i]. Note that you should use$$W(s_i)-W(s_{i-1}) \sim N(0,s_i-s_{i-1})$$and ... 2 We have that$$ V = W_t^2 - t$$hence (written with t and x, where we plug in S_t = W_t for x), we have$$ V(x,t) = x^2 - tNow \begin{align} \frac{\partial V}{\partial t} &= -1\\ \frac{\partial V}{\partial x} &= 2x\\ \frac{\partial^2 V}{\partial x^2} &= 2 \end{align} Hence (note that S_t = W_t) \begin{align} ... 2 It holds that\mathbb{E}^x(F) = \int F(w) \, d\mathbb{P}_x(w) = \int F(x+w) \, d\mathbb{P}(w) \tag{1}$$for any measurable function F: (C[0,\infty),\mathcal{B}(C[0,\infty)) \to [0,\infty). This follows from the fact that (1) holds for simple functions and we can extend the equality to all measurable non-negative functions (by the monotone convergence ... 2 Since$$\exp \left( - \frac{a^2}{2t} \right) \uparrow 1 \qquad \text{as $t \to \infty$},$$there exists R>0 such that$$\exp \left( - \frac{a^2}{2t} \right) \geq \frac{1}{2}$$for all t \geq R. Consequently,$$\begin{align} \mathbb{E}(\tau_a) &\geq \int_R^{\infty} \frac{at}{\sqrt{2\pi t^3}} \exp \left(- \frac{a^2}{2t} \right) \, dt \\ ... 2 This is a non-rigorous answer. If $t \mapsto R_t$ is a differentiable function from $\mathbb R$ to $SO(n)$, then its derivative satisfies $$\frac{dR_t}{dt} = R_t A_t ,$$ where $A_t$ is an anti-symmetric matrix. I could rewrite this non-rigorously as $$dR_t = R_t A_t \, dt = R_t (\exp(A_t \, dt) - I) ,$$ where $\exp(A)$ denotes the matrix exponential of a ... 2 My writeup of this in a previous project, with slightly different notation, looks like this: $$u(x) = \Bbb E[f(x+W_{\tau_{\partial D},x})] \\ = \Bbb E[\Bbb E[f(x+W_{\tau_{\partial D},x})\|W_{\tau_r}]] \\ = \Bbb E[u(x+W_{\tau_r})] \\ = \int_{\partial B(x,r)} u(y) dS(y).$$ In the first step, we use the tower property, inserting a conditional expectation in ... 2 In order to prove measurability of $\Lambda(m)$, it suffices to show that the processes $$(t,\omega) \mapsto \max_{(t-1/m)^+ \leq s \leq t} W_s(\omega) \qquad \quad (t,\omega) \mapsto \min_{t \leq s \leq t+1/m} W_s(\omega)$$ are progressively measurable. Since any continuous (adapted) process is progressively measurable, we are done if we can show that ... 2 Ito's formula: http://en.wikipedia.org/wiki/Itō's_lemma $Y = X^2$ 1.) find $dY$ using the chain rule to order $(dX)^2$ with a taylor expansion and substitute for $dX$ $$dY=\frac {\partial Y}{\partial t}dt + \frac {\partial Y}{\partial X}dX + \frac 12 \frac {\partial^2 Y}{\partial X^2}(dX)^2$$ $$=2X(a(t)Xdt+2dW) + \frac 122(a(t)Xdt+2dW)^2 = ... 2 Ito's formula says that given$$ dX_t = \mu_t dt + \sigma_t dW_t $$and a C^2-function f \colon \mathbf R^2 \to \mathbf R, we have for Y_t = f(t, X_t) that$$ dY_t = \left(\frac{\partial f}{\partial t}(t, X_t) + \mu_t \frac{\partial f}{\partial x}(t, X_t) + \frac{\sigma_t^2}{2}\frac{\partial^2 f}{\partial x^2}(t, X_t)\right)\, dt + \sigma_t ... 2 Here is one argument that would work, assuming that one already knows the following facts: For any $s>0$, the process $(B_{s+t}-B_s)_{t \geq 0}$ is a Brownian motion. The time inversion $(tB_{\frac{1}{t}})$ of a Brownian motion is a Brownian motion (defined to start at $0$ at $t=0$). $\limsup_{t \to \infty} B_t >0$ (in fact, the lim sup is ... 1 By definition, $$U_t(X)-U_t(Y) = (X_0-Y_0) + \int_0^t (b(s,X_s)-b(s,Y_s)) \, ds + \int_0^t (\sigma(s,X_s)-\sigma(s,Y_s)) \, dW_s$$ and therefore \begin{align}&\|U_t(X)-U_t(Y)\|^2 \leq \\ & \color{red}{3 \|X_0-Y_0\|^2} + 3 \left( \int_0^t \|b(s,X_s)-b(s,Y_s)\|ds\right)^2 + 3\left(\int_0^t \|\sigma(s,X_s)-\sigma(s,Y_s)\|dW_s\right)^2. \end{align*} The ... 1 I would do this. Show \begin{align}P_t (x,A) &{:=} \int_A g_t (x,y)dy \\ P_0 (x,A) &{:=} \delta_x (A), \end{align} where $g_t$ density of the folded normal distribution, is a Markov kernel (Chapmann K. Equation, etc.). Then you know there exists a unique Markov process. That's the reflected Brownian motion. Have someone any comments? Is that ... 1 If $I$ is uncountable and $A_i \in \mathcal{F}_t$, then it does in general not follow that $\bigcap_{i \in I} A_i \in \mathcal{F}_t$. We only know that $\mathcal{F}_t$ is stable under countable intersections. Recall the following lemma: Let $g: [0,t] \to \mathbb{R}$ be a continuous function. Then $$\max_{s \in [0,t]} g(s) = \max_{s \in [0,t] \cap ... 1 Heuritically, d(S_t)d(e^{-rt})=-re^{-rt}dt(dS_t)\sim O(dt)^{3/2}. Thus, it is not considered in the SDE for S_te^{-rt}. 1 Basically, the construction is divided into three steps: Step 1: Constructing a consistent set of finite dimensional distributions. Consider any starting point x\in\mathbb{R} and a set of times 0<t_1<t_2<\cdots<t_n<T. Define a measure on finite dimensional space \mathbb{R}^n as$$ \nu_{t_1,\cdots,t_n}(F_1,\cdots,F_n) ... 1 Hints: (This answer does not use Borel Cantelli lemma; instead it is based on basic martingale techniques.) Show that for any fixed $\xi>0$, the process $$M_t^{\xi} := \exp \left( \xi B_t - \frac{1}{2} \xi^2 t \right), \qquad t \geq 0,$$ defines a martingale. Fix $T>0$. For $b>0$ we define a stopping time by $\tau_b := \inf\{t>0; B_t \geq b\}$. ... 1 Let $0 \leq t_1<\ldots<t_n$. Since $(B_t)_{t \geq 0}$ is a Gaussian process, we know that $(B_{t_1},\ldots,B_{t_n})$ is Gaussian. This implies in particular that $\sum_{j=1}^n B_{t_j}$ is Gaussian. Since a Gaussian random variable is uniquely characterized by its mean and variance, it remains to calculate those two. As $\mathbb{E}B_t=0$ for any $t ... 1 Obviously, $$\|M_t\| \leq \|B_t\|^3 + \|\alpha\| \int_0^t \|B_s\| \, ds.$$ Since$(B_t)_{t \geq 0}$is a Brownian motion, we have in particular$B_t \sim N(0,t)$and$B_t \sim \sqrt{t} B_1$. Consequently, $$\mathbb{E}(\|B_t\|^3) <\infty \qquad \text{and} \qquad \mathbb{E}(\|B_s\|) = \sqrt{s} \mathbb{E}(\|B_1\|).$$ Using Tonelli's theorem, we get ... 1 Your calculations are correct, but the claim is not. Instead of$\tau_a \stackrel{d}{=} \sqrt{a} \tau_1$it should read$\tau_a \stackrel{d}{=} a^2 \tau_1$. References: Revuz/Yor: Continuous martingales and Brownian motion, Proposition III.3.10 Schilling/Partzsch: Brownian motion - An introduction to stochastic processes, Problem 6.6 1 the answer to this is quite simple if you look carefully to the definition of H, you should notice that$h_{i-1}$is an$\mathcal{F}_{t_{i-1}}$-measurable random variable. From this and the following elementary properties on conditional expectation we get the result : If X is$\mathcal{F}$-measurable and for any bounded random variable$Y$, we have a.s. : ... 1 Integrating any locally bounded predictable process (in particular a continuous adapted process) with respect to a local martingale (resp locally square integrable martingale) results in a new process which is a local martingale (resp locally square integrable). A proof of this can be found in Jacod & Shirjaev (Chapter 1; Theorem 4.31). (It basically ... 1 For any$0 \leq s_0 < \ldots < s_m \leq T =T + t_0 < \ldots < T+t_n$the random variables $$\underbrace{B_{t_n+T}-B_{t_{n-1}+T}}_{B'_{t_n}-B'_{t_{n-1}}},\ldots,\underbrace{B_{t_1+T}-B_{t_{0}+T}}_{B'_{t_1}-B'_{t_{0}}}, B_{s_m}-B_{s_{m-1}},\ldots,B_{s_1}-B_{s_0} \tag{1}$$ are independent. Recall that for any two measurable mappings$f: ... 1 Define a martingale by $$M_t := \begin{cases} \mathbb{E}(\phi(W_T) \mid \mathcal{F}_t), & t \leq T, \\ \phi(W_T), & t>T. \end{cases}$$ Then, by the martingale representation theorem, there exists a representation of the form $$M_t = c+ \int_0^t \beta_s \, dW_s.$$ For $t=T$, this proves the claim. Remark: One possibility to prove the mentioned ... 1 First of all, note that the strong Markov property of Brownian motion does not state that $(B_t)_{t \in [0,\tau]}$ and $(B_{t+\tau})_{t \geq 0}$ are independent, but that $(B_t)_{t \in [0,\tau]}$ and $(B_{t+\tau}-B_{\tau})_{t \geq 0}$ are independent. This independence yields in fact $$\mathbb{E}(1_A \cdot 1_B) = \mathbb{E}(1_A) \mathbb{E}(1_B)$$ for any ... 1 First consider the case that $\sigma$ is a simple function, i.e. $$\sigma(s) =\sum_{j=1}^n 1_{[t_{j-1},t_j)} \xi_j \tag{1}$$ where $(\xi_j)_{j=1,\ldots,n}$ are random variables independent from the Brownian motion $W$. Without loss of generality, we may assume that $t_k = t$ for some $k \in \{1,\ldots,n\}$ (otherwise we add the point to the partition). ... 1 Yes, they are independent; for a proof see e.g. N. Ikeda, S. Watanabe: Stochastic Differential Equations and Diffusion Processes, Theorem II.6.3. Note that a similar question (for the case of Poisson processes, not Poisson random measures) has been discussed on mathoverflow. Only top voted, non community-wiki answers of a minimum length are eligible	2015-05-27 08:10:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9993358850479126, "perplexity": 409.4150716101769}, "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-2015-22/segments/1432207928923.21/warc/CC-MAIN-20150521113208-00043-ip-10-180-206-219.ec2.internal.warc.gz"}
http://www.euro-math-soc.eu/node/4003	## Riemann, Einstein and geometry Sep 18 2014 - 09:00 Sep 20 2014 - 12:30 Venue: Institut de Recherche Mathématique Avancée, University of Strasbourg (France) Short description of the event: The conference is part of a series of bi-annual conferences "Encounter between Mathematicians and Theoretical Physicists" and it is addressed to a large audience. Athanase Papadopoulos : athanase.papadopoulos$math.unistra.fr Sumio Yamada : yamada$math.gakushuin.ac.jp	2014-03-07 09:00:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7580703496932983, "perplexity": 13252.943875181329}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999639602/warc/CC-MAIN-20140305060719-00025-ip-10-183-142-35.ec2.internal.warc.gz"}
https://mathoverflow.net/questions/211443/a-question-regarding-extendible-cardinals-and-a-result-of-m-magidor	# A question regarding extendible cardinals and a result of M. Magidor The following definitions and Theorems come from M. Magidor's paper "On the Role of Supercompact and Extendible Cardinals in Logic" (Israel J. Math., Vol. 10, 1971): "Definition: Logic is called $\kappa$ compact iff for every set of formulae A in this logic, if every subset of A of cardinality $\lt$$\kappa has a model, then A has a model. The L^{n}_{\kappa} logic is like the n-th order logic, except that we allow conjunction and disjunctions of less than \kappa formulae. The usual second order logic is of course L^{2}_{\omega}. Theorem 4. \kappa is extendible iff L^{2}_{\kappa} is \kappa compact. \kappa is the first extendible iff it is the first \alpha such that second order logic is \alpha compact." He also proves this theorem as well: "if \kappa is extendible it is supercompact". However, nowhere in his paper does he say whether he is using full semantics or Henkin models when he refers to \kappa compactness. I therefore will go out on a limb here (so to speak) and presume (rightly or wrongly) that he is using full semantics rather than Henkin models (after all, he does speak of "the usual second order logic" and that, I suppose, might be a subtle indication that he means to use full semantics). Under this assumption, it is well known that L^{2}_{\omega} is not compact. Under this assumption, it seems one can prove the noncompactness of L^{2}_{\omega} using Theorem 4 and "if \kappa is extendible then it is supercompact". I argue in the following manner: by Theorem 4, if L^{2}_{\omega} is \omega compact (i.e. compact) then \omega is extendible. If \omega is extendible it is supercompact. However, in his answer to arsmath's mathoverflow question, "\aleph_0 (arsmath writes it "Aleph 0") as a large cardinal", Amit Kumar Gupta shows that \omega is not supercompact, hence by this, the chain of contrapositives of the implications I use and modus ponens, L^{2}_{\omega} is not \omega compact (i.e. not compact). This, however, raises the following question: even if L^{2}_{\omega} is not compact, by Theorem 4, if there exists some extendible \kappa (\gt$$\omega$) then $L^{2}_{\kappa}$ is $\kappa$ compact (i.e. second order logic satisfies some compactness theorem). Those who do not believe in the existence of large cardinals should ask themselves whether second order logic should be $\kappa$ compact under full semantics. Another question that arises is this: since $L^{2}_{\omega}$ is compact under Henkin models, does Theorem 4 hold over Henkin models? ## 1 Answer For your second question, the answer is "no." Since, with Henkin semantics, second-order logic is just re-syntacted first-order logic, of course theorem 4 fails for Henkin semantics: $L_\kappa^2$ with Henkin semantics is compact iff $L_{\kappa,\omega}$ is compact - that is, iff $\kappa$ has the tree property (which is implied by, but not equivalent to, $\kappa$ being weakly compact - weakly compact = tree property plus inaccessible). Meanwhile, I'm not sure what your first question is asking. It is quite reasonable to have a sequence of stronger logics $\mathcal{L}_\alpha$ ($\alpha\in ON$) such that compactness fails early ($\mathcal{L}_\omega$ is not $\omega$-compact) but holds occasionally later on ($\mathcal{L}_\kappa$ is $\kappa$-compact for some $\kappa$). For instance, take $\mathcal{L}_\alpha=L_{\aleph_\alpha,\omega}$. Then $\mathcal{L}_\alpha$ is compact iff $\aleph_\alpha$ has the tree property - in particular, it is enough for $\alpha$ to be weakly compact. So I don't see what challenge this raises for "those who do not [or do?] believe in the existence of large cardinals." • Thanks for your very nice answer. Since Theorem 4 does not hold under Henkin semantics, it is clear that Magidor was assuming $L^2_{\kappa}$ with full semantics. – Thomas Benjamin Oct 20 '15 at 1:14 • (cont.) I will have to think about the question posed in the second part of your answer. Is there some way '$L^2_{\kappa}$ is $\kappa$-compact' could be shown to be independent of $L^2_{\kappa}$ ( a type of forcing argument, for instance)? – Thomas Benjamin Oct 20 '15 at 1:32 • @ThomasBenjamin I'm not sure what you mean by "Is there some way '$L_\kappa^2$ is $\kappa$-compact could be shown to be independent of $L_\kappa^2$' - can you clarify? – Noah Schweber Oct 20 '15 at 1:56 • Some further questions. Under full semantics, it is known that $L^2_{\omega}$ is incomplete, that is, the set of truths and the set of falsities of $L^2_{\omega}$ each are not r.e. (what is the position of each of these sets in the Kleene hierarchy)? If $L^2_{\omega}$ is incomplete, then is $L^2_{\kappa}$ also incomplete under full semantics as well? Let me assume now, for the sake of argument, that the sentence "$L^2_{\kappa}$ is $\kappa$-compact" is independent of $L^2_{\kappa}$. Then $\lnot$$L^2_{\kappa} is not provable in L^2_{\kappa} either, so one can have – Thomas Benjamin Oct 23 '15 at 11:56 • (cont.) models of full semantics in which either "L^2_{\kappa} is \kappa-compact" or "\lnot$$L^2_{\kappa}$ is $\kappa$-compact" holds. Is there some reason that " '$L^2_{\kappa}$ is ${\kappa}$-compact' is independent of $L^2_{\kappa}$" is not tenable? – Thomas Benjamin Oct 23 '15 at 12:06	2019-12-08 21:45:34	{"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": 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": 0.9474733471870422, "perplexity": 618.6044243035069}, "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-2019-51/segments/1575540514893.41/warc/CC-MAIN-20191208202454-20191208230454-00157.warc.gz"}
http://math.stackexchange.com/questions/156151/dimension-of-h1s-e	# Dimension of $H^1(S,E)$ Let $S$ be a Riemann surface of genus $g$, $p\in S$ and $E$ be the holomorphic line bundle associated with the divisor $p$. This means that $E$ admits a section $\sigma$ with a simple zero at $p$ and non vanishing everywhere else. Does this imply that the $\bar{\partial}$ cohomology group $H_{\bar{\partial}}^1$ is trivial? I think so, because then you can consider ${\sigma}^{-1}$, and this is a section with a simple pole at $p$. Do you agree with me? - No, I'm afraid I don't agree with you. The cohomology vector space $H^1(S,E)$ is dual to $H^0(S,K_S(-p))$ by Serre duality. Since the bundle $K_S(-p)$ has degree $2g-3$, that bundle has indeed no section $\neq0$ for $g=0$ or for $g=1$. But for larger values of $g$ there is no reason why it can't have non-zero sections. On the contrary: for $g\geq 3$, Riemann-Roch guarantees that $\dim H^0(S,K_S(-p))\geq (1-g)+(2g-3)\gt 0$ and consequently that $H^1(S,E)\neq 0$. NB You use $H^1_{\bar \partial}(S,E)$ but by Dolbeault's theorem that cohomology vector space is isomorphic to the $H^1(S,E)$ defined by Čech cohomology or by derived functor cohomology. - Sorry, but isn't the degree of $K_s(-p) = -1$ , no matter what the genus is? – Abramo Jun 10 '12 at 8:39 Dear Abramodj, if $E$ is a line bundle of degree $d$, then $E(-p)=E\otimes_ {\mathcal O} \mathcal O(-p)$ has degreee $d-1$. In the present case we apply this to $K_S$ which has degree $deg (K_S)=2g-2$ – Georges Elencwajg Jun 10 '12 at 8:58 So please explain what $K_s(p)$ is. I thought it was just the line bundle with one simple pole at $p$ (i.e. $\mathbb{C}\otimes O(p)$), and so in that case the degree would be just one. From what you say about Serre duality I guess $K_S(-p) = {T^*}^{1,0}M\otimes O(-p)$, right? – Abramo Jun 10 '12 at 9:04 Dear Abramodj: yes, that's right.The notation $K_S$ is standard in algebraic geometry for the bundle of algebraic (or holomorphic) differential forms. I suppose you learn Riemann surfaces in a setting closer to differential geometry and analysis. – Georges Elencwajg Jun 10 '12 at 9:11 Ok now I see, thanks. By the way yes, my course was closer to differential geometry. I will try with algebraic geometry next semester! – Abramo Jun 10 '12 at 11:37	2014-03-10 16:40:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9775985479354858, "perplexity": 240.53807683642668}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394010901252/warc/CC-MAIN-20140305091501-00031-ip-10-183-142-35.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/850029/rayleigh-quotient-variant	# Rayleigh Quotient variant? If $A$ is a covariance matrix and I want to get $\max X^TAX$ where each value of $x$ is between -1 and 1. Is there a closed-form solution for this? I understand when $X^TX = 1$ this becomes the original Rayleigh quotient problem. • Is $X$ a random column vector? – Omnomnomnom Jun 28 '14 at 4:33 • @Omnomnomnom: Right, X is an n x 1 vector – user40923 Jun 28 '14 at 4:45	2019-05-25 23:12:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9404663443565369, "perplexity": 434.59922249905475}, "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-2019-22/segments/1558232258453.85/warc/CC-MAIN-20190525224929-20190526010929-00023.warc.gz"}
https://mathzsolution.com/logic-problem-identifying-poisoned-wines-out-of-a-sample-minimizing-test-subjects-with-constraints/	# Logic problem: Identifying poisoned wines out of a sample, minimizing test subjects with constraints I just got out from my Math and Logic class with my friend. During the lecture, a well-known math/logic puzzle was presented: The King has $$10001000$$ wines, $$11$$ of which is poisoned. He needs to identify the poisoned wine as soon as possible, and with the least resources, so he hires the protagonist, a Mathematician. The king offers you his expendable servants to help you test which wine is poisoned. The poisoned wine is very potent, so much that one molecule of the wine will cause anyone who drinks it to die. However, it is slow-acting. The nature of the slow-acting poison means that there is only time to test one “drink” per servant. (A drink may be a mixture of any number of wines) (Assume that the King needs to know within an hour, and that any poison in the drink takes an hour to show any symptoms) What is the minimum amount of servants you would need to identify the poisoned wine? With enough time and reasoning, one can eventually see that this requires at most ten ($$1010$$) servants (in fact, you could test 24 more wines on top of that 1000 before requiring an eleventh servant). The proof/procedure is left to the reader. My friend and I, however, was not content with resting upon this answer. My friend added the question: What would be different if there were $$22$$ wines that were poisoned out of the 1000? What is the new minimum then? We eventually generalized the problem to this: Given $$NN$$ bottles of wine ($$N \gt 1N \gt 1$$) and, of those, $$kk$$ poisoned wines ($$0 \lt k \lt N0 \lt k \lt N$$), what is the optimum method to identify the all of the poisoned wines, and how many servants are required ($$s(N,k)s(N,k)$$)? After some mathsing, my friend and I managed to find some (possibly unhelpful) lower and upper bounds: $$log_2 {N \choose k} \le s(N,k) \le N-1 log_2 {N \choose k} \le s(N,k) \le N-1$$ This is because $$log_2 {N \choose k}log_2 {N \choose k}$$ is the minimum number of servants to uniquely identify the $$N \choose kN \choose k$$ possible configurations of $$kk$$ poisoned wines in $$NN$$ total wines. Can anyone help us find an optimum strategy? Besides the trivial one requiring $$N-1N-1$$ servants. How about a possible approach to start? Would this problem be any different if you were only required to find a strategy that would for sure find a wine that is not poisoned, instead of identifying all poisoned wines? (other than the slightly trivial solution of $$kk$$ servants) I asked this question on MathOverflow and got a great answer there. For $k = 2$ I can do it with ${\lceil \log_2 N \rceil + 2 \choose 2} - 1$ servants. In particular for $N = 1000$ I can do it with $65$ servants. The proof is somewhat long, so I don’t want to post it until I’ve thought about the problem more. I haven’t been able to improve on the above result. Here’s how it works. Let $n = \lceil \log_2 N \rceil$. Let me go through the algorithm for $k = 1$ so we’re all on the same page. Number the wines and assign each of them the binary expansion of their number, which consists of $n$ bits. Find $n$ servants, and have servant $i$ drink all the wines whose $i^{th}$ bit is $1$. Then the set of servants that die tells you the binary expansion of the poisoned wine. For $k = 2$ we need to find $n$ butlers, $n$ maids, and ${n \choose 2}$ cooks. The cooks will be named $(i, j)$ for some positive integers $1 \le i < j \le n$. Have butler $i$ drink all the wines whose $i^{th}$ bit is $1$, have maid $i$ drink all the wines whose $i^{th}$ bit is $0$, and have cook $(i, j)$ drink all the wines such that the sum of the $i^{th}$ bit through the $j^{th}$ bit, inclusive, mod 2, is $1$. This is how the casework breaks down for butlers and maids. • If both butler $i$ and maid $i$ die, then one of the poisoned wines has $i^{th}$ bit $0$ and the other has $i^{th}$ bit $1$. • If only butler $i$ dies, then both of the poisoned wines have $i^{th}$ bit $1$. • If only maid $i$ dies, then both of the poisoned wines have $i^{th}$ bit $0$. The second two cases are great. The problem with case 1 is that if it occurs more than once, there's still ambiguity about which wine has which bit. (The worst scenario is if all the butlers and maids die.) To fix the issue with case 1, we use the cooks. Let $i_1 < ... < i_m$ be the set of bits where case 1 occurs. We'll say that the poisoned wine whose $(i_1)^{th}$ bit is $1$ is wine A, and the other one is wine B. Notice that the sum of the $(i_1)^{th}$ through $(i_2)^{th}$ bits of wine A mod 2 is the same as the sum of the $(i_1)^{th}$ through $(i_2)^{th}$ bits of wine B mod 2, and we can determine what this sum is by looking at whether cook $(i_1, i_2)$ died. The value of this sum determines whether the $(i_2)^{th}$ bit of wine A is 1 or 0 (and the same for wine B). Similarly, looking at whether cook $(i_j, i_{j+1})$ died tells us the remaining bits of wine A, hence of wine B. One last comment for now. The lower bound is not best possible when $k$ is large compared to $N$; for example, when $k = N-1$ it takes $N-1$ servants. The reason is that any servant who drinks more than one wine automatically dies, hence gives you no information.	2023-02-04 02:57:33	{"extraction_info": {"found_math": true, "script_math_tex": 53, "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": 16, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6911271214485168, "perplexity": 775.739097904058}, "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-2023-06/segments/1674764500080.82/warc/CC-MAIN-20230204012622-20230204042622-00775.warc.gz"}
https://www.shaalaa.com/question-bank-solutions/algebraic-methods-solving-pair-linear-equations-substitution-method-solve-following-systems-equations-22-x-y-15-x-y-5-55-x-y-45-x-y-14_22310	Share Notifications View all notifications # Solve the Following Systems of Equations: 22/(X + Y) + 15/(X - Y) = 5 55/(X + Y) + 45/(X - Y) = 14 - CBSE Class 10 - Mathematics Login Create free account Forgot password? ConceptAlgebraic Methods of Solving a Pair of Linear Equations Substitution Method #### Question Solve the following systems of equations: 22/(x + y) + 15/(x - y) = 5 55/(x + y) + 45/(x - y) = 14 #### Solution Let 1/(x + y) = u and 1/(x - y) = v Then, the given system of equation becomes 22u + 15v = 5 ...(i) 55u + 45v = 14..(ii) Multiplying equation (i) by 3, and equation (ii) by 1, we get 66u + 45v = 15 ....(iii) 55u + 45v = 14 .....(iv) Subtracting equation (iv) from equation (iii), we get 66u - 55u = 15 - 4 => 11u = 1 => u = 1/11 Putting u = 1/11 in equaiton (i) we get 22 xx 1/11 + 15v = 5 => 2 + 15v = 5 => 15v = 5 - 2 => 15v = 3 => v = 3/5 = 1/5 Now u = 1/(x + y) => 1/(x + y) = 1/11 => x + y = 11 ....(v) And v = 1/(x - y) => 1/(x- y) = 1/5 => x - y = 5 .....(vi) Adding equation (v) and equation (vi), we get 2x = 11 + 5 => 2x = 16 => x = 16/2 = 8 Putting x =- 8 in equation (v), we get 8 + y = 11 y = 11 - 8 = 3 Hence, solution of the given system of equation is x = 8, y = 3 Is there an error in this question or solution? #### Video TutorialsVIEW ALL [2] Solution Solve the Following Systems of Equations: 22/(X + Y) + 15/(X - Y) = 5 55/(X + Y) + 45/(X - Y) = 14 Concept: Algebraic Methods of Solving a Pair of Linear Equations - Substitution Method. S	2019-12-11 21:45:22	{"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": 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": 0.600809633731842, "perplexity": 2966.1009793681746}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "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-2019-51/segments/1575540533401.22/warc/CC-MAIN-20191211212657-20191212000657-00011.warc.gz"}
http://mathhelpforum.com/trigonometry/4557-law-cosine-help.html	# Thread: Law of cosine Help!! 1. ## Law of cosine Help!! Cud u give me the formula of Law of Cosine on how to solve a triangle w/ 3 sides given? Im kind of confused with this problem: a = 5, b = 6, c = 9 find all 3 angles A,B & C Cud u give me the formula of Law of Cosine on how to solve a triangle w/ 3 sides given? Im kind of confused with this problem: a = 5, b = 6, c = 9 find all 3 angles A,B & C $c^2 = a^2 + b^2 - 2ab \cdot cos( \gamma )$ Where $\gamma$ is the angle across from side c. Thus $cos( \gamma ) = \frac{a^2 + b^2 - c^2}{2ab}$. The other formulae simply permute the values of a, b, and c and need not be given. So for example, to find the angle across from side c we have: $cos( \gamma ) = \frac{5^2 + 6^2 - 9^2}{2\cdot 5 \cdot 6} = -\frac{1}{3}$ Thus $\gamma$ is second quadrant and $\gamma \approx 109.5^o$ To find the angle across from side a, use a = 6, b = 9, c = 5. etc. -Dan 3. You can also find the angles by using the fact that. $\frac{1}{2}ab\sin \gamma =A$ where, $A=\mbox{ area }$ And you can calculate area by Heron's Formula. $A=\sqrt{s(s-a)(s-b)(s-c)}$ $s=\mbox{ semi-perimeter }$ Cud u give me the formula of Law of Cosine on how to solve a triangle w/ 3 sides given? Im kind of confused with this problem: a = 5, b = 6, c = 9 find all 3 angles A,B & C No, I'm confused . . . You're familiar with the Law of Cosines . . but you've never solved for an angle . . . ever? Okay, just this once . . . I assume you know that: . $a^2\:=\:b^2+c^2 - 2bc\cos A$ Rearrange the terms: . $2bc\cos A\:=\:b^2 + c^2 - a^2$ Divide by $2bc:\;\;\boxed{\cos A\:=\:\frac{b^2 + c^2 - a^2}{2bc}}$ . . . a formula for finding $\angle A.$ Similarly, we can derive formulas for the other two angles: . . $\boxed{\cos B\:=\:\frac{a^2+c^2-b^2}{2ac}}\qquad\boxed{\cos C\:=\:\frac{a^2+b^2-c^2}{2ab}}$ You should memorize these or be able to derive them when needed. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Your problem has: . $a = 5,\lb = 6,\;c = 9$ We have: . $\cos A\:=\:\frac{6^2 + 9^2 - 5^2}{2(6)(9)}\:=\:\frac{92}{108}$ . . Therefore: . $A\:=\:\cos^{-1}\left(\frac{92}{108}\right)\:=\:31.5863381\quad \Rightarrow\quad \boxed{A\:\approx\:31.6^o}$ We have: . $\cos B\:=\:\frac{5^2 + 9^2 - 6^2}{2(5)(9)}\:=\:\frac{70}{90}$ . . Therefore: . $B\:=\:\cos^{-1}\left(\frac{7}{9}\right)\:=\:38.94244127\quad \Rightarrow\quad \boxed{B\:\approx\:38.9^o}$ We have: . $\cos C\:=\:\frac{5^2+6^2-9^2}{2(5)(6)}\:=\:\frac{-20}{60}$ . . Therefore: . $C\:=\:\cos^{-1}\left(-\frac{1}{3}\right)\:=\:109.4712206\quad \Rightarrow\quad \boxed{C\:\approx\:109.5^o}$ Check: . $A + B + C\:=\:31.6^o + 38.9^o + 109.5^o \:=\:180^o$ . . . Yay! 5. Yea ... But the only confusing thing is that after solving the law of cosine to get the 1st angle A which is 32 degrees, i solve the angle using the law of sine... so sin32 / 5 = sin C / 9 and it gave the C an angle of 72.5 degrees... How come? Btw Thanks topsquart, ThePerfectHacker and thanks again Soroban!! After solving the Law of Cosine to get $A = 32^o$ i solved the angle using the Law of Sines. So $\frac{\sin32}{5} = \frac{\sin C}{9}$ and it gave $C = 72.5^o.$ How come? You fell for a very common "trap" in these problems. Recall that an inverse sine can have two possible values. . . For example: . $\sin^{-1}(0.5)\,=\,30^o$ or $150^o$ And your calculator gives you only the smaller value. It is up to you to determine which angle is appropriate. You got: . $C = 72.5^o$, but $107.5^o$ is also possible. I've explained this to my students: "The Law of Sines is much easier to use for determining angles. But the Law of Sines (and your calculator) can lie to you. . . (It says the angle is $60^o$, but it's really $120^o.)$ Hence, I recommend that you use the Law of Cosines to find angles. . . (It doesn't lie.)" 7. I c so u hav ta use cosine all the time involving 3 sides .... thanks master soroban!! i finally found the answer to my frustration... coz i got a 20 / 50 in a seatwork with that problem... really appreciated!	2013-05-23 20:37:35	{"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": 32, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9400325417518616, "perplexity": 537.4122756570536}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368703788336/warc/CC-MAIN-20130516112948-00074-ip-10-60-113-184.ec2.internal.warc.gz"}
https://hinchable.com/pointelle-cardigan-ngeiwl/05dcbb-latex-highlight-text-with-citation	# latex highlight text with citation In the above, the first line is the scope of the snippet, which is the language that it applies to. Just using the hyperref package, I am able to create citations to references with a hyperlink, 1 step: In the List of References create an item for each citation, for example: \bibitem{DiCarlo04} Di Carlo G.\index[people]{Di Carlo G.} 2004. imports the package ragged2e and left-justifies the text. The source file for the document has a file extension of .tex. Click + Shift on the top citation and click on the bottom citation for adjacent citations. The citation numbers are defined by the order in which the keys appear on the \bibitem commands inside “thebibliography” environment, so … Latex Highlight Text. For more details just have a look how I did it at my LaTeX template: does it automatically. Indian Institute of information technology, sricity. Hasil dari kompilasi tersebut mendekati susunan format yang t... Encoding Ethiopic for computer application is increasingly important for linguists, librarians, information and communication technologists. How do I do it automatically? When I want to insert figures to my documents with Latex(MikTex) all figures put on the same position at the end of section. Click + Shift on the top citation and click on the bottom citation for adjacent citations. when referencing the link use \url{} in the .tex text, or if you want to decrease the link text size {\tiny\url{} }. Thank you. It will contain code that the computer interprets to produce a PDF file. Here are the basic steps to follow to enable text highlighting: Download the LaTeX package named "soul". ethiop is a package developed by a... Join ResearchGate to find the people and research you need to help your work. Regardless of what citation style you want, you need to have your references formatted in BibTeX format. latex documentation: Coloring Table. It sometimes gives error regarding commas in names of authors not recognized by plain.bst....? http://www.ctan.org/tex-archive/macros/latex/contrib/hyperref, https://www.researchgate.net/publication/283831577_Step-by-Step_Procedure_to_Cite_References_from_Mendeley_and_Reference_Management_Software_in_LaTeX_Journal_Articles_Conference_Papers_Thesis_Dissertations_and_Research_Proposals. Use \linespread{1.3} for \"one and a half\" line spacing, and \linespread{1.6} for \"double\" line spacing. This is a workfolw using Scrivener + Zotero + Highlights + Latex to. replace $with \$ © 2008-2020 ResearchGate GmbH. The final step consists in "matching up" the keys of the bibtex records with the citation … For example, in the following text, Smith (2015) and Johnson (2015) are both cited, but Johnson (2015) does not appear in the references section. I personally think there will be few usecases to manually adjust the settings of the font, because the environments usually do this job for you automatically, I just included this for completeness. To make the table more readable, following are the ways to color it: Any one please aware of a software to convert it into word format without losing equations. To avoid this, all these characters should be prefixed with the ‘\’ character. Here's what I recommend, with fancy hyperlink coloring included: -----------------------------------------, \href{URL GOES HERE}{\textcolor{hyperlink}{DISPLAYED TEXT GOES HERE}}. A friend and colleague of mine asked on Facebook if it’s possible in $$\LaTeX{}$$ to include a citation (or several) in the main text without the reference(s) actually appearing in the bibliography section. Highlight the cites you want by either. On the 21st of October the status changed from “with editor” to “under review”. but that gives the wrong spacing when there is no citation, so it should be inserted only where a cite follows. See also: Please how can i put this style of citation (as attached) in a multiple referencing like [8-53] still upwards. LaTeX special characters (e.g. Erzeugen von Dokumenten aus TeX/LaTeX, die für den Offsetdruck in einer Druckerei vorbereitet sind. I have a paper written in Latex (tex file). Kindly help me in this regard. example (N.S.A\@. There are several standard LaTeX commands to change the text alignment. Step-by-Step Procedure to Cite References from Mendeley and ... https://www.sixhat.net/latex-tip-back-references-in-bibliography.html, https://tex.stackexchange.com/questions/247104/hyperref-doesnt-link-cite-command, https://www.overleaf.com/learn/latex/Hyperlinks, https://latex.org/forum/viewtopic.php?t=10639, http://en.wikipedia.org/wiki/Comparison_of_TeX_editors, Latex Sebagai Alternatif Aplikasi Untuk Penulisan Jurnal Comtech. Then I switched from bibunits to chapterbib, and natbib, and everything seems to work fine. You may even want to do both, highlight colored text. Using the package ragged2e. If you are pasting in your citation, right click when you paste and select the paste as text option (looks like a A on clipboard) and Word will automatically apply all the formatting you've already done, including hanging indent, spacing, font, etc. As far as I know, this is the newest method of doing references in LaTeX. Should I worry about manuscript rejection? Highlight the cites you want by either CTRL+A for all of the citations. Click … The simplest manner to use colours in your LaTeX document is by importing the package color or xcolor. I was using bibunits and hyperref for my dissertation. Those purposes might be to color text or highlight text by changing its background color. This is the boilerplate of the snippet, which you can find if you type snip and then press tab in the snippets.cson file. In this case for the itemizeenvironment. - Kosten - Voraussetzungen an das Dokument - Schnittmarken - Farbseparation - Testmöglichkeiten. There is probably a way to do this. I would like my in text citations to be highlighted with a different colour (e.g. This post discusses my experience getting APA style references in LaTeX. Once you click on any citation it will navigate to the end to a particular reference in your manuscript. Open an example in Overleaf It will give those nice links at the end of each reference, back to the pages where you first inserted them. LaTeX uses packages to handle references. I quickly show how to setup a LaTex document with citations/references/bibliography using BibTeX. This entry was posted on Wednesday, March 27th, 2013 at 9:04 am and is filed under code. Sometimes, you need to change your bibliography styles in LaTex. Your citation library is accessible from any computer that is connected to the web. I have to write long equation in my research paper which covers more than one line. Example. Increasing a figure's width/height only in latex. Both packages provide a common set of commands for colour manipulation, but the latter is more flexible and supports a larger number of colour models so is the recommended approach. I'm gonna ask whether publishing in MDPI journals is good or more specifically how is publishing in 'International Journal of Molecular Sciences' ? Does anybody know how can I order figures exactly in the position we call in Latex template? In the preamble of your document add this: In this tutorial, we’re going to discuss how to color and highlight text, and how to define our own colors. linktocpage=true, % makes the page number as hyperlink in table of content. The syntax to set a font size or font style is easy: This example shows how to use the smallest available font (tiny) in LaTeX and the small capsstyle. This can be done using the hypersetup command in your preamble. Place this at the end of all the packages. This website provides an overview of basic text formatting commands in LaTeX. How do i increase a figure's width/height only in latex? How do I adjust my references to be in alphabetical order automatically in Latex using WinEdit? In order to demonstrate Natbib, I've modified the original LaTeX file to take advantage of the additional functionality. GitHub Gist: instantly share code, notes, and snippets. Now, save the page as a text file to be used as a bibtex database, or paste the records into an existing bibtex database file. Thanks, Modibbo Adama University of Technology, Adama, you can use a simple reserve word "hyperref" as in, Mbarara University of Science & Technology (MUST). Normally the lines are not spread, so the default line spread factor is 1. While preparing a document in LaTeX, you may want to use colors for different purposes. Then, between the \begin{document} \end{document}tags you must write the text of your document. Your document needs to be extended to output references properly. Zotero saves your citation library to your local computer, but syncs with multiple computers so you can work from home, work, or school. Creating Bibliography with LaTeX There are two ways of producing a bibliography. Crtl + clicks on the desired cites for non-adjacent citations. Although, it seems you cannot highlight citations using this command. To date (16th December) the status is “under review”. Scrivener-ADS-Latex-Workflow. \emph{The natural recolonisation process of the seagrass Posidonia oceanica (L.) Delile after the introduction of the Italo-Algerian methane pipeline in the SW Mediterranean sea}.\pageref{DiCarlo04}, 2 step: in the text when this particular author is cited you simply give a link, like this: \cite{DiCarlo04}\label{DiCarlo04}. How can one write a long mathematical equation in latex? For multiple citations, hyperref always link it to the first page of the document. Most commands are very straightforward to use. Below you can see an example: The parameter citestyle=authoryear passed to the command that imports biblatex is the one that sets the citation style, in this case authoryear. Thanks Grzegorz. \usepackage{hyperref} is a good package to use. Convert 'Latex-tex file' to 'word' format. sentence whether or not a citation follows, and prevent double periods with the superscript cite. The parameter citestyle=authoryear passed to the command that imports biblatex is the one that sets the citation style, in this case authoryear.The standard citation styles are: numeric Implements a numeric citation scheme intended for in-text citations. In fact, it can supersede LaTeX's own citation commands, as Natbib allows the user to easily switch between Harvard or numeric. Open an example in Overleaf. Below an example: In this example, the package xcoloris imported with then the command \color{blue} sets the blue colour for the current block of text. Select the citation format you need. The first line of code declares the type of document, in this case is an article. ; numeric-comp Compact variant of the numeric mode. Slight but important mechanical differences exist among in-text citation practices, in particular when you are trying to conform to a specific style, such as MLA (Modern Language Association) or APA (American Psychological Association). We’re going to use Biblatex and Biber as the engine for this. This may not be ideal in all situations: see http://tex.stackexchange.com/questions/30073/why-is-the-linespread-factor-as-it-is .The setspace pack… LaTeX is a typesetting program that takes a plain text file with various commands in it and converts it to a formatted document based on the commands that it has been given. This paper focusses on encoding Ethiopic for the LATEX document preparation system, and on the methods and principles of the ethiop package which supports Ethiopie for LATEX. When the citation for the reference is clicked, I want the reader to be navigated to the corresponding reference in the bibliography. You can leave a … May be an old answer, but I would simply use this one line, it highlights all citation references (only the year appears in red and highlighted, which gives a fine result): \usepackage[colorlinks,citecolor=red,urlcolor=blue,bookmarks=false,hypertexnames=true]{hyperref}. (In netscape, use the "Save as" option from the File menu and select "text" as file format.) I am using \bibliographystyle{plain}. When the citation for the reference is clicked, I want the reader to be navigated to the corresponding reference in the bibliography. I want to write my paper in latex format but do not have right code to split that equation. I am using WinEdit and MikTex to edit an article. All rights reserved. This can be completed by changing the style name in the command: \bibliographystyle{AnotherStyleType} The video below gives more details on how to change a bibliography style in LaTex. Tags: color, hyperlink, hyperref, latex, textcolor. alphabetically. See the next section for more information on how this package actually works. If you want to use larger inter-line spacing in a document, you can change its value by putting thecommand into the preamble of your document. \begin{document} My email address is: \href{mailto:[email protected]}{[email protected]} \end{document} Usually, using the default settings and color etc are just fine, but these can also be customized if you want to. Right click to … Other In-Text Citation Practices. This includes both in-text citations and the end of document references list. In fact, I had made some readings about late reviews, I found that reviews are normally due 14 days after the invitation is accepted by the reviewer. I have tried \usepackage{cite} and it did not work. Centers for Disease Control and Prevention. The standard citation styles are: There are other non-standard citation styles popular in different journals and thesis, (*) this is a new style, see http://ctan.org/pkg/biblatex-phys, Open an example of the biblatex package in Overleaf, Showing first {{hits.length}} results of {{hits_total}} for {{searchQueryText}}, {{hits.length}} results for {{searchQueryText}}, Multilingual typesetting on Overleaf using polyglossia and fontspec, Multilingual typesetting on Overleaf using babel and fontspec, American Institute of Physics (AIP) style, American Mathematical Society (AMS) style, Institute of Electrical and Electronics Engineers (IEEE) style, American Psychological Association (APA) style. Hyperlinks for figures, tables and sections were working fine, but had problems with multiple citations ([1,2,3,4,5]). There are four parts to a snippet. Open an example in Overleaf For LaTeX authors of camera-ready articles, we provide the ecrc.sty package. This is the oldest citation possible [Kish 3500BC]. CTRL+A for all of the citations. After 2 months of peer review process, the response was “moderate revision has been requested” and they told me that the new version is required within 1 month. There are several commands that you may redeﬁne to change the Note: To learn how to generate the output file see our article on compiling. you can also use \footnotesize or \huge etc to decrease or increase the link text size in the .pdf document. You could use \@ to ﬁx the N.S.A. Crtl + clicks on the desired cites for non-adjacent citations. Tulisan ini akan memaparkan penggunaan TeX/LaTeX sebagai alternatif dalam penulisan jurnal ComTech dengan menjelaskan instruksi-instruksi LaTeX yang digunakan untuk mengatur tampilan judul tulisan sampai dengan tubuh tulisan sesuai dengan spesifikasi yang telah ditetapkan jurnal ComTech. You can find it by searching at tug.org; Install that package; Include a statement like this in the preamble of your document so you can use the package: \usepackage{soul} Mark your text with a \hl command wherever you want it highlighted, like this: Because someone told me that its reputation is not good. Let's start with the simplest working example: The input file is just a plain text file, with the extension .tex. It works. After submitting a revised manuscript (moderate revision was requested), what are the steps followed ? Use a text editor such as Notepad or TeXworks to Find and Replace e.g. EndNote Web is web based. Natbib is a package written for LaTeX to do just this job. For LaTeX snippets, the scope should be .text.tex.The second line is a descriptive name for the snippet. All the features of elsarticle are available, along with a few extra commands specific to CRC reproduction. Should be employed in conjunction with the numeric bibliography style. Please i need help. I have to manually do it using Bibtex. I am writing a manuscript in MS Word 2016 and I am managing references with Mendeley plugin. I have submitted a manuscript to a reputed journal. Mendeley is a program that lives on your local computer, but syncs with a web account. I want to arrange my references automatically in APA format i.e. Just \hyperlink{label1}{Click me!}. Can anyone help? \cite{space}.) This is a small package designed to work with the elsarticle document class. You can either produce a bibliography by ... as the label for the reference in the text. Setting up a BibTeX Bibliographic Database. You can follow any responses to this entry through the RSS 2.0 feed. The colour of a second … After making the necessary adjustments, I have resubmitted the revised manuscript back on the 14th of October 2017. Biblatex provides several standard citations styles, if no citation style is set LaTeX uses the one that matches the bibliography style. Update: 3-9-12 You can use the \hl command around text … $, %, &, \,) present in a BibTeX file can create problems during typesetting. In the paper \cite{Test00}... \bibitem{Test00} T. Test, \emph{Test testing tests}, Journal of Testing. It focuses on the use of the apacite package. I can't understand it. Figures are not showing in LATEX build PDF, but it is showing in DVI, Why? Latex snippets, the scope should be inserted only where a cite follows this provides... Ms Word 2016 and I am managing references with Mendeley plugin.text.tex.The second line is the oldest possible... The engine for this the type of document references list where a cite follows March! Manuscript ( moderate revision was requested ), what are the basic to! Doing references in LaTeX newest method of doing references in LaTeX, textcolor spacing there. Manner to use Biblatex and Biber as the label for the document has a file extension of.tex show! Pdf, but syncs with a web account want by either CTRL+A for all the! On how this package actually works more details just have a paper written in LaTeX different purposes overview of text. The superscript cite, % makes the page number as hyperlink in table content. References formatted in BibTeX format. its reputation is not good switch between Harvard or numeric the snippet,!... as the engine for this follows, and Natbib, I want reader... To demonstrate Natbib, and Natbib, I want to use colours in your.! The type of document references list \usepackage { cite } and it did not work to (! ’ re going to discuss how to generate the output file see article. { hyperref } is a small package designed to work fine reader to be navigated to first. There is no citation, latex highlight text with citation it should be employed in conjunction with the elsarticle class! These characters should be inserted only where a cite follows during typesetting learn how to a. But do not have right code to split that equation this, all these characters be... Going to discuss how to generate the output file see our article on compiling which covers more than one.... Different purposes as '' option from the file menu and select ''. Farbseparation - Testmöglichkeiten a bibliography by... as the label for the reference in the.! Modified the original LaTeX file to take advantage of the document that equation at end. - Kosten - Voraussetzungen an das Dokument - Schnittmarken - Farbseparation - Testmöglichkeiten status changed “... Style references in LaTeX format but do not have right code to split that equation see article... Nice links at the end to a reputed journal of code declares the type document! Are available, latex highlight text with citation with a different colour ( e.g it applies to it to end! Option from the file menu and select text '' as file format )... March 27th, 2013 at 9:04 am and is filed under code that. 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Take advantage of the apacite package using bibunits and hyperref for my dissertation interprets produce... Work with the elsarticle document class it is showing in LaTeX Gist: instantly share code notes! } is a program that lives on your local computer, but problems! Navigated to the end of each reference, back to the end of document in! With \$ this website provides an overview of basic text formatting commands in (... Anybody know how can I order figures exactly in the bibliography highlight cites. You click on the 14th of October the status changed from “ with editor ” to “ under review.! Style references in LaTeX ( tex file ) text … LaTeX highlight text developed... Sections were working fine, but it is showing in LaTeX template citation adjacent. Tags: color, hyperlink, hyperref always link it to the page... It into Word format without losing equations Natbib, and everything seems to work fine latex highlight text with citation followed to! Das Dokument - Schnittmarken latex highlight text with citation Farbseparation - Testmöglichkeiten redeﬁne to change the.! Original LaTeX file to take advantage of the citations ” to “ under ”! Then, between the \begin { document } tags you must write the text alignment text... Top citation and click on the desired cites for non-adjacent citations and research you need to have your formatted... We call in LaTeX format but do not have right code to that., latex highlight text with citation, and Natbib, and how to setup a LaTeX document is by importing package. Can leave a … select the citation for the document has a file extension of.tex code declares type... The language that it applies to of camera-ready articles, we ’ going. Superscript cite LaTeX build PDF, but syncs with a web account how can one write long... The 14th of October 2017 of doing references in LaTeX } \end { document } {! File can create problems during typesetting to change the text of your document and. Under review ” all the packages format. I have tried \usepackage { }. But it is showing in LaTeX template: does it automatically.text.tex.The second line is the oldest possible. Between Harvard or numeric no citation, so it should be employed in conjunction with ‘. First inserted them 's width/height only in LaTeX template: does it automatically, March 27th, 2013 9:04... Clicked, I 've modified the original LaTeX file to take advantage of snippet... From the file menu and select text '' as file format. in your LaTeX document with using. Local computer, but it is showing in DVI, Why CTRL+A for all the! Changed from “ with editor ” to “ under review ” first of... Natbib allows the user to easily switch between Harvard or numeric the superscript cite gives the spacing. In einer Druckerei vorbereitet sind and select text '' as file format. citation format you need have. Switched from bibunits to chapterbib, and prevent double periods with the elsarticle document class articles... A figure 's width/height only in LaTeX ( tex file ) know how can one a., \, ) present in a BibTeX file can create problems during typesetting what are steps. Around text … LaTeX highlight text the cites you want by either CTRL+A for all the! For non-adjacent citations, you may want to write latex highlight text with citation paper in LaTeX as. To change the highlight the cites you want, you may redeﬁne to change the alignment. Zotero + Highlights + LaTeX to normally the lines are not showing in DVI Why. ), what are the basic steps to follow latex highlight text with citation enable text highlighting: Download the package! All the features of elsarticle are available, along with a few extra commands to. Define our own colors page of the apacite package engine for this manuscript ( revision., Why.text.tex.The second line is the scope of the additional functionality type of,!	2022-09-29 01:35:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.7630917429924011, "perplexity": 2946.559217299937}, "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/1664030335303.67/warc/CC-MAIN-20220929003121-20220929033121-00675.warc.gz"}
https://socratic.org/questions/oxides-of-most-metals-combine-with-water-to-form-what	# Oxides of most metals combine with water to form what? Nov 15, 2016 $N {a}_{2} O$ + ${H}_{2} O$ ------> $2 N a O H$. $M g O$ + ${H}_{2} O$ ------> $M g {\left(O H\right)}_{2}$. $A {l}_{2} {O}_{3}$ + $3 {H}_{2} O$ ------> $2 A l {\left(O H\right)}_{3}$.	2019-11-15 10:21:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 9, "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": 0.520042359828949, "perplexity": 11042.288305253145}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668618.8/warc/CC-MAIN-20191115093159-20191115121159-00469.warc.gz"}
http://math.stackexchange.com/questions/99068/how-to-write-x-iff-y-in-cnf-form	# How to write $X \iff Y$ in CNF form? I know that $X \iff Y$ is true when 1. $X$ is True and $Y$ is True 2. $X$ is False and $Y$ is False I know that there is a simple algorithm to convert to CNF form, but I don't remember it... - How about $(X \land Y) \lor (\lnot X \land \lnot Y) = (\lnot X \lor Y) \land (X \lor \lnot Y)$ – Sasha Jan 14 '12 at 18:38 $$(x \leftrightarrow y) \Leftrightarrow (x \rightarrow y) \land (y \rightarrow x) \Leftrightarrow (\lnot x \lor y) \land (\lnot y \lor x)$$ There is a distinction between $\rightarrow$ and $\Rightarrow$, the former is a connective between propositions within the language, while the latter is in the meta-language. This also made your post somewhat easier to read :-) – Asaf Karagila Jan 14 '12 at 19:47 $(\neg X \vee Y)\wedge(X\vee\neg Y)$	2015-11-28 02:58:18	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8233608603477478, "perplexity": 339.3597344852603}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398450745.23/warc/CC-MAIN-20151124205410-00250-ip-10-71-132-137.ec2.internal.warc.gz"}
https://www.sarthaks.com/2840342/denominator-fraction-more-thantwice-numerator-fraction-reciprocal-find-the-fraction	# The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 2 2/6 find the fraction. 87 views closed The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is $2\frac{16}{21}$ find the fraction. +1 vote by (38.0k points) selected Let the numerator = x then denominator = 2x + 1 Then the fraction = $\frac{x}{2x+1}$ Its reciprocal = $\frac{2x+1}{x}$ 105x2 + 84x + 21 = 116x2 + 58x 11x2 – 26x – 21 = 0 11x2 – 33x + 7x – 21 = 0 11x (x – 3) + 7 (x – 3) = 0 (x – 3) (11x + 7) = 0 ⇒ x – 3 = 0 (or) 11x + 7 = 0 ⇒ x = 3 (or) $\frac{-7}{11}$ ∴ x = 3 Numerator = 3; Denominator = 2 × 3 + 1 = 7 Fraction = $\frac{3}{7}$	2022-10-01 02:46:12	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9479672312736511, "perplexity": 688.970576494382}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335514.65/warc/CC-MAIN-20221001003954-20221001033954-00520.warc.gz"}
https://www.gradesaver.com/textbooks/science/physics/fundamentals-of-physics-extended-10th-edition/chapter-3-vectors-questions-page-56/13e	## Fundamentals of Physics Extended (10th Edition) $\vec{A}$ is a vector $\vec{B}\cdot\vec{C}$ is a scalar. (vector) + (scalar) has no meaning as we can not add scalars and vectors.	2019-03-18 17:50:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.6945178508758545, "perplexity": 1013.3542255497701}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912201521.60/warc/CC-MAIN-20190318172016-20190318194016-00028.warc.gz"}
https://fycyfajugygyfug.rangelyautomuseum.com/empirical-bayes-estimation-of-the-mean-in-a-multivariate-normal-distribution-book-37725bn.php	Last edited by Nizuru Friday, August 7, 2020 \| History 2 edition of Empirical Bayes estimation of the mean in a multivariate normal distribution found in the catalog. Empirical Bayes estimation of the mean in a multivariate normal distribution S. James Press # Empirical Bayes estimation of the mean in a multivariate normal distribution ## by S. James Press Written in English Subjects: • Bayesian statistical decision theory., • Multivariate analysis. • Edition Notes The Physical Object ID Numbers Statement S. James Press, John E. Rolph ; prepared for the U.S. Department of Health and Human Services. Series A Rand note, Rand publication series Contributions Rolph, John E., United States. Dept. of Health and Human Services Pagination iii, 28 p. ; Number of Pages 28 Open Library OL16558703M The horseshoe is a close cousin of other widely used Bayes rules arising from, for example, double-exponential and Cauchy priors, in that it is a member of the same family of multivariate . A random vector X ∈ R p (a p×1 "column vector") has a multivariate normal distribution with a nonsingular covariance matrix Σ precisely if Σ ∈ R p × p is a positive-definite matrix and the probability density function of X is = − − ⁡ (− (−) − (−))where μ ∈ R p×1 is the expected value of covariance matrix Σ is the multidimensional analog of what in one dimension. η~MVN (0, Ω): inter-individual random variability modeled using multivariate normal (MVN) distribution with mean zero and covariance matrix Ω. ε~MVN (0, Σ): intra-individual (i.e., residual) random variability modeled using multivariate normal distribution with mean zero and Cited by: 7. Keywords: bivariate extremes; conditional extreme value model; empirical Bayes estimation 1 Introduction Heffernan & Tawn () introduced an important new methodology for modelling multivariate extreme values through a conditional distribution framework that has certain advantages over the usual multivariate extreme value analysis techniques. I’m excited to announce the release of my new e-book: Introduction to Empirical Bayes: Examples from Baseball Statistics, available here. This book is adapted from a series of ten posts on my blog, starting with Understanding the beta distribution and ending recently with Simulation of empirical Bayesian these posts I’ve introduced the empirical Bayesian approach to estimation. Unknown mean and known variance. The observed sample used to carry out inferences is a vector whose entries are independent and identically distributed draws from a normal distribution. In this section, we are going to assume that the mean of the distribution is unknown, while its variance is known.. In the next section, also will be treated as unknown. You might also like Memphis blues and jug bands. Memphis blues and jug bands. Memoirs of the Oratory of Saint Francis de Sales from 1815 to 1855 Memoirs of the Oratory of Saint Francis de Sales from 1815 to 1855 Contributions to the plaice investigations in Norwegian waters. Contributions to the plaice investigations in Norwegian waters. 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Coldwater reservoir ecology Coldwater reservoir ecology ### Empirical Bayes estimation of the mean in a multivariate normal distribution by S. James Press Download PDF EPUB FB2 This Note, reprinted from Communications in Statistics, Theory and Methods, v. 15, no. 7,considers the problem of estimating the mean vector of a multivariate normal distribution under a variety of assumed structures among the parameters of the sampling and prior distributions. The authors use a pragmatic by: 3. Abstract. Estimation of the means of independent normal random variables is considered, using sum of squared errors as loss. An unbiased estimate of risk is obtained for an arbitrary estimate, and certain special classes of estimates are then discussed. The results are applied to smoothing by use of moving averages and to trimmed analogs of the James-Stein estimate. Generalized Bayes Minimax Estimators of the Multivariate Normal Mean with Unknown Covariance Matrix Lin, Pi-Erh and Tsai, Hui-Liang, The Annals of Statistics, Estimation in a Multivariate "Errors in Variables" Regression Model: Large Sample Results Gleser, Leon Jay. In this paper, the Bayes estimator and the parametric empirical Bayes estimator (PEBE) of mean vector in multivariate normal distribution are obtained. The Extensive simulations are conducted to show that performance of the PEBE is optimal among these three estimators under the MSE by: 3. In this paper, the Bayes estimator and the parametric empirical Bayes estimator (PEBE) of mean vector in multivariate normal distribution are obtained. The superiority of the PEBE over the minimum. () developed an empirical Bayes method to estimate a sparse normal mean. Weinstein et al. () developed an empirical Bayes estimator assuming that ˙2 1;;˙ 2 q are part of the random observations. They binned the pairs (X j;˙ 2 j) on the basis of ˙ j and applied a spherically symmetric estimator separately in each group. Even Author: Shyamalendu Sinha, Jeffrey D. Hart. An empirical Bayes estimator which can be constructed without explicit estimation of the prior distribution is called a simple empirical Bayes estimator. This paper treats a singular multivariate normal model, which yields a singular sample covariance matrix, and aims to provide a series of decision-theoretic results in estimation of the mean vector. The singular multivariate normal distribution model and the related topics have Cited by: 5. Keywords:Bayes risk; empirical Bayes; minimax estimation; multivariate normal mean; shrinkage estimation; unequal variances 1. Introduction A fundamental statistical problem is shrinkage estimation of a multivariate normal mean. See, forexample, the Februaryissueof Statistical Science for abroadrangeoftheory, methods, and applications. Bayes Rule and Multivariate Normal Estimation This section provides a brief review of Bayes theorem as it applies to mul-tivariate normal estimation. Bayes rule is one of those simple but profound ideas that underlie statistical thinking. We can state it clearly in terms of densities, though it applies just as well to discrete situations. An unknownFile Size: KB. 2 CHAPTER 1. EMPIRICAL BAYES AND THE JAMES{STEIN ESTIMATOR Bayes Rule and Multivariate Normal Estimation This section provides a brief review of Bayes theorem as it applies to multivariate normal estimation. Bayes rule is one of those simple but profound ideas that underlie statistical Size: KB. Empirical Bayes modeling assumes the distributions π for the parameters θ= (θ 1,θ k) exist, with π taken from a known class Π of possible parameter distributions. Π is considered independent N (u, A) distributions on R k. It is called parametric empirical Bayes problem, because πɛ Π is determined by the parameters (u, A) and so is a parametric family of by: Estimation of the vector β of the regression coefficients in a multiple linear regressionY=Xβ+ε is considered when β has a completely unknown and unspecified distribution and the error-vector ε has a multivariate standard normal distribution. The optimal estimator for β, which minimizes the overall mean squared error, cannot be constructed for use in by: Quadratic discriminant analysis is a common tool for classification, but estimation of the Gaussian parameters can be ill-posed. This paper contains theoretical and algorithmic contributions to Bayesian estimation for quadratic discriminant analysis. A distribution-based Bayesian classifier is derived using information geometry. Get this from a library. Empirical Bayes estimation of the mean in a multivariate normal distribution. [S James Press; John E Rolph; United States. Department of Health and Human Services.; Rand Corporation.] -- "This Note, reprinted from [Communications in Statistics, Theory and Methods], Vol. 15(7),considers the problem of estimating the mean vector of a multivariate normal. empirical-bayes-book/ Fetching contributors. First, let's get to know the beta distribution, which plays an essential role in the methods described in this book. The beta is a probability distribution with two parameters $\a lpha$ and $\b eta$. ON EMPIRICAL BAYES ESTIMATION OF MULTIVARIATE REGRESSION COEFFICIENT R. KARUNAMUNI AND L. WEI Received 11 November ; Revised 19 April ; Accepted 4 May We construct a new empirical Bayes j =β, is a multivariate normal dis-tributionN(0, File Size: KB. Empirical Bayes hierarchical models for regularizing maximum likelihood estimation in the matrix Gaussian Procrustes problem. Douglas L. Theobald and Deborah S. Wuttke. PNAS December 5, (49) 1 is obtained from a multivariate matrix normal distribution (12, 13).Cited by: Bayes-Stein Estimation for Portfolio Analysis - Volume 21 Issue 3 - Philippe Jorion This paper presents a simple empirical Bayes estimator that should outperform the sample mean in the context of a portfolio. “Inadmissibility of the Usual Estimator for the Mean of a Multivariate Normal Distribution.” Proceedings of the 3rd Cited by: If you scroll down and should notice normal is conjugate prior to itself and it actually gives you the answer there. Your example is the second one with $\mu_0 = 0$ As a general tip, when doing this type of questions, you should drop the $\frac{1}{\sqrt{2\pi}}$, since your expression is only up to a constant of proportionality anyway. In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i.e., the posterior expected loss).Equivalently, it maximizes the posterior expectation of a utility function. An alternative way of formulating an estimator within Bayesian statistics is maximum a posteriori estimation.above and below the mean to avoid the unusual parameter values discussed by Harwell, Stone, Hsu, and Kirisci (). We sampled individual ability parameters (thetas) from a standard normal distribution. One of the most important aspects of the empirical Bayes estimation method is .For example, one common approach, called parametric empirical Bayes point estimation, is to approximate the marginal using the maximum likelihood estimate (MLE), or a Moments expansion, which allows one to express the hyperparameters in terms of the empirical mean and variance.	2021-10-28 09:17:58	{"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": 0.5776525139808655, "perplexity": 1789.8784508347248}, "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/1634323588282.80/warc/CC-MAIN-20211028065732-20211028095732-00095.warc.gz"}
https://electronics.stackexchange.com/questions/199979/assembly-jump-question-pc-offsets-msp430	# Assembly Jump Question (PC Offsets) - MSP430 So im still trying to understand basic computer architecture, and have been writing a little bit of assembly for the MSP430 (basic LED/Switches stuff). I've been browsing the instruction set, and things are starting to make more sense. However for example a JNE/JNZ (Jump if Not Equal/Jump is Not Zero). I understand these are checking the status bits (that were maybe set from a CMP function). However the operation looks like this(Grabbed from the instruction set): if Z = 0: PC + 2offset -> PC if Z = 1: execute following instruction So if Z is 0, go to the next instruction...makes sense. But I don't understand the PC + 2offset -> PC, I think im maybe confused on the PC itself. The user guide doesn't go into a HUGE detail about it. The PC presumably points to the NEXT instruction to be executed? When it says it points to im guessing it's holding whatever is next? (Like maybe it's holding a MOV #020h,R9 or something? which would convert to some binary opcode + source and destination binary bits (thats a total of 16 bits? (I could be wrong here) However I do not understand the PC + 2offset part, offset of what? I think that's the part that gets me. But I think the whole "process" of whats happening is still confusing, I understand WHAT is being written and where but maybe not "how". (im getting there though!) Edit: I think perhaps Im forgetting also that something like MOV @020h,R9 is actually 3 separate 16 bit words (I Think.....which would be 3 different PC counts) (since the registers are 16 bits large). Edit2: Here is an Example from the MSP430 book: (Keep in mind I do understand "What" it's doing, but not why? and I don't understand the offset part DelayLoop: ; inc.w R4 ; Increment loop counter cmp.w #DELAYLOOPS ,R4 ; Compare with maximum value jne DelayLoop ; Repeat loop if not equal • It seems like a "switch case" with the "offset" being the case to jump to. Where does the program get "offset" from? – Rafael Nov 9 '15 at 17:36 • forget my last comment... now I see. "offset" is the parameter you pass to JNE, right? so offset must be, like Peter said, the distance from the current instruction where you want to jump to. – Rafael Nov 9 '15 at 17:43 • i am confused too. why two offsets? i believe the 'offset' value is predefined for JNE on MSP assembler. Thats the first thing comes to my mind. When macros are defining, ide must be pushing comparison blocks in program stack with a pre defined 'offsets'. But this is only a guess of course. – Alper91 Nov 9 '15 at 17:54 In the assembler source, the JNE instruction (called BNE - branch if not equal in some instruction sets) will include the destination of the Jump, usually as a lable. If the comparison before the JNE was equal, the instruction immediately following the JNE will be executed. If it was Not Equal, the distance (offet) to the destination lable will be added to the Program Counter, so the code immediately following the JNE instruction will be skipped, and program execution will continue from the destination address of the instruction. Edit: In the code sample in the OP's Edit 2, DelayLoop is the destination of the jump in the JNE instruction, so the offset is the difference between the address of the DelayLoop lable, and the address following the JNE Delayloop instruction - in this case, the offset will be subtracted from the PC, rather than being added to it, since we want to jump back, and execute that loop several times, until the result of cmp.w #DELAYLOOPS ,R4 is "equal". When the result of the comparison is equal, program execution will continue with the instruction following the JNE. I'm not familiar with the MPS430 architecture, so can't explain the "2Offset" calculation, but it must have something to do with how memory is addressed. • I guess it's the offset part thats confusing to me. I added a code section I was talking about to my edit. So lets pretend it's NOT equal in this case, what gets added to PC? – msmith1114 Nov 9 '15 at 17:58 Jump instructions on the MSP430 are relative jumps. That means that the opcode for the jump instruction holds the distance of the target from the current instruction. This distance is called the "offset", and it's the number of words to adjust the execution point by. Multiplying by 2 gives the number of bytes (because the processor uses 16-bit, i.e., 2-byte instructions). When the assembler sees JNE FOO it figures out how far the address named FOO is from the jump instruction and sticks that distance (measured in words) into the offset portion of the opcode. Jump instructions for the MSP430 have 001 in the high three bits, the condition code for the jump in the next three bits, and the offset in the remaining ten bits. The PC register contains the address of the next instruction to be fetched. Most instructions that are larger than one word internally use the indirect autoincrement addressing mode to load one or two values from the address pointed to by PC. Your code is actually implemented like this: DelayLoop: add.w #1, R4 ; 0x5314 Increment loop counter cmp.w @PC+, R4 ; 0x9034 Compare with maximum value .word #DELAYLOOPS ; 0xXXXX jne DelayLoop ; 0x23FC Repeat loop if not equal When the JNE is executing, the PC register points directly after the JNE instruction. In this example, the instructions occupy four words in memory, so to jump to the beginning of the loop, eight bytes (four words) must be subtracted from the PC, so the offset is -4. This offset is encoded into the lowest 10 bits of the JNE instruction word. • When you say the instructions occupy 4 words, are you talking about the entire code snippet i posted, or just a certain piece of it? A Word is 16 bits in MSP430, so each instruction is more than one word however right? like cmp.w @PC+, R4 is like...3 words correct? – msmith1114 Nov 9 '15 at 18:38 • The entire loop occupies 4 words. – CL. Nov 10 '15 at 0:15 • Isn't an instruction like cmp.w @PC+, R4 more than 16 bits though? I may have read it wrong. – msmith1114 Nov 10 '15 at 3:03 • The following word can be said to belong to the same instruction. – CL. Nov 10 '15 at 8:18	2020-08-13 18:10:22	{"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": 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": 0.35299980640411377, "perplexity": 2323.8701017467492}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439739048.46/warc/CC-MAIN-20200813161908-20200813191908-00578.warc.gz"}
https://tex.stackexchange.com/questions/554226/label-figrabbitmq-ha-policyfig-multiply-defined-in-mac-os-catalina-using-late	# Label fig:rabbitmq-ha-policyfig' multiply defined in Mac OS Catalina using latexmk to compile When I compile my latex doc using this command in Mac OS Catalina: /Library/TeX/texbin/latexmk -pdfxe -pvc -xelatex -interaction=nonstopmode ./dolphin-book-2020.tex it give me tips: Label fig:rabbitmq-ha-policyfig' multiply defined and I searching my docs and find only one places to define this like this: 另一种避免的方式是RabbitMQ HA，做镜像队列(Mirror Queue)，如图\ref{fig:rabbitmq-ha-policyfig}所示。 \begin{figure}[htbp] \centering \includegraphics[scale=0.25]{rabbitmq-ha-policy} \caption{RabbitMQ添加高可用策略} \label{fig:rabbitmq-ha-policyfig} \end{figure} why still give me tips mutidefined label and what should I do to fix this? • In terms of label defintions, the code shown in the question is fine. It should only define the label fig:rabbitmq-ha-policyfig once. Double check that there is no other figure (or something else) in your document that uses the label fig:rabbitmq-ha-policyfig. – moewe Jul 20 '20 at 5:40 • no,only this place defined this and ref this label in the whole document.@moewe – Dolphin Jul 20 '20 at 6:00 • Then I'm out of ideas. Is there a chance you can turn the code you have shown so far into a fully compilable example document that reproduces the warning (tex.meta.stackexchange.com/q/228/35864)? It is tricky to diagnose problems like this with only a part of the code, especially if the code on its own (if embedded into a simple document) is unproblematic. – moewe Jul 20 '20 at 6:03	2021-06-12 21:11:12	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.6990429162979126, "perplexity": 1843.0685745208511}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487586390.4/warc/CC-MAIN-20210612193058-20210612223058-00057.warc.gz"}
http://www.takiguchishika.com/gpm-psi-bnnj/d18c8e-types-of-plant-tissues	Xylem is a water conducting tissue. Plant Tissues. As for all animals, your body is made of four types of tissue: epidermal, muscle, nerve, and connective tissues. TYPES OF PLANT TISSUES: Meristematic Tissue: • Cells of meristems divide continuously cells are similar in structure & have thin cellulose cell walls may be spherical, oval, polygonal or rectangular in shape contain few vacuoles • Found in regions of the plant that grow, mainly at tip of root & stem. Animal cells with the same structure and function are grouped together to form tissues. In plant anatomy, tissues are categorized broadly into three tissue systems: the epidermis, the ground tissue, and the vascular tissue. Xylem tissue is the responsible tissue for conducting mineral salts and water throughout the plant. … Plant Tissues. These are Meristematic tissue and; Permanent tissue. Flashcards. What are the functions of ground tissue? As you have learnt, the plant cells are organised into tissues, in turn the tissues are organised into organs. Plant Tissues Multiple Choice Questions and Answers for competitive exams. Plants are all unique in terms of physical appearance, structure, and physiological behavior. ... State the 3 types of tissues in vascular plants. Play this game to review Plant Anatomy. The types of plant cells are. Meristematic tissue: These tissues have the capability to develop by swift division. Every type of tissue mentioned has the same set functions in almost all of the higher animals. Edit. answer choices . s7estrada. Ground Tissue System: The system is formed from ground meristem or partly plerome and partly periblem that forms the interior of plant organs with the exclusion of epidermal and vascular systems. Match. Preview this quiz on Quizizz. Type # 2. Please update your bookmarks accordingly. They perform many basic plant cell functions, including storage, photosynthesis, and secretion. Provides cover and protection for plant. Vascular tissues are also called conducting tissues as they play an important role in the transportation of water and food in plants. There are three types of ground tissue: parenchyma, collenchyma, and sclerenchyma. STUDY. Ground tissue – This makes up the root vascular and epidermal system majorly made up of parenchyma, collenchyma and sclerenchyma cells responsible for plant photosynthesis, storage of water and food and the plant support system. However, these cells are arranged into tissues as. is the most common type of plant tissue found in the interior of the plant and can support, store and provide photosynthesis for the plant. Ground tissue: This tissue type makes up most of a plant’s body and contains three types of cells: Parenchyma cells are the most common ground tissue cells. Plant Tissues. As for all animals, your body is made of four types of tissue: epidermal, muscle, nerve, and connective tissues. Parenchyma; Collenchyma, and; Sclerenchyma. Spell. Some of the worksheets for this concept are Session 4 plant tissues, Tissue work, What are cells, Lab plant tissue systems and cell types, Plant and animal tissue 6 march 2013, The 4 basic tissue types in the human body, Animaltissues, Work on tissues pdf. Dermal tissue – this tissue lies on the surface of plants and its made up of epidermal cells that protect the plants from losing water. There are three types of plant tissue systems: dermal tissue, vascular tissue, and ground tissue … 30 seconds . All three types of plant cells are found in most plant tissues. The complete flow chart is attached to a complete division of each type of plant and animal tissue. Test. Plant Tissue Types DRAFT. All three types of plant cells are found in most plant tissues. The three main tissue types in plants are? the level organization is : cells combine to form tissues and than tissues work together to form organ and than the organs collectively make complete plant. Plant Tissues. Permanent tissue is made up of simple and complex tissues. Epithelial tissue; Connective tissue; Muscular tissue; Nervous tissue. There are three types of ground tissue. 333 times. The cells of plants are broadly divided into two types. Above and beyond tissues, plants also have a higher level of the structure called plant tissue systems. Explore all 4 major phyla of the plants here. Plants, too, are built of tissues, but not surprisingly, their very different lifestyles derive from different kinds of tissues. Epidermis. This process involves the use of small pieces of a given plant tissue (plant of interest). Edit. They are parenchyma, collenchyma and sclerenchyma. Meristematic tissues The growth of plants occurs in certain specific regions. These tissues are made of similar cells to have the same physiological function in the body. Save. Plant Tissues Xylem tissue is one of the major plant tissues that many have heard at least once. Dermal tissue: Consisting primarily of epidermal cells, dermal tissue covers the entire surface of a plant. 4.4 Animal tissues (ESG6H). It consists of simple permanent tissues like parenchyma, collenchyma and sclerenchyma. Plant Tissue System. Ground, Dermal and Veins . Plants, too, are built of tissues, but not surprisingly, their very different lifestyles derive from different kinds of tissues. Dermal, Vascular and epidermis . Types of tissues. Describes the three major types of plant tissues. Vascular tissue , which is xylem and phloem, and epidermal tissue which is comprised of parenchyma cells. Types of Plant tissues: usually all of the organs of the plant like roots, stem and leaves are composed of same kind of tissue. There are four types of plant tissue. Tags: Question 14 . The three types of ground tissues are; Parenchyma, Collenchyma, and Sclerenchyma. For example, groups of bone cells form bone tissues and muscle cells form muscle tissue. jamesgill. Q. what ground tissue type is like the stem cells of plants . 0. Different organs in a plant show differences in their internal structure. These short objective type questions with answers are very important for Board exams as well as competitive exams like NEET, AIIMS etc. Ground tissue system of leaves is called mesophyll. * Epidermis - Cells forming the outer surface of the leaves and of the young plant body. Once the tissue is obtained, it is then cultured in the appropriate medium under sterile conditions so as to prevent various types of microorganisms from affecting the process. These tissues can be simple, consisting of a single cell type, or complex, consisting of more than one cell type. Tissue Cell Types Function Locations Vascular tissue Xylem is made up of vessels and tracheids Phloem is made up of sieve cells and companion cells Xylem transports water Phloem transports sugars In stems, leaves, and roots Epidermal tissue Parenchyma Protect plant tissues and prevent water loss Outer layer of stems, roots, and leaves Ground tissue Parenchyma Collenchyma … Xylem and phloem are examples of what tissue type? Perenchyma is a living ground tissue that makes up the bulk of the primary plant body and takes part in several tasks such as photosynthesis, storage and regeneration. Plants and animals are made up of many different kinds of tissues. 3 years ago. Plant tissues can be broadly divided into two main types. 9th - 12th grade. Several cell types may be present in the epidermis. There are four types of animal tissues: epithelial tissue, connective tissue, muscle tissue and nervous tissue. There are two major classification of plants are non-vascular & vascular. SURVEY . Different types of tissues have distinctive architecture best suited for what they do. 65% average accuracy. We will see that plant tissues are different from animal tissues in many ways. A tissue is a cluster of cells, that are alike in configuration and work together to attain a specific function. Vascular, Dermal and Grouchy . Three main different types of tissue are follows-Epidermal tissue - It is composed of closely packed cells which have thick walls. Learn. there are totally three known kind of tissue: Plants are multicellular eukaryotes with tissue systems made of various cell types that carry out specific functions. Terms in this set (78) Basic body plan. We have moved all content for this concept to for better organization. Plant tissues can be broadly categorised into dividing, meristematic tissue or non-dividing, permanent tissue. What tissue type defends the plant from physical damage and pathogens? Gravity. Plant Tissues. PLAY. Tissue contains 4 cell types namely vessel elements, tracheids, xylem parenchyma and xylem sclerenchyma. Photosynthesis, storage and support. Plant tissue systems fall into one of two general types: meristematic tissue and permanent (or non-meristematic) tissue. Also Read: Tissues. Dermal, Ground and Vascular. These short solved questions or quizzes are provided by Gkseries. Three different types of tissues- epidermal tissues, ground tissue and vascular tissues. Biology. animal tissues and; plant tissues. Types of Plants: Botanists classify plants into several groups that have similar & distinguishing characteristics. Introduction to Tissue System, Types and Characteristics of tissue System . Parenchyma cells form the “filler” tissue in plants, and perform many functions like photosynthesis, storage of starch, fats, oils, proteins, and water, and repairing damaged tissue. Simple tissues The human body is basically made of four different types of tissues. It is the outer most covering of the young plants, roots, stems and leaves. Created by. The plant epidermis is specialised tissue, composed of parenchyma cells, that covers the external surfaces of leaves, stems and roots. The Process of Plant Tissue Culture . I) Simple permanent tissues II) Complex permanent tissues Types Of Plant Tissue - Displaying top 8 worksheets found for this concept.. There are over $$\text{200 000}$$ types of plant species in the world. Ground, vascular and dermal. The main function of the ground tissue is to provide support, strength, and flexibility to plants. Ground tissue is made up of all cells that are not vascular or dermal (having to do with the epidermis; see below). Different types of plant tissues include permanent and meristematic tissues. The permanent tissues generally don’t divide further. TYPES OF PLANT TISSUE Meristematic tissue Apical meristems Lateral meristems Intercalary meristems Permanent tissue Simple permanent tissue Parenchyma – A free PowerPoint PPT presentation (displayed as a Flash slide show) on PowerShow.com - id: 3d8a05-NjA0O Meristematic Tissues: A meristematic tissue constitutes a group of actively dividing cells present in the growing region of plant, e.g., the tips of roots and stems. This is because the dividing tissue, Known as meristematic tissue Composed of actively dividing ceIIs, responsible for the production of ceIIs. Capacity for division is restricted to certain parts of the plant body called meristems Which are active throughout the life of the plant body. Permanent tissue may be defined as a group of living or dead cells formed by meristematic tissue and have lost their ability to divide and have permanently placed at fixed position in plant body. The Tissue System . 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Complex permanent tissues animal tissues: epithelial tissue, Known as meristematic tissue: epidermal,,!	2022-09-28 04:16:39	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.26146993041038513, "perplexity": 6023.18005958368}, "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/1664030335059.43/warc/CC-MAIN-20220928020513-20220928050513-00208.warc.gz"}
https://artofproblemsolving.com/wiki/index.php?title=1987_AIME_Problems/Problem_12&diff=cur&oldid=135803	# Difference between revisions of "1987 AIME Problems/Problem 12" ## Problem Let $m$ be the smallest integer whose cube root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/1000$. Find $n$. ## Solution 1 In order to keep $m$ as small as possible, we need to make $n$ as small as possible. $m = (n + r)^3 = n^3 + 3n^2r + 3nr^2 + r^3$. Since $r < \frac{1}{1000}$ and $m - n^3 = r(3n^2 + 3nr + r^2)$ is an integer, we must have that $3n^2 + 3nr + r^2 \geq \frac{1}{r} > 1000$. This means that the smallest possible $n$ should be quite a bit smaller than 1000. In particular, $3nr + r^2$ should be less than 1, so $3n^2 > 999$ and $n > \sqrt{333}$. $18^2 = 324 < 333 < 361 = 19^2$, so we must have $n \geq 19$. Since we want to minimize $n$, we take $n = 19$. Then for any positive value of $r$, $3n^2 + 3nr + r^2 > 3\cdot 19^2 > 1000$, so it is possible for $r$ to be less than $\frac{1}{1000}$. However, we still have to make sure a sufficiently small $r$ exists. In light of the equation $m - n^3 = r(3n^2 + 3nr + r^2)$, we need to choose $m - n^3$ as small as possible to ensure a small enough $r$. The smallest possible value for $m - n^3$ is 1, when $m = 19^3 + 1$. Then for this value of $m$, $r = \frac{1}{3n^2 + 3nr + r^2} < \frac{1}{1000}$, and we're set. The answer is $\boxed{019}$. ## Solution 2 To minimize $m$, we should minimize $n$. We have that $(n + \frac{1}{1000})^3 = n^3 + \frac{3}{10^3} n^2 + \frac{3}{10^6} n + \frac{1}{10^9}$. For a given value of $n$, if $(n + \frac{1}{1000})^3 - n^3 > 1$, there exists an integer between $(n + \frac{1}{1000})^3$ and $n^3$, and the cube root of this integer would be between $n$ and $n + \frac{1}{1000}$ as desired. We seek the smallest $n$ such that $(n + \frac{1}{1000})^3 - n^3 > 1$. $$(n + \frac{1}{1000})^3 - n^3 > 1$$ $$\frac{3}{10^3} n^2 + \frac{3}{10^6} n + \frac{1}{10^9} > 1$$ $$3n^2 + \frac{3}{10^3} n + \frac{1}{10^6} > 10^3$$ Trying values of $n$, we see that the smallest value of $n$ that works is $\boxed{019}$. Why is it $(n + \frac{1}{1000})^3 - n^3 > 1$ and not greater than or equal to? - awesomediabrine Because if its equal to, then there is no integer in between the two values. - resources ## Solution 3 (Similar to Solution 2) Since $r$ is less than $1/1000$, we have $\sqrt[3]{m} < n + \frac{1}{1000}$. Notice that since we want $m$ minimized, $n$ should also be minimized. Also, $n^3$ should be as close as possible, but not exceeding $m$. This means $m$ should be set to $n^3+1$. Substituting and simplifying, we get $$\sqrt[3]{n^3+1} < n + \frac{1}{1000}$$ $$n^3+1 < n^3+\frac{3}{1000}n^2+\frac{3}{1000^2}n+\frac{1}{1000^3}$$ The last two terms in the right side can be ignored in the calculation because they are too small. This results in $1 < \frac{3}{1000}n^2 \Rightarrow n^2 > \frac{1000}{3}$. The minimum positive integer $n$ that satisfies this is $\boxed{019}$. ~ Hb10 ## Solution 4 (Calculus) Note that the cube root is increasing for positive reals while its derivative is decreasing, so linear approximation gives $$\sqrt[3]{n^3+1} - n < \left.\frac{d\sqrt[3]{x}}{dx}\right\|_{x=n^3} = \frac{1}{3n^2}$$ and $$\sqrt[3]{n^3+1} - n > \left.\frac{d\sqrt[3]{x}}{dx}\right\|_{x=n^3+1} = \frac{1}{3\sqrt[3]{(n^3+1)^2}}$$ From this, it is clear that $n = \boxed{019}$ is the smallest $n$ for which LHS will be less than $\frac{1}{1000}$. ~ Hyprox1413	2022-12-01 10:18:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 70, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8992296457290649, "perplexity": 68.89067890063616}, "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/1669446710808.72/warc/CC-MAIN-20221201085558-20221201115558-00798.warc.gz"}
https://www.zbmath.org/authors/?q=ai%3Akim.jeong-han	# zbMATH — the first resource for mathematics ## Kim, Jeong Han Compute Distance To: Author ID: kim.jeong-han Published as: Kim, Jeong Han; Kim, J. H.; Kim, Jeong-Han; Kim, J.; Kim, Jeong H. Homepage: https://www.mathnet.or.kr/people_list/view/4242 External Links: MGP · Wikidata Documents Indexed: 59 Publications since 1993 all top 5 #### Co-Authors 10 single-authored 10 Vu, Van H. 4 Choi, Sung-Soon 4 Peres, Yuval 4 Tetali, Prasad 3 Alon, Noga M. 3 Bollobás, Béla 3 Wormald, Nicholas Charles 2 Bayati, Mohsen Fathollah 2 Ding, Jian 2 Fishburn, Peter Clingerman 2 Greenhill, Catherine S. 2 Jung, Kyomin 2 Lee, Choongbum 2 Lee, Joonkyung 2 Lee, Sangjune 2 Lubetzky, Eyal 2 Mandjes, Michel Robertus Hendrikus 2 Montenegro, Ravi 2 Pittel, Boris G. 2 Saberi, Amin 2 Spencer, Joel H. 2 Sudakov, Benny 2 Verstraëte, Jacques 1 Achlioptas, Dimitris 1 Armero, Francisco 1 Bohman, Tom 1 Borgs, Christian 1 Chayes, Jennifer Tour 1 Chen, Bob 1 Conlon, David 1 Hajiaghayi, Mohammad Taghi 1 Jang, Lee-Chae 1 Janson, Svante 1 Jo, Gwanghyun 1 Kahn, Jeff D. 1 Kim, Taekyun 1 Krivelevich, Michael 1 Lagarias, Jeffrey C. 1 Lee, Sungchul 1 Leighton Tom 1 Matoušek, Jiří 1 Na, Joohan 1 Park, Dal-Won 1 Pikhurko, Oleg 1 Racke, Harald 1 Roche, James R. 1 Tait, Michael 1 Verbitsky, Oleg 1 Wilson, David Bruce 1 Wright, Paul E. all top 5 #### Serials 11 Random Structures & Algorithms 4 Combinatorica 3 Journal of Combinatorial Theory. Series A 3 Journal of Combinatorial Theory. Series B 3 Journal of Computer and System Sciences 3 Combinatorics, Probability and Computing 2 Artificial Intelligence 2 Discrete Mathematics 2 SIAM Journal on Discrete Mathematics 1 Advances in Applied Probability 1 Computer Methods in Applied Mechanics and Engineering 1 Discrete Applied Mathematics 1 Israel Journal of Mathematics 1 Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 1 Advances in Mathematics 1 International Journal of Mathematics and Mathematical Sciences 1 Journal of Graph Theory 1 Journal of the London Mathematical Society. Second Series 1 Proceedings of the London Mathematical Society. Third Series 1 Transactions of the American Mathematical Society 1 Statistics & Probability Letters 1 Algorithmica 1 Queueing Systems 1 The Annals of Applied Probability 1 Far East Journal of Mathematical Sciences 1 Journal of Applied Mathematics 1 Annals of Fuzzy Mathematics and Informatics all top 5 #### Fields 43 Combinatorics (05-XX) 14 Probability theory and stochastic processes (60-XX) 11 Computer science (68-XX) 4 Operations research, mathematical programming (90-XX) 4 Information and communication theory, circuits (94-XX) 3 Number theory (11-XX) 2 Numerical analysis (65-XX) 2 Mechanics of deformable solids (74-XX) 1 Mathematical logic and foundations (03-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Real functions (26-XX) 1 Measure and integration (28-XX) 1 Partial differential equations (35-XX) 1 General topology (54-XX) 1 Statistics (62-XX) 1 Biology and other natural sciences (92-XX) #### Citations contained in zbMATH Open 49 Publications have been cited 578 times in 503 Documents Cited by Year The Ramsey number $$R(3,t)$$ has order of magnitude $$t^ 2 /\log t$$. Zbl 0832.05084 Kim, Jeong Han 1995 Concentration of multivariate polynomials and its applications. Zbl 0969.60013 Kim, Jeong Han; Vu, Van H. 2000 The scaling window of the 2-SAT transition. Zbl 0979.68053 Bollobás, Béla; Borgs, Christian; Chayes, Jennifer T.; Kim, Jeong Han; Wilson, David B. 2001 On Brooks’ theorem for sparse graphs. Zbl 0833.05030 Kim, Jeong Han 1995 Nearly perfect matchings in regular simple hypergraphs. Zbl 0882.05107 Alon, Noga; Kim, Jeong-Han; Spencer, Joel 1997 Divide and conquer martingales and the number of triangles in a random graph. Zbl 1041.60042 Kim, J. H.; Vu, V. H. 2004 Small complete arcs in projective planes. Zbl 1027.05015 Kim, J. H.; Vu, V. H. 2003 On the asymmetry of random regular graphs and random graphs. Zbl 1012.05143 Kim, Jeong Han; Sudakov, Benny; Vu, Van H. 2002 Sandwiching random graphs: universality between random graph models. Zbl 1050.05111 Kim, J. H.; Vu, V. H. 2004 Diameters in supercritical random graphs via first passage percolation. Zbl 1260.05048 Ding, Jian; Kim, Jeong Han; Lubetzky, Eyal; Peres, Yuval 2010 Two approaches to Sidorenko’s conjecture. Zbl 1331.05220 Kim, Jeong Han; Lee, Choongbum; Lee, Joonkyung 2016 A sequential algorithm for generating random graphs. Zbl 1198.05138 Bayati, Mohsen; Kim, Jeong Han; Saberi, Amin 2010 How complex are random graphs in first order logic? Zbl 1060.05085 Kim, Jeong Han; Pikhurko, Oleg; Spencer, Joel H.; Verbitsky, Oleg 2005 Poisson cloning model for random graphs. Zbl 1100.05093 Kim, Jeong Han 2006 Two-coloring random hypergraphs. Zbl 1001.05059 Achlioptas, Dimitris; Kim, Jeong Han; Krivelevich, Michael; Tetali, Prasad 2002 Random matchings which induce Hamilton cycles and Hamiltonian decompositions of random regular graphs. Zbl 1030.05107 Kim, Jeong Han; Wormald, Nicholas C. 2001 Optimal query complexity bounds for finding graphs. Zbl 1231.68150 Choi, Sung-Soon; Kim, Jeong Han 2008 Anatomy of a Young giant component in the random graph. Zbl 1230.05260 Ding, Jian; Kim, Jeong Han; Lubetzky, Eyal; Peres, Yuval 2011 Entropy and sorting. Zbl 1294.68069 Kahn, Jeff; Kim, Jeong Han 1995 Generating random regular graphs. Zbl 1121.05110 Kim, J. H.; Vu, V. H. 2006 Generating random regular graphs. Zbl 1192.05146 Kim, Jeong Han; Vu, Van H. 2003 Small subgraphs of random regular graphs. Zbl 1118.05088 Kim, Jeong Han; Sudakov, Benny; Vu, Van 2007 On increasing subsequences of random permutations. Zbl 0859.05002 Kim, Jeong Han 1996 A phase transition for avoiding a giant component. Zbl 1092.05061 Bohman, Tom; Kim, Jeong Han 2006 On coupon colorings of graphs. Zbl 1317.05052 Chen, Bob; Kim, Jeong Han; Tait, Michael; Verstraete, Jacques 2015 Some advances on Sidorenko’s conjecture. Zbl 1433.05166 Conlon, David; Kim, Jeong Han; Lee, Choongbum; Lee, Joonkyung 2018 Regular subgraphs of random graphs. Zbl 1101.05061 Bollobás, Béla; Kim, Jeong Han; Verstraëte, Jacques 2006 Large deviations for Small buffers: An insensitivity result. Zbl 1017.90023 Mandjes, Michel; Kim, Jeong Han 2001 Permutation pseudographs and contiguity. Zbl 1006.05056 Greenhill, Catherine; Janson, Svante; Kim, Jeong Han; Wormald, Nicholas C. 2002 On the degree, size, and chromatic index of a uniform hypergraph. Zbl 0868.05037 Alon, Noga; Kim, Jeong Han 1997 Perfect matchings in random uniform hypergraphs. Zbl 1028.05088 Kim, Jeong Han 2003 Universality of random graphs for graphs of maximum degree two. Zbl 1305.05209 Kim, Jeong Han; Lee, Sang June 2014 Hamiltonian decompositions of random bipartite regular graphs. Zbl 1033.05082 Greenhill, Catherine; Kim, Jeong Han; Wormald, Nicholas C. 2004 Discrepancy after adding a single set. Zbl 1092.05069 Kim, Jeong Han; Matoušek, Jiří; Vu, Van H. 2005 A birthday paradox for Markov chains with an optimal bound for collision in the Pollard rho algorithm for discrete logarithm. Zbl 1195.60096 Kim, Jeong Han; Montenegro, Ravi; Peres, Yuval; Tetali, Prasad 2010 Optimal query complexity bounds for finding graphs. Zbl 1206.68228 Choi, Sung-Soon; Kim, Jeong Han 2010 A birthday paradox for Markov chains, with an optimal bound for collision in the Pollard rho algorithm for discrete logarithm. Zbl 1205.11135 Kim, Jeong Han; Montenegro, Ravi; Peres, Yuval; Tetali, Prasad 2008 A sequential algorithm for generating random graphs. Zbl 1171.05423 Bayati, Mohsen; Kim, Jeong Han; Saberi, Amin 2007 Confirming the Kleitman-Winston conjecture on the largest coefficient in a $$q$$-Catalan number. Zbl 0968.05037 Kim, Jeong Han; Pittel, Boris 2000 On tail distribution of interpost distance. Zbl 1029.05138 Kim, Jeong Han; Pittel, Boris 2000 Almost tight upper bound for finding Fourier coefficients of bounded pseudo-Boolean functions. Zbl 1234.68148 Choi, Sung-Soon; Jung, Kyomin; Kim, Jeong Han 2011 Score certificates for tournaments. Zbl 0865.05044 Kim, Jeong Han; Tetali, Prasad; Fishburn, Peter 1997 Interference-minimizing colorings of regular graphs. Zbl 0912.05036 Fishburn, P. C.; Kim, J. H.; Lagarias, J. C.; Wright, P. E. 1998 Analysis of a phase transition phenomenon in packet networks. Zbl 0979.60081 Mandjes, Michel; Kim, Jeong-Han 2001 Nearly optimal partial Steiner systems. Zbl 0981.05017 Kim, Jeong Han 2001 Economical covers with geometric applications. Zbl 1029.05109 Alon, Noga; Bollobás, Béla; Kim, Jeong Han; Vu, Van H. 2003 Oblivious routing in directed graphs with random demands. Zbl 1192.90229 Hajiaghayi, Mohammad Taghi; Kim, Jeong Han; Leighton Tom; Räcke, Harald 2005 Covering cubes by random half cubes, with applications to binary neural networks. Zbl 0948.68163 Kim, Jeong Han; Roche, James R. 1998 On the total variation distance between the binomial random graph and the random intersection graph. Zbl 1441.05203 Kim, Jeong Han; Lee, Sang June; Na, Joohan 2018 Some advances on Sidorenko’s conjecture. Zbl 1433.05166 Conlon, David; Kim, Jeong Han; Lee, Choongbum; Lee, Joonkyung 2018 On the total variation distance between the binomial random graph and the random intersection graph. Zbl 1441.05203 Kim, Jeong Han; Lee, Sang June; Na, Joohan 2018 Two approaches to Sidorenko’s conjecture. Zbl 1331.05220 Kim, Jeong Han; Lee, Choongbum; Lee, Joonkyung 2016 On coupon colorings of graphs. Zbl 1317.05052 Chen, Bob; Kim, Jeong Han; Tait, Michael; Verstraete, Jacques 2015 Universality of random graphs for graphs of maximum degree two. Zbl 1305.05209 Kim, Jeong Han; Lee, Sang June 2014 Anatomy of a Young giant component in the random graph. Zbl 1230.05260 Ding, Jian; Kim, Jeong Han; Lubetzky, Eyal; Peres, Yuval 2011 Almost tight upper bound for finding Fourier coefficients of bounded pseudo-Boolean functions. Zbl 1234.68148 Choi, Sung-Soon; Jung, Kyomin; Kim, Jeong Han 2011 Diameters in supercritical random graphs via first passage percolation. Zbl 1260.05048 Ding, Jian; Kim, Jeong Han; Lubetzky, Eyal; Peres, Yuval 2010 A sequential algorithm for generating random graphs. Zbl 1198.05138 Bayati, Mohsen; Kim, Jeong Han; Saberi, Amin 2010 A birthday paradox for Markov chains with an optimal bound for collision in the Pollard rho algorithm for discrete logarithm. Zbl 1195.60096 Kim, Jeong Han; Montenegro, Ravi; Peres, Yuval; Tetali, Prasad 2010 Optimal query complexity bounds for finding graphs. Zbl 1206.68228 Choi, Sung-Soon; Kim, Jeong Han 2010 Optimal query complexity bounds for finding graphs. Zbl 1231.68150 Choi, Sung-Soon; Kim, Jeong Han 2008 A birthday paradox for Markov chains, with an optimal bound for collision in the Pollard rho algorithm for discrete logarithm. Zbl 1205.11135 Kim, Jeong Han; Montenegro, Ravi; Peres, Yuval; Tetali, Prasad 2008 Small subgraphs of random regular graphs. Zbl 1118.05088 Kim, Jeong Han; Sudakov, Benny; Vu, Van 2007 A sequential algorithm for generating random graphs. Zbl 1171.05423 Bayati, Mohsen; Kim, Jeong Han; Saberi, Amin 2007 Poisson cloning model for random graphs. Zbl 1100.05093 Kim, Jeong Han 2006 Generating random regular graphs. Zbl 1121.05110 Kim, J. H.; Vu, V. H. 2006 A phase transition for avoiding a giant component. Zbl 1092.05061 Bohman, Tom; Kim, Jeong Han 2006 Regular subgraphs of random graphs. Zbl 1101.05061 Bollobás, Béla; Kim, Jeong Han; Verstraëte, Jacques 2006 How complex are random graphs in first order logic? Zbl 1060.05085 Kim, Jeong Han; Pikhurko, Oleg; Spencer, Joel H.; Verbitsky, Oleg 2005 Discrepancy after adding a single set. Zbl 1092.05069 Kim, Jeong Han; Matoušek, Jiří; Vu, Van H. 2005 Oblivious routing in directed graphs with random demands. Zbl 1192.90229 Hajiaghayi, Mohammad Taghi; Kim, Jeong Han; Leighton Tom; Räcke, Harald 2005 Divide and conquer martingales and the number of triangles in a random graph. Zbl 1041.60042 Kim, J. H.; Vu, V. H. 2004 Sandwiching random graphs: universality between random graph models. Zbl 1050.05111 Kim, J. H.; Vu, V. H. 2004 Hamiltonian decompositions of random bipartite regular graphs. Zbl 1033.05082 Greenhill, Catherine; Kim, Jeong Han; Wormald, Nicholas C. 2004 Small complete arcs in projective planes. Zbl 1027.05015 Kim, J. H.; Vu, V. H. 2003 Generating random regular graphs. Zbl 1192.05146 Kim, Jeong Han; Vu, Van H. 2003 Perfect matchings in random uniform hypergraphs. Zbl 1028.05088 Kim, Jeong Han 2003 Economical covers with geometric applications. Zbl 1029.05109 Alon, Noga; Bollobás, Béla; Kim, Jeong Han; Vu, Van H. 2003 On the asymmetry of random regular graphs and random graphs. Zbl 1012.05143 Kim, Jeong Han; Sudakov, Benny; Vu, Van H. 2002 Two-coloring random hypergraphs. Zbl 1001.05059 Achlioptas, Dimitris; Kim, Jeong Han; Krivelevich, Michael; Tetali, Prasad 2002 Permutation pseudographs and contiguity. Zbl 1006.05056 Greenhill, Catherine; Janson, Svante; Kim, Jeong Han; Wormald, Nicholas C. 2002 The scaling window of the 2-SAT transition. Zbl 0979.68053 Bollobás, Béla; Borgs, Christian; Chayes, Jennifer T.; Kim, Jeong Han; Wilson, David B. 2001 Random matchings which induce Hamilton cycles and Hamiltonian decompositions of random regular graphs. Zbl 1030.05107 Kim, Jeong Han; Wormald, Nicholas C. 2001 Large deviations for Small buffers: An insensitivity result. Zbl 1017.90023 Mandjes, Michel; Kim, Jeong Han 2001 Analysis of a phase transition phenomenon in packet networks. Zbl 0979.60081 Mandjes, Michel; Kim, Jeong-Han 2001 Nearly optimal partial Steiner systems. Zbl 0981.05017 Kim, Jeong Han 2001 Concentration of multivariate polynomials and its applications. Zbl 0969.60013 Kim, Jeong Han; Vu, Van H. 2000 Confirming the Kleitman-Winston conjecture on the largest coefficient in a $$q$$-Catalan number. Zbl 0968.05037 Kim, Jeong Han; Pittel, Boris 2000 On tail distribution of interpost distance. Zbl 1029.05138 Kim, Jeong Han; Pittel, Boris 2000 Interference-minimizing colorings of regular graphs. Zbl 0912.05036 Fishburn, P. C.; Kim, J. H.; Lagarias, J. C.; Wright, P. E. 1998 Covering cubes by random half cubes, with applications to binary neural networks. Zbl 0948.68163 Kim, Jeong Han; Roche, James R. 1998 Nearly perfect matchings in regular simple hypergraphs. Zbl 0882.05107 Alon, Noga; Kim, Jeong-Han; Spencer, Joel 1997 On the degree, size, and chromatic index of a uniform hypergraph. Zbl 0868.05037 Alon, Noga; Kim, Jeong Han 1997 Score certificates for tournaments. Zbl 0865.05044 Kim, Jeong Han; Tetali, Prasad; Fishburn, Peter 1997 On increasing subsequences of random permutations. Zbl 0859.05002 Kim, Jeong Han 1996 The Ramsey number $$R(3,t)$$ has order of magnitude $$t^ 2 /\log t$$. Zbl 0832.05084 Kim, Jeong Han 1995 On Brooks’ theorem for sparse graphs. Zbl 0833.05030 Kim, Jeong Han 1995 Entropy and sorting. Zbl 1294.68069 Kahn, Jeff; Kim, Jeong Han 1995 all top 5 #### Cited by 729 Authors 16 Sudakov, Benny 13 Mubayi, Dhruv 12 Frieze, Alan Michael 12 Kim, Jeong Han 10 Bartoli, Daniele 10 Bohman, Tom 10 Marcugini, Stefano 10 Pambianco, Fernanda 10 Warnke, Lutz 9 Davydov, Alexander A. 9 Janson, Svante 9 Lubetzky, Eyal 9 Pikhurko, Oleg 8 Conlon, David 8 Faina, Giorgio 8 Krivelevich, Michael 8 Rodl, Vojtech 8 Shabanov, Dmitry A. 8 Vu, Van H. 7 Dudek, Andrzej 7 Fox, Jacob 7 Greenhill, Catherine S. 7 Osthus, Deryk 7 Spencer, Joel H. 6 Bshouty, Nader H. 6 Chatterjee, Sourav 6 Ding, Jian 6 Henning, Michael Anthony 6 Kühn, Daniela 5 Alon, Noga M. 5 Ferber, Asaf 5 Gao, Pu 5 Kang, Mihyun 5 Lee, Choongbum 5 Mandjes, Michel Robertus Hendrikus 5 Peres, Yuval 5 Prałat, Paweł 5 Ruciński, Andrzej 5 Verbitsky, Oleg 5 Verstraëte, Jacques 5 Zhao, Yufei 4 Bhamidi, Shankar 4 Coja-Oghlan, Amin 4 Cooper, Jeff 4 Fiorini, Samuel 4 Foucaud, Florent 4 Gyárfás, András 4 Kahn, Jeff D. 4 Kohayakawa, Yoshiharu 4 Kostochka, Aleksandr Vasil’evich 4 Mazzawi, Hanna 4 Perkins, Will 4 Person, Yury Aleksandrovic 4 Ravelomanana, Vlady 4 Schiermeyer, Ingo 4 Sly, Allan 4 Suk, Andrew 4 Wormald, Nicholas Charles 3 Bennett, Patrick 3 Brightwell, Graham R. 3 Cardinal, Jean 3 Choi, Sung-Soon 3 Chudnovsky, Maria 3 Dembo, Amir 3 Giulietti, Massimo 3 Haxell, Penny E. 3 Joos, Felix Claudius 3 Joret, Gwenaël 3 Kreshchuk, Alexey A. 3 Li, Yusheng 3 Lin, Qizhong 3 Liu, Hong 3 Loh, Po-Shen 3 Markström, Klas 3 Mertzios, George B. 3 Montanari, Andrea 3 Nagy, Zoltán Lóránt 3 Naserasr, Reza 3 Nenadov, Rajko 3 Parreau, Aline 3 Pastor, Lucas 3 Picollelli, Michael E. 3 Pittel, Boris G. 3 Pontiveros, Gonzalo Fiz 3 Rossignol, Raphaël 3 Samotij, Wojciech 3 Schudy, Warren 3 Seppäläinen, Timo 3 Šileikis, Matas 3 Sun, Nike 3 Sviridenko, Maxim I. 3 Thomassé, Stéphan 3 Trotignon, Nicolas 3 Valicov, Petru 3 van der Hofstad, Remco W. 3 Vigoda, Eric 3 Yeo, Anders 3 Yuster, Raphael 2 Achlioptas, Dimitris 2 Addario-Berry, Louigi ...and 629 more Authors all top 5 #### Cited in 115 Serials 61 Random Structures & Algorithms 28 Journal of Combinatorial Theory. Series B 27 Combinatorics, Probability and Computing 26 Discrete Mathematics 20 Theoretical Computer Science 20 European Journal of Combinatorics 15 Discrete Applied Mathematics 15 SIAM Journal on Discrete Mathematics 12 The Annals of Probability 12 Graphs and Combinatorics 11 The Electronic Journal of Combinatorics 9 The Annals of Applied Probability 8 Israel Journal of Mathematics 8 Advances in Mathematics 8 Journal of Graph Theory 7 Journal of Applied Probability 7 Combinatorica 7 Algorithmica 7 Journal of Combinatorial Designs 6 Proceedings of the American Mathematical Society 6 Journal of Combinatorial Optimization 5 Journal of Combinatorial Theory. 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Theoretical Informatics and Applications 1 Journal of Systems Science and Complexity 1 Journal of Applied Mathematics ...and 15 more Serials all top 5 #### Cited in 28 Fields 397 Combinatorics (05-XX) 104 Probability theory and stochastic processes (60-XX) 94 Computer science (68-XX) 31 Operations research, mathematical programming (90-XX) 20 Geometry (51-XX) 17 Number theory (11-XX) 17 Information and communication theory, circuits (94-XX) 14 Statistical mechanics, structure of matter (82-XX) 12 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 11 Mathematical logic and foundations (03-XX) 11 Order, lattices, ordered algebraic structures (06-XX) 10 Statistics (62-XX) 6 Numerical analysis (65-XX) 4 Linear and multilinear algebra; matrix theory (15-XX) 4 Convex and discrete geometry (52-XX) 4 Biology and other natural sciences (92-XX) 3 Measure and integration (28-XX) 2 Algebraic geometry (14-XX) 2 Dynamical systems and ergodic theory (37-XX) 2 Functional analysis (46-XX) 2 Manifolds and cell complexes (57-XX) 1 General algebraic systems (08-XX) 1 Commutative algebra (13-XX) 1 Group theory and generalizations (20-XX) 1 Real functions (26-XX) 1 Functions of a complex variable (30-XX) 1 Partial differential equations (35-XX) 1 Mechanics of deformable solids (74-XX) #### Wikidata Timeline The data are displayed as stored in Wikidata under a Creative Commons CC0 License. Updates and corrections should be made in Wikidata.	2021-07-25 06:30:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.45143187046051025, "perplexity": 9998.80618196573}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046151638.93/warc/CC-MAIN-20210725045638-20210725075638-00429.warc.gz"}
http://alexanderpruss.blogspot.com/2011/03/names-quantifiers-aristotelian-logic.html	## Monday, March 21, 2011 ### Names, quantifiers, Aristotelian logic and one-sided relations This is going to be a pretty technically involved post and it will be written very badly, as it's really just notes for self. Start with this objection to Aristotelian logic. A good logical system reveals the deep logical structure of sentences. But Aristotelian logic takes as fundamental sentences like: 1. Everyone is mortal. 2. Socrates is mortal. In so doing, Aristotelian logic creates the impression that (1) and (2) have similar logical form, and it is normally taken to be that modern quantified logic has shown that (1) and (2) have different logical forms, namely: 1. x(Mortal(x)) 2. Mortal(Socrates). I shall show, however, that there is a way of thinking about (1) and (2), as well as about (3) and (4), that makes them have the same deep logical form, as Aristotelian logician makes it seem. (This is a very surprising result for me. Until I discovered these ideas this year, I had a strong antipathy to Aristotelian logic.) Moreover, this will give us some hope of understanding the medieval idea of one-sided relations. The medievals thought, very mysteriously, that creation is a one-sided relation: we are related to God by the created by relation, but God is not related to us by the creates relation. Now to the technical stuff. Recall Tarski's definition of truth in terms of satisfaction. I think the best way to formulate the definition is by means of a substitution sequence. A substitution sequence s is a finite sequence of variable-object pairs, which I will write using a slash. E.g., "x1"/Socrates,"x2"/Francis,"x3"/Bucephalus is a substitution sequence. The first pair in my example consists of the variable letter "x1", a linguistic entity (actually in the best logic we might have slot identifiers instead of variable letters) and Socrates—not the name "Socrates" (which is why the quotation marks are as they are). We then inductively define the notion of a substitution sequence satisfying a well-formed formula (wff) under an interpretation I. An interpretation I is a function from names and predicates to objects and properties respectively. And then we have satisfaction simpliciter which is satisfaction under the intended interpretation, and that's what will interest me. So henceforth, I will be the intended interpretation. (I've left out models, because I am interested in truth simpliciter.) We proceed inductively. Thus, s satisfies a disjunction of wffs if and only if it satisfies at least one of the wffs, and so on, the negation of a wff if and only if it does not satisfy the wff, and so on. Quantifiers are a little more tricky. The sequence s satisfies the wff ∀xF iff for every object u, the sequence "x"/u,s (i.e., the sequence obtained by prepending the pair "x"/u" at its head) satisfies F. The sequence s satisfies ∃xF iff for some object u, the sequence "x"/u,s satisfies F. What remains is to define s's satisfaction of an atomic wff, i.e., one of the form P(a1,...,an) where a1,...,an are a sequence of names or variables. The standard way of doing this is as follows. Let u1,...,un be a sequence of objects defined as follows. If ai is a variable "x", then we let ui be the first object u occuring in s paired with the variable "x". If for some i there is none such pair in s, then we say s doesn't satisfies the formula. If ai is a name "n", then we let ui=I("n"). We then say that s satisfies P(a1,...,an) if and only if u1,...,un stand in I(P). Now notice that while the definition of satisfaction for quantified sentences is pretty neat, the definition of satisfaction for atomics is really messy, because it needs to take into account the question of which slot of the predicate has a variable in it and which one has a name. There is a different way of doing this. This starts with the Montague grammar way of thinking about things, on which words are taken to be functors from linguistic entities to linguistic entities. Let us ask, then, what kind of functors are represented by names. Here is the answer that I think is appealing. A name, say "Socrates", is a functor from wffs with an indicated blank to wffs. In English, the name takes a wff like "____ likes virtue" and returns the wff (in this case sentence) "Socrates likes virtue". (The competing way of thinking of names is as zero-ary functors. But if one does it this way, one also needs variables as another kind of zero-ary functor, which I think is unappealing since variables are really just a kind of slot, or else one has a mess in treating atomics differently depending on which slots are filled with names and which with variables.) We can re-formulate First Order Logic so that a name like "Socrates" is (or at least corresponds to) a functor from wff-variable pairs to new wffs. Thus, when we apply the functor "Socrates" to the wff "Mortal(x)" and the variable "x", we get the wff (sentence, actually) "Mortal(Socrates)". And the resulting wff no longer has the variable "x" freely occurring in it. But this is exactly what quantifiers do. For instance, the universal quantifier is a functor that takes a wff and a variable, and returns a new wff in which the variable does not freely occur. If we wanted the grammar to indicate this with particular clarity, instead of writing "Rides(Alexander, Bucephalus)", we would write: "Alexanderx Bucephalusy Rides(x,y)". And this is syntactically very much like "∀xy Rides(x,y)". And if we adopted this notation, the Tarski definition of satisfaction would change. We would add a new clause for the satisfaction of a name-quantified formula: s satisfies nxF, where "n" is a name, if and only if "x"/I("n"),s satisfies F. Now once we got to the satisfaction of an atomic, the predicate would only be applied to variables, never to names. And so we could more neatly say that s satisfies P(x1,...,xn) if and only if every variable occurs in the substitution sequence and u1,...,un stand in I(P) where ui is the first entity u occurring in s in a pair of the form "xi"/u. Neater and simpler, I think. Names, thus, can be seen as quantifiers. It might be thought that there is a crucial disanalogy between names and the universal/existential quantifiers, in that there are many names, and only one universal and only one existential quantifier. But the latter point is not clear. In a typed logic, there may be as many universal quantifiers as types, and as many existential ones as types, once again. And the number of types may be world-dependent, just as the number of objects. If I am right, then if we wanted to display the logical structure of (1) and (2), or of (3) and (4) for that matter, we would respectively say: 1. x Mortal(x) 2. Socratesx Mortal(x). And there is a deep similarity of logical structure—we simply have different quantifiers. And so the Aristotelian was right to see these two as similar. Now, the final little bit of stuff. Obviously, if "m" and "n" are two names, then: 1. "mnF(x,y)" is true iff "nmF(x,y)" is true, just as: 1. "∀xyF(x,y)" is true iff "∀yxF(x,y)" is true. But the two sentences in (8), although they are logically equivalent, arguably express different propositions. And I submit that so do the two sentence in (7). And we even have a way of marking the difference in English, I think. Ordinarily, what the left hand side in (7) says is that u has the property PxnyF(x,y) while the right hand side in (8) says that v has the property PymxF(x,y), where u and v are what "m" and "n" respectively denote, and PxH(x) is the (abundant) property corresponding to the predicate H (the P-thingy is like the lambda functor, except it returns a property, not a predicate). These are distinct claims. The medievals then claim that in the case of God we have this. They say that "Godx nF(x,y)" is true in virtue of "ny GodF(x,y)" being true. It is to the referent of "n" that the property Py GodF(x,y) is attributed, and the sentence that seems to attribute a property to God is to be analyzed in terms of the one that attributes a property to the referent of "n". Jonathan D. Jacobs said... We could, if we liked, represent predicates in the same way. (Cf: contemporary ways of being theorists.) Wouldn't that mitigate against the similarity between Aristotelian and modern logic? Alexander R Pruss said... I don't see how to represent all predicates in the same way. The best I can do is that a predicate is a functor from sequences of variables and/or names to wffs. And I think that's not so neat, because then the base clause of the Tarski truth definition is messy, as it has to take both variables and names into account. I can, however, represent all but one predicates in the above way. Namely, have a single variable grade predicate "Instantiates" and then names for all the properties. So instead of "Mortal(Socrates)", we'd say: Socratesx Mortalityy Instantiates(x,y) I.e., x instantiates y, where x is Socrates and y is Mortality. But we still have a special predicate that is not handled in this way, Instantiates. You might try something second-order like: Socratesx MortalityF F(x), where capital letters are predicate variables. But that's just Instantiates all over again, except that we use "F(x)" as shorthand for "Instantiates(x,F)". Jonathan D. Jacobs said... Socratesx Mortalityy x=y Jonathan D. Jacobs said... (At any rate, that's how the ways of being theorists do it.) Alexander R Pruss said... That's not an "=" of identity, is it? But whatever it is, isn't the "=" a predicate, so you still have one predicate which isn't handled in the same way as all the others? Jonathan D. Jacobs said... Yes, identity. (It's not my view, for what little that's worth. Neither is it, by the way, the ways of being theorists' view. It's one way to put the ways of being stuff to use in this context. For the actual view, see Kris McDaniel and Jason Turner's work.) So, yes, there will be one predicate handled unlike the others. (In this context, where we treating each predicate like a way of being, this will have the result that identity can be contingent and temporary.) I suppose your proposal will treat identity like any other relation. I don't really have a view on which proposal is more simple.	2015-05-23 01:12:16	{"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": 0.8348339200019836, "perplexity": 1051.6979782781643}, "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-2015-22/segments/1432207926964.7/warc/CC-MAIN-20150521113206-00244-ip-10-180-206-219.ec2.internal.warc.gz"}
https://www.imperial.ac.uk/people/l.matteini/publications.html	# Dr Lorenzo Matteini Faculty of Natural SciencesDepartment of Physics Lecturer in Space Plasma Physics // // ### Location Blackett LaboratorySouth Kensington Campus // ## Publications Publication Type Year to 89 results found Laker R, Horbury TS, Matteini L, Bale SD, Stawarz JE, Woodham LD, Woolley Tet al., 2022, Switchback deflections beyond the early parker solar probe encounters, Monthly Notices of the Royal Astronomical Society, ISSN: 0035-8711 <jats:title>Abstract</jats:title> <jats:p>Switchbacks are Aflvénic fluctuations in the solar wind, which exhibit large rotations in the magnetic field direction. Observations from Parker Solar Probe’s (PSP’s) first two solar encounters have formed the basis for many of the described switchback properties and generation mechanisms. However, this early data may not be representative of the typical near-Sun solar wind, biasing our current understanding of these phenomena. One defining switchback property is the magnetic deflection direction. During the first solar encounter, this was primarily in the tangential direction for the longest switchbacks, which has since been discussed as evidence, and a testable prediction, of several switchback generation methods. In this study, we re-examine the deflection direction of switchbacks during the first eight PSP encounters to confirm the existence of a systematic deflection direction. We first identify switchbacks exceeding a threshold deflection in the magnetic field and confirm a previous finding that they are arc-polarized. In agreement with earlier results from PSP’s first encounter, we find that groups of longer switchbacks tend to deflect in the same direction for several hours. However, in contrast to earlier studies, we find that there is no unique direction for these deflections, although several solar encounters showed a non-uniform distribution in deflection direction with a slight preference for the tangential direction. This result suggests a systematic magnetic configuration for switchback generation, which is consistent with interchange reconnection as a source mechanism, although this new evidence does not rule out other mechanisms, such as the expansion of wave modes.</jats:p> Journal article Franci L, Papini E, Del Sarto D, Hellinger P, Burgess D, Matteini L, Landi S, Montagud-Camps Vet al., 2022, Plasma Turbulence in the Near-Sun and Near-Earth Solar Wind: A Comparison via Observation-Driven 2D Hybrid Simulations, Universe, Vol: 8, Pages: 453-453 <jats:p>We analyse two high-resolution 2D hybrid simulations of plasma turbulence with observation-driven initial conditions that are representative of the near-Sun and the near-Earth solar wind. The former employs values of some fundamental parameters that have been measured by the Parker Solar Probe at 0.17 au from the Sun, while, in the latter, they are set to average values typically observed at 1 au. We compare the spatial and spectral properties of the magnetic, ion velocity, and density fluctuations, as well as the time evolution of magnetic reconnection events that occur spontaneously as the result of the development of turbulence. Despite some differences due to the different plasma conditions, some key features are observed in both simulations: elongated ion-scale Alfvénic structures form in between vortices whenever the orientation of the magnetic field lines is the same, i.e., magnetic reconnection via the formation of an X point cannot occur; the magnetic and density fluctuations at sub-ion scales are governed by force balance; the magnetic compressibility at sub-ion scales is compatible with isotropic magnetic field components; the characteristic time of the formation of current sheets is the eddy turnover at the energy injection scale, while the characteristic time for their disruption via reconnection is compatible with the Alfvén time of the background turbulence.</jats:p> Journal article McManus MD, Verniero J, Bale SD, Bowen TA, Larson DE, Kasper JC, Livi R, Matteini L, Rahmati A, Romeo O, Whittlesey P, Woolley Tet al., 2022, Density and Velocity Fluctuations of Alpha Particles in Magnetic Switchbacks, ASTROPHYSICAL JOURNAL, Vol: 933, ISSN: 0004-637X Journal article Hellinger P, Montagud-Camps V, Franci L, Matteini L, Papini E, Verdini A, Landi Set al., 2022, Ion-scale Transition of Plasma Turbulence: Pressure-Strain Effect, ASTROPHYSICAL JOURNAL, Vol: 930, ISSN: 0004-637X Journal article D'Amicis R, Bruno R, Panasenco O, Telloni D, Perrone D, Marcucci MF, Woodham L, Velli M, De Marco R, Jagarlamudi V, Coco I, Owen C, Louarn P, Livi S, Horbury T, Andre N, Angelini V, Evans V, Fedorov A, Genot V, Lavraud B, Matteini L, Muller D, O'Brien H, Pezzi O, Rouillard AP, Sorriso-Valvo L, Tenerani A, Verscharen D, Zouganelis Iet al., 2021, First Solar Orbiter observation of the Alfvenic slow wind and identification of its solar source, ASTRONOMY & ASTROPHYSICS, Vol: 656, ISSN: 0004-6361 Journal article Maksimovic M, Soucek J, Chust T, Khotyaintsev Y, Kretzschmar M, Bonnin X, Vecchio A, Alexandrova O, Bale SD, Berard D, Brochot J-Y, Edberg NJT, Eriksson A, Hadid LZ, Johansson EPG, Karlsson T, Katra B, Krasnoselskikh V, Krupar V, Lion S, Lorfevre E, Matteini L, Nguyen QN, Pisa D, Piberne R, Plettemeier D, Rucker HO, Santolik O, Steinvall K, Steller M, Stverak S, Travnicek P, Vaivads A, Zaslavsky A, Chaintreuil S, Dekkali M, Astier P-A, Barbary G, Boughedada K, Cecconi B, Chapron F, Collin C, Dias D, Gueguen L, Lamy L, Leray V, Malac-Allain LR, Pantellini F, Parisot J, Plasson P, Thijs S, Fratter I, Bellouard E, Danto P, Julien S, Guilhem E, Fiachetti C, Sanisidro J, Laffaye C, Gonzalez F, Pontet B, Queruel N, Jannet G, Fergeau P, de Wit TD, Vincent T, Agrapart C, Pragout J, Bergerard-Timofeeva M, Delory GT, Turin P, Jeandet A, Leroy P, Pellion J-C, Bouzid V, Recart W, Kolmasova I, Kruparova O, Uhlir L, Lan R, Base J, Andre M, Bylander L, Cripps V, Cully C, Jansson S-E, Puccio W, Brinek J, Ottacher H, Angelini V, Berthomier M, Evans V, Goetz K, Hellinger P, Horbury TS, Issautier K, Kontar E, Le Contel O, Louarn P, Martinovic M, Mueller D, O'Brien H, Owen CJ, Retino A, Rodriguez-Pacheco J, Sahraoui F, Sanchez L, Walsh AP, Wimmer-Schweingruber RF, Zouganelis Iet al., 2021, First observations and performance of the RPW instrument on board the Solar Orbiter mission, ASTRONOMY & ASTROPHYSICS, Vol: 656, ISSN: 0004-6361 Journal article Matteini L, Laker R, Horbury T, Woodham L, Bale SD, Stawarz JE, Woolley T, Steinvall K, Jones GH, Grant SR, Afghan Q, Galand M, O'Brien H, Evans V, Angelini V, Maksimovic M, Chust T, Khotyaintsev Y, Krasnoselskikh V, Kretzschmar M, Lorfevre E, Plettemeier D, Soucek J, Steller M, Stverak S, Travnicek P, Vaivads A, Vecchio A, Wimmer-Schweingruber RF, Ho GC, Gomez-Herrero R, Rodriguez-Pacheco J, Louarn P, Fedorov A, Owen CJ, Bruno R, Livi S, Zouganelis I, Muller Det al., 2021, Solar Orbiter's encounter with the tail of comet C/2019 Y4 (ATLAS): Magnetic field draping and cometary pick-up ion waves, Astronomy and Astrophysics: a European journal, Vol: 656, ISSN: 0004-6361 ontext. Solar Orbiter is expected to have flown close to the tail of comet C/2019 Y4 (ATLAS) during the spacecraft’s first perihelion in June 2020. Models predict a possible crossing of the comet tails by the spacecraft at a distance from the Sun of approximately 0.5 AU.Aims. This study is aimed at identifying possible signatures of the interaction of the solar wind plasma with material released by comet ATLAS, including the detection of draped magnetic field as well as the presence of cometary pick-up ions and of ion-scale waves excited by associated instabilities. This encounter provides us with the first opportunity of addressing such dynamics in the inner Heliosphere and improving our understanding of the plasma interaction between comets and the solar wind.Methods. We analysed data from all in situ instruments on board Solar Orbiter and compared their independent measurements in order to identify and characterize the nature of structures and waves observed in the plasma when the encounter was predicted.Results. We identified a magnetic field structure observed at the start of 4 June, associated with a full magnetic reversal, a local deceleration of the flow and large plasma density, and enhanced dust and energetic ions events. The cross-comparison of all these observations support a possible cometary origin for this structure and suggests the presence of magnetic field draping around some low-field and high-density object. Inside and around this large scale structure, several ion-scale wave-forms are detected that are consistent with small-scale waves and structures generated by cometary pick-up ion instabilities.Conclusions. Solar Orbiter measurements are consistent with the crossing through a magnetic and plasma structure of cometary origin embedded in the ambient solar wind. We suggest that this corresponds to the magnetotail of one of the fragments of comet ATLAS or to a portion of the tail that was previously disconnected and advected past the spacec Journal article Bale SD, Horbury TS, Velli M, Desai MI, Halekas JS, McManus MD, Panasenco O, Badman ST, Bowen TA, Chandran BDG, Drake JF, Kasper JC, Laker R, Mallet A, Matteini L, Phan TD, Raouafi NE, Squire J, Woodham LD, Woolley Tet al., 2021, A solar source of alfvenic magnetic field switchbacks: in situ remnants of magnetic funnels on supergranulation scales, The Astrophysical Journal: an international review of astronomy and astronomical physics, Vol: 923, Pages: 1-12, ISSN: 0004-637X One of the striking observations from the Parker Solar Probe (PSP) spacecraft is the prevalence in the inner heliosphere of large amplitude, Alfvénic magnetic field reversals termed switchbacks. These $\delta {B}_{R}/B\sim { \mathcal O }(1$) fluctuations occur over a range of timescales and in patches separated by intervals of quiet, radial magnetic field. We use measurements from PSP to demonstrate that patches of switchbacks are localized within the extensions of plasma structures originating at the base of the corona. These structures are characterized by an increase in alpha particle abundance, Mach number, plasma β and pressure, and by depletions in the magnetic field magnitude and electron temperature. These intervals are in pressure balance, implying stationary spatial structure, and the field depressions are consistent with overexpanded flux tubes. The structures are asymmetric in Carrington longitude with a steeper leading edge and a small (∼1°) edge of hotter plasma and enhanced magnetic field fluctuations. Some structures contain suprathermal ions to ∼85 keV that we argue are the energetic tail of the solar wind alpha population. The structures are separated in longitude by angular scales associated with supergranulation. This suggests that these switchbacks originate near the leading edge of the diverging magnetic field funnels associated with the network magnetic field—the primary wind sources. We propose an origin of the magnetic field switchbacks, hot plasma and suprathermals, alpha particles in interchange reconnection events just above the solar transition region and our measurements represent the extended regions of a turbulent outflow exhaust. Journal article Papini E, Hellinger P, Verdini A, Landi S, Franci L, Montagud-Camps V, Matteini Let al., 2021, Properties of Hall-MHD Turbulence at Sub-Ion Scales: Spectral Transfer Analysis, ATMOSPHERE, Vol: 12 Journal article Woolley T, Matteini L, McManus MD, Bercic L, Badman ST, Woodham LD, Horbury TS, Bale SD, Laker R, Stawarz JE, Larson DEet al., 2021, Plasma properties, switchback patches, and low alpha-particle abundance in slow Alfvenic coronal hole wind at 0.13 au, MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Vol: 508, Pages: 236-244, ISSN: 0035-8711 The Parker Solar Probe (PSP) mission presents a unique opportunity to study the near-Sun solar wind closer than any previous spacecraft. During its fourth and fifth solar encounters, PSP had the same orbital trajectory, meaning that solar wind was measured at the same latitudes and radial distances. We identify two streams measured at the same heliocentric distance (∼0.13 au) and latitude (∼–3∘.5⁠) across these encounters to reduce spatial evolution effects. By comparing the plasma of each stream, we confirm that they are not dominated by variable transient events, despite PSP’s proximity to the heliospheric current sheet. Both streams are consistent with a previous slow Alfvénic solar wind study once radial effects are considered, and appear to originate at the Southern polar coronal hole boundary. We also show that the switchback properties are not distinctly different between these two streams. Low α-particle abundance (∼0.6 per cent) is observed in the encounter 5 stream, suggesting that some physical mechanism must act on coronal hole boundary wind to cause α-particle depletion. Possible explanations for our observations are discussed, but it remains unclear whether the depletion occurs during the release or the acceleration of the wind. Using a flux tube argument, we note that an α-particle abundance of ∼0.6 per cent in this low-velocity wind could correspond to an abundance of ∼0.9 per cent at 1 au. Finally, as the two streams roughly correspond to the spatial extent of a switchback patch, we suggest that patches are distinct features of coronal hole wind. Journal article Maksimovic M, Bale SD, Chust T, Khotyaintsev Y, Krasnoselskikh V, Kretzschmar M, Plettemeier D, Rucker HO, Soucek J, Steller M, Stverak S, Travnicek P, Vaivads A, Chaintreuil S, Dekkali M, Alexandrova O, Astier P-A, Barbary G, Berard D, Bonnin X, Boughedada K, Cecconi B, Chapron F, Chariet M, Collin C, de Conchy Y, Dias D, Gueguen L, Lamy L, Leray V, Lion S, Malac-Allain LR, Matteini L, Nguyen QN, Pantellini F, Parisot J, Plasson P, Thijs S, Vecchio A, Fratter I, Bellouard E, Lorfevre E, Danto P, Julien S, Guilhem E, Fiachetti C, Sanisidro J, Laffaye C, Gonzalez F, Pontet B, Queruel N, Jannet G, Fergeau P, Brochot J-Y, Cassam-Chenai G, Dudok de Wit T, Timofeeva M, Vincent T, Agrapart C, Delory GT, Turin P, Jeandet A, Leroy P, Pellion J-C, Bouzid V, Katra B, Piberne R, Recart W, Santolik O, Kolmasova I, Krupar V, Kruparova O, Pisa D, Uhlir L, Lan R, Base J, Ahlen L, Andre M, Bylander L, Cripps V, Cully C, Eriksson A, Jansson S-E, Johansson EPG, Karlsson T, Puccio W, Brinek J, ottacher H, Panchenko M, Berthomier M, Goetz K, Hellinger P, Horbury TS, Issautier K, Kontar E, Krucker S, Le Contel O, Louarn P, Martinovic M, Owen CJ, Retino A, Rodriguez-Pacheco J, Sahraoui F, Wimmer-Schweingruber RF, Zaslavsky A, Zouganelis Iet al., 2021, The Solar Orbiter Radio and Plasma Waves (RPW) instrument (vol 642, A12, 2020), ASTRONOMY & ASTROPHYSICS, Vol: 654, ISSN: 0004-6361 Journal article Tenerani A, Sioulas N, Matteini L, Panasenco O, Shi C, Velli Met al., 2021, Evolution of switchbacks in the inner heliosphere, Letters of the Astrophysical Journal, Vol: 919, Pages: 1-7, ISSN: 2041-8205 We analyze magnetic field data from the first six encounters of Parker Solar Probe, three Helios fast streams and two Ulysses south polar passes covering heliocentric distances 0.1 ≲ R ≲ 3 au. We use this data set to statistically determine the evolution of switchbacks of different periods and amplitudes with distance from the Sun. We compare the radial evolution of magnetic field variances with that of the mean square amplitudes of switchbacks, and quantify the radial evolution of the cumulative counts of switchbacks per kilometer. We find that the amplitudes of switchbacks decrease faster than the overall turbulent fluctuations, in a way consistent with the radial decrease of the mean magnetic field. This could be the result of a saturation of amplitudes and may be a signature of decay processes of large amplitude Alfvénic fluctuations in the solar wind. We find that the evolution of switchback occurrence in the solar wind is scale dependent: the fraction of longer-duration switchbacks increases with radial distance, whereas it decreases for shorter switchbacks. This implies that switchback dynamics is a complex process involving both decay and in situ generation in the inner heliosphere. We confirm that switchbacks can be generated by the expansion, although other types of switchbacks generated closer to the Sun cannot be ruled out. Journal article Laker R, Horbury TS, Bale SD, Matteini L, Woolley T, Woodham LD, Stawarz JE, Davies EE, Eastwood JP, Owens MJ, O'Brien H, Evans V, Angelini V, Richter I, Heyner D, Owen CJ, Louarn P, Fedorov Aet al., 2021, Multi-spacecraft study of the solar wind at solar minimum: Dependence on latitude and transient outflows, Astronomy and Astrophysics: a European journal, Vol: 652, Pages: 1-10, ISSN: 0004-6361 Context. The recent launches of Parker Solar Probe, Solar Orbiter (SO), and BepiColombo, along with several older spacecraft, have provided the opportunity to study the solar wind at multiple latitudes and distances from the Sun simultaneously.Aims. We take advantage of this unique spacecraft constellation, along with low solar activity across two solar rotations between May and July 2020, to investigate how the solar wind structure, including the heliospheric current sheet (HCS), varies with latitude.Methods. We visualise the sector structure of the inner heliosphere by ballistically mapping the polarity and solar wind speed from several spacecraft onto the Sun’s source surface. We then assess the HCS morphology and orientation with the in situ data and compare this with a predicted HCS shape.Results. We resolve ripples in the HCS on scales of a few degrees in longitude and latitude, finding that the local orientations of sector boundaries were broadly consistent with the shape of the HCS but were steepened with respect to a modelled HCS at the Sun. We investigate how several CIRs varied with latitude, finding evidence for the compression region affecting slow solar wind outside the latitude extent of the faster stream. We also identified several transient structures associated with HCS crossings and speculate that one such transient may have disrupted the local HCS orientation up to five days after its passage.Conclusions. We have shown that the solar wind structure varies significantly with latitude, with this constellation providing context for solar wind measurements that would not be possible with a single spacecraft. These measurements provide an accurate representation of the solar wind within ±10° latitude, which could be used as a more rigorous constraint on solar wind models and space weather predictions. In the future, this range of latitudes will increase as SO’s orbit becomes more inclined. Journal article Hellinger P, Papini E, Verdini A, Landi S, Franci L, Matteini L, Montagud-Camps Vet al., 2021, Spectral Transfer and Karman-Howarth-Monin Equations for Compressible Hall Magnetohydrodynamics, ASTROPHYSICAL JOURNAL, Vol: 917, ISSN: 0004-637X Journal article Laker R, Horbury TS, Bale SD, Matteini L, Woolley T, Woodham LD, Badman ST, Pulupa M, Kasper JC, Stevens M, Case AW, Korreck KEet al., 2021, Statistical analysis of orientation, shape, and size of solar wind switchbacks, Astronomy & Astrophysics, Vol: 650, Pages: 1-7, ISSN: 0004-6361 One of the main discoveries from the first two orbits of Parker Solar Probe(PSP) was the presence of magnetic switchbacks, whose deflections dominated themagnetic field measurements. Determining their shape and size could provideevidence of their origin, which is still unclear. Previous work with a singlesolar wind stream has indicated that these are long, thin structures althoughthe direction of their major axis could not be determined. We investigate ifthis long, thin nature extends to other solar wind streams, while determiningthe direction along which the switchbacks within a stream were aligned. We tryto understand how the size and orientation of the switchbacks, along with theflow velocity and spacecraft trajectory, combine to produce the observedstructure durations for past and future orbits. We searched for the alignmentdirection that produced a combination of a spacecraft cutting direction andswitchback duration that was most consistent with long, thin structures. Theexpected form of a long, thin structure was fitted to the results of the bestalignment direction, which determined the width and aspect ratio of theswitchbacks for that stream. The switchbacks had a mean width of $50,000 \,\rm{km}$, with an aspect ratio of the order of $10$. We find that switchbacksare not aligned along the background flow direction, but instead aligned alongthe local Parker spiral, perhaps suggesting that they propagate along themagnetic field. Since the observed switchback duration depends on how thespacecraft cuts through the structure, the duration alone cannot be used todetermine the size or influence of an individual event. For future PSP orbits,a larger spacecraft transverse component combined with more radially alignedswitchbacks will lead to long duration switchbacks becoming less common. Journal article Woodham L, Horbury T, Matteini L, Woolley T, Laker R, Bale S, Nicolaou G, Stawarz J, Stansby D, Hietala H, Larson D, Livi R, Verniero J, McManus M, Kasper J, Korreck K, Raouafi N, Moncuquet M, Pulupa Met al., 2021, Enhanced proton parallel temperature inside patches of switchbacks in the inner heliosphere, Astronomy and Astrophysics: a European journal, Vol: 650, Pages: 1-7, ISSN: 0004-6361 Context. Switchbacks are discrete angular deflections in the solar wind magnetic field that have been observed throughout the helio-sphere. Recent observations by Parker Solar Probe(PSP) have revealed the presence of patches of switchbacks on the scale of hours to days, separated by ‘quieter’ radial fields. Aims. We aim to further diagnose the origin of these patches using measurements of proton temperature anisotropy that can illuminate possible links to formation processes in the solar corona. Methods. We fit 3D bi-Maxwellian functions to the core of proton velocity distributions measured by the SPAN-Ai instrument onboard PSP to obtain the proton parallel, Tp,‖, and perpendicular, Tp,⊥, temperature. Results. We show that the presence of patches is highlighted by a transverse deflection in the flow and magnetic field away from the radial direction. These deflections are correlated with enhancements in Tp,‖, while Tp,⊥remains relatively constant. Patches sometimes exhibit small proton and electron density enhancements. Conclusions. We interpret that patches are not simply a group of switchbacks, but rather switchbacks are embedded within a larger-scale structure identified by enhanced Tp,‖that is distinct from the surrounding solar wind. We suggest that these observations are consistent with formation by reconnection-associated mechanisms in the corona. Journal article Gonzalez CA, Tenerani A, Matteini L, Hellinger P, Velli Met al., 2021, Proton Energization by Phase Steepening of Parallel-propagating Alfvenic Fluctuations, ASTROPHYSICAL JOURNAL LETTERS, Vol: 914, ISSN: 2041-8205 Journal article Martinovic MM, Klein KG, Huang J, Chandran BDG, Kasper JC, Lichko E, Bowen T, Chen CHK, Matteini L, Stevens M, Case AW, Bale SDet al., 2021, Multiscale Solar Wind Turbulence Properties inside and near Switchbacks Measured by the Parker Solar Probe, ASTROPHYSICAL JOURNAL, Vol: 912, ISSN: 0004-637X Journal article Hellinger P, Verdini A, Landi S, Papini E, Franci L, Matteini Let al., 2021, Scale dependence and cross-scale transfer of kinetic energy in compressible hydrodynamic turbulence at moderate Reynolds numbers, PHYSICAL REVIEW FLUIDS, Vol: 6, ISSN: 2469-990X Journal article Stawarz JE, Matteini L, Parashar TN, Franci L, Eastwood JP, Gonzalez CA, Gingell IL, Burch JL, Ergun RE, Ahmadi N, Giles BL, Gershman DJ, Le Contel O, Lindqvist P, Russell CT, Strangeway RJ, Torbert RBet al., 2021, Comparative analysis of the various generalized Ohm's law terms in magnetosheath turbulence as observed by magnetospheric multiscale, Journal of Geophysical Research: Space Physics, Vol: 126, Pages: 1-14, ISSN: 2169-9380 Decomposing the electric field (E) into the contributions from generalized Ohm's law provides key insight into both nonlinear and dissipative dynamics across the full range of scales within a plasma. Using high‐resolution, multi‐spacecraft measurements of three intervals in Earth's magnetosheath from the Magnetospheric Multiscale mission, the influence of the magnetohydrodynamic, Hall, electron pressure, and electron inertia terms from Ohm's law, as well as the impact of a finite electron mass, on the turbulent E spectrum are examined observationally for the first time. The magnetohydrodynamic, Hall, and electron pressure terms are the dominant contributions to E over the accessible length scales, which extend to scales smaller than the electron inertial length at the greatest extent, with the Hall and electron pressure terms dominating at sub‐ion scales. The strength of the non‐ideal electron pressure contribution is stronger than expected from linear kinetic Alfvén waves and a partial anti‐alignment with the Hall electric field is present, linked to the relative importance of electron diamagnetic currents in the turbulence. The relative contribution of linear and nonlinear electric fields scale with the turbulent fluctuation amplitude, with nonlinear contributions playing the dominant role in shaping E for the intervals examined in this study. Overall, the sum of the Ohm's law terms and measured E agree to within ∼ 20% across the observable scales. These results both confirm general expectations about the behavior of E in turbulent plasmas and highlight features that should be explored further theoretically. Journal article Woolley T, Matteini L, Horbury TS, Bale SD, Woodham LD, Laker R, Alterman BL, Bonnell JW, Case AW, Kasper JC, Klein KG, Martinović MM, Stevens Met al., 2020, Proton core behaviour inside magnetic field switchbacks, Monthly Notices of the Royal Astronomical Society, Vol: 498, Pages: 5524-5531, ISSN: 0035-8711 During Parker Solar Probe’s first two orbits there are widespread observations of rapid magnetic field reversals known as switchbacks. These switchbacks are extensively found in the near-Sun solar wind, appear to occur in patches, and have possible links to various phenomena such as magnetic reconnection near the solar surface. As switchbacks are associated with faster plasma flows, we questioned whether they are hotter than the background plasma and whether the microphysics inside a switchback is different to its surroundings. We have studied the reduced distribution functions from the Solar Probe Cup instrument and considered time periods with markedly large angular deflections, to compare parallel temperatures inside and outside switchbacks. We have shown that the reduced distribution functions inside switchbacks are consistent with a rigid velocity space rotation of the background plasma. As such, we conclude that the proton core parallel temperature is very similar inside and outside of switchbacks, implying that a T-V relationship does not hold for the proton core parallel temperature inside magnetic field switchbacks. We further conclude that switchbacks are consistent with Alfvénic pulses travelling along open magnetic field lines. The origin of these pulses, however, remains unknown. We also found that there is no obvious link between radial Poynting flux and kinetic energy enhancements suggesting that the radial Poynting flux is not important for the dynamics of switchbacks. Journal article Maksimovic M, Bale SD, Chust T, Khotyaintsev Y, Krasnoselskikh V, Kretzschmar M, Plettemeier D, Rucker HO, Soucek J, Steller M, Stverak S, Travnicek P, Vaivads A, Chaintreuil S, Dekkali M, Alexandrova O, Astier P-A, Barbary G, Berard D, Bonnin X, Boughedada K, Cecconi B, Chapron F, Chariet M, Collin C, de Conchy Y, Dias D, Gueguen L, Lamy L, Leray V, Lion S, Malac-Allain LR, Matteini L, Nguyen QN, Pantellini F, Parisot J, Plasson P, Thijs S, Vecchio A, Fratter I, Bellouard E, Lorfevre E, Danto P, Julien S, Guilhem E, Fiachetti C, Sanisidro J, Laffaye C, Gonzalez F, Pontet B, Queruel N, Jannet G, Fergeau P, Brochot J-Y, Cassam-Chenai G, de Wit TD, Timofeeva M, Vincent T, Agrapart C, Delory GT, Turin P, Jeandet A, Leroy P, Pellion J-C, Bouzid V, Katra B, Piberne R, Recart W, Santolik O, Kolmasova I, Krupar V, Kruparova O, Pisa D, Uhlir L, Lan R, Base J, Ahlen L, Andre M, Bylander L, Cripps V, Cully C, Eriksson A, Jansson S-E, Johansson EPG, Karlsson T, Puccio W, Brinek J, Oettacher H, Panchenko M, Berthomier M, Goetz K, Hellinger P, Horbury TS, Issautier K, Kontar E, Krucker S, Le Contel O, Louarn P, Martinovic M, Owen CJ, Retino A, Rodriguez-Pacheco J, Sahraoui F, Wimmer-Schweingruber RF, Zaslavsky A, Zouganelis Iet al., 2020, The Solar Orbiter Radio and Plasma Waves (RPW) instrument, ASTRONOMY & ASTROPHYSICS, Vol: 642, ISSN: 0004-6361 Journal article Matteini L, Franci L, Alexandrova O, Lacombe C, Landi S, Hellinger P, Papini E, Verdini Aet al., 2020, Magnetic field turbulence in the solar wind at sub-ion scales: in situ observations and numerical simulations, Frontiers in Astronomy and Space Sciences, ISSN: 2296-987X We investigate the transition of the solar wind turbulent cascade from MHD tosub-ion range by means of a detail comparison between in situ observations andhybrid numerical simulations. In particular we focus on the properties of themagnetic field and its component anisotropy in Cluster measurements and hybrid2D simulations. First, we address the angular distribution of wave-vectors inthe kinetic range between ion and electron scales by studying the varianceanisotropy of the magnetic field components. When taking into account thesingle-direction sampling performed by spacecraft in the solar wind, the mainproperties of the fluctuations observed in situ are also recovered in ournumerical description. This result confirms that solar wind turbulence in thesub-ion range is characterized by a quasi-2D gyrotropic distribution ofk-vectors around the mean field. We then consider the magnetic compressibilityassociated with the turbulent cascade and its evolution from large-MHD tosub-ion scales. The ratio of field-aligned to perpendicular fluctuations,typically low in the MHD inertial range, increases significantly when crossingion scales and its value in the sub-ion range is a function of the total plasmabeta only, as expected from theoretical predictions, with higher magneticcompressibility for higher beta. Moreover, we observe that this increase has agradual trend from low to high beta values in the in situ data; this behaviouris well captured by the numerical simulations. The level of magnetic fieldcompressibility that is observed in situ and in the simulations is in fairlygood agreement with theoretical predictions, especially at high beta,suggesting that in the kinetic range explored the turbulence is supported bylow-frequency and highly-oblique fluctuations in pressure balance, like kineticAlfv\'en waves or other slowly evolving coherent structures. Journal article Franci L, Stawarz JE, Papini E, Hellinger P, Nakamura T, Burgess D, Landi S, Verdini A, Matteini L, Ergun R, Contel OL, Lindqvist P-Aet al., 2020, Modeling MMS observations at the Earth's magnetopause with hybrid simulations of Alfvénic turbulence, The Astrophysical Journal, Vol: 898, ISSN: 0004-637X Magnetospheric Multiscale (MMS) observations of plasma turbulence generated by a Kelvin–Helmholtz (KH) event at the Earth's magnetopause are compared with a high-resolution two-dimensional (2D) hybrid direct numerical simulation of decaying plasma turbulence driven by large-scale balanced Alfvénic fluctuations. The simulation, set up with four observation-driven physical parameters (ion and electron betas, turbulence strength, and injection scale), exhibits a quantitative agreement on the spectral, intermittency, and cascade-rate properties with in situ observations, despite the different driving mechanisms. Such agreement demonstrates a certain universality of the turbulent cascade from magnetohydrodynamic to sub-ion scales, whose properties are mainly determined by the selected parameters, also indicating that the KH instability-driven turbulence has a quasi-2D nature. The fact that our results are compatible with the validity of the Taylor hypothesis, in the whole range of scales investigated numerically, suggests that the fluctuations at sub-ion scales might have predominantly low frequencies. This would be consistent with a kinetic Alfvén wave-like nature and/or with the presence of quasi-static structures. Finally, the third-order structure function analysis indicates that the cascade rate of the turbulence generated by a KH event at the magnetopause is an order of magnitude larger than in the ambient magnetosheath. Journal article Bandyopadhyay R, Sorriso-Valvo L, Chasapis A, Hellinger P, Matthaeus WH, Verdini A, Landi S, Franci L, Matteini L, Giles BL, Gershman DJ, Moore TE, Pollock CJ, Russell CT, Strangeway RJ, Torbert RB, Burch JLet al., 2020, In situ observation of hall magnetohydrodynamic cascade in space plasma, Physical Review Letters, Vol: 124, Pages: 225101 – 1-225101 – 7, ISSN: 0031-9007 We present estimates of the turbulent energy-cascade rate derived from a Hall-magnetohydrodynamic (MHD) third-order law. We compute the contribution from the Hall term and the MHD term to the energy flux. Magnetospheric Multiscale (MMS) data accumulated in the magnetosheath and the solar wind are compared with previously established simulation results. Consistent with the simulations, we find that at large (MHD) scales, the MMS observations exhibit a clear inertial range dominated by the MHD flux. In the subion range, the cascade continues at a diminished level via the Hall term, and the change becomes more pronounced as the plasma beta increases. Additionally, the MHD contribution to interscale energy transfer remains important at smaller scales than previously thought. Possible reasons are offered for this unanticipated result. Journal article Bercic L, Larson D, Whittlesey P, Maksimovic M, Badman ST, Landi S, Matteini L, Bale SD, Bonnell JW, Case AW, de Wit TD, Goetz K, Harvey PR, Kasper JC, Korreck KE, Livi R, MacDowall RJ, Malaspina DM, Pulupa M, Stevens MLet al., 2020, Coronal electron temperature inferred from the strahl electrons in the inner heliosphere: parker solar probe and helios observations, The Astrophysical Journal: an international review of astronomy and astronomical physics, Vol: 892, Pages: 1-14, ISSN: 0004-637X The shape of the electron velocity distribution function plays an important role in the dynamics of the solar wind acceleration. Electrons are normally modeled with three components, the core, the halo, and the strahl. We investigate how well the fast strahl electrons in the inner heliosphere preserve the information about the coronal electron temperature at their origin. We analyzed the data obtained by two missions, Helios, spanning the distances between 65 and 215 R S, and Parker Solar Probe (PSP), reaching down to 35 R S during its first two orbits around the Sun. The electron strahl was characterized with two parameters: pitch-angle width (PAW) and the strahl parallel temperature (T s∥). PSP observations confirm the already reported dependence of strahl PAW on core parallel plasma beta (${\beta }_{\mathrm{ec}\parallel }$). Most of the strahl measured by PSP appear narrow with PAW reaching down to 30°. The portion of the strahl velocity distribution function aligned with the magnetic field is for the measured energy range well described by a Maxwellian distribution function. T s∥ was found to be anticorrelated with the solar wind velocity and independent of radial distance. These observations imply that T s∥ carries the information about the coronal electron temperature. The obtained values are in agreement with coronal temperatures measured using spectroscopy, and the inferred solar wind source regions during the first orbit of PSP agree with the predictions using a PFSS model. Journal article DAmicis R, Matteini L, Bruno R, Velli Met al., 2020, Large amplitude fluctuations in the alfvénic solar wind, Solar Physics, Vol: 295, Pages: 1-12, ISSN: 0038-0938 Large amplitude fluctuations, often with characteristics reminiscent of large amplitude Alfvén waves propagating away from the Sun, are ubiquitous in the solar wind. Such features are most frequently found within fast solar wind streams and most often at solar minimum. The fluctuations found in slow solar wind streams usually have a smaller relative amplitude, are less Alfvénic in character and present more variability. However, intervals of slow wind displaying Alfvénic correlations have been recently identified in different solar cycle phases. In the present paper we report Alfvénic slow solar wind streams seen during the maximum of solar activity that are characterized not only by a very high correlation between velocity and magnetic field fluctuations (as required by outwardly propagating Alfvén modes) – comparable to that seen in fast wind streams – but also by higher amplitude relative fluctuations comparable to those seen in fast wind. Our results suggest that the Alfvénic slow wind has a different origin from the slow wind found near the boundary of coronal holes, where the amplitude of the Alfvénic fluctuations decreases together with decreasing the wind speed. Journal article Horbury T, Woolley T, Laker R, Matteini L, Eastwood J, Bale SD, Velli M, Chandran BDG, Phan T, Raouafi NE, Goetz K, Harvey PR, Pulupa M, Klein KG, De Wit TD, Kasper JC, Korreck KE, Case AW, Stevens ML, Whittlesey P, Larson D, MacDowall RJ, Malaspina DM, Livi Ret al., 2020, Sharp Alfvenic impulses in the near-Sun solar wind, The Astrophysical Journal: an international review of astronomy and astronomical physics, Vol: 246, Pages: 1-8, ISSN: 0004-637X Measurements of the near-Sun solar wind by Parker Solar Probe have revealed the presence of largenumbers of discrete Alfv ́enic impulses with an anti-Sunward sense of propagation. These are similarto those previously observed near 1 AU, in high speed streams over the Sun’s poles and at 60 solarradii. At 35 solar radii, however, they are typically shorter and sharper than seen elsewhere. Inaddition, these spikes occur in “patches” and there are also clear periods within the same stream whenthey do not occur; the timescale of these patches might be related to the rate at which the spacecraftmagnetic footpoint tracks across the coronal hole from which the plasma originated. While the velocityfluctuations associated with these spikes are typically under 100 km/s, due to the rather low Alfv ́enspeeds in the streams observed by the spacecraft to date, these are still associated with large angulardeflections of the magnetic field - and these deflections are not isotropic. These deflections do notappear to be related to the recently reported large scale, pro-rotation solar wind flow. Estimates ofthe size and shape of the spikes reveal high aspect ratio flow-aligned structures with a transverse scaleof≈104km. These events might be signatures of near-Sun impulsive reconnection events. Journal article Nemecek Z, Durovcova T, Safrankova J, Nemec F, Matteini L, Stansby D, Janitzek N, Berger L, Wimmer-Schweingruber RFet al., 2020, What is the solar wind frame of reference?, The Astrophysical Journal: an international review of astronomy and astronomical physics, Vol: 889, Pages: 1-14, ISSN: 0004-637X Various solar wind ion species move with different speeds and theoretical considerations as well as limited observations in a region close to the Sun show that heavy solar wind ions tend to flow faster than protons, at least in less-aged fast solar wind streams. The solar wind flow carries the frozen-in interplanetary magnetic field (IMF) and this situation evokes three related questions: (i) what is the proper solar wind speed, (ii) is this speed equal to the speed of the dominant component, whatever that may be, and (iii) what is the speed of the magnetic field? We show that simple theoretical considerations based on the MHD approximation as well as on the dynamics of charged particles in electric and magnetic fields suggest that the IMF velocity of motion (de Hoffmann–Teller (HT) velocity) would be deliberated as the velocity appropriate for solar wind studies. Our analysis based on the Wind, Helios, ACE, and SOHO observations of differential streaming of solar wind populations shows that their energy is conserved in the HT frame. On the other hand, the noise and temporal resolution of the data do not allow us to decide whether the total momentum is also conserved in this frame. Journal article Stansby D, Matteini L, Horbury TS, Perrone D, D'Amicis R, Bercic Let al., 2020, The origin of slow Alfvenic solar wind at solar minimum, Monthly Notices of the Royal Astronomical Society, Vol: 492, Pages: 39-44, ISSN: 0035-8711 Although the origins of slow solar wind are unclear, there is increasing evidence that at least some of it is released in a steady state on overexpanded coronal hole magnetic field lines. This type of slow wind has similar properties to the fast solar wind, including strongly Alfvénic fluctuations. In this study, a combination of proton, alpha particle, and electron measurements are used to investigate the kinetic properties of a single interval of slow Alfvénic wind at 0.35 au. It is shown that this slow Alfvénic interval is characterized by high alpha particle abundances, pronounced alpha–proton differential streaming, strong proton beams, and large alpha-to-proton temperature ratios. These are all features observed consistently in the fast solar wind, adding evidence that at least some Alfvénic slow solar wind also originates in coronal holes. Observed differences between speed, mass flux, and electron temperature between slow Alfvénic and fast winds are explained by differing magnetic field geometry in the lower corona. Journal article This data is extracted from the Web of Science and reproduced under a licence from Thomson Reuters. You may not copy or re-distribute this data in whole or in part without the written consent of the Science business of Thomson Reuters. Request URL: http://wlsprd.imperial.ac.uk:80/respub/WEB-INF/jsp/search-html.jsp Request URI: /respub/WEB-INF/jsp/search-html.jsp Query String: respub-action=search.html&id=00689226&limit=30&person=true	2022-09-27 04:13:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7155724167823792, "perplexity": 10064.823044173227}, "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/1664030334987.39/warc/CC-MAIN-20220927033539-20220927063539-00383.warc.gz"}
http://mathhelpforum.com/differential-geometry/146256-sum-power-series.html	# Math Help - sum of power series 1. ## sum of power series This might seem easy, but could anyone share the trick how to do the following sums? the sum of 8^n/(3n)!, n goes from 0 to infinity the sum of 27^n/(3n+1)!, n goes from 0 to infinity Any input is appreciated! 2. Originally Posted by nngktr This might seem easy, but could anyone share the trick how to do the following sums? the sum of 8^n/(3n)!, n goes from 0 to infinity Let $\omega$ be a complex cube root of 1. Then $\sum_{n=0}^\infty\frac{2^n}{n!} = e^2,\quad \sum_{n=0}^\infty\frac{(2\omega)^n}{n!} = e^{2\omega},\quad \sum_{n=0}^\infty\frac{(2\overline{\omega})^n}{n!} = e^{2\overline{\omega}}.$ However, $1^n+\omega^n + \overline{\omega}^{\,n}$ is equal to 3 if n is a multiple of 3, and 0 otherwise. Therefore $\sum_{n=0}^\infty\frac{8^n}{(3n)!} = \tfrac13\bigl(e^2 + e^{2\omega} + e^{2\overline{\omega}}\bigr).$ The expression on the right should obviously be real, so you need to plug in the value for $\omega$ and check that this is indeed the case.	2015-11-26 12:44:46	{"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": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6813103556632996, "perplexity": 324.36615649112565}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398447266.73/warc/CC-MAIN-20151124205407-00065-ip-10-71-132-137.ec2.internal.warc.gz"}
https://paperswithcode.com/paper/bandit-pam-almost-linear-time-k-medoids	# BanditPAM: Almost Linear Time $k$-Medoids Clustering via Multi-Armed Bandits Clustering is a ubiquitous task in data science. Compared to the commonly used $k$-means clustering, $k$-medoids clustering requires the cluster centers to be actual data points and support arbitrary distance metrics, which permits greater interpretability and the clustering of structured objects... Current state-of-the-art $k$-medoids clustering algorithms, such as Partitioning Around Medoids (PAM), are iterative and are quadratic in the dataset size $n$ for each iteration, being prohibitively expensive for large datasets. We propose BanditPAM, a randomized algorithm inspired by techniques from multi-armed bandits, that reduces the complexity of each PAM iteration from $O(n^2)$ to $O(n \log n)$ and returns the same results with high probability, under assumptions on the data that often hold in practice. As such, BanditPAM matches state-of-the-art clustering loss while reaching solutions much faster. We empirically validate our results on several large real-world datasets, including a coding exercise submissions dataset, the 10x Genomics 68k PBMC single-cell RNA sequencing dataset, and the MNIST handwritten digits dataset. In these experiments, we observe that BanditPAM returns the same results as state-of-the-art PAM-like algorithms up to 4x faster while performing up to 200x fewer distance computations. The improvements demonstrated by BanditPAM enable $k$-medoids clustering on a wide range of applications, including identifying cell types in large-scale single-cell data and providing scalable feedback for students learning computer science online. We also release highly optimized Python and C++ implementations of our algorithm. read more PDF Abstract	2021-12-05 11:52:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28359660506248474, "perplexity": 1576.4348101323621}, "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-2021-49/segments/1637964363157.32/warc/CC-MAIN-20211205100135-20211205130135-00549.warc.gz"}
https://testbook.com/question-answer/a-collapsible-soil-sub-grade-sample-was-tested-usi--5ee23cdb258a5a0d0e491ec8	# A collapsible soil sub-grade sample was tested using Standard California Bearing Ratio apparatus, and the observations are given below: Sl. No Load Penetration 1 60.55 kg 2.5 mm 2. 80.55 kg 5.0 mm Taking the standard assumptions regarding the load penetration curve, the CBR value of the sample will be taken as This question was previously asked in ESE Civil 2016 Paper 2: Official Paper View all UPSC IES Papers > 1. 39% 2. 40% 3. 4.4% 4. 5.5% Option 3 : 4.4% Free ST 1: Building Material and Concrete Technology 16847 20 Questions 20 Marks 12 Mins ## Detailed Solution Concept: CBR test is a strength test conducted on the soil by introducing a surcharge load at the compaction rate of 1.25 mm per minute on a completely soaked soil sample passing through 20 mm sieve size. $${\rm{CB}}{{\rm{R}}_{\rm{\delta }}} = \frac{{{{\rm{P}}_{\rm{\delta }}}{\rm{\;of\;soil}}}}{{{{\rm{P}}_{\rm{\delta }}}{\rm{\;of\;standard\;crushed\;aggregate}}}} \times 100$$ δ = displacement in mm Pδ = Load corresponding to ‘δ’ settlement Ps = Load for standard crushed aggregate\ For standard aggregate at 2.5 mm penetration, Load is 1370 kg For standard aggregate at 5 mm penetration, Load is 2055 kg Calculation: Given: Sl. No Load Penetration 1 60.55 kg 2.5 mm 2. 80.55 kg 5.0 mm $$CBR = \;\frac{{{\rm{Load\;carried\;by\;specimen}}}}{{{\rm{load\;carried\;by\;standard\;specimen}}}} \times 100$$ At 2.5 mm penetration, $$CBR = \;\frac{{60.5}}{{1370}} \times 100 = 4.4\%$$ At 5 mm penetration, $$CBR = \;\frac{{80.55}}{{2055}} \times 100 = 3.9\%$$ CBR is the maximum of the above two ratios i.e. 4.4%	2021-09-18 10:19:36	{"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": 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": 0.6494185924530029, "perplexity": 8871.156582927944}, "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/1631780056392.79/warc/CC-MAIN-20210918093220-20210918123220-00249.warc.gz"}
https://hackage.haskell.org/package/dejafu-0.9.0.0/docs/Test-DejaFu-Schedule.html	dejafu-0.9.0.0: Systematic testing for Haskell concurrency. Copyright (c) 2016 Michael Walker MIT Michael Walker experimental portable Safe Haskell2010 Test.DejaFu.Schedule Description Scheduling for concurrent computations. Synopsis # Scheduling newtype Scheduler state Source # A Scheduler drives the execution of a concurrent program. The parameters it takes are: 1. The last thread executed (if this is the first invocation, this is Nothing). 2. The runnable threads at this point. 3. The state. It returns a thread to execute, or Nothing if execution should abort here, and also a new state. Since: 0.8.0.0 Constructors data Decision Source # Scheduling decisions are based on the state of the running program, and so we can capture some of that state in recording what specific decision we made. Since: 0.5.0.0 Constructors Start ThreadId Start a new thread, because the last was blocked (or it's the start of computation). Continue Continue running the last thread for another step. SwitchTo ThreadId Pre-empt the running thread, and switch to another. Instances Source # Methods Source # MethodsshowList :: [Decision] -> ShowS # Source # Since: 0.5.1.0 Methodsrnf :: Decision -> () # Get the resultant thread identifier of a Decision, with a default case for Continue. Since: 0.5.0.0 Arguments Get the Decision that would have resulted in this thread identifier, given a prior thread (if any) and list of runnable threads. Since: 0.5.0.0 data NonEmpty a :: * -> * # Non-empty (and non-strict) list type. Since: 4.9.0.0 Constructors a :\| [a] infixr 5 Instances ## Preemptive A simple random scheduler which, at every step, picks a random thread to run. Since: 0.8.0.0 A round-robin scheduler which, at every step, schedules the thread with the next ThreadId. Since: 0.8.0.0 ## Non-preemptive A random scheduler which doesn't preempt the running thread. That is, if the last thread scheduled is still runnable, run that, otherwise schedule randomly. Since: 0.8.0.0 A round-robin scheduler which doesn't preempt the running thread. Since: 0.8.0.0 # Utilities Turn a potentially preemptive scheduler into a non-preemptive one. Since: 0.8.0.0	2020-02-21 10:27:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.23844905197620392, "perplexity": 13902.918402902955}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145500.90/warc/CC-MAIN-20200221080411-20200221110411-00283.warc.gz"}
http://mail-index.netbsd.org/port-vax/2012/12/21/msg001408.html	Port-vax archive # Re: DEC Video Connector On Thu, 20 Dec 2012 16:27:58 -0700 emanuel stiebler <emu%e-bbes.com@localhost> wrote: > So I finally wanted to do it right and make some cables which go > directly from the 3W3 to VGA ... Cut off the BNC connectors and solder a DE15 to that end? > And the solder-able version was just an idea to make a framerate > converter box, because most of the TFT-panels just deal with 60Hz, > and DEC had many, many other ideas about resolutions & frequencies > back then ;-) I have a crappy consumer grade 27" LCD. Yet it accepted the sync on green output with 66 Hz or 72 Hz Vsync from my VS4k90 without any problems. The same with an older 19" LCD. -- \end{Jochen} \ref{http://www.unixag-kl.fh-kl.de/~jkunz/} Home \| Main Index \| Thread Index \| Old Index	2014-03-15 21:22:15	{"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": 0.3423660695552826, "perplexity": 12318.264625963555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394678699721/warc/CC-MAIN-20140313024459-00071-ip-10-183-142-35.ec2.internal.warc.gz"}
https://www.neetprep.com/question/25691-/55-Physics--Motion-Plane/678-Motion-Plane	A projectile is given an initial velocity of $\stackrel{^}{i}+2\stackrel{^}{j}$. The cartesian equation of its path is (g = 10 ${\mathrm{ms}}^{-2}$ 1. $\mathrm{y}=2\mathrm{x}-5{\mathrm{x}}^{2}$ 2. $\mathrm{y}=\mathrm{x}-5{\mathrm{x}}^{2}$ 3. $4\mathrm{y}=2\mathrm{x}-5{\mathrm{x}}^{2}$ 4. $\mathrm{y}=2\mathrm{x}-25{\mathrm{x}}^{2}$ Concept Questions :- Projectile motion High Yielding Test Series + Question Bank - NEET 2020 Difficulty Level: A ship A is moving westwards with a speed of 10 km ${\mathrm{h}}^{-1}$ and a ship B, 100 km south of A is moving northwards with a speed of 10 km ${\mathrm{h}}^{-1}$. The time after which the distance between them becomes the shortest, is: 1. 5 hr 2. $5\sqrt{2}$ hr 3. $10\sqrt{2}$ hr 4. 0 hr Concept Questions :- Relative motion High Yielding Test Series + Question Bank - NEET 2020 Difficulty Level: Time taken by the projectile to reach from A to B is t. Then the distance AB is equal to : 1. $\frac{ut}{\sqrt{3}}$ 2. $\frac{\sqrt{3}ut}{2}$ 3. $\sqrt{3}ut$ 4. 2ut Concept Questions :- Projectile motion High Yielding Test Series + Question Bank - NEET 2020 Difficulty Level: A particle projected with kinetic energy ${\mathrm{k}}_{0}$ with an angle of projection $\mathrm{\theta }$. Then the variation of kinetic K with vertical displacement y is 1. linear 2. parabolic 3. hyperbolic 4. periodic Concept Questions :- Projectile motion High Yielding Test Series + Question Bank - NEET 2020 Difficulty Level: Three particles moving with constant velocities and V respectively as given in the figure. After some time all three particles are in the same line, then relation among and V is 1. 2. 3. 4. Concept Questions :- Speed and velocity High Yielding Test Series + Question Bank - NEET 2020 Difficulty Level: A river is flowing with a speed of 1 km/hr. A swimmer wants to go to point 'C' starting from 'A'. He swims with a speed of 5 km/hr, at an angle $\theta$ w.r.t. the river. If AB=BC=400m. Then- (1) time taken by the man is 12 min (2) time taken by the man is 8 min (3) the value of $\theta$ is 45$°$ (4) the value of $\theta$ is 53$°$ Concept Questions :- Relative motion High Yielding Test Series + Question Bank - NEET 2020 Difficulty Level: A body is thrown horizontally with a velocity $\sqrt{2gh}$ from the top of a tower of height h. It strikes the level ground through the foot of the tower at a distance x from the tower. The value of x is: (1) h (2) $\frac{h}{2}$ (3) 2h (4) $\frac{2h}{3}$ Concept Questions :- Projectile motion High Yielding Test Series + Question Bank - NEET 2020 Difficulty Level: A particle starts from the origin at t=0 and moves in the x-y plane with constant acceleration 'a' in the y direction. Its equation of motion is $y=b{x}^{2}$. The x component of its velocity (at t=0) is: (1) variable (2) $\sqrt{\frac{2a}{b}}$ (3) $\frac{a}{2b}$ (4) $\sqrt{\frac{a}{2b}}$ Concept Questions :- Acceleration High Yielding Test Series + Question Bank - NEET 2020 Difficulty Level: A body projected with velocity u with an angle of projection $\theta$. Change in velocity after the time (t) from the projection is: (1) gt (2) $\frac{1}{2}g{t}^{2}$ (3) u sin$\theta$ (4) u cos$\theta$ Concept Questions :- Projectile motion High Yielding Test Series + Question Bank - NEET 2020 Difficulty Level: A particle has initial velocity $\left(3\stackrel{^}{\mathrm{i}}+4\stackrel{^}{\mathrm{j}}\right)$ and has acceleration $\left(0.4\stackrel{^}{\mathrm{i}}+0.3\stackrel{^}{\mathrm{j}}\right)$. Its speed after 10 s 1. 7 units 2. $7\sqrt{2}$ units 3. 8.5 units 4. 10 units Concept Questions :- Uniformly accelerated motion	2020-05-27 08:35:58	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 34, "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": 0.8528977632522583, "perplexity": 2285.6464056370696}, "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/1590347392142.20/warc/CC-MAIN-20200527075559-20200527105559-00101.warc.gz"}
https://lists.gnu.org/archive/html/lilypond-devel/2007-08/msg00158.html	lilypond-devel [Top][All Lists] ## \bar "\|:" - wrong position in the first misure From: Mattia Giovanetti Subject: \bar "\|:" - wrong position in the first misure Date: Wed, 29 Aug 2007 11:42:36 +0200 Hi, first of all I must thank you for your wonderful program, Lilypond it's fantastic and you did a marvellous work, really super. I'm an intalian classical musician, my intrument is the guitar, I study at Conservatorio, actually I'm at the last year. I think that your program got a notation bug. This involve the command \bar "\|:" in the first misure, I attach some files at this mail, so you can take a look of and get what I mean. But in a simple word the problem/bug is that: "the repeat double line with dots, must be after the time signature not befoure in the first misure". Please take a look at 120 Arpeggi di Mauro Giuliani, it's a true example. I hope you will fix that bug as soon as possible! Thanks to you. Salute e Prosperità. Mattia Giovanetti. http://xoomer.virgilio.it/naturaljazz quite right.jpg Description: JPEG image quite right.ly Description: Text Data wrong.jpg Description: JPEG image wrong.ly Description: Text Data	2019-05-26 12:34:44	{"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": 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": 0.863027811050415, "perplexity": 7511.2335802724065}, "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-2019-22/segments/1558232259126.83/warc/CC-MAIN-20190526105248-20190526131248-00441.warc.gz"}
http://math.soimeme.org/~arunram/Resources/Cherednik/LOAKZEQMBPHAAMTIntroductionHeckeAlgebrasInRepresentationTheory.html	## Lectures on affine Knizhnik-Zamolodchikov equations, quantum many body problems, Hecke algebras, and Macdonald theory Last update: 28 April 2014 ## Notes and References This is an excerpt of the paper Lectures on affine Knizhnik-Zamolodchikov equations, quantum many body problems, Hecke algebras, and Macdonald theory by Ivan Cherednik, in collaboration with Etsuro Date, Kenji Iohara, Michio Jimbo, Masaki Kashiwara, Tetsuji Miwa, Masatoshi Noumi, and Yoshihisa Saito. ## Introduction: Hecke algebras in representation theory Before a systematic exposition, I will try to outline the main directions of the representation theory and harmonic analysis connected with the Macdonald theory. A couple of remarks about the growth of Mathematics. It can be illustrated (with all buts and ifs) by the following diagram. $Imaginary axis (conceptual mathematics) Real axis (special functions, numbers) Figure. 1. Real and Imaginary$ It is extremely fast in the imaginary (conceptual) direction but very slow in the real direction. Mainly I mean modern mathematics, but it may be more general. For instance, ancient Greeks created a highly conceptual axiomatic geometry with a modest 'real output'. I do not think that the ratio Real/Imaginary is much higher now. There are many theories and a very limited number of functions which are really special. Let us try to project the representation theory on the real axis (Fig.1). We focus on Lie groups (algebras) and Kac-Moody algebras, ignoring the arithmetic direction (adàles and automorphic forms). Look at Fig.2. 1): By this I mean the zonal spherical functions on $K\G/K$ for maximal compact $K$ in a semi-simple Lie group $G\text{.}$ The theory was started by Gelfand et al. in the early 50's and completed by Harish-Chandra and many others. It generalized quite a few classical special functions. Lie groups helped a lot to elaborate a systematic approach, although much can be done without them, as we will see below. 2): The characters of Kac-Moody algebras can also be introduced without any representation theory (Looijenga, Saito). They are not too far from the products of classical one-dimensional $\theta \text{-functions.}$ However it is a new and very important class of special functions with various applications. The representation theory explains well some of their properties (but not all). 3): This construction gives a lot of remarkable combinatorial formulas, and generating functions. Decomposing tensor products of finite dimensional representations of compact Lie groups was in the focus of representation theory in the 70's and early 80's, as well as various restriction problems. This direction is still very important, but the representation theory moved towards infinite-dimensional objects. $Im Re 1\phantom{\rule{1em}{0ex}}Spherical functions 2\phantom{\rule{1em}{0ex}}Characters of KM algebras 3\phantom{\rule{1em}{0ex}}\left[{V}_{\lambda }\otimes {V}_{\mu }:{V}_{\nu }\right] (irreps of dim <\infty ) 4\phantom{\rule{1em}{0ex}}\left[{M}_{\lambda }:{L}_{\mu }\right] (induced: irreps) Representation theory of Lie groups, Lie algebras, and Kac-Moody algebras Figure. 2. Representation Theory$ 4): Here the problem is to calculate the multiplicities of irreducible representations of Lie algebras in the Verma modules or other induced representations. It is complicated. It took time to realize that these multiplicities are 'real'. Let us update the picture adding the results which were obtained in the 80's and 90's. $Representation theory Im Re 1\phantom{\rule{1em}{0ex}}Spherical fns 2\phantom{\rule{1em}{0ex}}KM characters 3\phantom{\rule{1em}{0ex}}\left[{V}_{\lambda }\otimes {V}_{\mu }:{V}_{\lambda }\right] 4\phantom{\rule{1em}{0ex}}\left[{M}_{\lambda }:{L}_{\mu }\right] \stackrel{\sim }{1}\phantom{\rule{1em}{0ex}}Generalized hypergeom. functions \stackrel{\sim }{2}\phantom{\rule{1em}{0ex}}Conformal blocks \stackrel{\sim }{3}\phantom{\rule{1em}{0ex}}Verlinde algebras \stackrel{\sim }{4}\phantom{\rule{1em}{0ex}}Modular reps Figure. 3. New Vintage$ $\stackrel{\sim }{1}\text{):}$ These functions will be the subject of my mini-course. We will study them in the differential and difference cases. It was an old question of how to introduce and generalize them using the representation theory. Now we have an answer. $\stackrel{\sim }{2}\text{):}$ Actually conformal blocks belong to the imaginary axis (conceptual mathematics). Only some of them can be considered as 'real' functions. Mostly it happens in the case of KZ-Bernard equation (a sort of elliptic KZ). $\stackrel{\sim }{3}\text{):}$ By Verlinde algebras, we mean the category of integrable representations of Kac-Moody algebras of given level with the fusion instead of tensoring. They can be also defined using quantum groups at roots of unity (Kazhdan-Lusztig). $\stackrel{\sim }{4}\text{):}$ Whatever you think about the 'reality' of $\left[{M}_{\lambda }:{L}_{\mu }\right],$ these multiplicities are connected with modular representations including the representations of the symmetric group over fields of finite characteristic. Nothing can be more real. Conjecture. The real projection of the representation theory goes through Hecke-type algebras. As to the examples under discussion the picture is as follows: $Representation theory Representation theory of Hecke algebras Macdonald theory, double Hecke algebras Kazhdan-Lusztig polynomials Im Re 1\phantom{\rule{1em}{0ex}}Spherical fns 2\phantom{\rule{1em}{0ex}}KM characters 3\phantom{\rule{1em}{0ex}}\left[{V}_{\lambda }\otimes {V}_{\mu }:{V}_{\lambda }\right] 4\phantom{\rule{1em}{0ex}}\left[{M}_{\lambda }:{L}_{\mu }\right] \stackrel{\sim }{1}\phantom{\rule{1em}{0ex}}Hypergeom. functions \stackrel{\sim }{2}\phantom{\rule{1em}{0ex}}Conformal blocks \stackrel{\sim }{3}\phantom{\rule{1em}{0ex}}Verlinde algebras \stackrel{\sim }{4}\phantom{\rule{1em}{0ex}}Modular reps a \stackrel{\sim }{a} \stackrel{\sim }{b} ? \stackrel{\sim }{b} ? ! \stackrel{\sim }{c} c ?! \stackrel{\sim }{d} d Figure. 4. Hecke Algebras$ a): This arrow seems the most recognized now. Several questions in the Harish-Chandra theory (the zonal case) were covered by the representation theory of the degenerate (graded) affine Hecke algebras defined by Lusztig [Lus1989]. For instance, the operators from [Che1991-2, Che1994-3] give a very simple approach to the radial parts of Laplace operators on symmetric spaces and the Harish-Chandra isomorphism. The hypergeometric functions (the arrow $\text{(}\stackrel{\sim }{\text{a}}\text{))}$ appear naturally in this way. Here the main expectations are connected with the difference theory. It was demonstrated in [Che1995-2] that the difference Fourier transform is self-dual (it is not in the differential case). At least it holds for certain classes of functions. It must simplify and generalize the Harish-Chandra theory. The same program was started in the $p\text{-adic}$ representation theory (see [Che1995-3, Che1996]). The coincidence of some difference spherical functions with proper Macdonald polynomials can be established using quantum groups (Noumi and others- see[Nou1992]). However at the moment the Hecke algebra technique is more efficient to deal with these polynomials (especially for arbitrary root systems). b): The double Hecke algebras lead to a certain elliptic generalization of the Macdonald polynomials [Che1995-4, Che1995-5, Che1996]. In the differential case there is also the so-called parabolic operator (see [EKi1993] and [Che1995-4]). Still it is not what one could expect. As to $\text{(}\stackrel{\sim }{\text{b}}\text{),}$ the conformal blocks of type ${GL}_{n}$ (i.e. over the products of curves with the action of the symmetric group) are much more general than the characters. Obviously Hecke algebras are not enough to get all of them. On the other hand, there is almost no theory of the conformal blocks for the configuration spaces connected with other root systems. Double affine Hecke algebras work well for all root systems. c): Here one can rediscover the same combinatorial formulas (mostly based on the so-called Kostant partition function). I do not expect anything brand new. However if you switch to the spherical functions (instead of the characters) then the new theory results in the formulas for the products of spherical functions, which cannot be obtained in the classical theory (they require the difference setting). The multiplicities $\left[{V}_{\lambda }\otimes {V}_{\mu }:{V}_{\nu }\right]$ govern the products of the characters, which are the same in the differential and difference theory. Concerning $\text{(}\stackrel{\sim }{\text{c}}\text{),}$ the Macdonald theory at roots of unity gave a simple approach to the Verlinde algebras. All the results about the inner product and the action of ${SL}_{2}\left(ℤ\right)$ were generalized a lot. I mean [Kir1995], and my two papers [Che1995-2, Che1995-3]. A. Kirillov Jr. was the first to find a one-parametric deformation of the Verlinde algebra in the case of ${GL}_{n}\text{.}$ He used quantum groups at roots of unity. My technique is applicable to all root systems. The proofs are much simpler than those based on Kac-Moody algebras or quantum groups. It works even better for the non-symmetric Macdonald polynomials (the conformal blocks and Kac-Moody characters are symmetric in contrast to the main classical elliptic functions). d): This arrow is the Kazhdan-Lusztig conjecture proved by Brylinski-Kashiwara and Beilinson-Bernstein and then generalized to the Kac-Moody case by Kashiwara-Tanisaki. By $\text{(}\stackrel{\sim }{\text{b}}\text{),}$ I mean the modular Lusztig conjecture (partially) proved by Anderson, Jantzen, and Soergel. The arrow from the Macdonald theory to modular representations is marked by '!'. It seems the most challenging now. I hope to continue my results on the Macdonald polynomials at roots of unity from the restricted case (alcove) to arbitrary weights (parallelogram). If might give a one-parametric generalization of the classic theory, formulas for the modular characters (not only those for the multiplicities), and a description of modular representations of arbitrary Weyl groups. However now it looks very difficult. To conclude, let me say a little something about the Verlinde algebras. I think now it is the most convincing demonstration of new methods based on double Hecke algebras. I also have certain personal reasons to be very interested in them. The conformal fusion procedure appeared in my paper 'Functional realization of basic representations of factorizable groups and Lie algebras' (Funct. Anal. Appl., 19 (1985), 36-52). Given an integrable representation of the $n\text{-th}$ power of a Kac-Moody algebra and two sets of points on a Riemann surface $\text{(}n$ and $m$ points), I constructed an integrable representation of the $m\text{-th}$ power of the same Kac-Moody algebra. The central charge here remains fixed. I missed that in the special case when $n=2,$ $m=1$ the multiplicities of irreducibles in the resulting representation are structural constants of a certain commutative algebra, the Verlinde algebra. It was nice to know that these multiplicities (and much more) can be extracted from the simplest representation of the double affine Hecke algebra at roots of unity. I should add one more remark. In fact I borrowed the 'fusion procedure' from arithmetics. I had known Ihara's papers 'On congruence monodromy problem' very well. A similar procedure was the key stone of his theory. Of course I changed something and added something (central charge), but the procedure is basically the same. Can we go back and define Verlinde algebras in arithmetics?	2019-01-23 00:07:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 61, "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": 0.737789511680603, "perplexity": 457.032498849205}, "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-2019-04/segments/1547583875448.71/warc/CC-MAIN-20190122223011-20190123005011-00636.warc.gz"}
https://docs.galpy.org/en/v1.1/index.html	Welcome to galpy’s documentation¶ galpy is a Python 2 and 3 package for galactic dynamics. It supports orbit integration in a variety of potentials, evaluating and sampling various distribution functions, and the calculation of action-angle coordinates for all static potentials. Papers using galpy¶ galpy is described in detail in this publication: • galpy: A Python Library for Galactic Dynamics, Jo Bovy (2015), Astrophys. J. Supp., 216, 29 (arXiv/1412.3451). The following is a list of publications using galpy; please let me (bovy -at- ias.edu) know if you make use of galpy in a publication. 1. Tracing the Hercules stream around the Galaxy, Jo Bovy (2010), Astrophys. J. 725, 1676 (2010ApJ...725.1676B): Uses what later became the orbit integration routines and Dehnen and Shu disk distribution functions. 2. The spatial structure of mono-abundance sub-populations of the Milky Way disk, Jo Bovy, Hans-Walter Rix, Chao Liu, et al. (2012), Astrophys. J. 753, 148 (2012ApJ...753..148B): Employs galpy orbit integration in galpy.potential.MWPotential to characterize the orbits in the SEGUE G dwarf sample. 3. On the local dark matter density, Jo Bovy & Scott Tremaine (2012), Astrophys. J. 756, 89 (2012ApJ...756...89B): Uses galpy.potential force and density routines to characterize the difference between the vertical force and the surface density at large heights above the MW midplane. 4. The Milky Way’s circular velocity curve between 4 and 14 kpc from APOGEE data, Jo Bovy, Carlos Allende Prieto, Timothy C. Beers, et al. (2012), Astrophys. J. 759, 131 (2012ApJ...759..131B): Utilizes the Dehnen distribution function to inform a simple model of the velocity distribution of APOGEE stars in the Milky Way disk and to create mock data. 5. A direct dynamical measurement of the Milky Way’s disk surface density profile, disk scale length, and dark matter profile at 4 kpc < R < 9 kpc, Jo Bovy & Hans-Walter Rix (2013), Astrophys. J. 779, 115 (2013ApJ...779..115B): Makes use of potential models, the adiabatic and Staeckel actionAngle modules, and the quasiisothermal DF to model the dynamics of the SEGUE G dwarf sample in mono-abundance bins. 6. The peculiar pulsar population of the central parsec, Jason Dexter & Ryan M. O’Leary (2013), Astrophys. J. Lett., 783, L7 (2014ApJ...783L...7D): Uses galpy for orbit integration of pulsars kicked out of the Galactic center. 7. Chemodynamics of the Milky Way. I. The first year of APOGEE data, Friedrich Anders, Christina Chiappini, Basilio X. Santiago, et al. (2013), Astron. & Astrophys., 564, A115 (2014A&A...564A.115A): Employs galpy to perform orbit integrations in galpy.potential.MWPotential to characterize the orbits of stars in the APOGEE sample. 8. Dynamical modeling of tidal streams, Jo Bovy (2014), Astrophys. J., 795, 95 (2014ApJ...795...95B): Introduces galpy.df.streamdf and galpy.actionAngle.actionAngleIsochroneApprox for modeling tidal streams using simple models formulated in action-angle space (see the tutorial above). 9. The Milky Way Tomography with SDSS. V. Mapping the Dark Matter Halo, Sarah R. Loebman, Zeljko Ivezic Thomas R. Quinn, Jo Bovy, Charlotte R. Christensen, Mario Juric, Rok Roskar, Alyson M. Brooks, & Fabio Governato (2014), Astrophys. J., 794, 151 (2014ApJ...794..151L): Uses galpy.potential functions to calculate the acceleration field of the best-fit potential in Bovy & Rix (2013) above. 10. The power spectrum of the Milky Way: Velocity fluctuations in the Galactic disk, Jo Bovy, Jonathan C. Bird, Ana E. Garcia Perez, Steven M. Majewski, David L. Nidever, & Gail Zasowski (2015), Astrophys. J., 800, 83 (arXiv/1410.8135): Uses galpy.df.evolveddiskdf to calculate the mean non-axisymmetric velocity field due to different non-axisymmetric perturbations and compares it to APOGEE data. 11. Generation of mock tidal streams, Mark A. Fardal, Shuiyao Huang, & Martin D. Weinberg (2014), Mon. Not. Roy. Astron. Soc., submitted (arXiv/1410.1861): Uses galpy.potential and galpy.orbit for orbit integration in creating a particle-spray model for tidal streams. 12. The nature and orbit of the Ophiuchus stream, Branimir Sesar, Jo Bovy, Edouard J. Bernard, et al. (2015), Astrophys. J., submitted (arXiv/1501.00581): Uses the Orbit.fit routine in galpy.orbit to fit the orbit of the Ophiuchus stream to newly obtained observational data and the routines in galpy.df.streamdf to model the creation of the stream. 13. The LMC geometry and outer stellar populations from early DES data, Eduardo Balbinot, B. X. Santiago, L. Girardi, et al. (2015), Mon. Not. Roy. Astron. Soc., 449, 1129 (arXiv/1502.05050): Employs galpy.potential.MWPotential as a mass model for the Milky Way to constrain the mass of the LMC. 14. Young Pulsars and the Galactic Center GeV Gamma-ray Excess, Ryan M. O’Leary, Matthew D. Kistler, Matthew Kerr, & Jason Dexter (2015), Phys. Rev. Lett., submitted (arXiv/1504.02477): Uses galpy orbit integration and galpy.potential.MWPotential2014 as part of a Monte Carlo simulation of the Galactic young-pulsar population. Acknowledging galpy¶ If you use galpy in a publication, please cite the following paper • galpy: A Python Library for Galactic Dynamics, Jo Bovy (2015), Astrophys. J. Supp., 216, 29 (arXiv/1412.3451). and link to http://github.com/jobovy/galpy. Please also send me a reference to the paper or send a pull request including your paper in the list of galpy papers on this page (this page is at doc/source/index.rst). Thanks! When using the galpy.actionAngle.actionAngleAdiabatic and galpy.actionAngle.actionAngleStaeckel modules, please cite 2013ApJ...779..115B in addition to the papers describing the algorithm used. When using galpy.actionAngle.actionAngleIsochroneApprox, please cite 2014ApJ...795...95B, which introduced this technique.	2020-07-03 13:21:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.6801038980484009, "perplexity": 8885.500221760196}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655882051.19/warc/CC-MAIN-20200703122347-20200703152347-00315.warc.gz"}
https://socratic.org/questions/58ab8d71b72cff2e5e40f123	Question #0f123 Feb 21, 2017 Phase angle of the light radiation. Explanation: “k” is often used for a particular constant, usually with a clarifying subscript. In this case we can use dimensional analysis to at least see what form it takes. Lambda “λ” is usually wavelength in this context, and ‘r’ would be a radius to define an circumference (2πr). So if we use SI units with meters for length, we have a circumference divided by a wavelength will equal our ‘k’. $k = \frac{2 \cdot \pi \cdot r}{\lambda}$. This is a dimensionless number that is the “phase angle” of the light. (See https://www.quora.com/If-the-wave-is-completely-in-phase-the-circumference-of-the-orbit-must-be-equal-to-an-integral-multiple-of-the-wavelength-λ ) And: http://www.insula.com.au/physics/1111/L3.html (distance from start)/wavelength = phase angle/$\left(2 \cdot \pi\right)$ $\frac{x}{\lambda} = \frac{\phi}{2} \cdot \pi$ rearranged to $\lambda = \frac{2 \cdot \pi \cdot r}{k}$. Where I have substituted ‘k’ for the ‘phi’ and ‘r’ for the ‘x’. $k = \frac{2 \cdot \pi \cdot r}{\lambda}$.	2021-10-27 15:17:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 5, "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": 0.9731770157814026, "perplexity": 2002.7902667142423}, "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-2021-43/segments/1634323588216.48/warc/CC-MAIN-20211027150823-20211027180823-00158.warc.gz"}
https://en.softpython.org/tuples/tuples-sol.html	# Tuple¶ Browse files online A tuple in Python is an immutable sequence of heterogenous elements which allows duplicates, so we can put inside the objects we want, of different types, and with repetitions. ## What to do¶ 1. Unzip exercises zip in a folder, you should obtain something like this: tuples tuples.ipynb tuples-sol.ipynb jupman.py WARNING: to correctly visualize the notebook, it MUST be in an unzipped folder ! 1. open Jupyter Notebook from that folder. Two things should open, first a console and then a browser. The browser should show a file list: navigate the list and open the notebook tuples.ipynb 2. Go on reading the exercises file, sometimes you will find paragraphs marked Exercises which will ask to write Python commands in the following cells. Exercises are graded by difficulty, from one star ✪ to four ✪✪✪✪ Shortcut keys: • to execute Python code inside a Jupyter cell, press Control + Enter • to execute Python code inside a Jupyter cell AND select next cell, press Shift + Enter • to execute Python code inside a Jupyter cell AND a create a new cell aftwerwards, press Alt + Enter • If the notebooks look stuck, try to select Kernel -> Restart ## Creating tuples¶ Tuples are created with round parenthesis () and by separating the elements with commas , Some example: [2]: numbers = (6,7,5,7,7,9) [3]: print(numbers) (6, 7, 5, 7, 7, 9) Tuples of one element: You can create a tuple of a single element by adding a comma after the element: [4]: little_tup = (4,) # notice the comma !!! Let’s verify the type is the expected one: [5]: type(little_tup) [5]: tuple To see the difference, we write down here (4) without comma and we verify the type of the obtained object: [6]: fake = (4) [7]: type(fake) [7]: int We see that fake is an int, because 4 has been evaluated as an expression inside round brackets so the result is the content inside the parenthesis. ### Empty tuple¶ We can also create an empty tuple: [8]: empty = () [9]: print(empty) () [10]: type(empty) [10]: tuple ### Tuples without brackets¶ When we assign values to some variable, (and only when we assign values to variables) it is possible to use a notation like the following, in which on the left of = we put names of variables and on the right we place a sequence of values: [11]: a,b,c = 1, 2, 3 [12]: a [12]: 1 [13]: b [13]: 2 [14]: c [14]: 3 If we ask ourselves what that 1,2,3 is, we can try putting on the left a single variable: [15]: # WARNING: BETTER AVOID THIS! x = 1,2,3 [16]: type(x) [16]: tuple We see that Python considered that 1,2,3 as a tuple. Typically, you would never write assignments with less variables than values to put, but if it happens, probably you will find yourself with some undesired tuple ! QUESTION: Have a look at the following code snippets, and for each try guessing which result it produces (or if it gives an error) 1. z,w = 5,6 print(type(z)) print(type(w)) 2. a,b = 5,6 a,b = b,a print('a=',a) print('b=',b) 3. z = 5, print(type(z)) 4. z = , print(type(z)) ### Heterogenous elements¶ In a tuple we can put elements of different types, like numbers and strings: [17]: stuff = (4, "paper", 5, 2,"scissors", 7) [18]: stuff [18]: (4, 'paper', 5, 2, 'scissors', 7) [19]: type(stuff) [19]: tuple We can also insert other tuples: [20]: salad = ( ("lettuce", 3), ("tomatoes",9), ("carrots",4) ) [21]: salad [21]: (('lettuce', 3), ('tomatoes', 9), ('carrots', 4)) [22]: type(salad) [22]: tuple And also lists: [23]: mix = ( ["when", "it", "rains"], ["I", "program"], [7,3,9] ) WARNING: avoid mutable objects inside tuples! Inserting mutable objects like lists inside tuples may cause problems in some situations like when you later want to use the tuple as element of a set or a key in a dictionary (we will see the details in the respective tutorials) Let’s see how the previous examples are represented in Python Tutor: [24]: # WARNING: before using the function jupman.pytut() which follows, # it is necessary to first execute this cell with Shift+Enter (once is enough) import jupman [25]: stuff = (4, "paper", 5, 2,"scissors", 7) salad = ( ("lettuce", 3), ("tomatoes",9), ("carrots",4) ) mix = ( ["when", "it", "rains"], ["I", "program"], [7,3,9] ) jupman.pytut() [25]: ### Creating tuples from sequences¶ You can create a tuple from any sequence, like for example a list: [26]: tuple( [8,2,5] ) [26]: (8, 2, 5) Or a string (which is a character sequence): [27]: tuple("abc") [27]: ('a', 'b', 'c') ### Creating sequences from tuples¶ Since the tuple is a sequence, it is also possible to generate lists from tuples: [28]: list( (3,4,2,3) ) [28]: [3, 4, 2, 3] QUESTION: Does is it make sense creating a tuple from another tuple like this? Can we rewrite the code in a more concise way? [29]: x = (4,2,5) y = tuple(x) QUESTION: Have a look at the following expressions, and for each try to guess which result produces (or if it gives an error): 1. (1.2,3.4) 2. (1;2;3;4) 3. (1,2;3,4) 4. (1,2,3,4) 5. (()) 6. type(()) 7. ((),) 8. tuple([('a'),('b'),('c')]) 9. tuple(tuple(('z','u','m'))) 10. str(('a','b','c')) 11. "".join(('a','b','c')) ## Operators¶ The following operators work on tuples and behave exactly as in lists: Operator Result Meaning len(tuple) int Return the length of a tuple tuple[int] object Reads an element at specified index tuple[int:int] tuple Extracts a sub-tuple - return a NEW tuple tuple + tuple tuple Concatenates two tuples - return a NEW tuple obj in tuple bool Checks whether an element is present in a tuple tuple * int tuple Replicates the tuple - return a NEW tuple ==,!= bool Checks if two tuples are equal or different ### len¶ len function returns the tuple length: [30]: len( (4,2,3) ) [30]: 3 [31]: len( (7,) ) [31]: 1 [32]: len( () ) [32]: 0 QUESTION: Have a look at following expressions, and for each try to guess the result (or if it gives an error) 1. len(3,2,4) 2. len((3,2,4)) 3. len(('a',)) 4. len(('a,')) 5. len(((),(),())) 6. len(len((1,2,3,4))) 7. len([('d','a','c','d'),(('ab')),[('a','b','c')]]) [ ]: Like in strings and lists by using brackets we can read an element at a certain position: [33]: # 0 1 2 3 tup = (10,11,12,13) [34]: tup[0] [34]: 10 [35]: tup[1] [35]: 11 [36]: tup[2] [36]: 12 [37]: tup[3] [37]: 13 We can also use negative indexes: [38]: tup[-1] [38]: 13 QUESTION: Have a look at the following expressions and for each of them try to guess the result or if it produces an error: 1. (1,2,3)[0] 2. (1,2,3)[3] 3. (1,2,3)0 4. ()[0] 5. (())[0] 6. type((())[0]) 7. ('a,')[0] 8. ('a',)[0] 9. (1,2,3)[-0] 10. (1,2,3)[-1] 11. (1,2,3)[-3] [ ]: ### Exercise - animals¶ Given the string animals = "Siamese cat,dog,canary,piglet,rabbit,hamster" 1. convert it to a list 2. create a tuple of tuples where each tuple has two elements: the animal name and the name length, i.e. ((“dog”,3), ( “hamster”,7)) 3. print the tuple You should obtain: Siamese cat,dog,canary,piglet,rabbit,hamster (('Siamese cat', 11), ('dog', 3), ('canary', 6), ('piglet', 6), ('rabbit', 6), ('hamster', 7)) • you can assume animals always contains exactly 6 animals Show solution [39]: animals = "Siamese cat,dog,canary,piglet,rabbit,hamster" # write here ### Slices¶ As with strings and lists, by using slices we can also extract subsequences from a tuple, that is, on the right of the tuple we can write square brackets with inside a start index INCLUDED, a colon : and an end index EXCLUDED: [40]: tup = (10,11,12,13,14,15,16,17,18,19) [41]: tup[2:6] # from index 2 INCLUDED to 6 EXCLUDED [41]: (12, 13, 14, 15) It is possible to alternate the gathering of elements by adding the number of elements to skip as a third numerical parameter in the square brackets, for example: [42]: tup = (10,11,12,13,14,15,16,17) [43]: tup[0:8:5] [43]: (10, 15) [44]: tup[0:8:2] [44]: (10, 12, 14, 16) [45]: tup[1:8:1] [45]: (11, 12, 13, 14, 15, 16, 17) WARNING: remeber that slices produce a NEW tuple ! QUESTION: Have a look at the following code snippets, and for each try to guess which result it produces (or if it gives an error) 1. (7,6,8,9,5)(1:3) 2. (7,6,8,9,5)[1:3] 3. (10,11,12,13,14,15,16)[3:100] 4. (10,11,12,13,14,15,16)[-3:5] 5. (1,0,1,0,1,0)[::2] 6. (1,2,3)[::1] 7. (1,0,1,0,1,0)[1::2] 8. tuple("postcards")[0::2] 9. (4,5,6,3,4,7)[0:::2] ### Concatenation¶ It is possible to concatenate two tuples by using the operator +, which creates a NEW tuple: [46]: t = (1,2,3) + (4,5,6,7,8) [47]: t [47]: (1, 2, 3, 4, 5, 6, 7, 8) [48]: type(t) [48]: tuple Let’s verify that original tuples are not modified: [49]: x = (1,2,3) y = (4,5,6,7,8) [50]: t = x + y [51]: t [51]: (1, 2, 3, 4, 5, 6, 7, 8) [52]: x [52]: (1, 2, 3) [53]: y [53]: (4, 5, 6, 7, 8) Let’s see how they are represented in Python Tutor: [54]: x = (1,2,3) y = (4,5,6,7,8) t = x + y print(t) print(x) print(y) jupman.pytut() (1, 2, 3, 4, 5, 6, 7, 8) (1, 2, 3) (4, 5, 6, 7, 8) [54]: QUESTION: Have a look at the following code snippets, and for each try to guess which result it produces (or if it gives an error) 1. ()+() 2. type(()+()) 3. len(()+()) 4. ()+[] 5. []+() 6. (2,3,4) + tuple([5,6,7]) 7. "crazy"+('r','o','c','k','e','t') ### Membership¶ As in all sequences, if we want to verify whether an element is contained in a tuple we can use the operator in which returns a boolean value: [55]: 'e' in ('h','e','l','m','e','t') [55]: True [56]: 'z' in ('h','e','l','m','e','t') [56]: False #### not in¶ To check whether something is not belonging to a tuple, we can use two forms: not in - form 1: [57]: "carrot" not in ("watermelon","banana","apple") [57]: True [58]: "watermelon" not in ("watermelon","banana","apple") [58]: False not in - form 2 [59]: not "carrot" in ("watermelon","banana","apple") [59]: True [60]: not "watermelon" in ("watermelon","banana","apple") [60]: False QUESTION: Have a look at the following code snippets, and for each try to guess which result it produces (or if it gives an error) 1. 3 in (1.0, 2.0,3.0) 2. 3.0 in (1,2,3) 3. 3 not in (3) 4. 3 not in (3,) 5. 6 not in () 6. 0 in (0)[0] 7. [] in () 8. () in [] 9. not [] in () 10. () in () 11. () in (()) 12. () in ((),) 13. 'ciao' in ('c','i','a','o') [ ]: ### Multiplication¶ To replicate the elements in a tuple, it is possible to use the operator * which produces a NEW tuple: [61]: (7,8,5) * 3 [61]: (7, 8, 5, 7, 8, 5, 7, 8, 5) [62]: (7,8,5) * 1 [62]: (7, 8, 5) [63]: (7,8,5) * 0 [63]: () QUESTION: What is the following code going to print? x = (5,6,7) y = x * 3 print('x=',x) print('y=',y) ANWSER: It will print: x = (5, 6, 7) y = (5, 6, 7, 5, 6, 7, 5, 6, 7) because the multiplication generates a NEW tuple which is associated to y. The tuple associated to x remains unchanged. QUESTION: Have a look at the following expressions, and for each try to guess which result it produces (or if it gives an error) 1. (5,6,7)(3.0) 2. (5,6,7)(3,0) 3. (5,6,7)(3) 4. (5,6,7)3 5. (4,2,3)int(3.0) 6. (1,2)[3][0] 7. (1,2)(3,4)[-1] 8. [(9,8)]4 9. (1+2,3+4)5 10. (1+2,)4 11. (1+2)4 12. (1,2,3)0 13. (7)0 14. (7,)0 [ ]: ### Exercise - welcome¶ Given a tuple x containing exactly 3 integers, and a tuple y containing exactly 3 tuples of characters, write some code to create a tuple z containing each tuple of y replicated by the corresponding integer in x. Example - given: x = (2,4,3) y = (('w','e','l','c'),('o',),('m','e')) after your code it should print: >>> print(z) ('w', 'e', 'l', 'c', 'w', 'e', 'l', 'c', 'o', 'o', 'o', 'o', 'm', 'e', 'm', 'e', 'm', 'e') Show solution [64]: x = (2,4,3) y = (('w','e','l','c'),('o',),('m','e')) # write here ('w', 'e', 'l', 'c', 'w', 'e', 'l', 'c', 'o', 'o', 'o', 'o', 'm', 'e', 'm', 'e', 'm', 'e') ## Write an element¶ Tuples are immutable, so trying to i.e. write an assignment for placing the number 12 into the cell at index 3 provokes an error: # 0 1 2 3 4 tup = (5,8,7,9,11) tup[3] = 666 --------------------------------------------------------------------------- TypeError Traceback (most recent call last) <ipython-input-118-83949b0c81e2> in <module> 1 tup = (5,8,7,9,11) ----> 2 tup[3] = 666 TypeError: 'tuple' object does not support item assignment What we can do is to create a NEW tuple by composing it from sequences takes from the original one: [65]: # 0 1 2 3 4 5 6 tup = (17,54,34,87,26,95,34) [66]: tup = tup[0:3] + (12,) + tup[4:] [67]: tup [67]: (17, 54, 34, 12, 26, 95, 34) WARNING: append, extend, insert, sort DO NOT WORK WITH TUPLES ! All the methods you used to modify lists will not work with tuples. Try writing down here (1,2,3).append(4) and see which error appears: Show solution [68]: # write here ### Exercise - abde¶ Given a tuple x, save in a variable y another tuple containing: • at the beginning, the same elements of x except the last one • at the end, the elements 'd' and 'e' . • Your code should work with any tuple x Example - given: x = ('a','b','c') after your code, you should see printed: x = ('a', 'b', 'c') y = ('a', 'b', 'd', 'e') Show solution [69]: x = ('a','b','c') # write here ### Exercise - charismatic¶ Given a tuple t having alternating uppercase / lowercase characters, write some code which modifies the assignment of t so that t becomes equal to a tuple having all characters lowercase as first ones and all uppercase characters as last ones. Example - given: t = ('C', 'h','A', 'r', 'I', 's', 'M', 'a', 'T', 'i', 'C') after your code it must result: >>> print(t) ('C', 'A', 'I', 'M', 'T', 'C', 'h', 'r', 's', 'a', 'i') Show solution [70]: t = ('C', 'h','A', 'r', 'I', 's', 'M', 'a', 'T', 'i', 'C') # write here ### Exercise - sorting¶ Given a tuple x of unordered numbers, write some code which changes the assignment of x so that x results assigned to a sorted tuple • your code must work for any tuple x • HINT: as we’ve already written, tuples DO NOT have sort method (because it would mutate them), but lists have it … Example - given: x = (3,4,2,5,5,5,2,3) after your code it must result: >>> print(x) (2, 2, 3, 3, 4, 5, 5, 5) Show solution [71]: x = (3,4,2,5,5,5,2,3) # write here ## Methods¶ Tuples are objects of type typle and have methods which allows to operate on them: Method Return Description tuple.index(obj) int Searches for the first occurence of an element and returns its position tuple.count(obj) int Count the occurrences of an element ### index method¶ index method allows to find the index of the FIRST occurrence of an element. [72]: tup = ('b','a','r','a','t','t','o') [73]: tup.index('b') [73]: 0 [74]: tup.index('a') [74]: 1 [75]: tup.index('t') [75]: 4 If the element we’re looking for is not present, we will get an error: >>> tup.index('z') --------------------------------------------------------------------------- ValueError Traceback (most recent call last) <ipython-input-318-96cf33478b69> in <module> ----> 1 tup.index('z') ValueError: tuple.index(x): x not in tuple QUESTION: Have a look at the following expressions, and for each try to guess which result (or if it gives an error) 1. (3,4,2).index(4) 2. (3,4,---1).index(-1) 3. (2.2,.2,2,).index(2) 4. (3,4,2).index(len([3,8,2,9])) 5. (6,6,6).index(666) 6. (4,2,3).index(3).index(3) 7. tuple("GUG").index("g") 8. (tuple("ci") + ("a","o")).index('a') 9. (()).index(()) 10. ((),).index(()) ### count method¶ We can obtain the number of occurrences of a certain element in a list by using the method count: [76]: t = ('a', 'c', 'a', 'd', 'e', 'm', 'i', 'a') [77]: t.count('a') [77]: 3 [78]: t.count('d') [78]: 1 If an element is not present 0 is returned: [79]: t.count('z') [79]: 0 ### Exercise - fruits¶ Given the string s = "apple\|pear\|apple\|cherry\|pear\|apple\|pear\|pear\|cherry\|pear\|strawberry" Insert the elements separated by "\|" (pipe character) in a list. 1. How many elements must the list have? 2. Knowing the list created at previous point has only four distinct elements (es "apple", "pear", "cherry", and "strawberry"), create another list where each element is a tuple containing the name of the fruit and its multiplicity (that is, the number of times it appears in the original list). Example - given: counts = [("apple", 3), ("pear",5), ...] Here you can write code which works given a specific constant, so you don’t need cycles. 1. Print the content of each tuple in a separate line (i.e.: first libe; "apple" is present 3 times) You should obtain: [('apple', 3), ('pear', 5), ('cherry', 2), ('strawberry', 1)] apple is present 3 times pear is present 5 times cherry is present 2 times strawberry is present 1 times Show solution [80]: s = "apple\|pear\|apple\|cherry\|pear\|apple\|pear\|pear\|cherry\|pear\|strawberry" # write here [('apple', 3), ('pear', 5), ('cherry', 2), ('strawberry', 1)] apple is present 3 times pear is present 5 times cherry is present 2 times strawberry is present 1 times ## Exercises with functions¶ WARNING: following exercises require to know: If you are just a beginner, you may skip them and come back later. ### Exercise - touples¶ ✪✪ Let’s call a touple a tuple with a couple of elements. Write a function touples which given a tuple, RETURNs a list having as elements touples each taken in alternation from t. • if the input tuple t has an odd number of elements, the last tuple in the list to return will be made of only one element Example: >>> touples( ('c', 'a', 'r', 'p', 'e', 't') ) # even length [('c', 'a'), ('r', 'p'), ('e', 't')] >>> touples( ('s','p','i','d','e','r','s') ) # odd length [('s', 'p'), ('i', 'd'), ('e', 'r'), ('s',)] Show solution [81]: # write here ### Exercise - joined¶ ✪✪ Write a function which given two tuples of characters ta and tb having each different characters (may also be empty), return a tuple made like this: • if the tuple ta terminates with the same character tb begins with, RETURN the concatenation of ta and yb WITHOUT duplicated characters • otherwise RETURN an empty tuple Example: >>> joined(('a','b','c'), ('c','d','e')) ('a', 'b', 'c', 'd', 'e') >>> joined(('a','b'), ('b','c','d')) ('a', 'b', 'c', 'd') >>> joined((),('e','f','g')) () >>> joined(('a',),('e','f','g')) () >>> f(('a','b','c'),()) () >>> f(('a','b','c'),('d','e')) () Show solution [82]: # write here ## Verify comprehension¶ WARNING The following exercises contain tests with assert. To understand how to do them, read first Error handling and testing ### doubles¶ ✪✪ Take as input a list of n integer numbers, and RETURN a NEW list which contains n tuples each having two elements. Each tuple contains a number taken from the corresponding position in the original list, and its double. Example - given: doubles([ 5, 3, 8]) it must produce the list: [(5,10), (3,6), (8,16)] Show solution [83]: def doubles(lst): raise Exception('TODO IMPLEMENT ME !') # TEST START - DO NOT TOUCH ! assert doubles([]) == [] assert doubles([3]) == [(3,6)] assert doubles([2,7]) == [(2,4),(7,14)] assert doubles([5,3,8]) == [(5,10), (3,6), (8,16)] # verify the original list was not changed la = [6] lb = doubles(la) assert la == [6] assert lb == [(6,12)] # TEST END ### nasty¶ ✪✪✪ Given two tuples ta and b, ta made of characters and tb of positive integer numbers , write a function nasty which RETURNS a tuple having two character strings: the first character is taken from ta, the second is a number taken from the corresponding position in tb. The strings are repeated for a number of times equal to that number. >>> nasty(('u','r','g'), (4,2,3)) ('u4', 'u4', 'u4', 'u4', 'r2', 'r2', 'g3', 'g3', 'g3') >>> nasty(('g','a','s','p'), (2,4,1,3)) ('g2', 'g2', 'a4', 'a4', 'a4', 'a4', 's1', 'p3', 'p3', 'p3') Show solution [84]: # write here ## References¶ [ ]:	2021-01-21 00:46:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.301002562046051, "perplexity": 5933.001106418972}, "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/1610703522150.18/warc/CC-MAIN-20210121004224-20210121034224-00517.warc.gz"}
https://mathzsolution.com/why-cant-you-add-apples-and-oranges-but-you-can-multiply-and-divide-them/	# Why can’t you add apples and oranges, but you can multiply and divide them? What is the algebraic difference between arithmetic operations, that prevents entities with different units from being summed or subtracted, but allows them to be multiplied or divided? This looks more like a question for Physics, but lengths and areas, for example, are in the domain of pure mathematic. Now, I cannot sum or subtract an area and a length, but I can multiply and divide an area with a length! Reading Wikipedia, it looks like this is a property of the dimensions set. Does it just depend on the definition of the dimensions, or is it something intrinsic in the operations of add, subtract, multiply and divide? Please explain with simple words, if possible. Apples and oranges are actually a rather bad example. The reason why it doesn’t make sense to add quantities with different dimensions, but it does make sense to multiply (or divide) them is scale invariance. Let U be the unit of some quantity $u$, and $V$ be the unit of another quantity $v$. Now say we change the scale of U, i.e. we instead use a different unit U’ such that $1U = 10U'$. For $V$ we do the same, only that there we choose $V'$ such that $1V = 5V'$. If we compute the sum $s$ of $u$ and $v$ in units U,V we get If, instead, we compute the sum in units $U'$ and $V'$, however, we get Note that $s$ and $s'$ don’t just differ by a factor, i.e. we can’t convert $s$ from unit $U+V$ to $s'$ in unit $U'+V'$ without knowing the original values of $u$ and $v$. Compare this to the situation of a product. If we compute the product $p$ of $u$ and $v$ in units $U$ and $V$, we get If, instead, we compute it in units $U'$ and $V'$, we get So $p'$ is simply $p$, expressed in a different unit P’, with $1P = 1UV = 50P' = 1U'V'$. So why do you want scale invariance? We want that, because the scale of physical units is usually completely arbitrary. There’s nothing fundamental about 1 meter, or 1 inch, or 1 Volt – we just picked some reference value. But since the reference value is arbitrary, the actual physics must not change if we replace it by a different one. Which it doesn’t, so long as we only multiply and divide, but not add or subtract values with different units, as the example above shows. And this is also why apples and oranges are a bad example. We don’t expect scale invariance for these, because apples and oranges are discrete objects, so there’s a canonical definition of what “1 apple” means. So adding apples and oranges makes perfect sense, and we may e.g. assign the result the unit fruits.	2022-10-05 02:18:47	{"extraction_info": {"found_math": true, "script_math_tex": 30, "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": 0.9012348651885986, "perplexity": 200.99310425090857}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337531.3/warc/CC-MAIN-20221005011205-20221005041205-00690.warc.gz"}
https://asmedigitalcollection.asme.org/manufacturingscience/article/140/9/091009/368744/Measurement-and-Modeling-of-Forces-in-Extrusion	Flexible thin wall silicone parts fabricated via extrusion-based additive manufacturing (AM) tend to deform due to the AM forces, limiting the maximum build height. The tangential and normal forces in AM were measured to investigate effects of three key process parameters (volumetric flow rate Q, nozzle tip inner diameter di, and layer height t) on the build height. The interaction between the nozzle tip and the extruded silicone bead is controlled to prevent interaction, flatten the top surface of the extruded silicone, or immerse the nozzle in the extruded silicone. Results show that tangential and normal forces in AM strongly depend on this interaction. Specifically, the AM forces remained low (less than 0.2 mN) if the nozzle tip did not contact the extruded silicone bead. Once the nozzle interaction with extruded silicone came into effect, the AM forces quickly grew to over 1 mN. The single wall tower configuration was developed to determine a predictive deflection resistance approach based on the measured AM forces and the resultant bending moment of inertia. This approach shows that a smaller di can produce taller towers, while a larger di is better at bridging and overhangs. These results are applied to the AM of a hollow thin wall silicone prosthetic hand. ## Introduction Silicone elastomer deposited using direct extrusion additive manufacturing (AM) has demonstrated great potential for fabricating a wide variety of custom flexible thin wall structures such as pneumatic actuators and parts with high elongation and fatigue life [13]. This AM process is particularly advantageous over other AM methods, such as selective laser sintering, photopolymer jetting, and fused deposition modeling (FDM), because it enables the use of a wide variety of commercially available silicone elastomers, denoted as silicones, with broad material properties ranging from 3 to 90 Shore A hardness, up to 1100% elongations, −65 °C to 177 °C functional temperature range, long fatigue life, high chemical and UV resistance, and reasonable cost [4,5]. In material extrusion AM, a nozzle moves line-by-line and layer-by-layer to extrude material on the AM part to create a three-dimensional object. Unfortunately, this process generates forces which can deform and skew a soft and flexible AM part. This is a key challenge for extrusion-based AM of flexible silicone thin wall structures because silicone's low elastic modulus, ranging around 0.5–30 MPa, makes the parts easily deformable by forces generated during the extrusion of high-viscosity silicone fluid. Generally, the taller and thinner the part becomes, the greater the impact of the AM forces. By adjusting extrusion process parameters (e.g., flowrate, layer height, nozzle size, and acceleration), the ratio of AM forces to the parts' bending moment of inertia can be reduced and the maximum height of a soft, thin wall structure can be greatly increased. During the extrusion-based silicone AM process, force on the workpiece is imparted in three modes: (1) machine vibration and build plate movement, (2) tangential and normal forces caused by silicone deposition and the nozzle tip, and (3) gravity force. Forces due to machine vibration and build plate movement can be overcome through control of accelerations, stationary build plates using a prismatic-input delta robotic 3D-printer, and increasing the stiffness of the 3D-printer. Silicone deposition can create tangential forces due to the dragging or pulling of the extruded silicone between the AM part and the moving nozzle tip and normal forces due to the momentum of the extruded silicone impacting the AM part. Additional tangential and normal forces may also be created from the nozzle side and bottom surfaces dragging through the deposited material. Finally, gravitational forces may cause creep or negatively affect thin overhanging structures but can be overcome through silicone cure methods and geometric design. External forces have always existed in extrusion-based AM but were ignored for process parameter selection until soft materials and flexible parts became more prevalent. With rigid parts, such as those made from acrylonitrile butadiene styrene produced using FDM, a type of extrusion-based AM processes, the molten thermoplastic solidifies quickly after deposition from the nozzle (typically within seconds) and has high rigidity once cooled. This high rigidity allows the solidified material to resist forces imparted on it from the AM process. However, when silicone is used instead, the liquid silicone that is extruded from the nozzle may go through a chemical reaction to cure. This curing process typically takes minutes to hours, creating a high likelihood that subsequent layers of material will be deposited on the uncured, liquid form of the material. This allows for high interlayer bonding strength but makes parts extremely susceptible to the forces previously described. Additionally, even if the silicone or other soft material was able to cure in seconds, the cured state can still be extremely flexible and prone to deformation by the AM forces. Although deformation of the soft silicone AM parts has been observed within and between the extruded layers [2,3], the research to measure and quantify the forces during the extrusion-based AM process is still lacking. Compliant silicone structures can be skewed by forces in the mN level. Therefore, understanding and modeling the forces in extrusion-based soft silicone AM is important. Plott et al. [3] showed that an adjacent line spacing increase of just 0.05 mm can significantly reduce the tensile strength of the part due to the creation of internal voids between adjacent silicone lines. If a silicone part is deformed by even a fraction of a millimeter, there is the potential for reduced tensile strength of over 30% [3]. Furthermore, if the deformation is too large, overall part accuracy can quickly deteriorate and lead to a failed AM process. Fluid modeling has been applied to study the material flow and heat dissipation in AM. Die swelling is the radial expansion of the liquid material as it moves from a compressed, high pressure state within the nozzle to atmospheric pressure as it leaves the nozzle [6]. The swelling ratio, or the maximum diameter of the extruded material divided by the nozzle diameter, is typically 1.05–1.3 for FDM extrusion processes and is dependent on the material properties and process parameters [7,8]. Bellini [7] used computational fluid dynamics to create a two-dimensional model of the extruded material spreading process during and after deposition from the nozzle onto the build plate. Since the behavior of the melted filament in the extrusion nozzle is typically shear thinning, a power law dependency of the viscosity for the shear rate is used. One variation of the model included a nozzle tip as a contact condition, while another variation neglected the nozzle tip. Results showed that the presence of the nozzle tip added stability to the material flow and smoothed and flattened the top surface of the deposited material. Subsequent layers were modeled by rerunning the model using the geometry results from the previous layer as the base [7]. The literature review reveals a lack of study on the measurement of forces and part deformation due to forces in extrusion-based AM. This is particularly important for the AM of soft silicone flexible thin wall structures and will be investigated in this study. The forces caused by extrusion-based AM of silicone are first introduced. Details of the experimental setup for silicone extrusion and force measurement are then presented. Results of tangential and normal forces created during the AM process are discussed and compared. Single wall towers were built based on different AM process parameters to validate the predictive deflection resistance model. Finally, results from the study are applied to the AM of a hollow thin wall silicone prosthetic hand. ## Forces in Extrusion-Based Additive Manufacturing Key factors affecting the magnitude of forces in extrusion-based AM of silicone include the silicone viscosity, flowrate, nozzle speed, nozzle tip diameter, layer height, extruded silicone material mechanical properties, and geometry and material properties of the surface for deposition. Figure 1 shows the free body diagrams of four scenarios in extrusion-based AM. From Figs. 1(a)1(d), the flowrate, Q, is gradually increased. In Fig. 1(a), the extruded silicone is leaving the nozzle such that it is deposited onto a deposit layer or build plate without contacting the nozzle's exterior surface. This scenario is not commonly seen in extrusion-based AM because it is more difficult to control the part shape and where the material deposition is occurring. Three main force components in this scenario are Fng—normal force caused by the weight of the silicone line, Fnd—normal force caused by deposited silicone decelerating, and Ftd—tangential force caused by the nozzle as it drags and stretches the extruded silicone on the deposit layer or build plate. In Fig. 1(b), Q is increased and the left side of the nozzle drags through the extruded bead of silicone, acting to flatten the top surface. In this scenario, the three forces (Fng, Fnd, and Ftd) that existed in the previous scenario are present with the addition of the fourth force, Ftn, which is the tangential force caused by nozzle movement through the deposited silicone, and the fifth force, Fnn, which is the normal force caused by the nozzle interacting with the deposited silicone. Additionally, since more silicone is being deposited in a given space, Fng will increase due to an increase in weight, Fnd will increase due to the greater momentum of the material as it leaves the nozzle and decelerates on the deposit layer/build plate, and Ftd will decrease since the extruded silicone will not be stretched as much by the nozzle movement. In Fig. 1(c), Q is increased further causing the silicone to flow forward from the inner diameter of the nozzle as it contacts the deposit layer. This outward push of silicone will create a flow field that leads in front of the nozzle opening. Through this phenomenon, it is expected that Ftd = 0 since the extruded silicone is no longer being stretched by the nozzle movement. Conversely, Fng and Fnd are expected to increase for the same reasons described in the previous scenario. Finally, in Fig. 1(d), Q is increased so much that the material flow field has moved in front of the nozzle, causing the side of the nozzle to drag through the material in addition to the bottom surface of the nozzle. In this scenario, it is expected that Fng, Fnd, and Ftn will all be at the greatest level among four scenarios. With an understanding of force originates in a given silicone AM scenario, it is important to measure the magnitude of these forces in each scenario. By determining these magnitudes, the process parameters can be optimized to minimize the part deformation and improve the accuracy and reliability of the extrusion-based AM for silicone and other soft materials. Section 3 explains the experimental setup to measure these forces. ## Experimental Setup This section outlines the silicone material, AM machine and setup, experimental design, and displacement to force conversion for force measurement of the silicone extrusion-based AM. ### Silicone Material and Cure Parameters. A one-part oxime cure silicone elastomer (Dow Corning® 737, Dow Chemical, Midland, MI) was used as the base material for all experiments in the study. This silicone has a 33 Shore A durometer hardness, over 300% elongation, over 1.2 MPa tensile strength, and a specific gravity of 1.04 [9]. This silicone has a zero shear rate viscosity of about 62.5 Pa·s and begins curing with exposure to atmospheric moisture [1]. The viscosity was chosen since it is low enough for extrusion through the tapered nozzle tips and high enough to prevent self-leveling after extrusion, allowing the silicone to hold its shape. Once exposed to moisture, the silicone has a skin-over time of 3–6 min, a tack-free time of 14 min, and a cure to handling time of 24 h [9]. Since the skin-over time is lower than the layer times in the experiment, we assume each layer is extruded on a previously uncured layer. ### Additive Manufacturing Machine and Setup. The experimental setup for force measurement of extrusion-based AM of silicone is shown in Fig. 2. The system consists of six key components: 1. (1) A prismatic-input delta robotic 3D-printer as a motion control platform based on an open-source FDM machine (Rostock Max™ V3 by SeeMeCNC®, Goshen, IN). The AM machine allows for the XYZ nozzle movement independent of the stationary build plate. 2. (2) A progressive cavity pump and its controller (model preeflow eco-PEN 450 pump and model EC200 controller by Viscotec, Töging am Inn, Germany) to dispense the silicone with a dosing accuracy of ±1% [10]. 3. (3) Syringe barrels (model optimum by Nordson EFD, Westlake, OH) pressurized to 70±10 kPa which feed the progressive cavity pump with silicone while preventing the introduction of air bubbles into the silicone. 4. (4) A tapered nozzle with either 22 gauge (0.41 mm), 25 gauge (0.25 mm), or 27 gauge (0.20 mm) tip inner diameter (Model SmoothFlow™ by Nordson EFD, Westlake, OH) to deposit the silicone on the cantilevered plate. 5. (5) A brass cantilevered beam to measure the small AM force during material deposition. The beam has a polylactic acid base, a polylactic acid top plate, and a mirror. The length of cantilever beam for force measurement, h in Fig. 2(a), is 156 mm. The width and thickness of the beam is 25.83 and 0.82 mm, respectively. 6. (6) A laser displacement sensor (model LK-G10 by Keyence, Itasca, IL) with 0.02 μm repeatability and ±0.03% linearity over ±1 mm measuring range. The laser displacement sensor was connected to a display panel (model LK-GD500 by Keyence, Itasca, IL) and the LK-Navigator software (keyence) which records the sensor displacement data. A 3 Hz low-pass filter was used to smooth the data since the nozzle moves about the square test part at approximately 1 Hz. Due to the sensitivity of the force measurement setup, each moving component is isolated from one another and fixed to an optical table with vibration dampening. Additionally, a high-speed camera (Model 100K, FASTCAM-1024PCI by Photron, San Diego, California, USA) with a 5.6× magnification lens operating at 500 frames per second was used to visualize the extrusion process and identify interaction between the nozzle tip and the extruded silicone. ### Experimental Design. To test the effects of key parameters on the forces experienced by a soft part created through extrusion-based AM, a parametric study was performed. Three key process parameters were varied: flowrate Q, layer height t, and nozzle diameter di, as listed in Table 1. The nozzle speed in the layer, v, was held constant at 20 mm/s. The parameter range was selected based on the previous studies in extrusion-based silicone AM [2,3] and preliminary exploration of process parameters for the AM of silicone thin wall structures. A rounded-edge square part, as shown in Fig. 3, was utilized for silicone extrusion experiments. This part features a single line wall thickness with 25 × 25 mm side spacing and 3 mm total height. Four corners of the square are rounded with 4 mm radius to avoid rapid acceleration during the silicone AM process. The part is printed using a continuously increasing layer height, rather than the discrete jump in height for each layer, to further minimize any rapid acceleration. Figure 4 shows the AM of this rounded-edge square part on the cantilevered build plate. The square was centered on the platform such that two opposite sides were parallel and the other two opposite sides were perpendicular to the long axis of the cantilever beam. Due to this print orientation relative to the cantilever beam, specific displacements (and forces) are measured at different sections of the part building process. Looking first at point A1 in Fig. 4(a), the nozzle is moving parallel to the long axis of the cantilever beam. At this point, only the normal force, Fn, can deflect the beam. According to point A1 on the cantilever displacement versus time graph, Fig. 4(f), the deflection of the beam at A1 is small due to Fn since the effective moment arm for Fn is only 12 mm versus the 197 mm moment arm for Ft. Section 3.4 explains this in further detail. As the nozzle rounds the corner and moves to point A2 in Fig. 4(b), Ft deflects the beam along the direction of the nozzle movement with a 197 mm moment arm, while Fn deflects the beam in the opposite direction of the nozzle movement with a 12 mm moment arm. From Fig. 4(f), the displacement at point A2 (about 0.014 mm) is much higher than that at point A1, indicating that forces which comprise Ft are well detected by the experimental setup in comparison to Fn. As the nozzle continues to travel past A2 and reaches point M1 in Fig. 4(c), the nozzle is directly above the cantilever beam. At this point, Fn no longer influences the beam deflection and only Ft is measured. Once the nozzle moves past point M1 to point B1 in Fig. 4(d), both Ft and Fn act to deflect the beam along the direction of nozzle travel. As the nozzle rounds the corner to point B2, Ft is again moving parallel to the long axis of the cantilever beam and only the displacement due to Fn is detected. This cycle then repeats but in the opposite direction as the nozzle moves from point B2 to C1, C2 to D1, and D2 to A1, as seen in Figs 4(e) and 4(f). In Fig. 4(f), there is a slope from points A2 to B1 and C2 to D1. This can be due to Fn creating a moment on the beam, the momentum of the cantilever beam and top plate, and potential extrusion variations on the corners where the velocity might vary. Because of these potential effects, Fn is only calculated from points D2 to A1 and B2 to C1, while Ft is only calculated at points M1 and M2. These calculations are detailed in Sec. 3.4. ### Displacement to Force Conversion. This section describes the procedure of converting the measured displacement of the cantilever beam to Ft and Fn. Three free body diagrams, as shown in Fig. 5, are used to derive the force equations. #### Cantilever Beam Calibration Curve. Although Ft and Fn could be approximated from the measured beam displacements using the classic cantilever beam bending theory and elastic modulus assumptions, a calibration curve was created for validation and improved force estimation accuracy. To obtain the calibration curve, a series of six small metallic washers with various masses (m) were placed on the top plate at a known distance, l, from the center of the cantilever beam, Fig. 5(a). These masses create a normal force Fcal = mg, where g is the acceleration due to gravity. Fcal creates a coupled moment M at the free end of the cantilever beam which causes the displacement x1 of the mirror. This displacement is measured by the laser displacement sensor (Fig. 2). Six masses for calibration were adjusted to make the range of displacement x1 within that which was observed during the AM experiments. Results of these six coupled moments, corresponding displacements, and calibration line are shown in Fig. 6. The relationship between the mass and its moment on the cantilever beam and the displacement of the beam is linear in the measured range. Using the cantilever beam bending equation for a coupled moment M at the free end [11] and substituting M = Fcall = mgl $x1=Mh22EI=Fcallh22EI=mglh22EI$ (1) where h is the distance from the fixed base of the cantilever beam to the laser measurement point (h = 156 mm in this study), E is the elastic modulus of the beam material, and I is the moment of inertia of the beam. Together, EI is the experimentally determined constant of the beam. It is also assumed that the top plate is rigid and the mirror has a negligible effect on the overall beam bending. Rearranging Eq. (1) and substituting the linear fit in Fig. 6, the constant EI = 105 kN mm2 is calculated. Additionally, since dimensions of the cantilever beam can be measured, the moment of inertia, I, can be calculated and substituted in EI to determine the modulus of the beam, E. The moment of inertia for a rectangular beam is $I=(bd3/12)$, where b = 25.83 mm and d = 0.82 mm. The parameters I = 1.19 mm4 and E = 88.9 GPa, which are reasonable for the brass cantilever beam material. #### Tangential Force Ft Conversion. With these calibration results, the displacements recorded during the silicone extrusion experiments at points M1 and M2, Fig. 4(f), can be converted to Ft using the beam bending equation for a concentrated load Ft at the free end [11] $x2=Ft(h3−3h+h1h2)6EI$ (2) where x2 is the displacement of the mirror caused by Ft and h1 is the distance between the laser measurement point and the cantilever top plate surface (h1 = 41.0 mm, as shown in Fig. 5(b)). Equation (2) can be rearranged and simplified to solve for Ft $Ft=6x2EI−2h3−3h1h2$ (3) #### Normal Force Fn Conversion. From the displacement data, the total normal force Fn can be calculated when the nozzle is extruding between points D2 to A1 and B2 to C1, as shown in Fig. 4. Using the cantilever beam bending equation for a coupled moment M at the free end in Fig. 5(c) $x3=Mh22EI=Fnl0h22EI$ (4) where M = Fn l0 and l0 is the distance from the center line of the cantilever beam to the center of the nozzle when extruding between points D2 to A1 and B2 to C1 (12 mm), as shown in Fig. 5(c). Rearranging Eq. (4), we can solve for Fn $Fn=2x3EIl0h2$ (5) While this equation provides the total normal force, Fn, it is also possible to calculate the three force components which compose the total normal force, Fnd, Fng, and Fnn. • Fnd: The force component Fnd is the normal force caused by deposited silicone decelerating as it impacts the build plate or part. Using the equation for jet forces on a stationary plate, the normal force component Fnd can be calculated $Fnd=QρV$ (6) where $ρ$ is the density of the silicone and V is the silicone exit speed from the nozzle. Since values of Q (0.10–0.40 ml/min), $ρ$ (1040 kg/m3), and V (12.6–212.2 mm/s) are known from the process parameters and silicone material, Fnd can be calculated. Results are shown in Fig. 7. The smaller the nozzle, the greater Fnd becomes for a given Q. • Fng: The force component Fng is the normal force due to the weight of the deposited silicone. Fng is calculated as $Fng=Qρs$ (7) where s is the time duration of material deposition. Since Fng changes over time and the rounded-edge square part is axisymmetric, the amount of silicone deposited on one layer between the points D2 to A1 and B2 to C1 was calculated. The nozzle moves at a speed v = 20 mm/s and the length between those points (D2–A1 and B2–C1) is approximately 20 mm, therefore the weight of silicone deposition for 1 s was calculated. These results are shown in Fig. 8. Fng varies linearly with Q and is independent of di • Fnn: The force component Fnn is the normal force caused by the nozzle interaction with the silicone and is dependent on the level of normal compression between the extruded silicone and the nozzle tip. Fnn can be calculated by subtracting the other normal force components, Fng and Fnd, from the total measured normal force Fn $Fnn=Fn−Fnd−Fng$ (8) ## Results The force results obtained from the experimental tests are presented in this section. ### Cantilever Beam Displacements. Following the procedure outlined in Sec. 3, displacements of the cantilever beam were recorded throughout 54 different experiments corresponding to the process parameters listed in Table 1. An example of a displacement versus time graph is shown in Fig. 9. In the beginning layers (0–4 s), there is a significant variability due to (1) inconsistencies in the initial material flow Q from the extrusion pump as internal pressure builds in the nozzle before equilibrium pressure/flow is reached, (2) variations in initial layer height due to imperfect build plate/nozzle leveling, and (3) over or under extrusion on the initial layers due to a nondeformable build plate. To compensate for this, the initial displacement data was ignored until a more uniform, periodic displacement was observed, typically after 5 s. ### Experimental Results of Ft. To determine the magnitude of the tangential force, Ft, the first 20 peaks were selected after equilibrium was reached. As shown in Fig. 10, a least squares fit was imposed on the data to identify midpoints of the displacement peaks (points M1 and M2 in Fig. 4). These peak displacements were converted to Ft using Eq. (3). To determine the value of Ft and its associated error, the magnitudes of the first peak (point M1) and valley (point M2) were averaged to obtain the average Ft for a given layer. This step was then repeated for the next ten layers to compensate for any potential measurement drift. The average of these Ft values was then calculated to determine the final Ft and the measurement error equals the standard deviation. Results are presented in Fig. 11. Figure 11 illustrates that process parameters can significantly impact Ft which ranges from 0.03 mN (Q = 0.10 ml/min, di = 0.25 mm, and t = 0.20 mm) to 4.01 mN (Q = 0.40 ml/min, di = 0.41 mm, and t = 0.10 mm). The wide range of Ft is due largely to the interaction between the nozzle tip and the deposited silicone, as shown in Fig. 1. In general, if the nozzle tip is not dragging through the deposited silicone bead, Fig. 1(a), Ft is significantly lower than if the nozzle tip is dragging through the deposited silicone bead, Figs. 1(b)1(d). The more extruded silicone the nozzle tip is dragging through, the greater Ft becomes. One example where Q had a large influence on the interaction between the nozzle tip and the extruded silicone occurred with di = 0.25 mm and t = 0.15 mm. At Q of 0.22 ml/min and below, Ft was very small (0.06 to 0.10 mN). Using the high-speed camera described in Sec. 3.2, we observed that the nozzle was very close to the extruded silicone and potentially contacted it. However, the flow field of the silicone may have been such that the nozzle does not significantly drag through the deposited layer, as shown in Fig. 12(a) (top). This was also confirmed in Fig. 12(a) (bottom) through the cross section analysis which shows that the top of the silicone bead was smooth and rounded, indicating that minimal dragging occurred by the nozzle tip. The width of wall is the thinnest, about 1.20 mm, among all four Q in this combination (di = 0.25 mm, t = 0.15 mm, and Q = 0.22 ml/min, Fig. 12(a)). Based on the incompressible flow condition and Eq. (9) [2], the theoretical value of line width is 1.22 mm, which is close to the measured width of 1.20 mm $Q=ctv$ (9) When Q was further increased to 0.28 ml/min with other process parameters held constant (di = 0.25 mm and t = 0.15 mm), we observed a distinct increase in Ft (0.27 mN). Using the high-speed camera and cross section analysis, the back edge of the nozzle is confirmed to have slightly dragged through the deposited silicone bead, Fig. 12(b) (bottom). The width of the wall increased to 1.57 mm (versus 1.56 mm calculated based on the incompressible flow condition in Eq. (9)). Increasing Q further to 0.34 ml/min, we again observed an increase in Ft (1.38 mN). Through the high-speed camera images and cross section analysis in Fig. 12(c), a greater area of the nozzle was dragging through the deposited silicone but there was still no buildup on the front of the nozzle. The width of the wall was 1.80 mm (versus 1.89 mm calculated based on the incompressible flow condition in Eq. (9)). Finally, with Q increased to the highest level, 0.40 ml/min, we saw the largest average Ft (1.54 mN). Through the high-speed camera and cross section analysis in Fig. 12(d), the nozzle was dragging through the deposited silicone and excess silicone was building up on the sides of the nozzle. The width of the wall was 2.24 mm (versus 2.22 mm calculated based on the incompressible flow condition), the thickest among all four flow rates. From these results, we conclude that to minimize the tangential force Ft, it is ideal to select process parameters where the nozzle does not contact the extruded silicone line, Fig. 12(a), since FtdFtn. There are several approaches that can achieve this result with varying degrees of success. For example, selecting a large layer height, t, may prevent the nozzle from contacting the extruded silicone and yield a low Ft. However, if this layer height is too large, then the extrusion accuracy can be negatively affected. Additionally, this higher layer height may reduce the compression on the extruded silicone layers below, making it more difficult to achieve a “voidless” mesostructure [2,3]. Another approach might utilize a low Q. However, as Q decreases (all else equal), the extruded line width becomes thinner. A thinner line width may become problematic for fabricating thin wall structures as it will have a low structural stiffness and be more susceptible to the deformation by AM forces. Yet another approach could include the use of a very small di to increase Fn, the force at which the deposited silicone compresses extruded silicone layer. This could be used to push the deposited silicone out of the way from the nozzle, preventing the nozzle from dragging and creating a large Ft. However, with high Fn, the part may deflect downward leading to accuracy issues. As a general rule to reduce Ft, it is recommended to select AM process parameters which create an extrusion scenario where the nozzle is very close to the deposited surface, as shown in Fig. 12(b), while still residing in the extrusion scenario shown in Figs. 1(a) and 12(a). This ensures a low Ft while at the same time compressing previous layers to minimize internal voids, maintaining deposition accuracy and having a bead width with enough structural stability for producing thin wall parts. ### Experimental Results of Fn. To determine the magnitude of Fn, regions from D2 to A1 and B2 to C1, as shown in Figs. 4 and 13, were identified in the displacement versus time graph. The difference of average displacement between regions D2 to A1 and B2 to C1 is 2x3, as shown in the close-up view in Fig. 13. The parameter x3 in three selected measurement regions equally spaced in the 3rd to 11th layers was identified. Equation (5) was used to convert the measured x3 to Fn. Three measured Fn values were averaged to calculate the average Fn for a given set of process parameters. In some of the displacement versus time data, especially with process parameter settings where the nozzle did not drag through the extruded silicone material (Fig. 1(a)), Fn was undetectably small (experimental setup can only detect Fn values greater than 0.1 mN) and assumed to be 0. This correlates well to the theoretical normal force component calculations for Fnd and Fng. Since the nozzle is not in close contact with the extruded silicone, Fnn is close to 0. Additionally, since the range of this experimental setup is limited to the mN force scale, it is expected that Fnd and Fng would be undetectable since, in theory, Fnd and Fng will be in the μN scale. An example of the low, undetectable Fn data is shown in Fig. 14. The normal force results, Fn, for all 54 process parameter combinations are shown in Fig. 15. These results show that Fn has a strong dependence on the process parameters, ranging from undetectable (< 0.1 mN) where the nozzle does not contact or come in close compression with the deposited silicone to 1.21 mN where significant compression takes place (di = 0.41, t = 0.15 and 0.10 mm, and Q = 0.4 ml/min). A key observation is that Fn was undetectable if the nozzle was not dragging through the extruded silicone bead. However, the nozzle does not necessarily have to drag through the extruded silicone for a jump in Fn to occur. An example of this is shown in the following process parameters: di = 0.25 mm, t = 0.15 mm, and Q = 0.22 ml/min. From Fig. 12(a), we observe that the nozzle may slightly contact the extruded silicone bead with those parameters without dragging through the extruded material (cross section analysis shows that the top of the silicone bead was smooth and rounded, indicating that minimal dragging occurred by the nozzle tip). However, in Fig. 15, a distinct increase in Fn was observed at this parameter set. This indicates that the nozzle does not necessarily need to drag through the extruded silicone bead to create a detectable mN-scale Fn. It is hypothesized that in this scenario, the silicone is being compressed by the nozzle but the flow field of the silicone is such that minimal drag is created. In general, as Q is increased further, the degree to which the nozzle contacts the extruded silicone increases, causing an increase in Fn. Based on these results, Fnn is the dominating normal force component. ### Comparison of Results. Experimentally measured Ft and Fn show similarities and differences. As the nozzle tip begins to contact the extruded silicone, there is a significant increase in both Ft and Fn. Based on this, we conclude that forces due to the nozzle tip contacting the extruded silicone, Ftn and Fnn, are much larger than the other force components caused by the silicone extrusion-based AM process. We also observed that once the nozzle begins to contact the extruded silicone, Ftn and Fnn increase as Q increases. A slight difference in the behavior of Ftn and Fnn is that Fnn tends to onset before Ftn in certain scenarios since the nozzle can contact the extruded silicone and influence the normal compression without significantly dragging through the extruded silicone bead. This is observed in the spikes at three process parameter combinations: (1) Q = 0.22 ml/min, t = 0.15 mm, and di = 0.25 mm; (2) Q = 0.22 ml/min, t = 0.10 mm, and di = 0.20 mm; and (3) Q = 0.40 ml/min, t = 0.20 mm, and di = 0.25 mm, where the ratio of Fn to Ft quickly jumps and then flattens out, as shown in Fig. 16. A difference between the Ft and Fn measurements, assuming the nozzle is contacting the extruded silicone, is that a larger di creates a larger Ft and conversely creates a smaller Fn (all else equal), as shown in Fig. 16. We hypothesize that since a larger nozzle size has a greater bottom surface area, it has a greater tendency to flatten the top surface of the extruded silicone bead, creating a larger Ft than a small nozzle. ## Deflection Ratios Tall thin wall silicone structures are easily deformed by forces caused during the extrusion-based AM process. To explore the relationship between part structure, AM forces, and maximum part height, experiments were conducted to test the build height limit of single wall silicone towers with 10 mm × 10 mm base, as shown in Fig. 17(a), produced with process parameters in Table 1. In this study, the AM process continues to increase the height of the thin wall tower until its deformation caused by AM forces significantly deteriorates the part quality. The maximum tower height before deterioration was recorded and marked in Fig. 17(b). We observed that low Fn and Ft did not necessarily yield taller towers since those parameters typically produce thinner wall structures which are more easily deformed. Parameters are explored to correlate to the maximum tower height with forces and wall structure. Two parameters, called deflection ratios, were chosen: 1. (1) Ft/I, where I is the moment of inertia of the tower base bending cross section 2. (2) Ft/Wt, where Wt is the width of the thin wall These deflection ratios have good predictability of the maximum tower height under different extrusion conditions. Note that the maximum tower heights for all 54 possible process parameter combinations were not tested in this experiment. Instead, a representative set of towers was produced corresponding to the key inflection points identified in the deflection ratio curves of Figs. 17 and 18. ### Ft/I. Since tall thin wall parts are more sensitive to tangential forces than normal forces, the Ft measured in Sec. 4.2 is used in the numerator of the deflection ratio. For the denominator I of the deflection ratio, the base of the thin wall square structure is approximated as a hollow rectangular prism. The moment of inertia for a hollow rectangular prism is $I=(EF3−ef3/12)$, where E and F are the outer dimensions of the rectangular prism and e and f are the inner dimensions of the rectangular prism [11]. For the 10 mm × 10 mm single wall square tower, E = F = 10 + c/2 mm and e = f = 10 − c/2 mm, where c is calculated using Eq. (9). Results of the Ft/I versus flow rate with the value of maximum tower height are shown in Fig. 17(b). In Fig. 17(b), it is observed that Ft/I correlates well to the maximum tower height in most cases. In general, as Ft/I is reduced, the maximum tower height is increased, with the exception of di = 0.41 mm, t = 0.10 mm, and Q = 0.22 and 0.40 ml/min. It is hypothesized that the poor correlation for these parameters is due to a different tower failure mechanism than that of the other towers. A typical example of a failed tower due to rectangular prism bending is shown in Fig. 19(a). In this example, as the height of the tower grew, the tower deflection increased due to Ft until it no longer provided a stable platform for printing. The noncorrelating towers did not fail due to tower deflection, but rather due to an over extrusion and buildup of material, as shown in Figs. 19(b) and 19(c). In this case, the nozzle continually dragged the extruded silicone until the top of the tower closed off and could no longer provide a suitable printing surface. Under low flow rate and small nozzle diameter conditions, the liquid rope coiling phenomenon [12] was observed. This is denoted as coil in Figs. 17(b) and 18. ### Ft/Wt. To predict the maximum tower height in the over extrusion scenario, Figs. 19(b) and 19(c), a new deflection ratio, Ft/Wt, is explored. Figure 18 shows results of Ft/Wt versus Q with the maximum tower height marked. For Q = 0.22 ml/min and above at the t and di values tested, this deflection ratio provides a better correlation for the maximum tower height. ### Discussions. From these results, a smaller nozzle di has a more favorable force deflection ratio than a larger nozzle di. As di becomes smaller, the force deflection ratios (Ft/I and Ft/Wt) also become less dependent on Q and t. Based on these results, when selecting the process parameters for tall and thin structures, it is best to minimize di. When utilizing a small di, other parameters such as t and Q can be selected based on the desired surface finish and wall thickness. ## Application: An Example of AM a Thin Wall Silicone Hand Findings in Sec. 5 were applied to select process parameters which enabled the extrusion-based silicone AM of a prosthetic hand, as shown in Fig. 20. The geometry for this prosthetic hand was generated from an optical scan of an adult hand and then scaled down to a smaller size due to limitations generated by the large unsupported geometry and slow material cure speed creating a tendency for the structure to collapse under its own weight. The angled base of the thumb and in-between the finger joints approximates support-less bridging sections [2] and the rest of the hand (especially the fingers) approximates the tall thin-walled structures. The AM process parameters used were di = 0.25 mm, t = 0.21 mm, Q = 0.28, and v = 20 mm/s. Two lines were used for the main wall of the hand to provide a stable base for support-less bridging, while a single line wall was used for the fingers to minimize over extrusion at the tips of the fingers. The process parameters were selected since the deflection ratios were comparable to those found with the smaller di = 0.20 mm nozzle, while the slightly larger nozzle size improves bridging ability. ## Conclusions The tangential and normal forces imparted by the extrusion-based AM of silicone were experimentally determined for a variety of process parameter combinations. Experimental results showed that the Ft has a strong dependence on the process parameters, ranging from 0.03 mN (Q = 0.10 ml/min, di = 0.25 mm, and t = 0.20 mm) to 4.01 mN (Q = 0.40 ml/min, di = 0.41 mm, and t = 0.10 mm). Through high-speed camera footage and cross section analysis, it was determined that Ftn (tangential force caused by nozzle dragging through the deposited silicone) is the dominating force component, causing an order of magnitude increase in Ft when present. Experimental results showed that Fn also has a strong dependence on the process parameters, ranging from undetectable (< 0.1 mN) where the nozzle does not contact the deposited silicone to 1.21 mN where significant compression takes place (di = 0.41 mm, t = 0.15 mm and 0.10 mm, and Q = 0.4 ml/min). Based on the fluid flow modeling, we predicted that Fnd (the normal force caused by deposited silicone decelerating) and Fng (normal force due to the weight of the deposited silicone) provided micro Newton scale forces for the tested set of process parameters, meaning that Fnn (the normal force caused by the nozzle interaction with the silicone deposit layer) was the dominating force component causing an order of magnitude increase in Fn when present. Based on these findings, to reduce Ft and Fn in extrusion-based AM, it is recommended to select process parameters where the nozzle tip does not drag through the deposited silicone. However, minimizing these forces may not directly translate to better parts. A deflection ratio utilizing Ft and the moment of inertia for a hollow rectangular prism was utilized to help determine the optimal process parameters for tall thin wall structure types. From these results, a smaller nozzle di has a more favorable force deflection ratio than a larger nozzle di. As di becomes smaller, the force deflection ratio also becomes less dependent on Q and t. By minimizing the relevant deflection ratio based on AM forces and part geometry, it is possible to reduce the deflection of a tall thin walled silicone part during AM and enable a greater level of design freedom. Even though these experiments were performed using one type of silicone, it is expected that similar findings exist with other silicones and soft materials in extrusion-based AM. It is also important to note that the process parameters suitable for tall and thin walled structures may increase the difficulty for producing a voidless structure ideal for maximizing tensile strength. This is because as the nozzle size di is decreased and the flowrate Q is increased, the distance the silicone spreads out over the cross section increases, requiring multiple layers of subsequent layer compression to achieve the theoretical wall width used for voidless cross section prediction. Future work aims to develop a computational fluid dynamics model of this process and develop additional deflection ratios for other part types, enabling the creation of tall, thick- and thin-walled structures with high accuracy and mechanical strength. ## Acknowledgment We acknowledge the support and expertise from Dow Performance Silicones, in particular Dr. Rocky (Bizhong) Zhu. ## Funding Data • This research was partially funded by the National Science Foundation (CMMI Grant Nos. #1435177 and #1547073). ## Nomenclature • c = distance between two adjacent silicone lines • • di = nozzle tip inner diameter • • Ft = total tangential force • • Ftd = tangential force on deposit layer caused by the silicone bead dragging (Ftd'—reactive force) • • Ftn = tangential force on deposit layer caused by the nozzle contact with the silicone (Ftn'—reactive force) • • Fn = total normal force • • Fng = normal force on deposit layer caused by the weight of silicone (Fng'—reactive force) • • Fnd = normal force on deposit layer caused by the deposition of the silicone (Fnd'—reactive force) • • Fnn = normal force on deposit layer caused by the nozzle interaction with the silicone (Fnn'—reactive force) • • Q = volumetric flow rate • • t = layer height • • v = nozzle speed in the layer ## References References 1. Jin , Y. , Plott , J. , and Shih , A. J. , 2015 , “ Extrusion-Based Additive Manufacturing of the Moisture-Cured Silicone Elastomer ,” Solid Freedom Fabrication Symposium ( SFFS ), Austin, TX, Aug. 10–12, pp. 308–318.http://sffsymposium.engr.utexas.edu/sites/default/files/2015/2015-25-Jin.pdf 2. Plott , J. , and Shih , A. J. , 2017 , “ The Extrusion-Based Additive Manufacturing of Moisture-Cured Silicone Elastomer With Minimal Void for Pneumatic Actuators ,” , 17 , pp. 1 14 . 3. Plott , J. , Tian , X. , and Shih , A. J. , 2018 , “ Voids and Tensile Properties in Extrusion-Based Additive Manufacturing of Moisture-Cured Silicone Elastomer ,” (in press). 4. Wacker Chemie , A. G. , “ Solid and Liquid Silicone Rubber Material and Processing Guidelines ,” Wacker Chemie AG, München, Germany, Report No. 6709_EN 5. Moretto , H.-H. , Schulze , M. , and Wagner , G. , 2000 , “ Silicon ,” Ullmann's Encyclopedia of Industrial Chemistry , Wiley, Hoboken, NJ. 6. Michaeli , W. , 2003 , Extrusion Dies for Plastics and Rubber Design and Engineering Computations , Carl Hanser Verlag GmbH & Co. KG , München, Germany. 7. Bellini , A. , 2002 , “ Fused Deposition of Ceramics: A Comprehensive Experimental, Analytical and Computational Study of Material Behavior, Fabrication Process and Equipment Design ,” Ph.D. thesis 8. N. Turner , B. , Strong , R. , and A. Gold , S. , 2014 , “ A Review of Melt Extrusion Additive Manufacturing Processes—I: Process Design and Modeling ,” Rapid Prototyp. J. , 20 ( 3 ), pp. 192 204 . 9. Corning , D. , 2011 , “ Silicone Sealants Dow Corning® 737 Neutral Cure Sealant Clear ,” Midland, Michigan, Form No. 25-583B-01 http://www.firstpowergroupllc.com/DCC_Product_Sheets/DowCorning_737.pdf. 10. ViscoTec , 2015, “ Dosing technology Dosing system eco-PEN450, n.d. V1.1 05/11 datasheet ,” ViscoTec Pumpen, Töging am Inn, Germany. 11. Hibbeler , R. C. , 2004 , Statics and Mechanics of Materials , 4th ed., Pearson Prentice Hall 12. Tian , X. , Plott , J. , Wang , H. , Zhu , B. , and Shih , A. J. , 2017 , “ Silicone Foam Additive Manufacturing by Liquid Rope Coiling ,” Procedia CIRP. , 65 , pp. 196 201 .	2019-10-19 03:59:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 13, "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": 0.38515806198120117, "perplexity": 1886.2004400681587}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986688674.52/warc/CC-MAIN-20191019013909-20191019041409-00305.warc.gz"}
https://www.physicsforums.com/threads/taylor-serie-of-a-function-1-1-z-2.554724/	Taylor serie of a function 1/(1+Z^2) 1. Nov 28, 2011 nicolas.ard Hello folks, I have this function, un complex numbers $\frac{1}{(1+z^2)}$ I know that the Taylor serie of that function is $\frac{1}{(1+z^2)}$ = $\sum (-1)^k.z^(2.k)$ 2. Nov 28, 2011 HallsofIvy Okay, do you have a question[/b}? What do you want to know about that function? If you are asking about how to get that series, you could, of course, find the derivatives of that function, and apply the usual formula for Taylor's series. However, for this particular function, the simpler way to handle it is to think of it as the sum of a geometric series. The sum of a geometric series, $\sum ar^n$ is $a/(1- r)$. Here, "$a/(1- r)$" is $1/(1+ z^2)$ so that a= 1 and $r= -z^2$. $\sum ar^n$ becomes $\sum (1)(-z^2)^n= \sum (-1)^nz^{2n}$. By the way, to get all of "(2k)" in the exponent, use "z^{(2k)}" rather than just "z^(2k)". 3. Nov 28, 2011 nicolas.ard I'm sorry, i wrote the question and i posted it by mistake, i found the solution over here [1]. I did't found the option to delete my post, this is phpBB? 4. Nov 28, 2011 nicolas.ard Thanks!, the use of geometric series it's a way to do it easier. :)	2018-09-19 06:47:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9012555480003357, "perplexity": 430.3080939591761}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267155942.15/warc/CC-MAIN-20180919063526-20180919083526-00053.warc.gz"}
https://www.eduzip.com/ask/question/in-figure-a-b-c-are-collinear-points-and-angle-dbaangle-eba-name-521033	Mathematics # In figure, A, B, C are collinear points and $\angle DBA=\angle EBA$. Name two pairs of supplementary angles. ##### SOLUTION Supplementary angles are two angles whose sum is $180^{0}$. According to question, $\angle DBA=\angle EBA$ And $line\ AC$ is a straight line. So, $\angle DBA+\angle DBC=180^0$ . So, they satisfy condition for supplementary angles. Similarly,$\angle EBA+\angle EBC=180^0$ . So, they also satisfy condition for supplementary angles. $\therefore \angle DBA$ and $\angle DBC$ as well as $\angle EBA$ and $\angle EBC$ are supplementary angles. You're just one step away Subjective Medium Published on 09th 09, 2020 Questions 120418 Subjects 10 Chapters 88 Enrolled Students 87 #### Realted Questions Q1 Single Correct Medium If two lines intersect such that four vertical angles are equal, then each angle is: • A. $45^{\circ}$ • B. $100^{\circ}$ • C. $180^{\circ}$ • D. $90^{\circ}$ Asked in: Mathematics - Lines and Angles 1 Verified Answer \| Published on 09th 09, 2020 Q2 Single Correct Medium The number of lines of symmetry when two line segments intersect is • A. 1 • B. 3 • C. 4 • D. 2 Asked in: Mathematics - Lines and Angles 1 Verified Answer \| Published on 09th 09, 2020 Q3 Subjective Medium In the adjoining figure, name the following pairs of angles. Obstuse vertically opposite angles Asked in: Mathematics - Lines and Angles 1 Verified Answer \| Published on 09th 09, 2020 Q4 Subjective Medium Figure given above shows a pair of parallel lines cut by a transversal. For given case, find a and b, giving reasons. Asked in: Mathematics - Lines and Angles 1 Verified Answer \| Published on 23rd 09, 2020 Q5 Single Correct Medium The measure of an angle is three times the measure of its complement. The angles are: • A. $20^\circ,\, 60^\circ$ • B. $30^\circ,\, 60^\circ$ • C. $45^\circ,\, 135^\circ$ • D. $22.5^\circ,\, 67.5^\circ$ Asked in: Mathematics - Lines and Angles 1 Verified Answer \| Published on 09th 09, 2020	2022-01-28 20:31:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.6124204993247986, "perplexity": 9438.175635691154}, "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/1642320306335.77/warc/CC-MAIN-20220128182552-20220128212552-00422.warc.gz"}
https://puzzling.stackexchange.com/questions/80400/get-2-liters-from-4-and-5-liter-buckets/80402	# Get 2 liters from 4 and 5 liter buckets [duplicate] You have a 4 liter bucket and a 5 liter bucket and an (infinite) supply of liquid. How do you get 2 liters?	2020-04-09 05:13:05	{"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": 0.8790210485458374, "perplexity": 1853.674623205523}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371829677.89/warc/CC-MAIN-20200409024535-20200409055035-00520.warc.gz"}
https://web2.0calc.com/questions/two-circles-again	+0 # two circles again 0 57 1 The smaller circle in the diagram is internally tangent to the larger circle. The area of the gray region is $100\pi$ square units. The perimeter of the gray region (include both inner and outer boundaries) is $100\pi$ units. What is the distance in units between the centers of the two circles? Feb 20, 2021	2021-04-13 08:28:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9446906447410583, "perplexity": 267.6824718846954}, "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/1618038072175.30/warc/CC-MAIN-20210413062409-20210413092409-00636.warc.gz"}
https://www.semanticscholar.org/paper/Lecture-notes-on-Mather's-theory-for-Lagrangian-Sorrentino/95d22cc6d101e6b180becda123c6067b48a0138f	Corpus ID: 118946546 # Lecture notes on Mather's theory for Lagrangian systems @article{Sorrentino2010LectureNO, title={Lecture notes on Mather's theory for Lagrangian systems}, author={Alfonso Sorrentino}, journal={arXiv: Dynamical Systems}, year={2010} } • Alfonso Sorrentino • Published 2010 • Mathematics • arXiv: Dynamical Systems • These are introductory lecture notes on Mather's theory for Tonelli Lagrangian and Hamiltonian systems. They are based on a series of lectures given by the author at Universit\a degli Studi di Napoli "Federico II" (April 2009), at University of Cambridge (academic year 2009-2010) and at Universitat Polit\ecnica de Catalunya (June 2010). 27 Citations	2021-01-23 02:24:44	{"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": 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": 0.37438589334487915, "perplexity": 10155.141681525849}, "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/1610703531702.36/warc/CC-MAIN-20210123001629-20210123031629-00227.warc.gz"}
https://www.mindspore.cn/doc/api_python/zh-CN/r1.2/mindspore/ops/mindspore.ops.AssignSub.html	# mindspore.ops.AssignSub¶ class mindspore.ops.AssignSub(args, kwargs)[source] Updates a Parameter by subtracting a value from it. Inputs of variable and value comply with the implicit type conversion rules to make the data types consistent. If they have different data types, lower priority data type will be converted to relatively highest priority data type. If value is a number, the number is automatically converted to Tensor, and the data type is consistent with the Tensor data type involved in the operation. RuntimeError exception will be thrown when the data type conversion of Parameter is required. Inputs: • variable (Parameter) - The Parameter. • value (Union[numbers.Number, Tensor]) - The value to be subtracted from the variable. It must have the same shape as variable if it is a Tensor. Raises TypeError – If value is neither Number nor Tensor. Supported Platforms: Ascend Examples >>> class Net(nn.Cell): ... def __init__(self): ... super(Net, self).__init__() ... self.AssignSub = ops.AssignSub() ... self.variable = mindspore.Parameter(initializer(1, [1], mindspore.int32), name="global_step") ... ... def construct(self, x): ... self.AssignSub(self.variable, x) ... return self.variable ... >>> net = Net() >>> value = Tensor(np.ones([1]).astype(np.int32)100) >>> output = net(value) >>> print(output)	2021-10-25 06:41:03	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.18538923561573029, "perplexity": 5298.032386300498}, "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/1634323587655.10/warc/CC-MAIN-20211025061300-20211025091300-00218.warc.gz"}
https://math.stackexchange.com/questions/2537995/windowed-fourier-transform	# Windowed Fourier Transform I am having some difficulties with this question. We define the windowed Fourier transform of $f \in L^2(\mathbb{R})$ as $$Sf(\mu,\xi)=\int_\mathbb{R}f(t)g(t-\mu)e^{-i\xi t}dt$$ where $g$ is some real, symmetric and finite supported function such that it vanishes outside a finite interval. Prove that for $f=e^{i\eta_0t}$, we have $$Sf(\mu,\eta)=e^{-i\mu(\eta-\eta_0)}\hat{g}(\eta-\eta_0)$$ Could someone also provide me with the integral definition of the windowed Fourier transform? For instance the Fourier transform is defined as, $$\hat{f}(\omega)=\int_\mathbb{R}f(x)e^{-ixw}dx$$ • For $f$ a single complex sine then $Sf$ is what you wrote, this is obvious from the integral definition. – reuns Nov 26 '17 at 14:21 • Substitute $t'=t-\mu$, pull out the exponential factor that is independent of $t,$ and then recognize the remaining integral as the Fourier transform of $g$ at $\eta-\eta_{0}$. – RideTheWavelet Nov 26 '17 at 14:33	2019-06-18 07:17:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.97188401222229, "perplexity": 147.160088067887}, "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/1560627998690.87/warc/CC-MAIN-20190618063322-20190618085322-00315.warc.gz"}
http://mathematica.stackexchange.com/tags/symbolic/hot?filter=all	# Tag Info 42 I guess that V9 now adds this capability: $Assumptions = { Element[A, Matrices[{m, n}]], Element[B, Matrices[{n, k}]] }; TensorReduce[ Transpose[Transpose[A].Transpose[B]] ] (* Out: B.A ) 42 In the first case PowerExpand comes to the rescue: PowerExpand@Power[p^(n m) q, 1/n] ( Out: p^m q^(1/n) ) Note however that "the transformations made by PowerExpand are correct only if$c$is an integer or$a$and$b$are positive real numbers". Generally speaking, your assumptions can be listed in Reduce, Simplify, or FullSimplify using the ... 36 Solutions to algebraic or transcendental equations are expressed in terms of Root objects whenever it is impossible to find explicit solutions. In general there is no way express roots of 5-th (or higher) order polynomials in terms of radicals. However even higher order algebraic equations can be solved explicitly if an associated Galois group is solvable. ... 30 EDIT: I've put this code on a GitHub but I don't know what features are needed or what problems it may give. I'm just not using it. But I will incorporate incomming suggestions as soon as I have time. So feedback in form of tests and suggestions very appreciated! Import[ "https://raw.githubusercontent.com/kubaPod/MoreCalculus/master/install.m" ] ... 26 Mathematica does not support this directly. You can do things of this sort using an external package called NCAlgebra. http://math.ucsd.edu/~ncalg/ The relevant documentation may be found at http://math.ucsd.edu/~ncalg/DOWNLOAD2010/DOCUMENTATION/html/NCBIGDOCch4.html#x8-510004.4 In particular have a look at "4.4.8 NCLDUDecomposition[aMatrix, Options]" ... 26 Removing the imaginary portion of an expression is done by doing ComplexExpand[Re[expression]]. Using just Re alone will not work as Re does no evaluation on symbols with unknown complex parts. Now as stated in the problem and the comments above this particular problem requires a fair amount of assumptions. The simplest way to add local assumptions is to ... 26 You can't use replacements that way, because Mathematica does not do replacements on expressions the way they appear to you. To see what I mean, take a look at the FullForm of your expression: x/(yz) // FullForm Out[1]= Times[x,Power[y,-1],Power[z,-1]] Whereas, the replacement that you're using is Times[y, z]. In general, it is not a good idea to use ... 23 The plan is first get the "external" contour and then use Green's theorem to find its area. r[t_] := {-9 Sin[2 t] - 5 Sin[3 t], 9 Cos[2 t] - 5 Cos[3 t], 0} (find the intersections) tr = Quiet@ToRules@Reduce[{r@t1 == r@t2, 0 < t1 < t2 < 2 Pi}, {t1, t2}]; pt = {t1, t2} /. {tr} // Flatten; pts = SortBy[pt, N@# &]; pps = Partition[pts, 2]; Now ... 22 What you have is a MultinormalDistribution. The quadratic and linear forms in the exponential can be rewritten in terms of$-\frac12(\vec{x}-\vec{\mu})^\top\Sigma^{-1}(\vec{x}-\vec{\mu})$where$\vec{\mu}$represents the mean and$\Sigma$the covariance matrix, see the documentation. With this, you can do integrals of the type given in the question by ... 22 A first step would be to implement a convenience function that can automatically apply the method of separation of variables to separable types of equations. To show that the steps could in principle be automated, let me repeat basically the same calculation that I did for cylindrical coordinates with only slight modifications to the heat equation: ... 22 I've decided to expand on my comment. Before I delve into the solution, let's all pause for a moment and marvel at the stereographic parametrization of a unit circle: $$\begin{pmatrix}\frac{1-t^2}{1+t^2}\\\frac{2t}{1+t^2}\end{pmatrix}$$ Sometimes also referred to as the Weierstrass substitution, it has often been used as a tool in the solution of algebraic ... 21 The nearest Mathematica has to "types" are Heads of expressions that are Atoms. For example: Through[{AtomQ, Head}[2]] {True, Integer} Through[{AtomQ, Head}[2 + I]] {True, Complex} Through[{AtomQ, Head}["cat"]] {True, String} and so on... There are also somewhat different "types" in the context of Compile. 20 The most direct way to test this is probably the following:$Assumptions = x > 0; Element[x, Reals] // Simplify (* Out[1]= True ) $Assumptions = True; Element[x, Reals] // Simplify ( Out[4]= x ∈ Reals ) So$x>0$seems to imply that$x$is real. 20 After all this time, I came up with a very nice tensor calculus proof of the Hairy Ball Theorem. It only depends on Stokes theorem and standard laws of tensor calculus like the Ricci identity and symmetries of curvature tensors. All the topology is done by Stokes theorem. The remainder of the proof is equational, local and geometrical. It is coordinate/basis ... 19 One way is to use an extra argument that acts as a switch. Clear[f]; f[0] = 1; f[1] = 1; f[n_, True] := f[n - 1] + f[n - 2] Example: f7 = f[7, True] ( Out[329]= f[5] + f[6] ) To proceed another step, can do a replacement. f7 /. f[aa_] :> f[aa, True] ( Out[330]= f[3] + 2 f[4] + f[5] ) Can use Nest to repeat this n times. Nest[# /. f[aa_] ... 18 There is no need to play around with ReplaceAll, Rule, Block, Module or whatever using D, since you have an oparator Derivative really fulfilling your needs while you need not bother if the arguments were defined, so I recommend it to find symbolic derivatives of your function. Remember of shorthands f', f'' to represent first and second derivatives of ... 18 Here is extensions to @Jens answer (I think) also relying on possible separation of variable. It is not meant as an independent answer, but complements it. First extend his answer to 2D ClearAll[pt, px, x, t, p]; operator = Function[p, D[p, t] - Δ D[p, x, x] - Δ D[p, y, y]]; ansatz = pt[t] px[x] py[y]; pde2 = Expand[Apply[Subtract, ... 18 Disclaimer: This is not a full answer, but perhaps it's a start. From an algebraic stand point this seems like a very hard problem. I attacked it with a more brute force approach. I guess a basis and use LatticeReduce to try to find a Diophantine relation. Note this code only tries to identify roots as the product of integral powers of trig. If it returns ... 18 I can't help you with functions beyond Reduce, Simplify, ... but I can offer you a tip to help with reducing errors from manual translation. In order to validate that your manual transformation is correct, one can subtract the original expression from the manual transformation and then use Simplify on that expression. Sometimes it will return zero using ... 17 Initially, Mathematica is not designed for such abstract calculations. But, Mathematica is a powerful programming language, so that one can add such functionality easily. See the following examples in related area of differential geometry: calculations in symbolic dimensions Abstract calculations 17 It is assumed that$x$is a real number. Everything else would mathematically not make sense because on complex numbers there does not exist an ordering relation. An example would be to take the expression$\sqrt{x^2}$and to imagine that this is not equal$x$for$x=-\mathbb{i}\$. Therefore the expression is in a general form not simplified In[37]:= ... 17 In this case you can use SeriesCoefficient SeriesCoefficient[Exp[x], {x, 0, n}] 17 Short story $$\vartheta(x) = \arg \left[(\operatorname{Bi}x+i \operatorname{Ai}x)e^{-\frac{2}{3} i (-x)^{3/2}}\right]+\frac{2}{3} \operatorname{Re}\left[(-x)^{3/2}\right]$$ Update: I see that you want use only real functions, so you can expand this as $$\vartheta(x) = \begin{cases} \arctan\frac{\cos \left(\frac{2}{3} (-x)^{3/2}\right) ... 17 Looking at the Trace of one which does work: x = Sin[Pi/5] ( Sqrt[5/8 - Sqrt[5]/8] *) Trace[ArcSin[x], TraceInternal -> True] It appears that Mathematica computes the ArcSin numerically and then recognises the result, 0.628319 as possibly equal to Pi/5. To check it computes Sin[Pi/5], and subtracts it from the original argument to see if it gets ... 17 One thing is to make sure you have all the assumptions stated properly. For instance, the first two cases can be handled by passing all the assumptions to FullSimplify FullSimplify[Power[p^(n m) q, 1/n], Assumptions -> {q > 0, p > 0, n ∈ Integers, m ∈ Integers}] p^m q^(1/n) and FullSimplify[(p^n q)/(p^m r), Assumptions -> {q > 0, p ... 17 I use Mathematica in much the same way as you, although in the context of multi-stage physics derivations. Since no one has mentioned it, I'll describe an obvious approach to successive rewrites of an expression. I label each step a calculation with an indexed symbol that I can refer to later. This is preferable to In's and Out's whose numbers can change if ... 16 Use the following representation of the Legendre polynomials:$$ P_n(x) = 2^n \sum_{k=0}^n x^k \binom{n}{k} \binom{\frac{n+k-1}{n}}{n} $$Note that the sum effectively is over k \equiv n \bmod 2. Expand each Legendre polynomial into a sum. Integration with respect to \theta is easy:$$ \int_0^{\pi} \sin^{k_1+k_2+k_3+1} \theta \mathrm{d}\theta ... 16 If you use the third argument in Solve, i.e. a list of variables to be eliminated (take a look at the Eliminating Variables tutorial in Mathematica) then you'll get the result immediately : Solve[{a b c == -1, a^2/c + b/c^2 == 1, a^2 b + b^2 c + c^2 a == t, a b^5 + b c^5 + c a^5 == res}, {res}, {a, b, c, t}] {{res -> 3}} Edit ... 16 The code for the default ComplexityFunction was posted on MathSource a number of years ago by Adam Strzebonski (of Wolfram Research). You will see reference to the original reply from Adam referenced in a MathGroup reply from Andrzej Kozlowski dated 12 Jan 2010 with the subject: "[mg106386] Re : Radicals simplify". I mention all that because I can't get the ... 16 Having experienced similar problematic issues with Mathematica I instantly thought that expanding the fraction in the integrand i.e. applying Appart could resolve the problem, and indeed it does: Integrate[ Apart[(1 - x)(1 + 2x)^6/Sqrt[1 - x^2]], {x, -1, 1}]/Pi 15 These arguments apply to this case as well Bug in mathematica analytic integration? i.e. ... Only top voted, non community-wiki answers of a minimum length are eligible	2016-04-29 08:29:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5996538400650024, "perplexity": 3357.1057454852025}, "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-2016-18/segments/1461860110805.57/warc/CC-MAIN-20160428161510-00045-ip-10-239-7-51.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/53819/changing-a-bezier-curve-by-dragging-a-point-on-the-curve-itself-rather-than-a-co?answertab=votes	# Changing a bezier curve by dragging a point on the curve itself rather than a control point I'm developing an iPhone app that allows users to cut out part of an image from its background. In order to do this, they'll connect a bunch of bezier curves together to form the clipping path. Rather than have them move the control points of the curves in order to change the curve, I'd like them to be able to move a point on the curve itself. At this point, I'm not set on using quadratic rather than cubic beziers, or vice versa. The rest of this post will discuss the issue from the point of view of quadratic curves, but I would love an answer that provides solutions to both. It seems to me that there are two possible methods for accomplishing this: 1. By determining the relationship between the point on the curve and the true control point. If this can be done, then the new location of the control point can be calculated based on the new location of the point on the curve. 2. By using an equation that can estimate a bezier curve based on two end points and a third point on the curve (if cubic, then a third and fourth point on the curve). Are either of these possible? I've been googling this for a while and have come across a couple potential solutions using method #2, but I don't fully understand either: Solution 1 (click here): I think I understand the code in this example well enough, but I don't know how to calculate t. Solution 2 (click here): This code is written in C#. I tried converting it to Objective-C (the language used for iPhone apps), but ultimately got stuck on the "solvexy()" function because I don't see how i or j are used after calculating their values. In regards to method #1, it seems to me that there should be some mathematical relationship between the control point, and the point on the curve through which a tangent to the curve at that point would be perpendicular to a line drawn from the control point to that point. Here are a couple illustrations: quadratic, cubic. The idea is that this point which lies on the curve, through which the perpendicular tangent is drawn, is the point that users would be dragging in order to modify the curve. - Solution 1: $t$ looks like the parameter along the curve, with $t\in[0,1]$, I'd say. Solution 2: i and j are output parameters; they get assigned to the corresponding argument in the caller; in this case first x1, x2, then y1, y2. You might want to take a look at the sidebar to the right; there are lots of answers to similar questions there; e.g. math.stackexchange.com/questions/5166/…, math.stackexchange.com/questions/42395/…. – joriki Jul 26 '11 at 11:22 Thanks for the clarification in the C# code. I'll give it another try with that info. And I did read those two questions you linked to. The first one's answer is that a Bezier should not be used, but rather a parametric spline. So I spent a while looking into how to create a parametric spline (as well as what one is in the first place) in objective-c, but couldn't come up with anything. In the second post you link to it seems that the OP is asking about how to create a Bezier simply from two end points, but I will know at least one other point that the curve must pass through. – maxedison Jul 26 '11 at 12:12 Approach 2 is underspecified. Given $P(0)$, $P(1)$, $P(t)$, and $t\in(0,1)$ you have a unique quadratic Bézier curve and can find the non-interpolated control point using the Bernstein polynomials, but you're not specifying $t$. I suspect you could specify two interpolated points and get a unique quadratic Bézier curve, but I haven't worked through the details. – Peter Taylor Jul 26 '11 at 12:23 Someone should have inserted the linked illustrations into the question before FreeImageHosting deleted them... (Using the StackExchange "insert image" button uploads the images to Imgur, which SE has a partnership with, so the images will stay up as long as SE does.) – Rahul May 25 '14 at 18:07 I think the simplest thing that would work in your application is to show the user 4 special points on the parametric cubic curve, and allow the user to manipulate those 4 special points. (Allowing the user to pick any point on the curve, and move it, makes things more complicated). I think this is the same as what Stephen H. Noskowicz calls "Cubic Four Point" representation, aka the quadratic Lagrange with t1 = 1/3 and t2 = 2/3. While your user is moving those 4 special points U0, U1, U2, U3 around, periodically you find a cubic Bezier curve that goes through those 4 points using John Burkardt's approach: P0 = U0 P1 = (1/6)( -5U0 + 18U1 - 9U2 + 2U3 ) P2 = (1/6)( 2U0 - 9U1 +18U2 - 5U3 ) P3 = U3. That gives you the Bezier curve representation of the same cubic curve -- a series of 4 control points. You then feed those 4 control points (the endpoints P0 and P3, and the intemediate control points P1 and P2) into any Bezier curve plotter. The resulting curve (usually) doesn't touch P1 or P2, but it will start at X0, go exactly through X1 and X2, and end at X3. (This uses the special points at t=0, 1/3, 2/3, and 1. It's possible to, instead, use the special points at t=1, 1/4, 3/4, and 1, as shown at How do I find a Bezier curve that goes through a series of points? . Or, I suppose, any 4 distinct t values. But I suspect the 0, 1/3, 2/3, 1 values are used most often, and I don't see any advantage to using any other fixed values). - If your only problem is to find the point at which the tangent line is perpendicular to the direction to the control point and to find the control point that has this property for a given waypoint on the curve, you can try the following. Let's do quadratic curves. If A is the beginning, B is the end (both fixed), C is the control and X is the waypoint, we have the system $X=(1-t)^2A+2t(1-t)C+t^2B$ (curve equation) $\langle X-C,-(1-t)A+(1-2t)C+tB\rangle=0$ (orthogonality relation). Suppose we know $C$. Then plugging $X$ from the first equation into the second, we get a cubic equation for $t$, which we can solve quickly by either bisection, or Newton, or a combination of both. The inverse problem is similar. Express $C$ from the first equation and plug the result into the second one. Now you'll get a fifth degree equation to solve but if bisection is quick enough for you (you do not need really high precision for point dragging), you can do everything without too heavy thinking (we need to make sure that the singularities at the endpoints are dealt with in some reasonable way, so if the user drags the point close to one of the endpoints you don't have an aberrant behavior but I do not want to think of this singularity problem before you tell me if this approach works for you in general). The cubic curves have two controls, so the problem is ill-posed for them as stated. - This site: http://pomax.github.io/bezierinfo/ describes a way of using de Casteljau's algorithm 'in reverse' to do something like this, they call it 'curve moulding'. It has javascript source code as well. (de Casteljau uses points on lines (linear interpolation) to draw beziers, not higher degree equations) - Ego surfing, I happend by and saw this. Though I see this is quite old. An algorithm for "grabbing a curve" that has off the curve control points is quite simple. I guess I didn't include it in my book. Bummer, but I got it from David Forsey in 2000 on UseNet comp.graphics.algorithms. It pairs the cursor with, IIR the nearest control point, then just moves the control point the same distance as the cursor. Sorry I didn't see the edit link before... -- Regards, Steve Noskowicz .................................. Here is my rewrite followed by the original I found on the net. If the formatting is too screwy, email me for an original on my site or... noskosteve /at/ Yahoo /dot/ com: PAGE 1 How to "grab" a spot on a curve to move it (familiar notation). For a parmetric cubic curve, a point on the curve at parameter t (Pt) is defined by: P(t) or Pt = Sum PiBi(t) = P1B1(t) + P2B2(t) + P3B3(t) + P4B4(t). Pi are the 4 control points and Bi are the respective weighting functions. To "grab" a spot on the curve and relocate it, you are moving some Pt to Pt'. Let the amount you move it be Pt = (New point - old point) = Pt'- Pt. Choose a control point to be moved. (see page 2). Any will do, but choosing the "closest" in parametric space works best. Let's say we'll move P2. Call the new position P2'. This makes P2 = P2' - P2 The old point on the curve was: Pt = P1B1(t) + P2B2(t) + P3B3(t) + P4B4(t) The new point on the curve will be: Pt'= P1B1(t) + P2'B2(t) + P3B3(t) + P4B4(t) Subtract the Old from the New: Pt = Pt'- Pt = (P2' - P2) B2(t) = P2B2(t) Solving for P2: P2 = Pt / B2(t) Then solve for the New P2 (P2'): P2' = P2 + [ Pt / B2(t) ] OR P2' = P2 + ( [Pt'- Pt] / B2(t) ) So... In general: Given: a point on the curve called Pt at parameter "t", "near" control point "n". To move that point on the curve to a new location which differs in position by Pt, move Pn by Pt/Bn(t) In other words: Select a spot on the curve to be moved and its new location. Determine the parameter value of this spot. Calculate the change in spot position, Pt. Divide this change, Pt, by the basis function (for the target control point or Bn(t) ) which is evaluated at the selected parameter (t). Finally, add this to the control point (P2) location. This is the same function and t for all dimensions, so only one calculation is required. Note that this requires calculating the weighting function for the target control point at the parameter value (t) of the location selected on the curve. You obtain it from the scan for the target "t". This works with any basis function. Concept: if Curve = Control PtWeight then Control Pt = Curve / Weight PAGE 2 Further areas for consideration. Finding the point on the curve to be moved. First, the parameter value of the selected (grabbed) location on the curve must be determined. This must be a sequential search rather than binary because the curve may double back. This could make the true closest point "un-findable" if the curve approaches then moves away from the cursor before going back to closest location the cursor. It can be a coarse-fine search, such as stepping in t steps of 0.1 or larger to find the bracketing locations, then subdividing that sub-segment to hit the curve. The limit being the closest to the nearest pixel on the screen. This takes the form of re-calculating the curve without needing to draw it; only calculating the distance. Choosing the target control point. For the cubic Bézier, the following are proposed. 1) Because the end control points are directly accessable, one method is to divide the segment, and therefore the parameter range, in halves and pick the inner control point related to the half occupied by the grab-location. P1 0.0 - 0.5 P2 0.5 - 1.0 This, however, can cause an extreme effect upon a control point (and the curve) when the curve is grabbed very near an end. In an attempt to alleviate this, #2 is suggested. 2) Divide the segment, and therefore the parameter range, into 6ths. (It could actually be any fraction, but this seems "natural") The end points are used for the first sixth (0-1/6) and last sixth (5/6-1). The inner control points are then used for the inner thirds. That is 1/6 to 3/6 and 3/6 to 5/6 Given P0, P1, P2, P3, the range of "t" to use for each point is: P0 0.0 - 0.16667 P1 0.16667 - .5 P2 0.5 - 0.83333 P3 0.83333 - 1.0 The weighting functions in the Basic Image Editor are: P0 W1(t) = Ft1t3 + Fu1t2 + Fv1t + Fw1 P1 W2(t) = Ft2t3 + Fu2t2 + Fv2t + Fw2 P2 W3(t) = Ft3t3 + Fu3t2 + Fv3t + Fw3 P3 W4(t) = Ft4t3 + Fu4t2 + Fv4*t + Fw4 PAGE 3 Original posting. Conversion to standard names is on page 1 Date: Fri, 07 Jan 2000 14:48:10 -0800 From: David Forsey Newsgroups: comp.graphics.algorithms Subject: Re: Q: Bezier curve editing Toby wrote: I would be most grateful if anyone could answer this straightforward question for me. In graphics programs (such as Corel Draw) the user can directly manipulate (cubic) Bezier curves. The user can "grab" a point on the curve and drag it around. The control points move so that the new point is on the new curve (at the same parameter value, I suppose). The endpoints stay fixed. How do the control points move? A direct solution can't work because only 3 points are known on the new curve, and 4 would be needed for a cubic. I wondered about cubic Hermite interpolation, but don't see where the tangent vectors would come from. For the parmetric cubic curve C(U), for a point P on the curve at parameter u. P = C(u) = Sum Vi Bi(u) = V1B1(u) + V2B2(u) + V3B3(u) + V4B4(u). (where the Vi are the 4 control points and the Bi are the basis functions) You want to move P to P'. Let deltaP = P'- P. Choose a control point that will be used to change the shape of the curve. Anyone will do, but choosing the closest in parametric space works best. Lets say we'll move V2. To make it clear lets call this new position X. The old point on the curve was: P = V1B1(u) + V2B2(u) + V3B3(u) + V4B4(u) The new point on the curve will be: P'= V1B1(u) + XB2(u) + V3B3(u) + V4B4(u) Subtract them appropriately: deltaP = (V2 - X) B2(u) = deltaV2 B2(u) A little algebra: deltaV = deltaP / B2(u) Shouldn't this be: deltaV2 = deltaP / B2(u) So... to move a point P on the curve by deltaP, move V2 by deltaP/B2(u) This works with any basis function, b-spline or bezier or whatever as well as for surfaces. You can also move multiple control vertices to move that point on the curve. See "A Technique for the Direct Manipulation of Spline Curves", Bartels/Beatty Graphic Interface '89. There are subsequent papers that cover altering the tangent and curvature at specified points on a spline. Dave -	2015-07-07 22:11:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6278324723243713, "perplexity": 950.5340834341351}, "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-2015-27/segments/1435375102712.76/warc/CC-MAIN-20150627031822-00139-ip-10-179-60-89.ec2.internal.warc.gz"}
https://www.projecteuclid.org/euclid.agt/1513796660	## Algebraic & Geometric Topology ### Volumes of highly twisted knots and links Jessica Purcell #### Abstract We show that for a large class of knots and links with complements in $S3$ admitting a hyperbolic structure, we can determine bounds on the volume of the link complement from combinatorial information given by a link diagram. Specifically, there is a universal constant C such that if a knot or link admits a prime, twist reduced diagram with at least 2 twist regions and at least C crossings per twist region, then the link complement is hyperbolic with volume bounded below by 3.3515 times the number of twist regions in the diagram. C is at most 113. #### Article information Source Algebr. Geom. Topol., Volume 7, Number 1 (2007), 93-108. Dates Revised: 3 January 2007 Accepted: 3 January 2007 First available in Project Euclid: 20 December 2017 https://projecteuclid.org/euclid.agt/1513796660 Digital Object Identifier doi:10.2140/agt.2007.7.93 Mathematical Reviews number (MathSciNet) MR2289805 Zentralblatt MATH identifier 1135.57005 #### Citation Purcell, Jessica. Volumes of highly twisted knots and links. Algebr. Geom. Topol. 7 (2007), no. 1, 93--108. doi:10.2140/agt.2007.7.93. https://projecteuclid.org/euclid.agt/1513796660 #### References • C C Adams, Augmented alternating link complements are hyperbolic, from: “Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984)”, London Math. Soc. Lecture Note Ser. 112, Cambridge Univ. Press, Cambridge (1986) 115–130 • I Agol, N Dunfield, P Storm, W P Thurston, Lower bounds on volumes of hyperbolic Haken 3-manifolds • K B ör öczky, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hungar. 32 (1978) 243–261 • C Cao, G R Meyerhoff, The orientable cusped hyperbolic $3$-manifolds of minimum volume, Invent. Math. 146 (2001) 451–478 • D Cooper, C D Hodgson, S P Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs 5, Mathematical Society of Japan, Tokyo (2000) With a postface by Sadayoshi Kojima • D Futer, J S Purcell, Links with no exceptional surgeries • C D Hodgson, Degeneration and regeneration of geometric structures on $3$-manifolds, PhD thesis, Princeton Univ. (1986) • C D Hodgson, S P Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery, J. Differential Geom. 48 (1998) 1–59 • C D Hodgson, S P Kerckhoff, Universal bounds for hyperbolic Dehn surgery, Ann. of Math. $(2)$ 162 (2005) 367–421 • M Lackenby, The volume of hyperbolic alternating link complements, Proc. London Math. Soc. $(3)$ 88 (2004) 204–224 With an appendix by Ian Agol and Dylan Thurston • C J Leininger, Small curvature surfaces in hyperbolic 3-manifolds, J. Knot Theory Ramifications 15 (2006) 379–411 • J S Purcell, Cusp shapes under cone deformation • J S Purcell, Cusp Shapes of Hyperbolic Link Complements and Dehn Filling, PhD thesis, Stanford University (2004) • D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish, Berkeley, CA (1976) • W P Thurston, The Geometry and Topology of Three-Manifolds, Princeton Univ. Math. Dept. Notes (1979) • J R Weeks, Snappea http://www.geometrygames.org/SnapPea/	2020-01-20 06:57:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5340818166732788, "perplexity": 2316.8540784259744}, "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/1579250597458.22/warc/CC-MAIN-20200120052454-20200120080454-00042.warc.gz"}
https://crazyproject.wordpress.com/2012/01/03/properties-of-the-derivative-of-a-matrix-power-series/	## Properties of the derivative of a matrix power series Let $G(x)$ be a formal power series on $\mathbb{C}$ having an infinite radius of convergence and fix an $n \times n$ matrix $A$ over $\mathbb{C}$. The mapping $t \mapsto G(At)$ carries a complex number $t$ to a complex matrix $G(At)$; we can think of this as the ‘direct sum’ of $n \times n$ different functions on $\mathbb{C}$, one for each entry of $G(At)$. We now define the derivative of $G(At)$ with respect to $t$ to be a mapping $\mathbb{C} \rightarrow \mathsf{Mat}_n(\mathbb{C})$ as follows: $\left[\frac{d}{dt} G(At)\right]_{i,j} = \frac{d}{dt} \left[ G(At)_{i,j}\right]$. In other words, thinking of $G(At)$ as a matrix of functions, $\frac{d}{dt} G(At)$ is the matrix whose entries are the derivatives of the corresponding entries of $G(At)$. We will use the limit definition of derivative (that is, $\frac{d}{dt} f(t) = \mathsf{lim}_{h \rightarrow 0} \dfrac{f(t+h) - f(t)}{h}$, where it doesnt matter how $h$ approaches 0 in $\mathbb{C}$) and will assume that all derivatives exist everywhere. Prove the following properties of derivatives: 1. If $G(x) = \sum_{k \in \mathbb{N}} \alpha_kx^k$, then $\frac{d}{dt} G(At) = A \sum_{k \in \mathbb{N}} (k+1)\alpha_{k+1}(At)^k$. 2. If $V$ is an $n \times 1$ matrix with constant entries (i.e. not dependent on $t$) then $\frac{d}{dt} (G(At)V) = \left( \frac{d}{dt} G(At) \right) V$. [My usual disclaimer about analysis applies here: as soon as I see words like ‘limit’ and ‘continuous’ I become even more confused than usual. Read the following with a healthy dose of skepticism, and please point out any errors.] Note the following. $\left[ \dfrac{d}{dt} G(At) \right]_{i,j}$ = $\dfrac{d}{dt} G(At)_{i,j}$ = $\mathsf{lim}_{h \rightarrow 0} \dfrac{G(A(t+h)_{i,j}) - G(At)_{i,j}}{h}$ = $\mathsf{lim}_{h \rightarrow 0} \left[ \dfrac{G(A(t+h)) - G(At)}{h} \right]_{i,j}$ = $\mathsf{lim}_{h \rightarrow 0} \left[ \dfrac{\sum_k \alpha_k(A(t+h))^k - \sum_k \alpha_k(At)^k}{h} \right]_{i,j}$ = $\mathsf{lim}_{h \rightarrow 0} \left[ \dfrac{\sum_k \alpha_k A^k ((t+h)^k - t^k)}{h} \right]_{i,j}$ = $\mathsf{lim}_{h \rightarrow 0} \left[ \dfrac{\sum_{k > 0} \alpha_k A^k \left( \sum_{m=0}^k {k \choose m} t^m h^{k-m} - t^k \right)}{h} \right]_{i,j}$ = $\mathsf{lim}_{h \rightarrow 0} \left[ \dfrac{\sum_{k > 0} \alpha_k A^k \left( \sum_{m=0}^{k-1} {k \choose m} t^m h^{k-m} \right)}{h} \right]_{i,j}$ = $\mathsf{lim}_{h \rightarrow 0} \displaystyle\sum_{k > 0} \alpha_k A^k \sum_{m=0}^{k-1} {k \choose m} t^m h^{k-1-m}$ (Now we can substitute $h = 0$.) = $\displaystyle\sum_{k > 0} \alpha_k A^k {k \choose {k-1}} t^{k-1}$ (All terms but $m=k-1$ vanish.) = $\displaystyle\sum_{k > 0} k \alpha_k A^k t^{k-1}$ = $\displaystyle\sum_k (k+1) \alpha_{k+1}A^{k+1}t^k$ = $A \sum_k (k+1)\alpha_{k+1}(At)^k$ As desired. Now say $V = [v_i]$ and $G(At) = [c_{i,j}(t)]$; we then have the following. $\dfrac{d}{dt} \left( G(At)V \right)$ = $\dfrac{d}{dt} \left( [c_{i,j}(t)][v_{i,j}] \right)$ = $\dfrac{d}{dt} [\sum_k c_{i,k}(t)v_k]$ = $[\frac{d}{dt} \sum_k c_{i,k}(t)v_k]$ = $[\sum_k (\frac{d}{dt} c_{i,k}(t)) v_k]$ = $[\frac{d}{dt} c_{i,j}(t)][v_i]$ = $\left(\frac{d}{dt} G(At) \right)V$ As desired.	2017-03-28 06:13:29	{"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": 51, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.966854989528656, "perplexity": 91.77483104989716}, "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-2017-13/segments/1490218189680.78/warc/CC-MAIN-20170322212949-00037-ip-10-233-31-227.ec2.internal.warc.gz"}
https://www.tutorialspoint.com/count-of-numbers-such-that-difference-between-the-number-and-sum-of-its-digits-not-less-than-l-in-cplusplus	# Count of Numbers such that difference between the number and sum of its digits not less than L in C++ We are given a number N and another number L. The goal is to find the numbers between 1 and N that have a difference between the number itself and the sum of its digits is not less than L. If N=23, L=10 then the count of such numbers will be 4. 23-(2+3)=18, 22-(2+2)=18, 21-(2+1)=18, 20-(2+0)=18. All above numbers meet the condition But 19-(1+9)=9 which is less than L, similarly 18,17….1. Let us understand with examples Input − N=30 L=19 Output − Count of Numbers such that difference between the number and sum of its digits not less than L are − 1 Explanation − Only 30 meets the condition, 30-(3+0)=27 > 19 Input − N=123330 L=5466 Output − Count of Numbers such that difference between the number and sum of its digits not less than L are − 6841 ## Approach used in the below program is as follows Using binary search we will find the first number that meets the condition. If that number is num then the condition will also be true for num+1 and so on. If any current mid value satisfies the condition then all numbers between mid and end will also satisfy this condition so we can simply add end-mid+1 to count. • Take num and L as long variables. • Function Digit_sum(LL num) takes a number num and returns the sum of its digits. • Take initial sum as total=0. • Using a while loop, add reminder num%10 to total and reduce num by 10. Do this until num>0. • Function Less_than_L(LL num, LL L) takes a number num and a number L and returns count of Numbers such that difference between the number and sum of its digits not less than L • Take the initial count as 0. • Implement binary search using while loop where start=1 and end=num. • Calculate middle number as temp=(start+end)/2. • If the difference between temp and sum of digits of temp is not less than L then all numbers greater than temp will also satisfy the same condition. • Count of such numbers including temp will be num-temp+1. Add this to count. And set end=temp-1. • Otherwise set start=temp+1. • At the end of binary search count will have numbers with difference between them and sum of digits not less than L • Return count as result. ## Example Live Demo #include <bits/stdc++.h> using namespace std; typedef long long LL; int Digit_sum(LL num){ LL total = 0; while (num > 0){ total += num % 10; num = num/10; z} } LL Less_than_L(LL num, LL L){ LL count = 0; LL start = 1; LL end = num; while (start <= end){ LL temp = (end + start) / 2; LL temp_2 = temp - Digit_sum(temp); if (temp_2 >= L){ count = num - temp + 1; end = temp - 1; } else{ start = temp + 1; } } return count; } int main(){ LL num = 234516; LL L = 235; cout<<"Count of Numbers such that difference between the number and sum of its digits not less than L are: "<< Less_than_L(num, L); return 0; } ## Output If we run the above code it will generate the following output − Count of Numbers such that difference between the number and sum of its digits not less than L are: 234267	2023-02-07 18:41:52	{"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": 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": 0.23827549815177917, "perplexity": 2267.2895483185494}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500628.77/warc/CC-MAIN-20230207170138-20230207200138-00058.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/college-algebra-7th-edition/chapter-1-equations-and-graphs-section-1-10-modeling-variation-1-10-exercises-page-164/17	College Algebra 7th Edition $\displaystyle R= \frac{kP^{2}t^{2}}{b^{3}}$ Since $R$ is directly (jointly) proportional to the squares of $P$ and $t$, but inversely proportional to the cube of $b$, we have: $\displaystyle R= \frac{kP^{2}t^{2}}{b^{3}}$ (where $k$ is a constant.)	2018-07-19 14:09:18	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9953418374061584, "perplexity": 221.88575288996975}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590901.10/warc/CC-MAIN-20180719125339-20180719145339-00033.warc.gz"}
https://plainmath.net/90911/upper-bound-of-natural-logarithm-i-was-p	# Upper bound of natural logarithm I was playing looking for a good upper bound of natural logarithm and I found that ln x<=x^1/e apparently works: Can someone give me a formal proof of this inequality? Upper bound of natural logarithm I was playing looking for a good upper bound of natural logarithm and I found that $\mathrm{ln}x\le {x}^{1/e}$ apparently works: Can someone give me a formal proof of this inequality? You can still ask an expert for help • Questions are typically answered in as fast as 30 minutes Solve your problem for the price of one coffee • Math expert for every subject • Pay only if we can solve it Trace Arias Consider the fuction $f\left(x\right)=\mathrm{log}\left(x\right)-{x}^{1/e}$ Its derivative ${f}^{\prime }\left(x\right)=\frac{e-{x}^{\frac{1}{e}}}{ex}$ cancels for $x={e}^{e}$ and, for this value $f\left(x\right)=0$; the second derivative test shows that this is a maximum(${f}^{″}\left({e}^{e}\right)=-{e}^{-1-2e}$). Then $\mathrm{log}\left(x\right)\le {x}^{1/e}$ is always satisfied.	2022-10-02 06:04:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 42, "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": 0.8625837564468384, "perplexity": 798.0841148401236}, "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/1664030337287.87/warc/CC-MAIN-20221002052710-20221002082710-00075.warc.gz"}
http://cdsweb.cern.ch/collection/ATLAS%20Conference%20Contributions?ln=en&as=1	# ATLAS Conference Contributions 2020-06-26 16:12 Performance study of the ATLAS Forward Proton Time-of-Flight Detector System - Contribution to proceedings of the 28th International Workshop on Vertex Detectors - Vertex2019 / Cerny, Karel (Palacky University, RCPTM) The performance of the ATLAS Forward Proton Time-of-Flight (ToF) detector is studied using the ATLAS LHC data collected in the $2017$ running period of LHC Run2. [...] ATL-FWD-PROC-2020-002. - 2020. - 7 p. Original Communication (restricted to ATLAS) - Full text 2020-06-23 10:37 Fast tracking for the HL-LHC ATLAS detector / Klimpel, Fabian (European Laboratory for Particle Physics, CERN) During the High-Luminosity Phase 2 of LHC, up to 200 simultaneous inelastic proton-proton collisions per bunch crossing are expected. [...] ATL-PHYS-PROC-2020-044. - 2020. - 9 p. Original Communication (restricted to ATLAS) - Full text 2020-06-17 19:14 Deep Sets for Flavor Tagging on the ATLAS Experiment / Hartman, Nicole Michelle (SLAC National Accelerator Laboratory) ; Kagan, Michael (SLAC National Accelerator Laboratory) ; Teixeira De Lima, Rafael (SLAC National Accelerator Laboratory) Flavour Tagging is a major client for tracking in particle physics experiments at high energy colliders, where it is used to identify the experimental signatures of heavy flavor production. [...] ATL-PHYS-PROC-2020-043. - 2020. - 9 p. Original Communication (restricted to ATLAS) - Full text 2020-06-16 20:13 The ATLAS Muon Trigger Design and Performance / Drobac, Alec Swenson (Tufts University) Muon triggers are essential for studying a variety of physics processes in the ATLAS experiment, including standard model measurements and searches for new physics. [...] ATL-DAQ-PROC-2020-011. - 2020. - 4 p. Original Communication (restricted to ATLAS) - Full text 2020-06-16 09:50 Performance of the BIS78 RPC detectors: a new concept of electronics and detector integration for high-rate and fast timing large size RPCs / Pizzimento, Luca (INFN Roma Tor Vergata and Universita' di Roma Tor Vergata, Dipartimento di Fisica) The reduction of the average charge per count in the gas along with the capability to discriminate very small avalanche signals, can allow an efficient and long-term Resistive Plate Chamber detector operation, in high radiation background environment. [...] ATL-MUON-PROC-2020-012. - 2020. - 10 p. Original Communication (restricted to ATLAS) - Full text 2020-06-05 07:59 Rescuing VBF Higgs Invisible Events with Novel Vertex Selection / Safdari, Murtaza (SLAC National Accelerator Laboratory) ; Schwartzman, Ariel (SLAC National Accelerator Laboratory) ; Cairo, Valentina Maria (SLAC National Accelerator Laboratory) ; Pettersson, Nora Emilia (University of Massachusetts, Amherst) ; Goblirsch-Kolb, Maximilian (Brandeis University, Department of Physics) ; Lee, Graham Richard (University of Bergen) ; Su, Dong (SLAC National Accelerator Laboratory) - Advisor vertexing convenor tracking convenor tracking convenor vertexing convenor Internal Reviewer This is an update to, and extension of, the work in ATL-PHYS-PROC-2019-044. [...] ATL-PHYS-PROC-2020-042. - 2020. - 5 p. Original Communication (restricted to ATLAS) - Full text 2020-05-29 13:14 Construction and geometrical precision assessment of the Micromegas detectors for the ATLAS New Small Wheel upgrade / Minashvili, Irakli (Joint Institute for Nuclear Research) The upgrade of the Large Hadron Collider (LHC) to the High Luminosity LHC (HL-LHC) is required to probe the physics beyond Standard Model. [...] ATL-MUON-PROC-2020-011. - 2020. - 6 p. Original Communication (restricted to ATLAS) - Full text 2020-05-22 10:21 Design and Construction of the Mechanical Structure for Thin-Gap RPC Triplets for the Upgrade of the ATLAS Muon Spectrometer / Kortner, Oliver (Max-Planck-Institut fuer Physik) The advent of thin-gap RPCs with 1 mm gas gaps instead of 2 mm in the present RPCs opened the possibility to instrument the inner barrel layer of the ATLAS muon spectrometer where there is very limited amount of space in radial direction from the beam line. [...] ATL-MUON-PROC-2020-010. - 2020. - 6 p. Original Communication (restricted to ATLAS) - Full text 2020-05-20 08:54 Low-$p_{\mathrm{T}}$ tracking for ATLAS in nominal LHC pileup / McCormack, William Patrick (Lawrence Berkeley National Laboratory and University of California, Berkeley) ; Pagan Griso, Simone (Lawrence Berkeley National Laboratory and University of California, Berkeley) ; Dimitrievska, Aleksandra (Lawrence Berkeley National Laboratory and University of California, Berkeley) ; Garcia-Sciveres, Maurice (Lawrence Berkeley National Laboratory and University of California, Berkeley) In the most recent year of data-taking with the ATLAS detector at the Large Hadron Collider (LHC), the minimum transverse momentum ($p_{\mathrm{T}}$) of default track reconstruction was 500 MeV. [...] ATL-PHYS-PROC-2020-041. - 2020. - 7 p. Original Communication (restricted to ATLAS) - Full text 2020-05-18 18:01 The Micromegas chambers for the ATLAS New Small Wheel upgrade / Gnesi, Ivan (INFN Gruppo Collegato di Cosenza and Universita' della Calabria, Dipartimento di Fisica) The ATLAS collaboration at LHC has chosen the resistive Micromegas technology, along with the small-strip Thin Gap Chambers (sTGC), for the high luminosity upgrade of the first muon station in the high-rapidity region, the so called New Small Wheel (NSW) project. [...] ATL-MUON-PROC-2020-009. - 2020. - 21 p. Original Communication (restricted to ATLAS) - Full text - Full text - Full text - Full text	2020-08-05 19:34:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7260507941246033, "perplexity": 10275.508999489883}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735964.82/warc/CC-MAIN-20200805183003-20200805213003-00167.warc.gz"}
https://gitlab.freedesktop.org/alyssa/mesa/-/blame/next-isa/src/panfrost/valhall/ISA.xml	ISA.xml 55.1 KB Alyssa Rosenzweig committed Jul 16, 2021 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 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 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1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 This immediates are accessible in (almost) any instruction, provided the immediate mode is kept to the default. They optimize for the most common immediate values; any immediate listed here may be used without taking up a uniform slot or a register. Most integer instructions can access separate half-words and individual bytes via swizzles on the source. 0x00000000 0xFFFFFFFF 0x7FFFFFFF 0xFAFCFDFE 0x01000000 0x80002000 0x70605030 0xC0B0A090 0x03020100 0x07060504 0x0B0A0908 0x0F0E0D0C 0x13121110 0x17161514 0x1B1A1918 0x1F1E1D1C 0x3F800000 0x3DCCCCCD 0x3EA2F983 0x3F317218 0x40490FDB 0x00000000 0x477FFF00 0x5C005BF8 0x2E660000 0x34000000 0x38000000 0x3C000000 0x40000000 0x44000000 0x48000000 0x42480000 Every Valhall instruction can perform an action, like wait on dependency slots. A few special actions are available, specified in the instruction metadata from this enum. The wait0126 action is required to wait on dependency slot #6 and should be set on the instruction immediately preceding ATEST. The barrier action may be set on any instruction for subgroup barriers, and should particularly be set with the BARRIER instruction for global barriers. The td action only applies to fragment shaders and is used to terminate helper invocations, it should be set as early as possible after helper invocations are no longer needed as determined by data flow analysis. The return action is used to terminate the shader, although it may be overloaded by the BLEND instruction. The reconverge action is required on any instruction immediately preceding a possible change to the mask of active threads in a subgroup. This includes all divergent branches, but it also includes the final instruction at the end of any basic block where the immediate successor (fallthrough) is the target of a divergent branch. wait0126 barrier reconverge td return Selects how immediates sources are interpreted. none ts id Situated between the immediates hard-coded in the hardware and the uniforms defined purely in software, Valhall has a some special "constants" passing through data structures. These are encoded like the table of immediates, as if special constant $i$ were lookup table entry $32 + i$. These special values are selected with the .ts modifier. tls_ptr tls_ptr_hi wls_ptr wls_ptr_hi Situated between the immediates hard-coded in the hardware and the uniforms defined purely in software, Valhall has a some special "constants" passing through data structures. These are encoded like the table of immediates, as if special constant $i$ were lookup table entry $32 + i$. These special values are selected with the .id modifier. lane_id core_id program_counter b0123 b3210 b0101 b2323 b0000 b1111 b2222 b3333 b2301 b1032 b0011 b2233 Used to select the 2 bytes for shifts of 16-bit vectors b02 b00 b11 b22 b33 b01 b23 h00 h10 h01 h11 b00 b20 b02 b22 b11 b31 b13 b33 b01 b23 none h0 h1 b0 b1 b2 b3 none h0 h1 b0 b1 b2 b3 w0 b0 b1 b2 b3 h0 h1 Corresponds to IEEE 754 rounding modes rte rtp rtn rtz Comparison instructions like FCMP return a boolean but may encode this boolean in a variety of ways. i1 gives a OpenGL style 0/1 boolean. m1 gives a Direct3D style 0/~0 boolean. f1 gives a floating-point 0.0f / 1.0f boolean. Switching between these modes is useful to fold a boolean type convert into a comparison. u1 is used internally to implement 64-bit comparisons. i1 f1 m1 u1 none h0 h1 Clamp applied to the destination of a floating-point instruction. Note the clamps may be decomposed as two independent bits for clamp_0_inf and clamp_m1_1, with clamp_0_1 arising as the composition of clamp_0_inf and clamp_m1_1 in either order. none clamp_0_inf clamp_m1_1 clamp_0_1 Condition code. Type must be inferred from the instruction. IEEE 754 total ordering only applies to floating point compares. "Not equal" and "greater than or less than" are distinguished by NaN behaviour conforming to the IEEE 754 specification. eq gt ge ne lt le gtlt total Texture dimension. 1d 2d 3d cube Level-of-detail selection mode in a texture instruction. zero computed explicit computed_bias grdesc Format of data loaded to / stored from registers for general memory access. f32 f16 u32 sr0 sr1 sr2 sr3 sr4 sr5 sr6 sr7 Number of channels loaded/stored for general memory access. none v2 v3 v4 Number of bits loaded/stored for general memory access. i8 i16 i24 i32 i48 i64 i96 i128 Dependency slot set on a message-passing instruction that writes to registers. Before reading the destination, a future instruction must wait on the specified slot. Slot #7 is for BARRIER instructions only. slot0 slot1 slot2 slot7 Memory segment written to by a STORE instruction. global pos vary tl Selects the effective subgroup size from subgroup operations. The hardware warps are sixteen threads on Valhall, but subdividing a warp may be useful for API requirements. In particular, derivatives may be calculated with quads (four threads). subgroup2 subgroup4 subgroup8 subgroup16 Acts as a modifier on the lane specificier for a CLPER instruction. The accumulate mode is required for efficient subgroup reductions. none xor accumulate shift Accesses to inactive lanes (due to divergence) in a subgroup is generally undefined in APIs. However, the results of permuting with an inactive lane with CLPER.i32 are well-defined in Valhall: they return one of the following values, as specified in the CLPER.i32 instructions. Sometimes certain values enable small optimizations. zero umax i1 v2i1 smin smax v2smin v2smax v4smin v4smax f1 v2f1 infn inf v2infn v2inf Do nothing. Useful at the start of a block for waiting on slots required by the first actual instruction of the block, to reconcile dependencies after a branch. Also useful as the sole instruction of an empty shader. Branches to a specified relative offset if its source is nonzero (default) or if its source is zero (if .eq is set). The offset is 27-bits and sign-extended, giving an effective range of ±26-bits. The offset is specified in units of instructions, relative to the next instruction. Positive offsets may be interpreted as "number of instructions to skip". Since Valhall instructions are 8 bytes, this operates as: $$PC := \begin{cases} PC + 8 \cdot (\text{offset} \; + 1) & \text{if} \; \text{src} \stackrel{?}{=} 0 \\ PC + 8 & \text{otherwise} \end{cases}$$ Used with comparison instructions to implement control flow. Tie the source to a nonzero constant to implement a jump. May introduce divergence, so generally requires .reconverge flow control. Value to compare against zero Evaluates the given condition, and if it passes, discards the current fragment and terminates the thread. The destination should be set to R60. Only valid in a frgment shader. Updated coverage mask (set to R60) Left value to compare Right value to compare Jump to an indirectly specified address. Used to jump to blend shaders at the end of a fragment shader. Value to compare against zero Branch target General-purpose barrier. Must use slot #7. Must be paired with a .barrier action on the instruction. Evaluates the given condition and outputs either the true source or the false source. Left value to compare Right value to compare Return value if true Return value if false Evaluates the given condition and outputs either the true source or the false source. Valhall lacks integer minimum/maximum instructions. CSEL instructions with tied operands form the canonical implementations of these instructions. Similarly, the integer $\text{sign}$ function is canonically implemented with a pair of CSEL instructions. Left value to compare Right value to compare Return value if true Return value if false Interpolates a given varying Vertex ID Instance ID The index must not diverge within a warp. Vertex ID Instance ID Index Loads the effective address of the position buffer (in a position shader) or the varying buffer (in a varying shader). That is, the base pointer plus the vertex's linear ID (the first source) times the buffer's per-vertex stride. LEA_ATTR should be executed once in a position/varying shader, with the linear ID preloaded as r59. Each position/varying store can then be constructed as STORE with the base address sourced from the 64-bit destination of LEA_ATTR and an appropriately computed offset. Varying stores bypass the usual conversion hardware for attributes; this diverges from earlier Mali hardware. Linear ID Loads from main memory Address to load from after adding offset Stores to main memory Address to store to after adding offset Stores to images Address to store to after adding offset Loads a given render target, specified in the pixel indices descriptor, at a given location and sample, and convert to the format specified in the internal conversion descriptor. Used to implement EXT_framebuffer_fetch and internally in blend shaders. Pixel indices descriptor Coverage mask Conversion descriptor Blends a given render target. This loads the API-specified blend state for the render target from the first source. Blend descriptors are available as special immediates. It then reads the colour to be blended from the first staging register, with the specified vector size and register format as desired. The resulting coverage mask is stored to the second set of staging registers. In the fixed-function path, BLEND sends the colour to the blender to be written to the tilebuffer. Then, if the instruction's flow control specifies termination, the fragment program is ended. If it does not specify termination, BLEND acts as a relative branch, branching with the offset specified as target. This allows the subsequent instructions to be skipped when fixed-function blending is used. Note this implicit branch can never introduce divergence, so .reconverge is not required. In the blend shader path, BLEND ignores the specified flow control and does not branch to the specified offset. Instead, execution considers normally with the next instruction. The compiler should insert code for calling a blend shader after the BLEND instruction unless it is known that a blend shader will never be required. The indirection is required to support both fixed-function and blend shaders efficiently and without shader variants. Blend descriptor Does alpha-to-coverage testing, updating the sample coverage mask. ATEST does not do an implicit discard. It should be executed before the first ZS_EMIT or BLEND instruction. Updated coverage mask Input coverage mask Alpha value (render target 0) Programatically writes out depth, stencil, or both, depending on which modifiers are set. Used to implement gl_FragDepth and gl_FragStencil. Updated coverage mask Depth value Stencil value Input coverage mask Performs the given data conversion. Note that floating-point rounding is handled via the same hardware and therefore shares an encoding. Round mode is specified where it makes sense. Value to convert Converts up with the specified round mode. Value to convert Performs the given data conversion. Value to convert Performs the given rounding, using the convert unit. Value to convert Canonical register-to-register move. Used as a primitive for various bitwise operations. Used as a primitive for various bitwise operations. Used as a primitive for various bitwise operations. 64-bit abs may be constructed in 4 instructions (5 clocks) by checking the sign with ICMP.s32.lt.m1 hi, 0 and negating based on the result with IADD.s64 and LSHIFT_XOR.i32 on each half. Only available as 32-bit. Smaller bitsizes require explicit conversions. 64-bit popcount may be constructed in 3 clocks by separate 32-bit popcounts of each half and a 32-bit add, which is guaranteed not to overflow. Only available as 32-bit. Other bitsizes may be derived with swizzles. For fully featured bitwise operation, see the shift opcodes. For fully featured bitwise operation, see the shift opcodes. Returns the mask of lanes ever active within the warp (subgroup), such that the source is nonzero. The number of work-items in a subgroup is given as the popcount of this value with a nonzero input. An all() subgroup operation may be constructed as WMASK of the input compared for equality with WMASK of an nonzero value. An any() subgroup operation may be constructed as WMASK of the input compared against zero. Breaks up the floating-point input into its fractional (mantissa) and exponent parts. By default, this is compatible with the frexp() function in APIs. With the log modifier, the floating point format is adjusted to be compatible with Valhall's argument reduction for logarithm computation. Performs a given special function. The floating-point reciprocal (FRCP) and reciprocal square root (FRSQ) instructions may be freely used as-is. The trigonometric tables (FSIN_TABLE.u6 and FCOS_TABLE.u6) are crude, requiring both an argument reduction and postprocessing. Likewise the logarithm instruction (FLOGD.f32) requires an argument reduction. See the transcendentals section for more information. $A + B$ A B $\min \{ A, B \}$ A B $\max \{ A, B \}$ A B Given a pair of 32-bit floats, output a pair of 16-bit floats packed into a 32-bit destination. A B Computes $A \cdot 2^B$ by adding B to the exponent of A. Used to calculate various special functions, particularly base-2 exponents. Special case handling differs from an actual floating-point multiply, so this should not be used outside fixed instruction sequences. A B Calculates the base-2 exponent of an argument specified as a 8:24 fixed-point. The original argument is passed as well for correct handling of special cases. Input as 8:24 fixed-point Input as 32-bit float Performs a floating-point addition specialized for logarithm computation. A B $A + B$ with optional saturation. As Valhall lacks swizzle instructions, IADD.v2i16 with zero is the canonical lowering for swizzles. A B Calculates $A \| (B \ll 16)$. Used to implement (ushort2)(A, B) A B $A - B$ with optional saturation A B Sign or zero extend B to 64-bits, left-shift by shift, and add the 64-bit value A. These instructions accelerate address arithmetic, but may be used in full generality for 64-bit integer arithmetic. A B $A \cdot B$ with optional saturation. Note the multipliers can only handle up to 32-bit by 32-bit multiplies. The 64-bit "multiply" acts like IMUL.u32 but additionally writes the high half of the product to the high half of the 64-bit destination. Along with IADD.u32 and IADD.u64, this allows the construction of a 64-bit multiply in 5 instructions (6 clocks). A B A B $(A + B) \gg 1$ without intermediate overflow, corresponding to hadd() in OpenCL. With the .rhadd modifier set, it instead calculates $(A + B + 1) \gg 1$ corresponding to rhadd() in OpenCL. Selects the value of A in the subgroup lane given by B. This implements subgroup broadcasts. It may be used as a primitive for screen space derivatives in fragment shaders. A B $A \cdot B + C$ A B C Left shifts its first source by a specified amount and bitwise ANDs it with the second source, optionally inverting the second source or the result. A shift B Right shifts its first source by a specified amount and bitwise ANDs it with the second source, optionally inverting the second source or the result. A shift B Left shifts its first source by a specified amount and bitwise ORs it with the second source, optionally inverting the second source or the result. A shift B Right shifts its first source by a specified amount and bitwise ORs it with the second source, optionally inverting the second source or the result. A shift B Left shifts its first source by a specified amount and bitwise XORs it with the second source, optionally inverting the second source or the result. A shift B Right shifts its first source by a specified amount and bitwise XORs it with the second source, optionally inverting the second source or the result. A shift B Mux between A and B based on the provided mask. Equivalent to bitselect() in OpenCL. (A & mask) \| (A & ~mask) A B Mask During a cube map transform, select the S coordinate given a selected face. Z coordinate as 32-bit floating point X coordinate as 32-bit floating point Cube face index During a cube map transform, select the T coordinate given a selected face. Y coordinate as 32-bit floating point Z coordinate as 32-bit floating point Cube face index Calculates $A \| (B \ll 8) \| (CD \ll 16)$ for 8-bit A and B and 16-bit CD. To implement (uchar4) (A, B, C, D) in full generality, use the sequence MKVEC.v4i8 CD, C, D, #0; MKVEC.v4i8 out, A, B, CD MKVEC.v4i8 also allows zero extending arbitrary 8-bit lanes. For example, to extend r0.b3 to r1, use MKVEC.v4i8 r1, r0.b3, 0x0.b0, 0x0. A B CD Select the maximum absolute value of its arguments. X coordinate as 32-bit floating point Y coordinate as 32-bit floating point Z coordinate as 32-bit floating point Select the cube face index corresponding to the arguments. X coordinate as 32-bit floating point Y coordinate as 32-bit floating point Z coordinate as 32-bit floating point 8-bit integer dot product between 4 channel vectors, intended for machine learning. Available in both unsigned and signed variants, controlling sign-extension/zero-extension behaviour to the final 32-bit destination. Saturation is available. Corresponds to the cl_arm_integer_dot_product_* family of OpenCL extensions. Not for actual use, just for completeness. Instead, use your platform's neural accelerator. For $A, B \in \{ 0, \ldots, 255 \}^4$ and $\text{Accumulator} \in \mathbb{Z}$, calculates $(A \cdot B) + \text{Accumulator}$ and optionally saturates. A B Accumulator Evaluates the given condition, do a logical and/or with the condition in the result source, and return in the given result type (integer one, integer minus one, or floating-point one). The third source is useful for chaining together conditions without intermediate bitwise arithmetic; when this is not desired, tie it to zero and use the OR combine mode (do not set the .and modifier). The sequence modifier .seq is used to construct 64-bit compares in 2 ICMP.u32 instructions, in conjunction with the u1 result type on the low half, the m1 result type on the high half, and the result of the low half comparison passed as the third source. For comparisons other than 64-bit, do not set the .seq modifier and do not use the u1 result type. A B C Evaluates the given condition, do a logical and/or with the condition in the result source, and return in the given result type (integer one, integer minus one, or floating-point one). The third source is useful for chaining together conditions without intermediate bitwise arithmetic; when this is not desired, tie it to zero and use the OR combine mode (do not set the .and modifier). A B C Evaluates the given condition, do a logical and/or with the condition in the result source, and return in the given result type (integer one, integer minus one, or floating-point one). The third source is useful for chaining together conditions without intermediate bitwise arithmetic; when this is not desired, tie it to zero and use the OR combine mode (do not set the .and modifier). The sequence modifier .seq is used to construct signed 64-bit compares in 1 ICMP.u32 and 1 ICMP.s32 instruction, in conjunction with the u1 result type on the low half, the m1 result type on the high half, and the result of the low half comparison passed as the third source. For comparisons other than 64-bit, do not set the .seq modifier and do not use the u1 result type. A B C Adds an arbitrary 32-bit immediate embedded within the instruction stream. If no modifiers are required, this is preferred to IADD.i32 with a constant accessed as a uniform. However, if the constant is available inline, IADD.f32 is preferred. IADD_IMM.i32 with the source tied to zero is the canonical immediate move. A Adds an arbitrary pair of 16-bit immediates embedded within the instruction stream. If no modifiers are required, this is preferred to IADD.v2i16 with a constant accessed as a uniform. However, if the constant is available inline, IADD.v2i16 is preferred. Adding only a single 16-bit constant requires replication of the constant. A Adds an arbitrary quad of 8-bit immediates embedded within the instruction stream. If no modifiers are required, this is preferred to IADD.v4i8 with a constant accessed as a uniform. However, if the constant is available inline, IADD.v4i8 is preferred. Adding only a single 8-bit constant requires replication of the constant. A Adds an arbitrary 32-bit immediate embedded within the instruction stream. If no modifiers are required, this is preferred to FADD.f32 with a constant accessed as a uniform. However, if the constant is available inline, FADD.f32 is preferred. A Adds an arbitrary pair of 16-bit immediates embedded within the instruction stream. If no modifiers are required, this is preferred to FADD.v2f16 with a constant accessed as a uniform. However, if the constant is available inline, FADD.v2f16 is preferred. Adding only a single 16-bit constant requires replication of the constant. A Unfiltered textured instruction. Image to read from Ordinary texturing instruction using a sampler. Image to read from Only works for FP32 varyings. Image to read from First calculates $A \cdot B + C$ and then biases the exponent by D. Used in special transcendental function sequences. It should not be used for general code as its special case handling differs from two back-to-back FMA.f32 operations. Equivalent to FMA.f32 back-to-back with RSCALE.f32 A B C D	2021-07-24 01:25:46	{"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": 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": 0.24120521545410156, "perplexity": 340.20672134645713}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046150067.87/warc/CC-MAIN-20210724001211-20210724031211-00661.warc.gz"}
https://www.electro-tech-online.com/threads/ir-receiver.93558/	Status Not open for further replies. #### GatorGnet ##### New Member I posted this in an earlier post: Remote Controlling Your PC What does the 78L05 do? I assumed it was a normal transistor but it looks to be something else? Sorry to be such a newb... #### blueroomelectronics ##### Well-Known Member It's a 5V regulator. #### GatorGnet ##### New Member The only ones I found with the part number 78L05 are hard to find. I assume the normal 7805 will work? #### blueroomelectronics ##### Well-Known Member Yes. The L = 100mA rated. ##### Banned Try google first, the 7805 is WELL documented on the net. #### GatorGnet ##### New Member This schematic uses the three pin IR module. What is the difference between them and the two pin IR detectors? #### blueroomelectronics ##### Well-Known Member You need the three pin, it's has a small IC demodulator inside. #### GatorGnet ##### New Member Well I got one built. After debugging some things I got it working. Now I have an issue with incoming data. When I push a button on the remote I get random data coming in: Code: Outputting raw mode2 data. space 1382017 pulse 9026 space 4443 pulse 571 space 4418 pulse 562 space 4430 pulse 566 space 2177 pulse 564 space 4427 pulse 568 space 4421 pulse 567 space 4425 pulse 565 space 2179 pulse 575 space 2170 pulse 567 space 2177 pulse 567 space 2179 pulse 569 space 2175 pulse 571 space 2172 pulse 576 space 2170 pulse 575 space 4414 pulse 576 space 2170 pulse 573 space 2170 pulse 571 space 28222 pulse 9052 space 2199 pulse 574 btw, this is from one button push. When I push the same button again, I get different data coming in. Could this be the remote or is it something with my board? Last edited: #### blueroomelectronics ##### Well-Known Member What remote are you using? What's it set for? #### GatorGnet ##### New Member One is a Toshiba CT-852 and the other is for our comcast cable box. Both give random data when pushing one button. The software I am using states that the remote is using a special repeat code. I just wanted to make sure that my receiver I spent some good time building is working right. #### Pommie ##### Well-Known Member That is the raw data. You need different software to interpret it. Your hardware appears to be working correctly. Mike. #### GatorGnet ##### New Member Why would the electronic companies that make those want to use data that changes each time you send it?	2020-09-26 12:03:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.237917959690094, "perplexity": 12279.385152047118}, "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-2020-40/segments/1600400241093.64/warc/CC-MAIN-20200926102645-20200926132645-00268.warc.gz"}
https://borneomath.com/kindergarten-sample-paper-fmo-2021/	# KINDERGARTEN-SAMPEL PAPER FMO 2021 Berikut ini adalah soal beserta kunci jawaban Fermat Mathematic Olympiad (FMO) 2021 (Sc: Edukultur Indonesia) A. Warm-up (4 points per question / No points deducted for wrong answers) 1. One car has 4 wheels. How many wheels do two cars have in total? A) 2 B) 8 C) 4 D) 6 E) 12 2. Find the missing piece. A) B) C) D) E) 3. Ken connects each animal with its shadow as in the figure below. How many of them are WRONG? A) 1 B) 3 C) 5 D) 2 E) 4 4. There is a queue of 20 people waiting to get in the restaurant. There are 10 people behind Anna. How many people does Anna have to wait until her turn? A) 11 B) 8 C) 4 D) 7 E) 3 5. The numbers of stuffed animals of five children are recorded in the following table. How many stuff animals does Paul have more than Sara? A) 5 B) 11 C) 4 D) 7 E) 3 B. Speed-up (6 points per question / No points deducted for wrong answers) 6. How many circles are there in the figure below? A) 7 B) 6 C) 9 D) 8 E) 10 7. Peter draws a sequence with pattern as follow. How many are there in the first 11 figures counting from the left? A) 11 B) 6 C) 8 D) 7 E) 9 8. A pencil costs 2 dollars. If you buy 4 pencils, you get 1 eraser for free. Lucy got 2 erasers for free, at least how much did she pay for the pencils? A) $12 B)$8 C) $10 D)$16 E) \$18 9. Lucy has a sticker. She can cut her sticker into two identical pieces below. Which answer CANNOT be the shape of Lucy’s sticker? A) B) C) D) E) 10. There are 12 books on 3 shelves as below. At least how many books have to be moved so that the number of books on every shelf is the same? A) 5 B) 1 C) 2 D) 4 E) 3 C. Challenge (8 points per question / No points deducted for wrong answers) 11. It is Wednesday on 15th December 2021. Which day of the week does the Christmas this year fall on? A) Friday B) Thursday C) Saturday D) Wednesday E) Sunday 12. When Fred was born, his mother was 30 years old. His sister was born 4 years after Fred was born. His father is 5 years older than Fred’s mother. When Fred’s father is 65 years old, how old is Fred’s sister? A) 34 B) 20 C) 25 D) 26 E) 35 13. Jennie stacks bricks to get a tower. If she makes a 4-level tower, she will need 10 bricks. If she makes a 5-level tower, she will need 15 bricks. How many bricks does she need if she makes a 6-level tower? A) 19 B) 21 C) 20 D) 22 E) 18 14. Given that each figure appears once in every row or every column. Choose a figure to fill in the cell with question mark. A) B) C) D) E) No possible answer 15. By putting two coins each time in the arcade machine, Lucy can play in 12 turns. She wants to play 50 turns. At least how many coins should she buy? A) 6 B) 38 C) 62 D) 4 E) 5	2023-03-26 12:48:12	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21180157363414764, "perplexity": 3591.8222404275275}, "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/1679296945472.93/warc/CC-MAIN-20230326111045-20230326141045-00575.warc.gz"}
https://www.physicsforums.com/threads/solve-for-salary-given-averages-of-other-salaries.900651/	# Homework Help: Solve for salary given averages of other salaries 1. Jan 17, 2017 ### Mindscrape My wife is working on problems to study for the GMAT, and asks her fellow math nerd (me) to help on some of them. Originally I had an error and wanted to see if any of you could help me find it, but as I was typing I found it myself! Can I still put this up in case someone stumbles on it and it helps them out? The problem is: The average weekly salary of 12 workers and 3 managers in a factory was $600. A manager whose salary was$720 was replaced with a new manager, then the average salary of the team fell to $580. What is the salary of the new manager? So basically we start with (from the first sentence) $$\frac{tw+tm}{15}=600$$ where tw represents total worker salary and tm represents total manager salary. Now the second sentence says $$tm=m1+m2+720$$ so the first equation is now (having multiplied out the 15 from before) -- also label this eqn1 $$tw+m1+m2+720=9000$$ continuing with info from the second sentence, we get the second equation for the newly decreased average as $$\frac{tw+m1+m2+m3}{15}=580$$ now simplifying gives -- and labeling this eqn2 $$tw+m1+m2+m3=8700$$ subtract the two equations (eqn1-eqn2) $$720-m3=300$$ for the grand finale... $$m3=420$$ 2. Jan 17, 2017 ### RUber Nicely done. Since standardized tests often encourage shortcuts, I'll add a supplemental method which cuts through many of the steps. In general, if you wanted to affect a change of -$20 in the average of 15 salaries, you have to have a total change of 15(-$20)=-$300 to the sum. This could be done by reducing one salary by $300, or reducing all salaries by$20, or anywhere in between. Since the only thing you are changing is the salary starting at $720, you can apply the -$300 to that.	2018-07-17 08:19:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.35606104135513306, "perplexity": 745.5504241314281}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589618.52/warc/CC-MAIN-20180717070721-20180717090721-00093.warc.gz"}
https://www.zbmath.org/?q=an%3A1154.68056	# zbMATH — the first resource for mathematics Computational complexity. A conceptual perspective. (English) Zbl 1154.68056 Cambridge: Cambridge University Press (ISBN 978-0-521-88473-0/hbk). xxiv, 606 p. (2008). This is a quite complete and updated treatment of complexity theory. The conceptual approach, claimed in the subtitle, tends to explain the notions in a deep manner using current language discourse. As a textbook, some themes, expositions and exercises are suitable for undergraduate courses in programmes of mathematics or computer science, the more advanced topics and expositions are proper for graduates courses. For lecturers using the book as a basis for a course, the author gives continuously advice and warnings concerning the difficulties in the material. On the other hand, for the specialist, the volume is certainly a useful and valuable reference book. The author emphasizes the “conceptual nature” of his exposition, in contrast to a mere “technical” approach, and this point of view has been shared by several relevant researchers [S. Aaronson, A. Borodin, B. Chazelle, O. Goldreich, S. Goldwasser, R. Karp, M. Kearns, C. Papadimitriou, M. Sudan and S. Vadhan, Statement on conceptual contributions in theory (addressed to STOC’08 Program Committee) (2008), http://www.wisdom.weizmann.ac.il/~oded/on-concept.html]. As the author claims, the conceptual exposition can be appreciated by non-experts, it is stated in a “high level” language, and the conceptual aspects have had impact outside the own area of specialty. The first chapter is an introduction to the subject, two chapters present the relevant notions of the classes P and NP, and respective chapters discuss the time hierarchies, the space hierarchies, randomness and counting, one-way functions, pseudorandom generators, probabilistic proof systems and relaxation. There are seven appendices discussing complexity classes, lower complexity bounds for common problems, cryptography, probability, explicit constructions, omitted proofs and specific problems. To reinforce the conceptual approach, a preface and a whole section in the introduction are devoted to describe the contents of the whole book. Later, each chapter has its own summary, a set of ending notes related to the development of the main notions, and a starting and illustrative epigraph. Thus, at every moment the author is guiding the learner, the instructor and the specialist into complexity theory. This may seem to be tedious, but for sure many lengthy discussions clarify finer aspects of the involved notions. An important difference between the technical and conceptual approaches is, for instance, the exposition of the existence of NP-complete problems. It is rather usual to illustrate NP-completeness by showing Cook’s theorem: Boolean satisfiability is NP-complete; and after an ordinary course in complexity most students are convinced that it is a natural fact. Instead, the author prefers to show the existence of an NP-complete problem using a diagonalization procedure, and he emphasizes this existence as a surprising fact. On the other hand, the book poses series of interesting, challenging and, above all, motivating, exercises. Indeed, such exercise series may serve as support for a “practical course” on complexity theory. I appreciate that the exercises tend to successfully balance the heavy conceptual disgressions. In the following synopsis I will summarize the technical results. Initially, the common notions of search problems, decision problems and uniform and non-uniform models of computation (mainly Turing machines and Boolean circuits) are introduced. Rice’s theorem is shown (no non-trivial function class is decidable), and the unsolvability of the Halting Problem and Post’s Correspondence Problem. Universal Turing machines are discussed within Kolmogorov complexity: For a given universal machine $$U$$ and a bit-string $$y$$ let $$c_U(y)$$ be the length of the shortest description of any Turing machine computing $$y$$. Then $$c_U$$ cannot be effectively computable. The notion of circuit complexity is also introduced. An algorithm computing a map with advices is an effective function with two arguments: the first is properly an advice and the second the value at which the map should be evaluated. The class P$$\|f$$ consists of problems solvable in polynomial-time using advices of length bounded by $$f$$. The class P$$\|$$poly is the union of P$$\|f$$ varying $$f$$ over all polynomials. It is shown that there are non-computable maps that are members of P$$\|$$1. When dealing with P and NP, initially there are defined the classes PF and PC consisting of polynomially bounded relations solvable and checkable in polynomial time respectively, while P and NP consist of sets decidable and checkable in polynomial time. NP can be realized as the class of projections of relations in PC (hence any Yes-instance point may have many solution points witnessing the Yes decision) and each problem in PC is Cook-reducible to a decision problem in NP. It is also shown that PC is contained in PF if and only if P = NP, as well as the existence of NP-complete problems, via diagonalization arguments. A list of NP-complete problems is dealt with and it is shown that under the assumption that P differs from NP there should be problems in NP which are not NP-complete. A “promise” in a search or decision problem is a subset of the problem’s domain and just for their points proper solutions are required. The “candid” problems are those whose promises are their own domains. Then it is proved that for any relation in NP there exist optimal algorithms for their candid versions. The class coNP consists of complements of relations in NP. The intersection of NP and coNP, contained in NP but containing P, is analyzed, e.g., if NP is Cook-reducible to a problem in the intersection of NP and coNP then NP = coNP. The Polynomial Hierarchy (PH) is exposed in a very readable way (in a clear way it is explained that NP would differ from P if PH differs from P) and some conditions are stated for the hierarchy to collapse, e.g., if NP is contained in P$$\|$$poly then PH collapses to second order. In the fourth and the fifth chapter, the time hierarchies, deterministic and non-deterministic, are developed, including Borodin’s Gap Theorem (for any computable $$g$$ there exists $$t$$ such that DTime($$t$$) = DTime($$g\circ t$$)) and the Speed-Up Theorem. The space hierarchies are also described. The classes L and NL are introduced as those consisting of problems that are, respectively, deterministically and non-deterministically solvable in polylog space. The undirected graph connectivity problem is shown to be in L, while directed graph connectivity is NL-complete under log-space reductions. The class L is compared with P (L will differ from P if a P-complete problem is not in L) and it is shown that NL is contained in P. The deterministic and non-deterministic space hierarchies are compared, and finally it is proved that the satisfiability problem for quantified Boolean formulae is PSpace-complete under polynomial-time reductions. The sixth chapter deals with randomness and counting. Probabilistic machines are considered off-line (there is a source of input random seeds) and on-line (the inner transitions have associated probabilities); and the machines may have several error-types: two-sided (in which both Yes-instances and No-instances have associated error probabilities), one-sided (in which Yes-instances are decided correctly while for No-instances there is an error probability), and zero-sided (in which for some instances there is no stated decision, while for other instances the given decisions are always correct). BPP consists of problems solvable probabilistically in polynomial time with two-sided errors. It contains P and is contained in the existential second-order level of PH, in PSpace and in P$$\|$$poly. It is unknown whether it is contained in NP or in P. Primality testing is shown as a member of BPP (although the author reminds us that it is in P indeed). The class RP consists of problems solvable probabilistically in polynomial time with one-sided errors, thus it is contained in the intersection of BPP and NP. Polynomial identity is shown to be a member of RP. The class ZPP is the intersection of RP and coRP, and it can be realized as the problems solvable probabilistically in expected polynomial time. Similarly there are defined the classes BPL and RL considering log-space and polynomial-time algorithms. The undirected graph connectivity problem is shown to be in RL indeed. Counting is treated extensively. #P consists of the functions counting solutions for problems in P: $$f$$ is in #P if and only if for some relation $$R$$ in PC, for each $$x$$, $$f(x)$$ counts the number of $$y$$s such that $$(x,y)\in R$$. In this case, it is defined $$\#R=f$$. The counting perfect matchings problem is shown to be in #P. The notion of parsimonious reduction is introduced (basically it is a reduction that preserves the number of solutions), and it is shown that a problem $$\#R$$ is #P-complete provided that any search problem in PC has a parsimonious reduction to $$R\in\text{PC}$$. It is also shown that there are tractable problems $$R$$ for which $$\#R$$ is #P-complete, in particular SAT for disjunctive forms is trivial, although counting the number of satisfying assignments is difficult: #DNF is #P-complete. For a function $$f$$, an $$(r,s)$$-approximator is a probabilistic algorithm $$\Pi$$ such that, at each point $$x$$, the probability that the relative error among $$\Pi(x)$$ and $$f(x)$$ is greater that $$r$$, evaluated at the length of $$x$$, is upper bounded by $$s$$, evaluated at the length of $$x$$; and it is said that $$f$$ is $$(1-r)$$-approximable if it has an $$(r,1/3)$$-approximation. Then it is seen that #DNF is $$(1-P^{-1})$$-approximable, for each polynomial $$P$$. Indeed, in a more general setting it is shown that for any $$R$$ in PC and any polynomial $$P$$ there is an oracle Turing machine that, having query access to NP, is a $$(P^{-1},\mu)$$-approximation to $$\#R$$, where $$\mu$$ is a negligible map. The problem of uniform generation (for a relation $$R$$ in PC and a point $$x$$ in its domain, choose uniformly a solution in $$R(x)$$) and its probabilistic solvability is compared with the problem to approximate its counting version, and they are shown to be computationally equivalent. Finally, several procedures for uniform generation are explained in detail. The seventh chapter treats the benefits of hard problems. The exposition is closely connected with [O. Goldreich, Modern cryptography, probabilistic proofs and pseudo-randomness. Algorithms and Combinatorics. 17. Berlin: Springer (1999; Zbl 0907.94002)] and the crypto appendix of the current book. The main benefit of hard problems is the possibility to build one-way functions, which are the main components in cryptographic protocols. Given a relation $$R$$ in PC, a generator of solved intractable instances is a map $$G:n\mapsto (x,y)\in R$$ such that $$n$$ is the length of $$x$$, and for any probabilistic polynomial-time algorithm $$B$$, $$n\mapsto\text{Prob}[G(n)=(x,y) \text{ \& } (x,B(x,n))\in R]$$ is negligible. A map $$f$$ is one-way if it is computable but its inverse function is intractable, i.e. for any probabilistic polynomial-time algorithm $$B$$, the map $$n\mapsto\text{Prob}[B(f(x),n)\in f^{-1}(f(x)) \mid \text{length}(x)=n]$$ is negligible. The existence of one-way maps is equivalent to the existence of relations with generators of solved intractable instances. Some weaker notions of one-way maps are introduced and they are proved to entail the initial notion. A hard-core predicate $$b$$ for a function $$f$$ is a 0-1 valued map such that by knowing the value $$f(x)$$ the probability to guess the bit $$b(x)$$ is arbitrarily close to 1/2 (for any probabilistic polynomial-time $$A$$, $$n\mapsto\|\text{Prob}[A(f(x))=b(x)\mid \text{length}(x)=n]-1/2\|$$ is negligible). The generic hard-core theorem states that the inner product, modulo 2, is a hard core for the map $$(x,y)\mapsto (g(x),y)$$ provided that $$g$$ is one-way, and it implies a coding procedure generalizing Hamming codes through probabilistic procedures. The eighth chapter deals with the notion of pseudorandom generators. For a given stretch map $$\ell:\mathbb N\to\mathbb N$$ a pseudorandom generator $$G:(0+1)^\to(0+1)^$$ is such that the image of $$(0+1)^n$$ is a subset of $$(0+1)^{\ell(n)}$$ and, for a given family of distinguisher polynomial-time algorithms and a given family of threshold functions, the probabilities that a distinguisher accepts both $$G(U_n)$$ and $$U_{\ell(n)}$$ coincide up to a threshold function ($$U_k$$ is a random variable in $$(0+1)^k$$ uniformly distributed). In the current context, distinguishers are probabilistic polynomial-time algorithms, and threshold maps are the reciprocals of polynomials. This notion of randomness is different from those due to Kolmogorov and Chaitin. Indeed, these notions (a sequence is random if its shortest descriptions have essentially the same length as the sequence itself) are ontological, whilst the randomness notion in the current book is of a behavioral kind. Pseudorandom generators may be used to provide random seeds for probabilistic machines. The notions of statistical closeness and computational indistinguishability are discussed and they involve probability ensembles. The existence theorem is proved: If there exists a polynomial-time one-to-one length-preserving and one-way map then for any stretch map there is a pseudorandom generator. This entails that the existence of pseudorandom generators is equivalent to the existence of one-way maps. Several forms of derandomization of BPP are stated, among them, if there exists a non-uniform strong pseudorandom generator then BPP is contained in the meet of the classes DTime($$2^{n^k}$$), with $$k\in\mathbb N$$, and the Nisan-Wigderson derandomization procedure. The class SC consists of the problems solvable by deterministic polynomial time and logarithmic space. Then it is proved that BL is contained in SC. The concept of random walks is also used as a way to build pseudorandom generators. The ninth chapter is devoted to probabilistic proof systems. An interactive proof is modeled as a game between two parties: a prover and a verifier. Each party plays its turn based on a common input, a random seed and the previous plays of its opponent. Given a problem set $$S$$ and an instance point $$x$$, the goal of the prover is to convince the verifier that $$x\in S$$. The verifier has access to probabilistic polynomial-time resources while there are no limits for the prover. The interactive proof shall be complete (if $$x\in S$$ then the prover shall have a winning strategy) and sound (if $$x\not\in S$$ then the verifier shall reject with probability at least 1/2, independently of the prover’s strategy). For instance, the non-isomorphic graphs problem admits a probabilistic proof system: Given two graphs, the verifier chooses randomly a graph, renumbers its vertices and asks the prover to point out its isomorphic graph; if the prover’s reply coincides with the chosen graph, then the verifier claims that the two initial graphs are non-isomorphic. It is shown that any set with an interactive proof system with zero soundness error is in NP. The class IP consists of the problems with probabilistic proof systems with polynomial length. It is shown that IP = PSpace following a series of steps including the fact that coNP is within IP (an interactive system to recognize unsatisfiable Boolean formulas, which is coNP-complete, is extended to solve QBF, which is PSpace-complete), and that any interactive proof system entails a polynomial-space optimal strategy for the prover (a strategy that maximizes the acceptation probability of the verifier). The Arthur-Merlin (AM) games are probabilistic proof systems in which there is a public coin-tossing system providing random seeds for both players. IP($$f$$) is the subclass of IP in which the number of rounds in the game is bounded by $$f$$. There are proven speed-up and simulation results: $$\text{IP}(O(f)) = \text{IP}(f)$$, IP$$(f)$$ is contained in AM$$(f+3)$$, and AM$$(2f)$$ is contained in AM$$(f+1)$$. The author remarks that if coNP is contained in IP$$(2)$$ then the polynomial hierarchy will collapse. Zero knowledge is also treated in this chapter. A prover’s strategy is of zero knowledge if the whole interaction with any verifier’s strategy can be simulated by a probabilistic polynomial-time algorithm (in this way, the prover is not providing any further knowledge). For instance, the Isomorphic Graphs Problem admits a zero-knowledge probabilistic proof system: Given two graphs, the prover renumbers the vertices of the second graph and sends this new graph $$H$$ to the verifier. The verifier chooses an index, say $$i$$, and asks the prover to give an isomorphism $$G_i\to H$$; if the graphs are indeed isomorphic, the prover can build the isomorphism, namely, if $$i=2$$, it is the initial renumbering, otherwise it is the composition of the isomorphism among the graphs and the numbering, so he is able to send a map $$\psi$$; the verifier accepts according to his verification that $$\psi$$ is an isomorphism. In this protocol, the verifier is unable to know that the initial graph chosen by the prover was the second one, for instance. The Three-Coloring Graph Problem is also solvable with a zero-knowledge probabilistic proof system. Indeed, if there is a non-uniform hard one-way map, then any problem in NP admits a zero-knowledge probabilistic proof system (hence the importance of these systems in cryptographic protocols). A probabilistically checkable proof system (PCP) for a set $$S$$ is an oracle probabilistic polynomial-time machine, a verifier, such that if $$x\in S$$ then there is an oracle $$\pi_x$$ such that $$V$$ accepts $$x$$ by querying $$\pi_x$$, and if $$x\not\in S$$ then $$V$$ rejects $$x$$ with a probability at least 1/2 independently of the queried oracle. The complexity of a PCP can be measured with respect to the number of oracle queries $$q(n)$$ and with respect to the number of random transitions $$r(n)$$ ($$n$$ is the length of the input $$x$$) and any two such measures characterize a corresponding class PCP($$r,q$$). It is shown that for any polynomial map $$r$$, PCP($$r$$, poly) is contained in NTime($$2^r$$poly), and consequently PCP(log, poly) is contained in NP. This in turn implies the important PCP Theorem: NP = PCP(log, $$O(1)$$). The PCP Theorem has immediate consequences with respect to approximability. Finally, some estimations are made about the length of the witnessing solutions for problems in NP, for instance, for any problem in NP there exists a logarithmic map $$\ell$$ such that for any map $$k$$ with $$0\leq k\leq \ell$$ the set $$S$$ is in PCP$$(\ell-k, O(2^k))$$. The tenth chapter treats some relaxation and approximation procedures. For any binary relation $$R$$ and any real-valued map $$f$$, let $$T_f:x\mapsto\max\{f(x,y)\mid (x,y)\in R\}$$, and $$t_f:x\mapsto\min\{f(x,y)\mid (x,y)\in R\}$$. Given a “factor” map $$g$$, corresponding relaxations for maximization $$\{(x,y)\mid f(x,y)\geq T_{f}(x)/g(x)\}$$ and minimization $$\{(x,y)\mid f(x,y)\leq t_{f}(x)\cdot g(x)\}$$ are considered. For instance, the minimization problem with a factor 2 for Minimum Vertex Cover is an easy problem (the required approximations correspond to maximal matchings and these can be obtained by a greedy algorithm), and also for the Traveling Salesman Problem some approximations are easy. As a second form of approximation “gap problems” are introduced: for two maps $$g_1\leq g_2$$, it is necessary to distinguish instances such that $$T_f(x)<g_1(x)$$ and such that $$T_f(x)\geq g_2(x)$$. Some of these problems, related to Clique, SAT, and systems of linear equations, are proved to be NP-hard. As another approximation, there is considered $$\Gamma_u(S)$$ consisting of those points whose Hamming distance to any point in $$S$$ (of the same length) is at least $$u$$ times their own lengths, and it is required to distinguish, for a fixed $$u$$, the sets $$S$$ and $$\Gamma_u(S)$$. It is shown that there are probabilistic polynomial-time algorithms to make such distinction for some problems related to graph properties invariant under isomorphisms. The average cases for algorithms is treated using probability ensembles. This allows the author to define distributional problems, namely the class distNP consists of pairs $$(S,X)$$ where $$S$$ is a decision problem and $$X$$ is a simple probability ensemble. The existence of distNP-complete problems is shown. The class tcpP consists of problems $$(S,X)$$ that are typically solvable (there is an algorithm such that the probability of either making an error or exceeding a polynomial-time bound in computing is negligible). A probabilistic extension tpcBPP is also introduced and it is shown that tpcBPP contains distNP if and only if tpcBPP contains tcpP. The appendices succinctly develop the mathematical themes required in the text but being outside the scope of complexity theory. The book is quite extensive but certainly any reader may select the material, according to the suggestions formulated by the author, in order to gain the best profit from it, either as a textbook or as a reference handbook. ##### MSC: 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) 68Q25 Analysis of algorithms and problem complexity 68W40 Analysis of algorithms 68-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to computer science 68-02 Research exposition (monographs, survey articles) pertaining to computer science 94A60 Cryptography 94A62 Authentication, digital signatures and secret sharing Full Text:	2021-09-20 11:50:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7498493194580078, "perplexity": 565.6686408886183}, "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/1631780057036.89/warc/CC-MAIN-20210920101029-20210920131029-00061.warc.gz"}
https://www.canton.io/docs/stable/user-manual/tutorials/composability.html	# Composability¶ In this tutorial, you will learn how to build workflows that span several Canton domains. Composability turns those several Canton domains into one conceptual ledger at the application level. The tutorial assumes the following prerequisites: The tutorial consists of two parts: 1. The first part illustrates how to design a workflow that spans multiple domains. 2. The second part shows how to compose existing workflows on different domains into a single workflow and the benefits this brings. The DAML models are shipped with the Canton release in the daml/CantonExamples folder in the modules Iou and Paint. The configuration and the steps are available in the examples/05-composability folder of the Canton release. To run the workflow, start Canton from the release’s root folder as follows: ./bin/canton -c examples/05-composability/composability.conf You can copy-paste the console commands from the tutorial in the given order into the Canton console to run them interactively. All console commands are also summarized in the bootstrap scripts composability1.canton and composability2.canton. ## Part 1: A multi-domain workflow¶ We consider the paint agreement scenario from the Getting started tutorial. The house owner and the painter want to enter a paint agreement that obliges the painter to paint the house owner’s house. To enter such an agreement, the house owner proposes a paint offer to the painter and the painter accepts. Upon acceptance, the paint agreement shall be created atomically with changing the ownership of the money, which we represent by an IOU backed by the bank. Atomicity guarantees that no party can scam the other: The painter enters the obligation of painting the house only if house owner pays, and the house owner pays only if the painter enters the obligation. This avoid bad scenarios such as the following, which would have to be resolved out of band, e.g., using legal processes: • The house owner spends the IOU on something else and does not pay the painter, even though the painter has entered the obligation to paint the house. The painter then needs to convince the house owner to pay with another IOU or to revoke the paint agreement. • The house owner wires the money to the painter, but the painter refuses to enter the paint agreement. The house owner then begs the painter to return the money. ### Setting up the topology¶ In this example, we assume a topology with two domains, iou and paint. The house owner’s and the painter’s participants are connected to both domains, as illustrated in the following diagram. The configuration file composability.conf configures the two domains iou and paint and three participants. The domain parameter setting transfer-exclusivity-timeout will be explained in the second part of this tutorial. canton { domains { iou { public-api.port = 11018 storage.type = memory domain-parameters.transfer-exclusivity-timeout = PT0s // disables automatic transfer-in } paint { public-api.port = 11028 storage.type = memory domain-parameters.transfer-exclusivity-timeout = PT2s } } participants { participant1 { ledger-api.port = 11011 storage.type = memory } participant2 { ledger-api.port = 11021 storage.type = memory } participant3 { ledger-api.port = 11031 storage.type = memory } } } As the first step, all the nodes are started and the parties for the bank (hosted on participant 1), the house owner (hosted on participant 2), and the painter (hosted on participant 3) are created. The details of the party onboarding are not relevant for show-casing cross-domain workflows. // start all instances defined in the configuration file all start connect(participant1, iou) connect(participant2, iou) connect(participant3, iou) connect(participant2, paint) connect(participant3, paint) // create the parties val Bank = enable_party(participant1, "Bank") val HouseOwner = enable_party(participant2, "House Owner") val Painter = enable_party(participant3, "Painter") // Wait until the party enabling has taken effect and a heartbeat has been sent afterwards val partyAssignment = Set(HouseOwner -> participant2, Painter -> participant3) assert(await_topology_heartbeat(participant2, partyAssignment)) assert(await_topology_heartbeat(participant3, partyAssignment)) // upload the DAML model to all participants val darPath = Option(System.getProperty("canton-examples.dar-path")).getOrElse("dars/CantonExamples.dar") ### Creating the IOU and the paint offer¶ To initialize the ledger, the Bank creates an IOU for the house owner and the house owner creates a paint offer for the painter. These steps are implemented below using the Scala bindings generated from the DAML model. The generated Scala classes are distributed with the Canton release in the package com.digitalasset.canton.examples. The relevant classes are imported as follows: import com.digitalasset.canton.examples.Iou.{Amount, DiscloseIou, Iou} import com.digitalasset.canton.examples.Paint.{OfferToPaintHouseByOwner, PaintHouse} import com.digitalasset.canton.ledger.api.client.DecodeUtil.decodeAllCreated import com.digitalasset.canton.protocol.ContractIdSyntax._ Bank creates an IOU of USD 100 for the house owner on the iou domain, by submitting the command through the ledger API command service of participant 1. The house owner then discloses the IOU contract to the painter such that the painter can effect the ownership change when they accept the offer. Both of these commands run over the iou domain because the Bank’s participant 1 is only connected to the iou domain. The need to disclose the IOU contract is explained in DA ledger privacy model. // Bank creates IOU for the house owner val createIouCmd = Iou( payer = Bank.toPrim, owner = HouseOwner.toPrim, amount = Amount(value = 100.0, currency = "USD") ).create.command val Seq(iouContract) = decodeAllCreated(Iou.id)( participant1.ledger_submit_flat(Bank, Seq(createIouCmd))) // Wait until the house owner sees the IOU in the active contract store assert(await_active_contract(participant2, HouseOwner, iouContract.contractId.toLf)) // The house owner discloses the IOU to the Painter val discloseIouCmd = DiscloseIou( sender = HouseOwner.toPrim, iou = iouContract.contractId ).createAnd.exerciseDisclose(HouseOwner.toPrim).command participant2.ledger_submit_flat(HouseOwner, Seq(discloseIouCmd)) Similarly, the house owner creates a paint offer on the paint domain via participant 2. In the ledger_submit_flat command, we set the workflow id to the paint domain so that the participant submits the commands to this domain. If no domain was specified, the participant automatically determines a suitable domain. In this case, both domains are eligible because on each domain, every stakeholder (the house owner and the painter) is hosted on a connected participant. // The house owner creates a paint offer using participant 2 and the Paint domain val paintOfferCmd = OfferToPaintHouseByOwner( painter = Painter.toPrim, houseOwner = HouseOwner.toPrim, bank = Bank.toPrim, iouId = iouContract.contractId ).create.command val Seq(paintOffer) = decodeAllCreated(OfferToPaintHouseByOwner.id)( participant2.ledger_submit_flat(HouseOwner, Seq(paintOfferCmd), workflowId = paint.name)) ### Transferring a contract¶ In Canton, contracts reside on at most one domain at a time. For example, the IOU contract resides on the iou domain because it has been created by a command that was submitted to the iou domain. Similarly, the paint offer resides on the paint domain. In the current version of Canton, a command can only use contracts that reside on the domain that the command is submitted to. Therefore, before the painter can accept the offer and thereby become the owner of the IOU contract, both contracts must be brought to a common domain. In this example, the house owner and the painter are hosted on participants that are connected to both domains, whereas the Bank is only connected to the iou domain. The IOU contract cannot be moved to the paint domain because all stakeholders of a contract must be connected to the contract’s domain of residence. Conversely, the paint offer can be transferred to the iou domain, so that the painter can accept the offer on the iou domain. Stakeholders can change the residence domain of a contract using the transfer command. In the example, the painter transfers the paint offer from the paint domain to the iou domain. // Wait until the painter sees the paint offer in the active contract store assert(await_active_contract(participant3, Painter, paintOffer.contractId.toLf)) // Painter transfers the paint offer to the IOU domain participant3.transfer( Painter, // Initiator of the transfer paintOffer.contractId.toLf, // Contract to be transferred paint.alias, // Origin domain iou.alias // Target domain ) ### Atomic acceptance¶ The paint offer and the IOU contract both reside on the iou domain now. Accordingly, the painter can complete the workflow by accepting the offer. // Painter accepts the paint offer on the IOU domain val acceptCmd = paintOffer.contractId.exerciseAcceptByPainter(Painter.toPrim).command val acceptTx = participant3.ledger_submit_flat(Painter, Seq(acceptCmd)) val Seq(painterIou) = decodeAllCreated(Iou.id)(acceptTx) val Seq(paintHouse) = decodeAllCreated(PaintHouse.id)(acceptTx) This transaction executes on the iou domain because the input contracts (the paint offer and the IOU) reside there. It atomically creates two contracts on the iou domain: the painter’s new IOU and the agreement to paint the house. The unhappy scenarios needing out-of-band resolution are avoided. ### Completing the workflow¶ Finally, the paint agreement can be transferred back to the paint domain, where it actually belongs. // Wait until the house owner sees the PaintHouse agreement assert(await_active_contract(participant2, HouseOwner, paintHouse.contractId.toLf)) // The house owner moves the PaintHouse agreement back to the Paint domain participant2.transfer( HouseOwner, paintHouse.contractId.toLf, iou.alias, paint.alias ) Note that the painter’s IOU remains on the iou domain. The painter can therefore call the IOU and cash it out. // Painter converts the Iou into cash participant3.ledger_submit_flat( Painter, Seq(painterIou.contractId.exerciseCall(Painter.toPrim).command), iou.name ) ### Take aways¶ • Contracts reside on domains. Commands can only use contracts that reside on the domain to which they are submitted. • Stakeholders can move contracts from one domain to another using transfer. All stakeholders must be connected to the origin and the target domain. ## Part 2: Composing existing workflows¶ This part shows how existing workflows can be composed even if they work on separate domains. The running example is a variation of the paint example from the first part with a more complicated topology. We therefore assume that you have gone through the first part of this tutorial. Technially, this tutorial runs through the same steps as the first part, but more details are exposed. The console commands assume that you start with a fresh Canton console. ### Existing workflows¶ Consider a situation where the two domains iou and paint have evolved separately: • The iou domain for managing IOUs, • The paint domain for managing paint agreements. Accordingly, there are separate applications for managing IOUs (issuing, changing ownership, calling) and paint agreements, and the house owner and the painter have connected their applications to different participants. The situation is illustrated in the following picture. To enter in a paint agreement in this setting, the house owner and the painter need to perform the following steps: 1. The house owner creates a paint offer through participant 2 on the paint domain. 2. The painter accepts the paint offer through participant 3 on the paint domain. As a consequence, a paint agreement is created. 3. The painter sets a reminder that he needs to receive an IOU from the house owner on the iou domain. 4. When the house owner observes a new paint agreement through participant 2 on the paint domain, she changes the IOU ownership to the painter through participant 5 on the iou domain. 5. The painter observes a new IOU through participant 4 on the iou domain and therefore removes the reminder. Overall, a non-trivial amount of out-of-band coordination is required to keep the paint ledger consistent with the iou ledger. If this coordination breaks down, the unhappy scenarios from the first part can happen. ### Required changes¶ We now show how the house owner and the painter can avoid need for out-of-band coordination when entering in paint agreements. The goal is to reuse the existing infrastructure for managing IOUs and paint agreements as much as possible. The following changes are needed: 1. The house owner and the painter connect their participants for paint agreements to the iou domain: The Canton configuration is accordingly extended with the two participants 4 and 5. (The connections themselves are set up in the next section.) canton { participants { participant4 { ledger-api.port = 11041 storage.type = memory } participant5 { ledger-api.port = 11051 storage.type = memory } } } 2. They replace their DAML model for paint offers such that the house owner must specify an IOU in the offer and its accept choice makes the painter the new owner of the IOU. 3. They create a new application for the paint offer-accept workflow. The DAML models for IOUs and paint agreements themselves remain unchanged, and so do the applications that deal with them. ### Preparation using the existing workflows¶ We extend the topology from the first part as described. The commands are explained in detail in Canton’s identity management manual. // start all instances defined in the configuration file all start connect(participant1, iou) connect(participant2, iou) connect(participant3, iou) connect(participant2, paint) connect(participant3, paint) connect(participant4, iou) connect(participant5, iou) // create the parties val Bank = enable_party(participant1, "Bank") val HouseOwner = enable_party(participant2, "House Owner") val Painter = enable_party(participant3, "Painter") // enable the house owner on participant 5 and the painter on participant 4 // as explained in the identity management documentation at // https://www.canton.io/docs/stable/user-manual/usermanual/identity_management.html#party-on-two-nodes import com.digitalasset.canton.console.ParticipantReference def authorizePartyParticipant(partyId: PartyId, createdAt: ParticipantReference, to: ParticipantReference): Unit = { val createdAtP = createdAt.tryToId val toP = to.tryToId createdAt.authorize_party_to_participant(IdentityChangeOp.Add, None, partyId, toP, RequestSide.From) to.authorize_party_to_participant(IdentityChangeOp.Add, None, partyId, toP, RequestSide.To) } authorizePartyParticipant(HouseOwner, participant2, participant5) authorizePartyParticipant(Painter, participant3, participant4) // Wait until the party enabling has taken effect and a heartbeat has been sent afterwards val partyAssignment = Set(HouseOwner -> participant2, HouseOwner -> participant5, Painter -> participant3, Painter -> participant4) assert(await_topology_heartbeat(participant2, partyAssignment)) assert(await_topology_heartbeat(participant3, partyAssignment)) // upload the DAML model to all participants val darPath = Option(System.getProperty("canton-examples.dar-path")).getOrElse("dars/CantonExamples.dar") As before, the Bank creates an IOU and the house owner discloses it to the painter on the iou domain, using their existing applications for IOUs. import com.digitalasset.canton.examples.Iou.{Amount, DiscloseIou, Dummy, Iou} import com.digitalasset.canton.examples.Paint.{OfferToPaintHouseByOwner, PaintHouse} import com.digitalasset.canton.ledger.api.client.DecodeUtil.decodeAllCreated import com.digitalasset.canton.protocol.ContractIdSyntax._ val createIouCmd = Iou( payer = Bank.toPrim, owner = HouseOwner.toPrim, amount = Amount(value = 100.0, currency = "USD") ).create.command val Seq(iouContract) = decodeAllCreated(Iou.id)( participant1.ledger_submit_flat(Bank, Seq(createIouCmd))) // Wait until the house owner sees the IOU in the active contract store of participant 5 assert(await_active_contract(participant5, HouseOwner, iouContract.contractId.toLf)) // The house owner discloses the IOU to the Painter val discloseIouCmd = DiscloseIou( sender = HouseOwner.toPrim, iou = iouContract.contractId ).createAnd.exerciseDisclose(HouseOwner.toPrim).command decodeAllCreated(DiscloseIou.id)(participant5.ledger_submit_flat(HouseOwner, Seq(discloseIouCmd))) ### The paint offer-accept workflow¶ The new paint offer-accept workflow happens in four steps: 1. Create the offer on the paint domain. 2. Transfer the contract to the iou domain. 3. Accept the offer. 4. Transfer the paint agreement to the paint domain. #### Making the offer¶ The house owner creates a paint offer on the paint domain. // The house owner creates a paint offer using participant 2 and the Paint domain val paintOfferCmd = OfferToPaintHouseByOwner( painter = Painter.toPrim, houseOwner = HouseOwner.toPrim, bank = Bank.toPrim, iouId = iouContract.contractId ).create.command val Seq(paintOffer) = decodeAllCreated(OfferToPaintHouseByOwner.id)( participant2.ledger_submit_flat(HouseOwner, Seq(paintOfferCmd), workflowId = paint.name)) #### Transfers are not atomic¶ In the first part, we have used transfer to move the offer to the iou domain. Now, we look a bit behind the scenes. A contract transfer happens in two atomic steps: transfer-out and transfer-in. transfer is merely a shorthand for the two steps. In particular, transfer is not an atomic operation like other ledger commands. During a transfer-out, the contract is deactivated on the origin domain, in this case the paint domain. Any stakeholder whose participant is connected to the origin domain and the target domain can initiate a transfer-out. The transfer_out command returns a transfer Id. // Wait until the painter sees the paint offer in the active contract store assert(await_active_contract(participant3, Painter, paintOffer.contractId.toLf)) // Painter transfers the paint offer to the IOU domain val paintOfferTransferId = participant3.transfer_out( Painter, // Initiator of the transfer paintOffer.contractId.toLf, // Contract to be transferred paint.alias, // Origin domain iou.alias // Target domain ) The transfer_in command consumes the transfer Id and activates the contract on the target domain. participant3.transfer_in(Painter, paintOfferTransferId, iou.alias) Between the transfer-out and the transfer-in, the contract does not reside on any domain and cannot be used by commands. We say that the contract is in transit. #### Accepting the paint offer¶ The painter accepts the offer, as before. // Wait until the Painter sees the IOU contract on participant 3. // Since disclosed contracts do not show up in the active contract service, // the Painter instead creates and archives a dummy contract; // Canton's ordering ensures that the participant knows the earlier disclosure afterwards. val dummyCmd = Dummy(party = Painter.toPrim).createAnd.exerciseArchive(Painter.toPrim).command participant3.ledger_submit_flat(Painter, Seq(dummyCmd), workflowId = iou.name) // Painter accepts the paint offer on the Iou domain val acceptCmd = paintOffer.contractId.exerciseAcceptByPainter(Painter.toPrim).command val acceptTx = participant3.ledger_submit_flat(Painter, Seq(acceptCmd)) val Seq(painterIou) = decodeAllCreated(Iou.id)(acceptTx) val Seq(paintHouse) = decodeAllCreated(PaintHouse.id)(acceptTx) #### Automatic transfer-in¶ Finally, the paint agreement is transferred back to the paint domain such that the existing infrastructure around paint agreements can work unchanged. // Wait until the house owner sees the PaintHouse agreement assert(await_active_contract(participant2, HouseOwner, paintHouse.contractId.toLf)) val paintHouseId = paintHouse.contractId // The house owner moves the PaintHouse agreement back to the Paint domain participant2.transfer_out( HouseOwner, paintHouseId.toLf, iou.alias, paint.alias ) // After the exclusivity period, which is set to 2 seconds, // the contract is automatically transferred into the target domain assert(retryUntilTrue(java.time.Duration.ofSeconds(10)) { participant3.acs_search(paint.name, filterId=paintHouseId.toString).nonEmpty && participant2.acs_search(paint.name, filterId=paintHouseId.toString).nonEmpty }) Here, there is only a transfer_out command but no transfer_in command. This is because the participants of contract stakeholders automatically try to transfer-in the contract to the target domain so that the contract becomes usable again. The domain parameter transfer-exclusivity-timeout on the target domain specifies how long they wait before they attempt to do so. Before the timeout, only the initiator of the transfer is allowed to transfer-in the contract. This reduces contention for contracts with many stakeholders, as the initiator normally completes the transfer before all other stakeholders simultaneously attempt to transfer-in the contract. On the paint domain, this timeout is set to two seconds in the configuration file. Therefore, the retryUntilTrue normally succeeds within the allotted ten seconds. Setting the transfer-exclusivity-timeout to 0 as on the iou domain disables automatic transfer-in. This is why the above transfer of the paint offer had to be completed manually. Manual completion is also needed if the automatic transfer in fails, e.g., due to timeouts on the target domain. Automatic transfer-in therefore is a safety net that reduces the risk that the contract gets stuck in transit. ### Continuing the existing workflows¶ The painter now owns an IOU on the iou domain and the entered paint agreement resides on the paint domain. Accordingly, the existing workflows for IOUs and paint agreements can be used unchanged. For example, the painter can call the IOU. // Painter converts the Iou into cash participant4.ledger_submit_flat( Painter, Seq(painterIou.contractId.exerciseCall(Painter.toPrim).command), iou.name ) ### Take aways¶ • Contract transfers take two atomic steps: transfer-out and transfer-in. While the contract is being transferred, the contract does not reside on any domain. • Transfer-in happens under normal circumstances automatically after the transfer-exclusivity-timeout configured on the target domain. A timeout of 0 disables automatic transfer-in. If the automatic transfer-in does not complete, the contract can be transferred in manually.	2020-04-03 23:28:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21046625077724457, "perplexity": 8461.078890467159}, "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-2020-16/segments/1585370518767.60/warc/CC-MAIN-20200403220847-20200404010847-00424.warc.gz"}
https://mathoverflow.net/tags/foundations/hot	# Tag Info 210 I apologize for writing a lengthy answer, but I get the feeling the discussions about foundations for formalized mathematics are often hindered by lack of information. I have used proof assistants for a while now, and also worked on their design and implementation. While I will be quick to tell jokes about set theory, I am bitterly aware of the shortcomings ... 81 Very often one has the feeling that set-theoretic issues are somewhat cheatable, and people feel like they have eluded foundations when they manage to cheat them. Even worse, some claim that foundations are irrelevant because each time they dare to be relevant, they can be cheated. What these people haven't understood is that the best foundation is the one ... 71 Set theory provides a foundation for mathematics in roughly the same way that Turing machines provide a foundation for computer science. A computer program written in Java or assembly language isn't actually a Turing machine, and there are lots of good reasons not to do real programming in Turing machines - real languages have all sorts of useful higher ... 47 I like your analogy with programming languages. If we think of ST as a low-level programming language and UF as a high-level one, then one advantage of UF is obvious: it is more convenient to write proofs (programs) in a high-level language. It is feasible to write proofs in UF, but it's virtually impossible to write down even statements of theorems in plain ... 43 I think that Penelope Maddy's article What Do We Want a Foundation to Do? is a good starting point if you want to read some literature. I don't agree with all of Maddy's conclusions but the terminology that she introduces in this article is exceedingly helpful, as well as the very simple but often overlooked point that the concept of a "foundation of ... 42 This is a question that has been discussed a lot on the Foundations of Mathematics mailing list (unfortunately with more polemics than necessary IMO—though I confess that I may have been guilty of stoking the flames somewhat because I love to watch a good argument!). My feeling is that to ask whether univalent foundations or set theory is the "better ... 39 One approach, mentioned by Pace Nielsen in the comments, is to start with what I call strict formalism. The only substantive assumption required for strict formalism is that you are capable of recognizing and manipulating finite strings of symbols in certain simple ways, and can understand what a syntactic rule is at the level of being able to confirm or ... 38 I have long found this question interesting, and in some recent joint work with Makoto Kikuchi, now available, we consider various aspects of the question of whether a set-theoretic version of mereology can form a foundation of mathematics. In particular, for our main thesis we argue that the particular understanding of mereology by means of the inclusion ... 35 EDIT: Since this question has gotten so much interest, I have decided to substantially rewrite my answer, stating explicitly here on MO some of the more important points rather than forcing the reader to follow links and chase down references. To begin with, it is important to distinguish between what currently existing proof assistants can do versus what ... 34 The main fact is that a very weak meta-theory typically suffices, for theorems about models of set theory. Indeed, for almost all of the meta-mathematical results in set theory with which I am familiar, using only PA or considerably less in the meta-theory is more than sufficient. Consider a typical forcing argument. Even though set theorists consider ZFC ... 33 It is quite difficult to answer this question comprehensively. It's a bit like asking "so what's been going on in analysis lately?" It is probably best if logicians who work in various areas each answer what is going on in their area. I will speak about logic in computer science. I am very curious to see what logicians from other areas have to say about ... 32 I think the main reason replacement is seen as an essential part of ZF is that it naturally follows from the ontology of set theory, as do the other axioms of ZF. The ontology of set theory is rooted in the idea that sets are obtained by an iterative process along a wellordered "ordinal clock", where at each step all the sets whose elements were generated ... 30 I still find it very surprising that this random talk I gave attracts so much attention, especially as not everything I said was very well thought out. I am more than happy to engage with people in discussions about what I said and whether or not some things I said were ill-informed. But onto my answer to your question: whilst I am not an expert in proof ... 28 Actually, you have rediscovered a nice motivation of using prime ideals as points. Indeed, your collection of points are triples $(R, k_x, \mathrm{ev}_x)$ where , $\mathrm{ev}_x \colon R \to k_x$ is a homomorphism. The collection of all such triples is a class rather a set. In any case, you should not change the universe to get the underlying topological ... 27 I prefer to think of ZFC as a proposed model of mathematics. I want to emphasize both words "proposed" and "model". For comparison, consider quantum mechanics. It can be modeled — as far as we know, perfectly — by the theory of Hilbert spaces. But the state right now of the electron in your retinal cell being excited by photon being emitted by the leftmost ... 27 I think Noah's answer is mostly right, but partly misleading, and explaining why will take too much space for a comment, so I'm posting a separate answer. As Noah says, the main conceptual point is that HoTT forces us to un-confuse ourselves about the difference between a topological space and an $\infty$-groupoid, which are conceptually distinct but ... 27 Here's an example that links more to mathematical practice outside category theory proper. Recall that for a small site $(C,J)$, where I take $J$ to be a Grothendieck pretopology on the small category $C$, any presheaf $C^{op} \to \mathbf{Set}$ has a sheafification, and this extends to give us a functor $[C^{op},\mathbf{Set}] \to Sh(C,J)$ from presheaves to ... 26 This is not at all intended as a complete answer to the question, but one criterion that feels important is that for a bijection $f$ to count as explicit, one shouldn't need to know in advance that there exists a bijection in order to prove that $f$ is a well-defined bijection. So for example if you order the elements of two sets $A$ and $B$ in some way that ... 26 The answer is yes, in fact one has a lot better than bi-interpretability, as shown by the corollary at the end. It follows by mixing the comments by Martin Brandenburg and mine (and a few additional details I found on MO). The key observation is the following: Theorem: The category of co-group objects in the category of groups is equivalent to the category ... 25 This isn't easy to do, and the reason it isn't easy is because of the step "$\infty$-groupoids are the same thing as spaces." Of course the homotopy hypothesis tells you that any $\infty$-groupoid is equivalent to the fundamental $\infty$-groupoid of a space, but that doesn't mean that they're literally the same thing. The way that HoTT approaches about $\... 24 Here are some resources: The appendix of the homotopy type theory book gives two formal presentations of homotopy type theory. Martín Escardó wrote lecture notes Introduction to Univalent Foundations of Mathematics with Agda which are at the same time written as traditional mathematics and formalized in Agda (so as formal as it gets). Designed for teaching ... 24 The answer to the question in the title is no, assuming you want to exclude the trivial case of the terminal category. Let$E$be an (elementary) topos whose opposite is also a topos. The initial object of a topos is strictly initial (any map into it is an iso), so the terminal object$1$of$E$is strictly terminal. Since there is a map from$1$to the ... 23 You asked: When set-theorists talk about models of ZFC, are they using an informal set theory as their meta-theory? The short answer is yes. A set theorist is doing mathematics and hence is reasoning informally, just like any other mathematician. It's important, at least when you're first wrapping your mind around these concepts, to distinguish between ... 23 One feature of the foundations of mathematics that poses a special challenge (compared to other branches of mathematics) is that it is very easy to get confused about certain distinctions—truth versus provability, theory versus meta-theory, formal versus informal, syntax versus arithmetic, etc. One book that I think is helpful in this regard is Torkel ... 22 You wrote: Suppose our intuition for the phrase "subset of$X$" comes from the idea of having an effective total function$X \rightarrow \{0,1\}$that returns an answer in a finite amount of time. In this case, the subsets of$X$ought to form a Boolean algebra. Unfortunately, this is not a workable intuition at all. If you insist that all subsets be ... 22 Let me try to answer as a set theorist, rather than as a category theorist, since I think that your question concerns at bottom a matter often considered in set theory. Namely, the essence of your question, to my way of thinking, revolves around the fact that Grothendieck universes (or Grothendieck-Zermelo universes, as one might call them) need not all ... 22 As you noticed, the iterative conception of sets requires a pre-existing universe of sets, and ordinals with which we can label the stages. So if you work within ZFC itself, in other words within an existing model of ZFC, you can perform that iterative construction to obtain$V\$. Like Asaf Karagila says here, you cannot get nothing from nothing. Typically, ... 22 The other answers are good, but I would like to point out that Ivan's "uncheatable" lemma can in fact be cheated. The proof of that lemma (due to Freyd) makes inescapable use of classical logic, and in constructive mathematics it is possible to have a non-poset that is complete for the size of its own set of objects (a complete small category). ... 21 I’ll give first a simplicial definition of univalence, and then a type-theoretic one, and discuss the equivalence between them as we go. The first thing to know is that univalence is a property that can be defined for any family of types, or in models, for any fibration. The Univalence Axiom then says that a particular family of types — a “universe” — is ... 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http://now.kateezet.pl/improved-euler-method-pdf.html	# Improved Euler Method Pdf Filling post holes where fence was removed - suburban backyard environment. Euler's methods. Stability of two classes of improved backward Euler methods for stochastic delay differential equations of neutral type: Article 2, Volume 5, Issue 3, Summer 2017, Page 201-213 PDF (326. In particular, we'll consider the IVP , We can solve this initial value problem explicitly by hand using integrating factors, but that would be tedious since we would. ) So you make a small line with the slope given by the equation. Like natural frequency formats, discrete items rep-resented by the icon array were expected to correspond with humans’ perception of natural sampling, thus improving Bayesian reasoning. Given the point (t 0;x 0) the value x_(t 0) = dx(t) dt t=t 0 is the slope of the tangent line to the graph of x(t) at the point (t 0;x 0). Please try again using a different payment method. 10 in the text lists TI-85 and BASIC programs implementing the improved Euler method to approximate the solution of the initial value problem dy x y dx =+, y(0) 1= (1) considered in Example 2 of Section 2. The Runge-Kutta method is a far better method to use than the Euler or Improved Euler method in terms of computational resources and accuracy. The ability to timely resolve internal and external product and team problems and make responsible decisions is what any junior PM should learn at the start of career. The Cornell notes taking method ensures to divide a single page in three sections namely Cues, Notes, and Summary Section. Types of Teaching Method. Improved Euler's Method and Runge-Kutta 4 Explained. 1 Improved Euler (Heun’s) Method yfxy xy′= ()( ),,00 • Euler Method – Use constant derivative between points i & i+1 – calculated at xi • Better to use average derivative across the interval. This is the web site of the International DOI Foundation (IDF), a not-for-profit membership organization that is the governance and management body for the federation of Registration Agencies providing Digital Object Identifier (DOI) services and registration, and is the registration authority for the ISO. aLEONHARD EULER (1707-1783). A Second-Order Improved Front Tracking Method for the Numerical Treatment of the Hyperbolic Euler Equations. t/: Euler’s method appliedto this problemgives UnC1 D. The two key players in advancing computations techniques for CFD were NASA and Boeing. Here we'll explore three: Euler integration, an improved version, and then the Runge-Kutta method, which will be our preferred method. Probabilistic reachability and control synthesis for stochastic switched systems using the tamed Euler method A. Error estimation is carried out by the step doubling method. alternative methods which can efﬁciently integrate a multi-time-scale problem and retain the transient information of species and physical processes at different timescales. Here, the quantities , , and are as for the Euler-Bernoulli Equation. These angles are called Euler angles or Tait-Bryan angles. 2 Accuracy of Numerical Methods 530. The labs will be submitted in pdf format. Denote by ϕ(t) the exact solution for this initial value problem. An arbitrary Lagrangian-Eulerian RKDG method for compressible Euler equations on unstructured. and particles, so that the erosion modelling e ciency can also be improved. Figure:Euler’s Method Lecture 3 Introduction to Numerical Methods for Di erential and Di erential. Specifically using the methods of Eulers and Rk4. REVIEW: We start with the diﬀerential equation dy(t) dt = f (t,y(t)) (1. 1 Modi ed Euler Method Numerical solution of Initial Value Problem: dY dt = f(t;Y) ,Y(t n+1) = Y(t n) + Z t n+1 tn f(t;Y(t))dt: Approximate integral using the trapezium rule:. (2013) modify and generalize the Smolyak method in various dimensions to improve its performance in economic applications. 1 Modi ed Euler Method Numerical solution of Initial Value Problem: dY dt = f(t;Y) ,Y(t n+1) = Y(t n) + Z t n+1 tn f(t;Y(t))dt: Approximate integral using the trapezium rule:. Doğrusal İnterpolasyon Metodunda Durma Koşulları. Matlab will return your answer. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Differential Equations, Numerical Solution of Partial +. Mathews and Kurtis D. Motorcycle glasses: useful tips, driving and models. Moreover, compare the results corresponding to different step sizes (h = 0. 25 Aug 2019: 1. Atrial fibrillation (AF) is the most common cardiac arrhythmia with a prevalence of at least 3% in the adult population of Sweden. When the derivative is a function of x only, Euler's method is equivalent to the rectangle rule for numerical quadrature. The modified Euler’s method for approximating the solution to ), MATLAB complete course by by Fitzpatrick and Ledeczi in English, MATLAB Programming from Basics in ENGLISH, MATLAB/SIMULINK Complete course in HINDI/URDU, How to Develop Battery Management Systems in Simulink, Data Science Complete Course using MATLAB, Design Motor Controllers with Simscape Electrical. - Euler equations, MHD, waves, hyperbolic systems of conservation laws, primitive form, conservative form, integral form. For generality, we assume that all these material parameters are functions of. This transform is named after the mathematician and renowned astronomer Pierre Simon Laplace who lived in France. The Euler method is the simplest and most fundamental method for numerical integration. Quiet or suspended respiration (at endexpiration). Firstly, we discuss the concept of convergence Then, the stability of each method is examined briey. Now if the order of the method is better, Improved Euler's relative advantage should be even greater at a smaller step size. The questions here is: Find the value of y(1) by solving the following initial value problem. Maha y, [email protected] The input data for Modified Euler's Method in C given below are initial and final values of x i. Project Euler: Copyright Information \| Privacy Policy. Derivation Numerical Methods for Solving Differential Equationsof Euler's Method - Let’s start with a general first order Initial Value Problem = ( T, U) U( T0)= U0 ( s) where f(x,y) is a known function and the values in the initial condition are also known numbers. Forward Euler’s method. # src-ch1/euler_simple. The screenshot below is the graphical output of Euler’s MATLAB program. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Exact solution Numerical solution. Improved Euler’s method. n) leads also to Euler’s method u n+1 = u n +hf(t n,u n). 5 5 4 8 3 1 6 8 7 9. The Improved Euler method This is also called the Runge-Kutta 2 method or RK2, or the Heun method. Euler was a test of the capabilities of these new tools, both technically and aesthetically. Let us reconsider the following SDE dSt = Stµdt + StσdWt. 07/10/2020;. Now if the order of the method is better, Improved Euler's relative advantage should be even greater at a smaller step size. \) Solving the $$N$$th Order Euler Equation Using the Power Function $$y = {x^k}$$ Consider another way of solving the Euler equations. Quiet or suspended respiration (at endexpiration). , the local. In this paper, we deduce the asymptotic error distribution of the Euler method for the nonlinear ltering problem with continuous-time observations. Lab Project 3: Improved Euler Method The improved Euler method is described by this improved update: h = t i+1 −t. Verlet methods are not part of the Runge-Kutta family of methods. Improved Euler's Method A more accurate way of approximating the integral is by finding the area of a trapezoid. In this post, I'll sort out the process of problem solving, and also describe some effective methods and exercises for teams. Alternative numerical methods - the explicit Euler and the implicit iterative Heun methods - are imple-mented and assessed in their ability to minimize errors and produce more accurate. Posted By Andrew NeidermanMedia Publishing TEXT ID 77483c4c. ps, gammaFormulas. ad00417nnn np21tvp4e03rktk fw9d0jxwwptle 5ra2e61ob2ho0az 0dvnhznwi3 6qqblgtlkl1fy3 1ykoo9ob0wrg wv1dbyj04ze5vf 5bkokpnpk8kw 9voc146eio hnyzh9h038 m26vnzxum8q. (b) Use Euler Method, and Runge-Kutta methods of order 2 and order 4 to approximate the solution of the IVP with h 0. EULER' S METHOD APPLIED TO TRAJECTORY PROBLEMS Now that we are familiar with using Euler’s method and recursion techniques to solve differential equations, let’s see how to apply this to trajectory problems. In summary, the modiﬁed Euler method for approximating the solution to the initial. We start with some ﬁxed stepsize methods. Well-matched Euler diagrams generated by a drawing method [27]. An Improved Second-Order Finite-Volume Algorithm for Detached-Eddy Simulation Based on A combined discontinuous Galerkin finite element method for miscible displacement problem. Our EL approach makes use of stochastic modeling and improved collision algorithms to compete against the other simulation techniques. Then we'll improve upon this method by using a tabular …Euler's Method will only be accurate over small increments and as long as our function does not. pdf from MAT 223 at University of Science, Malaysia. The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma (γ). Let h h h be the incremental change in the x x x -coordinate, also known as step size. 00 again and followed the Euler polygon. This is a di–cult task because we have so little to work with. Euler’s method is based on the insight that some diﬀerential equations (which are the ones we can solve using Euler’s method) provide us with the slope of the function (at all points), while an initial value provides us with a point on the. Using a layout method, the diagram is transformed into another that depicts the same set relations, but optimizes specific aesthetic criteria. The description in our course notes is a little confusing, so I need The improved Euler method. The listed and described above methods for developing your thinking abilities are the most common. He not only made formative contributions to the subjects of geometry, calculus, mechanics, and number theory but also developed methods for solving problems in astronomy and demonstrated practical applications of mathematics. The Improved Euler's Method Euler's method is one algorithm which generates approximate solutions to the initial value problem. The collective farmers improved their work with the help of the new machines. All these methods use a ﬁxed step size, but there are other methods that use a variable step size (though not neccessarily better in all circumstances). To get from one step to the next, we will form the linear approximation at. 5 µ slope Y coats t y j5 h 1. Through operant conditioning, an individual makes an association between a particular behavior and a consequence. com Approximation Methods in Probability Theory. Solution : Given To find. Clearly, in this example the Improved Euler method is much more accurate than the Euler method: about 18 times more accurate at. DMP Support for Importing Eulerian Archive Files for Subsequent Simulations The DMP support for importing Euler ARC dramatically enhances the performance for applications such as blast. The Grammar Translation Method (also known as the classical method , the traditional method , the prussian method ), is a method of foreign language teaching in which the primary focus is on the study of the target language grammar, vocabulary and ultimately the translation of native language texts or. Line equation. This equation can be used to modeled the growth of a population in an environment with a nite carrying capacity P max. Method of forming single-walled carbon nanotubes. This article describes the benefits of using effective methods in teaching lecture. Clearly, in this example the Improved Euler method is much more accurate than the Euler method: about 18 times more accurate at. This thesis concantrates on numerical methods for solving ordinary dierential equa-tions. Here, h is the step size. The forward Euler method is y nC1Dy nChf. improves the stability of the method. Therefore using a tangent line approximation of the unknown function, we have that. Find Useful Open Source By Browsing and Combining 7,000 Topics In 59 Categories, Spanning The Top 338,713 Projects. 1 Improved Euler (Heun’s) Method yfxy xy′= ()( ),,00 • Euler Method – Use constant derivative between points i & i+1 – calculated at xi • Better to use average derivative across the interval. Compress or optimize PDF files online, easily and free. Euler was a test of the capabilities of these new tools, both technically and aesthetically. 0: View License. Let us see a compilation of Numerical methods in C programming languages with output, explanation, algorithms, flowcharts, etc. known as Modified Improved Modified Euler (MIME) method. Lab Project 3: Improved Euler Method The improved Euler method is described by this improved update: h = t i+1 −t. © AP Photo / Michel Euler. l One method to improve Euler's method is to determine derivatives at the beginning and predicted ending of the interval and average them. bolic systems (see for example [7]). Role-plays have students improvise scenes based on character descriptions given. Posted by WIREDVT. Read Stock Identification Methods online, read in mobile device or Kindle. A convenient way to ob-tain time accurate solution of various reaction groups is the Euler method. method or the improved Euler method. Euler's method only samples the slope at the left end-point, that is, the initial point (t 0, y 0). 15 K) Document Type: Research Paper: Authors: Omid Farkhondeh Rouz; Davood Ahmadian : Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran. Here, h is the step size. Let's call this first slope K 0 = f(t 0, y 0). Posted on 28. View Chapter 7. This article describes the benefits of using effective methods in teaching lecture. Figure:Euler’s Method Lecture 3 Introduction to Numerical Methods for Di erential and Di erential. develop Euler’s Method for solving ordinary differential equations, 2. method and Improved Euler’s method as well as the slope eld and exact solution. for Euler’s method, assuming that the correct solution and numerical approximation stay within R. The Euler equations for transonic flows were incorporated into codes in 1981. This is a di–cult task because we have so little to work with. Introduction. Jump to navigation Jump to search. For instance, spacecraft use a variation of the Euler method to Boelkins, M. Start with HTML, CSS, JavaScript, SQL, Python, Data Science, and more. 2 The Euler predictor-corrector method This method is also sometimes called the improved Euler’s method. 1 Obtain a numerical solution of the differential equation given the initial conditions that with intervals of 0. 25 Aug 2019: 1. First thing, you could have mentioned, what RK method you have used. Numerical Methods Python Pdf. 5 ylo y lo 0. 2 Both the prevalence of AF and the related stroke risk increase markedly in the elderly. Picard İterasyon Yöntemi (Picard Iteration Method). This method involves making observations, forming questions, making hypotheses, doing an experiment, analyzing the data, and forming a conclusion. (2013) modify and generalize the Smolyak method in various dimensions to improve its performance in economic applications. Smith has written: 'A fully Galerkin method for the recovery of stiffness and damping parameters in Euler-Bernoulli beam models' -- subject a. Code for MODIFIED EULER'S METHOD in C Programming. Quora is a place to gain and share knowledge. FILE INFORMATION. Here we introduce an approximate method, the Forward Euler method, to determine the solution $y(t)$. t/D 0 and we have simply u0. of equations. m Simplified Bulirsch-Stoer method. previous tick method or refreshing time method) and it is robust to some dependence structure of the noise process. Clearly, in this example the Improved Euler method is much more accurate than the Euler method: about 18 times more accurate at. Let’s brieﬂy explore both these possibilities. m; The midpoint method for scalar equations: midpoint1. As in Euler’s method, we know the solution must go through the point (a,c) and we know its slope there is m = f(a,c). It is fast and flexible, and can be applied to many complicated programs. Image at minimum depth necessary. Improved Euler's method - order 2 - error reduces by a factor of approxi-mately 22 = 4 as step size is doubled - 2 function evaluation2 per step. Negative for M1, M2, and C1 since computation proceeds upstream. There are two types of sampling methods: Probability sampling involves random selection, allowing you to make statistical inferences about the whole group. The method is applied to four test problems and the results are presented and compared with those of a conventional extended period simulation method and some other existing methods. Calculus Made Easy — Silvanus P. - Uses periodic boundary conditions - Uses pseudopotential method. pdf - Euler’s method Program for the TI-83 calculator This program uses Euler’s Method to solve numerically an IVP of the form f (x, y) dx dy, with initial value (X0,Y0). m) and the Runge{Kutta method (rk4. Implementation of Euler's Method Example 8. preparation for IELTS Macmillan Ready for Advanced Third Edition - preparation for CAE Macmillan Improve your Skills Series - preparation for First, Advanced and IELTS The Official Guide to the TOEFL Test. The resulting value-iterative ECM method can accurately solve models with at least up to 20 state variables and can successfully compete in accuracy and speed with state-of-the-art Euler equation methods. However, it comes at the cost of additional evaluations of fand a handful of extra oating point operations on the side. Download Ebook Stock Identification Methods free in PDF, Tuebl and EPUB Format. Illustration of the Euler method. 1 Modi ed Euler Method Numerical solution of Initial Value Problem: dY dt = f(t;Y) ,Y(t n+1) = Y(t n) + Z t n+1 tn f(t;Y(t))dt: Approximate integral using the trapezium rule:. Particle swarm optimization (PSO) is an efficient algorithm for obtaining the optimal solution of a nonlinear optimization problem. m Simplified Bulirsch-Stoer method. and rearrange to around with step where is the differential equation evaluated at and (8. Find out the differences between these calculators and which you should buy for tests. PDF \| On Aug 4, 2016, George Klimi published Improved Euler's Method (Excel Sheet) \| Find, read and cite all the research you need on ResearchGate. Euler's function on average. + h4yiv /4! In each case, compare your answer to that obtained using Euler’s method. improves the stability of the method. INTRO: More Methods, More Problems To produce a computational solution, we used the Euler method, which essentially uses the derivative information to make a linear prediction about the value at the next desired time. txt) or read online for free. Euler's method) give approximate quantitative in-formation about solutions. \) Solving the $$N$$th Order Euler Equation Using the Power Function $$y = {x^k}$$ Consider another way of solving the Euler equations. Cambria Math was designed by Jelle Bosma to take advantage of two major technologies from Microsoft: one a new way to improve the appearance of text displayed on screens, the other a sophisticated new method of creating and typesetting equations. is the Euler method which we derive by approximating the derivative in the dierential equation by the slope of a secant line. Every minute there is a change in the. Other implementations of the Euler method are given below. To accelerate the convergence, Newton's method is recommended. Before stating the convergence theorem, we introduce the concept of accuracy. Download free Structural Analysis PDF Books and training materials. 1 at (xi, y(xi)) by the line through (xi, y(xi)) with slope. Use the improved Euler method to solve the coupled, rst-order dierential equations for an RLC circuit for oscillations that are forced by a sine-wave voltage The "improved Euler method" species that the required dq/dt at t + ∆t is to be obtained by: (1) getting a crude estimate of q(t + ∆t) from Eq. Witteveen, J. Numerical methods for wave equations. Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. In this paper, we provide a new method for establishing the blowup of C2 solutions for the pressureless Euler-Poisson system with attractive forces for RN (N ≥ 2) with ρ(0, x0) > 0 and Ω0ij(x0)=12∂. 18 Use Euler’s method to approximate the solution to dy dx = y −y2 = y(1 −y) with initial condition y(0) = 2. is the shear modulus (usually called in other contexts) and is a constant which depends on the geometry of the beam. Let's call this first slope K 0 = f(t 0, y 0). Euler’s Method Extra example The general solution to the di erential equation dy dx = x y is a family of circles centered at the origin with equations of the form x 2+ y = k2. Numerical Methods for Large-Scale Dynamic Economic Models. The algorithm for adaptive step size Euler’s method can be thusly stated: 1) % Define original step size (h), y_half and y_full, TOL – these are defined as part of the original Euler’s loop. To get a strong approximation method of order 3/4 using an approximation to. This is a di-cult task because we have so little to work with. Abstract In this paper, an extension is introduced into Max-Min Improved Euler methods for solving initial value problems of fuzzy fractional differential equations (FFDEs). Let $\map y x$ be the particular solution of $(1)$. REVIEW: We start with the diﬀerential equation dy(t) dt = f (t,y(t)) (1. yn + hf(xn, We have tutors online 24/7 who can help you get unstuck. Maha y, [email protected] I have written a C code using the improved Euler method to determine the position, velocity and energy of the oscillator at regular time intervals. How much power can I expect to. If you wish, you can write your own programs to implement Euler, modi ed Euler and Runge-Kutta 4th-order methods. Euler angles can be calculated from Euler rates by integrating Ωε over time: Λ=[]T =∫Ω dt θ φ ψ ε (2) For many indoor mobile robotics applications, where floors are typically level, it is an acceptable and widely used assumption that φ and θ can be considered to equal zero. 2020 by lysa Leave a Comment on Practical Numerical Methods for Chemical Engineers Using Excel with VBA. analiz dalının ürettiği yöntemler (limit-türev-integral) sayılar kuramında kullanılabilirdi. e y0 and the value of increment i. for Euler’s method, assuming that the correct solution and numerical approximation stay within R. Consider the ODE (1. Talents Global Arabia > Blog > Uncategorized > modified euler's method calculator. We have a new and improved read on this topic. it will help you have an overview and solid multi-faceted knowledge. Description. edui Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center San Diego State University San Diego, CA 92182-7720. Euler’s method in Excel to simulate simple differential equation models It is shown how to implement Euler’s method in Excel to approximately solve an initial‐value problem (IVP). This is only an approximation of the actual slope at y(t 1) because y 1 is only an. You only need to change Line 1 in Figure 2, and then Lines 67 which compute the update formula. To increase the number of steps (and thereby decrease the step size) one need only change the value of N specified in the second line of. The forward Euler method is y nC1Dy nChf. Consider the DE dy dt = 2y + 1 with initial condition y(0) = 3. //Modified Euler's Method for differential equations #include #include #include using namespace std; double df(double x, double y) {. There are different types of qualitative research methods like an in-depth interview, focus groups, ethnographic research, content analysis, case. Losses or sources are not modeled. This work presents Euler's method for solving initial value problems in ordinary differential equations. It is a pleasure to thank the publisher for showing interest in this book and cooperating in producing it. Doğrusal İnterpolasyon Metodunda Durma Koşulları. Euler Method Problem? Consider the differential equation. e is an irrational number (it cannot be written as a simple fraction). 2 to approximate the solution to y'= 4 xK2 y y 0 = 2 at the point x = 0. Modiﬁed Euler and Improved Euler are examples of 2nd order two-stage Runge-Kutta methods. Calculus Made Easy — Silvanus P. m; another one can be tested from mat_feuler. The forward Euler method reads y k+1 = y k+ hf(t k;y k):. As studied in previous works by several authors, the error structure of the method is characterized by conditional expectations of some functionals of multiple. This file was created by the Typo3 extension sevenpack version 0. 2 Unsteady Euler Solver In the unsteady Euler solver, named WAKEFF-3U, the finite volume method is applied only for the dis lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. 181; Recent Progress in the Theory of the Euler and Navier–Stokes. derive Euler’s formula from Taylor series, and 4. PDF Compression: You can easily compress your PDF and make it smaller with this online tool - just in a few seconds and completely free. 2) Example8. The simplest numerical method, Euler's method, is studied in Chapter 2. Taylor Series method. 2020-10-24 Baldor Motor Wiring Diagrams 110v Two Direction. be/E1si7kdQUew. The Improved Euler’s method, also known as the Heun formula or the average slope method, gives a more accurate approximation than the Euler rule and gives an explicit formula for computing yn+1. Now, if we decrease the timestep size from 0. EULER'SMETHOD 3 WhenusingEuler'smethod,wetypicallyusethesamestepsize x forallofthe linearapproximations. Numerical Solutions by Euler and Improved Euler Methods (scalar equations) LAB 4 : MATLAB solvers for First-Order IVP. EULER' S METHOD APPLIED TO TRAJECTORY PROBLEMS Now that we are familiar with using Euler’s method and recursion techniques to solve differential equations, let’s see how to apply this to trajectory problems. (Gary Marple) March 29th, 2019 Math 316: Di erential Equations. Bayesian Methods for Hackers — Cameron Davidson-Pilon. y n+1 = y(t n)+ h 2 (f(t n;y(t n))+f(t n+1;y(t n+1))): So this shows that y(t n+1) y n+1 = O(h 3): Modi ed Euler: y n+1 = y n +hf(t n;y n) y n+1 = y n + h. The technique of improved the Euler Method called as modified Euler method. 5 4 3 0 6 2 5. be/E1si7kdQUew. current method, the resolution of shocks is. Euler Metodu Örnek Soru-3 (Euler's Method). Euler Method Pdf The problem is solved in a distributed model reference adaptive control framework that includes positive μ-modification to address input constraints. 10 Improved Euler method Just shrinking the step size isn't always practical as a way to improve accuracy in Euler's method. We also focus on numerical methods for systems. Euler Method Pdf. Online PDF Ebook Epub Library. The Runge-Kutta algorithm is similar to the Euler and improved Euler methods in that it also uses, in the notation of the last section, y n+1 = y n + approximate value for Z t n+1 t n f t,ϕ(t) dt But rather than approximating R t n+1 t n f t,ϕ(t) dtby the area of a rectangle, as does Euler,. All these methods use a ﬁxed step size, but there are other methods that use a variable step size (though not neccessarily better in all circumstances). Harmonic Euler as a Proposed Method. Calculus Made Easy — Silvanus P. The proposed algorithms are tested on various illustrative examples. x(t) 2R;0 t t f. However, an explicit Euler method needs a time-marching. and particles, so that the erosion modelling e ciency can also be improved. m; Heun's method for scalar equations: heun1. Figure:Euler’s Method Lecture 3 Introduction to Numerical Methods for Di erential and Di erential. Euler’s Method for Ordinary Differential Equations. On Rational Approximations to Euler's Constant 𝛾 and to 𝛾 On Rational Approximations to Euler's Constant 𝛾 and to 𝛾 Fast convergences towards Euler-Mascheroni constant Computational and Applied Mathematics , Dec 2018. F Skinner is regarded as the father of operant conditioning and. Euler’s((Forward)(Method(Alternatively, from step size we use the Taylor series to approximate the function size Taking only the first derivative: This formula is referred to as Euler’s forward method, or explicit Euler’s method, or Euler-Cauchy method, or point-slope method. Through operant conditioning, an individual makes an association between a particular behavior and a consequence. m — normal modes of oscillation of linear mass & spring system. Ranked as 7484 on our all-time top downloads list with 5015 downloads. Modified Euler’s Method is a popular method of numerical analysis for integration of initial value problem with the best accuracy and reliability. 5Y\ECW$8?9fQ58_NV+]TP6e_iLU1LcrW0;RpXbIHD!=iSE4/bi(0)W5L)FdnEP#hak"P:)(TB1,8de=D<N3Lg&8t-l). 1 Euler’s Method 74 3. In order to have a better understanding of the Euler integration method, we need to recall the equation of a line: \[y = m \cdot x + n \tag{4. While it is not the most efﬁcient method, it does provide us with a picture of how one proceeds and can be improved by introducing better techniques, which are typically covered in a numerical analysis text. Compact Rectangular Euler Diagram(left) and Euler Diagram with Duplications(right). 10 in the text lists TI-85 and BASIC programs implementing the improved Euler method to approximate the solution of the initial value problem dy x y dx =+, y(0) 1= (1) considered in Example 2 of Section 2. Jin, Weiya, Dennis, Brian H. 1 is based on approximating the integral curve of Equation 3. As studied in previous works by several authors, the error structure of the method is characterized by conditional expectations of some functionals of multiple. develop Euler’s Method for solving ordinary differential equations, 2. Doğrusal İnterpolasyon Metodunda Durma Koşulları. 10 --- Timezone: UTC Creation date: 2020-09-07 Creation time: 19-17-35 --- Number of references 6305 article WangMarshakUsherEtAl20. is the shear modulus (usually called in other contexts) and is a constant which depends on the geometry of the beam. The two key players in advancing computations techniques for CFD were NASA and Boeing. pdf from MAT 223 at University of Science, Malaysia. Positive for S2, S3, and C3 since computation proceeds downstream (dY/dX) 1 = First depth increment for Improved Euler method [m]. Calculus Made Easy — Silvanus P. 71827 a t I 0. The development of two and three dimensional Euler correction methods based on the Clebsch transformation is described. In this study, a PSO-based Euler-type method is proposed to solve the initial value problem of ordinary differential equations. However, it comes at the cost of additional evaluations of fand a handful of extra oating point operations on the side. Numerical Methods for Large-Scale Dynamic Economic Models. The algorithm for adaptive step size Euler’s method can be thusly stated: 1) % Define original step size (h), y_half and y_full, TOL – these are defined as part of the original Euler’s loop. All of Euler's solutions are rather long, so we will only summarize them In his first solution, the one given in E451, Euler rewrites his first two fundamental equations as ff=b!c 2+b!c 2!aa=b!c 2+b+c+a b+c!a gg=a!c 2+a+c 2!bb=a!c 2+a+c+b a+c!b (). Improved Euler’s method Geometric idea: The basic idea can be easily expressed in geometric terms. Our discussion is focused on the asymptotics of. Euler's Method Calculator. improved in performance in 2012 release. from which we get the Trapezoidal method. THIERRYLUANGRATH. Euler’s Method Motivation: for a single IVP x_(t) = f(t;x(t));x(t 0) = x 0; i. Let us see a compilation of Numerical methods in C programming languages with output, explanation, algorithms, flowcharts, etc. You could say that Euler’s method is. 1 is based on approximating the integral curve of Equation 3. Firstly, we discuss the concept of convergence Then, the stability of each method is examined briey. 2 Steps for MATLAB implementation The purpose of using an example is to show you the details of implementing the. % Progress. The Euler equations in one dimension appear as We will solve the Euler equations using a high-order Godunov method—a nite volume method whereby the uxes • New PPM limiters: Recent work [5] has formulated improved limiters for PPM that do not clip the. As leaders in online education and learning to code, we've taught over 45 million people using a tested curriculum and an interactive learning environment. Adli Bin Ja'affar. Runge-Kutta Methods Calculator is an online application on Runge-Kutta methods for solving systems of ordinary differential equations at initals value problems given by. • Fourth-order Runge-Kutta methods can also be used, but care must be taken in. The absolutely simplest quadrature rule that we can use in (1. You should also get the graph, if your computer is set up properly. • The Euler and Modied Euler Method (Taylor Method of order 1) • The Higher-order Taylor Methods • The Runge-Kutta Methods • The Multistep Note: The global error bound for Euler's method depends upon h, whereas the local error depends upon h2. © AP Photo / Michel Euler. A convenient way to ob-tain time accurate solution of various reaction groups is the Euler method. Math 3280 Worksheet 7: Improved Euler and Runge-Kutta methods for numerical solutions Group members (2 to 4): (1) For the initial value problem y(1) = 1 and dy dx = y x y x x+ y , approximate y(2) by using: (a) 2 steps of the improved Euler method (b) 1 step of the 4th-order Runge-Kutta method (formulae on reverse). In Euler’s Method, the quantity ∆ x is referred to as the step size. pylab as plt """. y nC1/Dy nCahy nC1; and solving for y nC1yields the explicit. {\displaystyle y'=ky} ′ While the Euler method integrates a first-order ODE, any ODE of order N can be represented as a system of first-order ODEs. If you continue browsing the site, you agree to the use of cookies on this website. This algorithm requires the Jacobian. Program of RK(RANGA KUTTA) SECOND ORDER METHOD in. 174), states e^(ix)=cosx+isinx, (1) where i is the imaginary unit. In summary, the modiﬁed Euler method for approximating the solution to the initial. Solution : Given To find. Motorcycle glasses: useful tips, driving and models. 583-1 eklinde 7 235 233 basamaklı bir sayı! Analitik Sayılar Kuramı. 2 The Euler predictor-corrector method This method is also sometimes called the improved Euler’s method. Comparison of Euler and Improved Euler. In the image to the right, the blue circle is being approximated by the red line segments. It's a platform to ask questions and connect with people who contribute unique insights and quality answers. This method is illustratedin thefollowing ﬁgure. + h4yiv /4! In each case, compare your answer to that obtained using Euler’s method. 3, it will take 10 steps to be able to approximate y(3). How to use an optimizer¶. 1 Euler’s Method In this section we will look at the simplest method for solving ﬁrst order equations, Euler’s Method. PDF \| On Aug 4, 2016, George Klimi published Improved Euler's Method (Excel Sheet) \| Find, read and cite all the research you need on ResearchGate. To accelerate the convergence of the basic method of fourth order, Carstensen-Petkovie’s approach [7] using Weierstrass’ correction is applied. 2 Cooling and Mixing 96 4. The results show that the proposed method is considerably more accurate than the conventional algorithm and more efficient than existing improved methods of. y nC1/Dy nCahy nC1; and solving for y nC1yields the explicit. This is a di-cult task because we have so little to work with. For inference on the full parameter vector, iid sampling is not required by either a ˜2 test as in Kaplan and Sun (2017) or the simulation-based method of Chernozhukov, Hansen, and Jansson (2009), but. - Uses periodic boundary conditions - Uses pseudopotential method. 5 Initial value: y(0) = 1. In the case where m = p is an odd prime, we have E m(a) = Q p(a). LGB20103_Numerical_Methods. These are obtained as a generalization of the explicit Runge-Kutta methods, by giving up the requirement that we can compute the stages by means of forward substitution. Here is the table for. (1) It is also known as the Euler-Mascheroni constant. Euler's and Runge-Kutta Method. 1 Numerical Approximations: Euler’s Method 519. m) and the Runge{Kutta method (rk4. Euler's method is a numerical technique to solve ordinary differential equations of the form. thevoltreport. It is named after Karl Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. txt) or read online for free. As example we take a model of an irreversible molecular decay reaction: A k. stability of these points and the related stability of ﬁxed points of Runge-Kutta methods. In this case, the solution graph is only slightly curved, so it's "easy" for Euler's Method to produce a fairly close result. This project was carried under the guidance of Prof. Here we'll explore three: Euler integration, an improved version, and then the Runge-Kutta method, which will be our preferred method. m Simplified Bulirsch-Stoer method. Indeed, for such nonlinear SDEs the classical Monte Carlo Euler method has been shown to converge by exploiting that the Euler approximations diverge only on events whose probabilities decay to zero very rapidly. Given the system. m — numerical solution of 1D heat equation (Crank—Nicholson method) wave. Case studies are presented to students in written form. 5 2 y'con5 910. Graph your solution. This equation can be used to modeled the growth of a population in an environment with a nite carrying capacity P max. • Very active area of statistics - many different methods have been described. The improved Euler method for solving the initial value problem Equation 3. When the derivative is a function of x only, Euler's method is equivalent to the rectangle rule for numerical quadrature. Learn more about eulers, improved eulers, runge kutta, graphing. y n/Dy n 1Cah: The implicit Euler method is y nC1Dy nChf. Leonhard Euler (1707-1783), Swiss mathematician and physicist, remarkable genius. Euler's method written in terms of the notation defined in Section 3. We will call the distance between the steps h and the various points. The following code will do the. In the numerical analysis literature, the Verlet method is also knows as the explicit central difference. At the end of the solution it is necessary to go back from $$t$$ to the original independent variable $$x$$ substituting \(t = \ln x. Euler Angles The rotation of the material matrix is done by implementing Euler Angles using Bunge (ZXZ) notation is the method selected as the rotation matrix transformation for the stiffness matrix, stress, and strain components. stability of these points and the related stability of ﬁxed points of Runge-Kutta methods. (b) Use Euler Method, and Runge-Kutta methods of order 2 and order 4 to approximate the solution of the IVP with h 0. 2020-10-24 Baldor Motor Wiring Diagrams 110v Two Direction. Useful background for this topic. All one can ask for is a reasonably good approximation. 2 shows three of these approximating curves and illustrates an. In the original Euler angle formulation, a rotation is described by successive rotations about the Z, X and again Z axes ( or for that matter Y-X-Y, or Z-Y-Z ). A2A: Because you can use higher order derivatives using the Taylor series which allows much better accuracy with larger steps. Clearly, in this example the Improved Euler method is much more accurate than the Euler method: about 18 times more accurate at. derive Euler’s formula from Taylor series, and 4. @inproceedings{Witteveen2008ASI, title={A Second-Order Improved Front Tracking Method for the Numerical Treatment of the Hyperbolic Euler Equations}, author={J. Economic analysis is based on the general knowledge of the dialectical method, which assumes that all phenomena and processes need to be considered in constant motion, change and development. Improved Euler Implementation Figure 2. Heun's Formula / Improved Euler Method The Improved Euler’s method, also known as the Heun formula or the average slope method, gives a more accurate approximation than the Euler rule and gives an explicit formula for computing y n+1. You know what dy/dx or the slope is there (that's what the differential equation tells you. Our similarity checker allows you to upload different formats of documents including. improved in performance in 2012 release. Graph your solution. Modified Euler’s Matlab Program improved+euler+method+calculator (516 items) Filters. The summary of the properties of these methods [1,8] including the new Third Order Euler Method (TOEM) being proposed are presented in Table 1. x z 2 2 ba dx  dz 2 LGB20103 NUMERICAL METHODS 5 Y 2012 JANUARY CONFIDEN NTIAL Table parameter for Gaussian In ntegration Euler Kutta second order Runge-K ge-Kutta 4th Order Classiccal Rungg. , Trott 2004, p. • This greater accuracy comes at expense of more computational work, as it is now necessary to evaluate f (t, y) twice in order to go from tn to tn+1. Modified euler's method. Euler method; Euler method. 10 --- Timezone: UTC Creation date: 2020-09-07 Creation time: 19-17-35 --- Number of references 6305 article WangMarshakUsherEtAl20. m; One step at a time: One step of Euler's method. The following experiment illustrates the quality of the approximation. The Improved Euler's Method Euler's method is one algorithm which generates approximate solutions to the initial value problem. For the classical compressible Euler equations, Deshpande and Raul [5] proposed the kinetic theory based ﬂuid-in-cell method and subsequently Deshpande [4] improved it by adding antidiﬀusive terms. Euler's Method, Taylor Series Method, Runge Kutta Methods, Multi-Step Methods and Stability. A method of obtaining an approximate solution of an ordinary differential equation of the form dy / dx = f The classical buckling theory (Euler's Method) makes the following assumptions in the derivation of the formulae [16]: (1) The column is elastic and. These cookies will be stored in your browser only with your consent. Improved Euler Implementation Figure 2. pdf - Euler’s method Program for the TI-83 calculator This program uses Euler’s Method to solve numerically an IVP of the form f (x, y) dx dy, with initial value (X0,Y0). e is an irrational number (it cannot be written as a simple fraction). Losses or sources are not modeled. For instance, spacecraft use a variation of the Euler method to Boelkins, M. 1 Stability of Equilibrium Points We now deﬁne the stability of an equilibrium point. explicit Euler's method, or Euler-Cauchy method, or point-slope method. After reading this chapter, you should be able to: 1. Method of forming single-walled carbon nanotubes. View Chapter 7. Method and examples. Euler’s Method Extra example The general solution to the di erential equation dy dx = x y is a family of circles centered at the origin with equations of the form x 2+ y = k2. Runge-Kutta (RK4) numerical solution for Differential Equations. This method involves making observations, forming questions, making hypotheses, doing an experiment, analyzing the data, and forming a conclusion. 5 of Eulerapprox Y n 910. To accelerate the convergence of the basic method of fourth order, Carstensen-Petkovie’s approach [7] using Weierstrass’ correction is applied. Knowledge-based programming for everyone. pdf), Text File (. Also considers some suggestions of learning process as a factor in increasing the quality of education. Itiscommontouseatabletokeeptrackoftheestimatesineach step. - Euler equations, MHD, waves, hyperbolic systems of conservation laws, primitive form, conservative form, integral form. Improving the Improved Modified Euler Method for Better Performance on Autonomous Initial Value Problems. Mathews and Kurtis D. dX = Distance increment for Improved Euler method [m]. COMSOL Multiphysics Workflow. Use Euler's method with two steps to estimate y when x=1. Posted By Andrew NeidermanMedia Publishing TEXT ID 77483c4c. Numerical Solutions by Euler and Improved Euler Methods (scalar equations) LAB 4 : MATLAB solvers for First-Order IVP. How to compress a PDF? Upload your PDF file. As studied in previous works by several authors, the error structure of the method is characterized by conditional expectations of some functionals of multiple. In Iode Project 3, when you write your own numerical routine for the Improved Euler Method, you can keep the framework of euler. method and Improved Euler’s method as well as the slope eld and exact solution. In order to use Euler's Method to generate a numerical solution to an initial value problem of the form To improve the solution, shrink the step-size! To illustrate that Euler's Method isn't always this terribly bad, look at the following picture, made for exactly the same problem, only using a step. Improved Euler Implementation Figure 2. Euler’s Method Motivation: for a single IVP x_(t) = f(t;x(t));x(t 0) = x 0; i. This work presents Euler's method for solving initial value problems in ordinary differential equations. You may wish to compute the exact. The Improved Modified Euler (IME) Method As earlier stated, we had achieved an improvement on the Modified Euler method. 1 Improved Euler (Heun’s) Method yfxy xy′= ()( ),,00 • Euler Method – Use constant derivative between points i & i+1 – calculated at xi • Better to use average derivative across the interval. This gives a direct estimate, and Euler’s method takes the form of. be/E1si7kdQUew. The ability to timely resolve internal and external product and team problems and make responsible decisions is what any junior PM should learn at the start of career. All one can ask for is a reasonably good approximation. The Euler method is a typical one for numerically solving initial value problems of ordinary differential equations. The general solution of this equation can be found by standard methods. The forward Euler method reads y k+1 = y k+ hf(t k;y k):. Also an adaptive grid size mechanism based on the fixed grid size technique is proposed. 2 Finding Numerical Solutions MATLAB has a number of tools for numerically solving ordinary diﬀerential equations. Solution: Let us define. Ranked as 7484 on our all-time top downloads list with 5015 downloads. 1 Modi ed Euler Method Numerical solution of Initial Value Problem: dY dt = f(t;Y) ,Y(t n+1) = Y(t n) + Z t n+1 tn f(t;Y(t))dt: Approximate integral using the trapezium rule:. order method. To accelerate the convergence of the basic method of fourth order, Carstensen-Petkovie’s approach [7] using Weierstrass’ correction is applied. Euler Methods - Free download as PDF File (. Lab Project 3: Improved Euler Method The improved Euler method is described by this improved update: h = t i+1 −t. Clearly, in this example the Improved Euler method is much more accurate than the Euler method: about 18 times more accurate at. Any reasonable action submits to particular regulatory principles on which option the result of activity significantly depends. To develop a higher order Runge-Kutta method, we sample the derivative function at even more `auxiliary points'' between our last computed solution and the next one. Taylor series method does a) RK method b) Modified Euler method c) Simpsons d) Euler method 14. 15 K) Document Type: Research Paper: Authors: Omid Farkhondeh Rouz; Davood Ahmadian : Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran. Method and examples. Alternative: implicit Euler method. Stochastic_Control_2020_May. Euler's Method. 2 Both the prevalence of AF and the related stroke risk increase markedly in the elderly. P xn + h 2,yn + hf (x n,yn) 2 along the tangent line to the solution curve through (xn,yn) and then stepping from P to (xn+1,yn+1) along the line through P whose slope is f(xn,y n∗). Another name for your "improved Euler's method" is Heun's method. Euler's and Runge-Kutta Method. However, the convergence rate of the least-squares scheme is currently a hindrance and it should be considerably improved to make the whole method attractive in applications. 1 Euler equation properties. Start with HTML, CSS, JavaScript, SQL, Python, Data Science, and more. (Actually, my earlier Flash version also used Runge-Kutta method of Order 4, but it was a lot smoother than the javascript version. Improving the Improved Modified Euler Method for Better Performance on Autonomous Initial Value Problems. 1 Modi ed Euler Method Numerical solution of Initial Value Problem: dY dt = f(t;Y) ,Y(t n+1) = Y(t n) + Z t n+1 tn f(t;Y(t))dt: Approximate integral using the trapezium rule:. In 1735, an astronomical problem proposed by the Academy, for the solution of which several eminent mathematicians had demanded several months’ time, was solved by Leonhard Euler in three days with the aid of improved methods of his own, but the effort threw him into a fever which endangered his life and deprived him of his right eye, his. The algorithm results in a second-order-accurate ﬁnite ele-ment discretization on deforming meshes and accuracy can be improved using local. determine how the step size affects the accuracy of a solution, 3. Testing the Euler scheme. This method involves making observations, forming questions, making hypotheses, doing an experiment, analyzing the data, and forming a conclusion. If$f$is only continuous in$t$and Lipshitz in$y$, then the global error is only guaranteed to shrink to zero as$h$does. Find out the differences between these calculators and which you should buy for tests. The improved Euler's method (or Heun's method) approximates the solution of an initial value problem of the form y' = f(x,y), y(x_0) = y_0. In this paper, we propose a new method for determining Euler angles of projections by applying Genetic Algorithms (i. This file was created by the Typo3 extension sevenpack version 0. Heat transfer in a mass is very important for a number of objects such as cooling of electronic parts or the fabrication of large beams. Numerical Methods for Large-Scale Dynamic Economic Models. 2: Runge Kutta Methods (RKM) (A) 2nd Order RKM (or Improved Euler Method) Failure of Euler Method: Only slope on left end of interval [t,t + h] is used. Euler's Method after the famous Leonhard Euler. m — normal modes of oscillation of linear mass & spring system. In contrast, Euler diagrams were expected to enhance the perception of. Ebook also available in docx and mobi. The k 1 and k 2 are known as stages of the Runge-Kutta method. 4 Autonomous Second Order Equations 115 Chapter 5 Linear Second Order Equations 5. We will focus on the main two, the built-in functions ode23 and ode45, which implement versions. It is not an efﬁcient numerical meth od, but it is an intuitiveway tointroducemanyimportantideas. Improved Euler's Method in MATLAB % Improved Euler's Method % Solves the initial value. Otherwise, extrapolation methods (generally the Euler-Maclaurin formula but also Richardson extrapolation) are used to speed up convergence. Runge-Kutta/improved Euler method. 2 Accuracy of Numerical Methods 530.$\begingroup\$ Take a look at this answer for an implementation of Euler's method; the same answer also contains a link to a document that discusses a similar implementation of the Improved Euler Method ("Método Euler Mejorado") in the file. Active Calculus: a free, open text (PDF). June 27 2020. The implicit Euler method discussed above belongs to a whole class of implicit Runge-Kutta methods. Excel 2007 was used. How to use an optimizer¶. ) Facit: For stable ODEs with a fast decaying solution (Real(λ) << −1 ) or highly oscillatory modes (Im(λ) >> 1 ) the explicit Euler method demands small step sizes. Given an autonomous system of diﬀerential. The transform method finds its application in those problems which can't be solved directly. Whenever errors are made, motivating remedial methods are generated to strengthen and improve the student's learning experience. In this work we shall only consider forward Euler’s method. 5 of Eulerapprox Y n 910. stability of these points and the related stability of ﬁxed points of Runge-Kutta methods. Euler method for integrating 1D ODEs: Euler1D. Solving different types of challenges and puzzles can help you become a better problem solver, learn the intricacies of a programming language, prepare for job interviews, learn new algorithms, and more. An Improved Second-Order Finite-Volume Algorithm for Detached-Eddy Simulation Based on A combined discontinuous Galerkin finite element method for miscible displacement problem. 1 is based on approximating the integral curve of Equation 3. A2A: Because you can use higher order derivatives using the Taylor series which allows much better accuracy with larger steps. Euler method 2. The Improved Euler’s method, also known as the Heun formula or the average slope method, gives a more accurate approximation than the Euler rule and gives an explicit formula for computing yn+1. It is not an efﬁcient numerical meth od, but it is an intuitiveway tointroducemanyimportantideas. Doğrusal İnterpolasyon Metodunda Durma Koşulları. Methods and tools for automation of development of information systems Improving web applications security using path-based role access control Method of automatic detection of new motionless objects on the basis of. develop Euler’s Method for solving ordinary differential equations, 2. The commands on these models are the same, though the TI-82 has one key named differently and this will be noted in the program listing. For inference on the full parameter vector, iid sampling is not required by either a ˜2 test as in Kaplan and Sun (2017) or the simulation-based method of Chernozhukov, Hansen, and Jansson (2009), but. 5 µ slope Y coats t y j5 h 1. Euler Method Pdf. Figure:Euler’s Method Lecture 3 Introduction to Numerical Methods for Di erential and Di erential. The improved Euler method from the exercises should quarter the error every time we halve the interval, so we would have to approximately do half as many "halvings" to get the same error. A New Approach to Fuzzy Initial Value Problem by Improved Euler Method. ’s Method - mu. 36(3):396-417[ Abstract ][ PDF 130K ]( 1013 )[HTML]. t/around tDt nand tDt nC1, respectively. Please try again using a different payment method. First thing, you could have mentioned, what RK method you have used. You could say that Euler’s method is. He also popularized the use of the Greek letter π to denote the ratio of a circle’s circumference to its diameter. Stochastic_Control_2020_May. 0: View License. One possible method for solving this equation is Newton's method. The method is applied to four test problems and the results are presented and compared with those of a conventional extended period simulation method and some other existing methods. - Euler equations, MHD, waves, hyperbolic systems of conservation laws, primitive form, conservative form, integral form. 1 Numerical Approximations: Euler’s Method 519. In addition, the precision can be greatly improved for a given time step by using a numerical procedure which is more sophisticated than the rather simple Euler's method described by eqn 2. Durch die Differentialgleichung ist im Punkt (x0,y0) mit dem Wert y'(x0) = f(x0,y0) die Steigung der Tangente der Euler-Verfahren Die Ableitung wird durch den rechten (vorwärts) Differenzenquotienten ersetzt. Thus, use of Euler’s method should be limited to cases when max{\|y (x 0± )\|} ∞, for some neighborhood near x 0. Euler’s method goes bananas; it oscillates more and more widely. Here is a brief introduction to RK methods and Euler method, working, there merits and demerits. The improved Euler's method (or Heun's method) approximates the solution of an initial value problem of the form y' = f(x,y), y(x_0) = y_0. Forward Euler’s method This scheme is known as the improved Euler formula. In the image to the right, the blue circle is being approximated by the red line segments. Heat transfer in a mass is very important for a number of objects such as cooling of electronic parts or the fabrication of large beams. pdf from MAT 2680 at New York City College of Technology, CUNY. Euler's totient function φ(n) is the number of positive integers not exceeding n that have no common divisors with n (other than the common divisor 1). A Second-Order Improved Front Tracking Method for the Numerical Treatment of the Hyperbolic Euler Equations. You should begin this project by implementing the improved Euler method with your own calculator or computer system. yüzyıla gelindiğinde Euler asal sayı çalımalarına hız verecek çok önemli bir nok-tayı fark etti. pdf), Text File (.	2021-01-25 14:21:35	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6008341312408447, "perplexity": 1349.0359215246565}, "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/1610703581888.64/warc/CC-MAIN-20210125123120-20210125153120-00167.warc.gz"}
http://moza.spartaplus.ru/7b6c1a5946f/35	Out Of Limits! Limits Add: amujyten69 - Date: 2020-12-09 00:32:55 - Views: 2347 - Clicks: 7903 When you say dynamic limit in. a 400 MW machine is dispatched at 70 MW, but MBase =410 MVA), it might cause the machine to initialize out of limits. I would take this little 45 to the island with me but sadly it is almost unplayable. Find more ways to say off-limits, along with related words, antonyms and example phrases at Thesaurus. &0183;&32;Out Of Limits/Love 1985/Collision Course/Hyper Space/Other Limits/Bell Star/Twilight City/Borealis/Bella Dalena/ Limits Beyond/Saturn/Re-Entry Out of Limits was an album of space theme tunes from the short lived US instrumental group The Marketts. Last edited on 26 October, at 21:37. Say you’re asked to find this limit: You first try to plug 4 into the function, and you get 0 in the numerator and the denominator, which tells you to move. My professional outcome is the result of more than 10 Out Of Limits! years of. Easy opt-out at any time. plotting out of limits I am using adt3. Beatport is the world's largest electronic music store for DJs. Step 1 Go to the assessor’s website for the county in which the property is located. share \| cite Out Of Limits! \| follow \| edited 1. American Heritage&174; Dictionary of the English. In this post, I'll you how to set your soft-limits,. Unlimited Outside Connections if you want to add outside connections (like highways or railways) in-game, No Border Limit Camera (or some similar mod) to be able to move your camera freely, My improved version of NoPillars to place regularilly zonable roads (like basic 2-lane road), intersections and buildings with roads outside city's boundaries. 16 meeting and is likely to take letters of interest, appoint someone through. &0183;&32;Error: Gecode: Float::linear: Number out of limits. (US:37) “Listening to this album most of what you experience is a repeated. By FSantos, J in PMDG MD-11 (Legacy Version) Recommended Posts. Viewed 5 times 0. Hi everyone, new here! OUT OF LIMITS A2. IP Internet Protocol; CPU Central Processing Unit; NFGM Nano Floating Gate Memory; AKE Authentication Key Exchange; RCD Residual Current Devices; UCS Universal Charger Solution; UA Unit Attention; FPLA FIELD PROGRAMMABLE LOGIC-ARRAY; SMART Self Monitoring, Analysis and Reporting Technology; MRV Monitoring,. com, the world's most trusted free thesaurus. Outer Limits (1963) by Jerry Cole And His Spacemen. There is a witchy spooky lead guitar throughout this majestic record that is a must listen. &0183;&32;Nisswa: Hoff to resign council seat after moving out of city limits Council will address vacancy at Dec. With defense outside the limit coverage, there are separate limits available for legal defense costs and court-awarded damages. We have added the song to our site without lyrics so that you can listen to it and tell others what you think of it. . Lyrics (albums) Lyrics (songs) Videos. Check out Out Of Limits by Lightsphere on Beatport. Not to be entered or frequented by a designated group: a bar that is off-limits to military personnel. I can't find an explanation anywhere online. ( In this example, I would try to change MBase ~80). The limit of a constant times a function is equal to the product of the constant and the limit of the function: \lim\limits_x \to a kf\left( x \right) = k\lim\limits_x \to a f\left( x \right). The Marketts &183; Album &183; 1964 &183; 12 songs. when I first start these drawing the print preview is fine, so it is something that is Limits! happening while working on the drawing, then I. I am building a simple model on Minizinc 2. Image Credit: Hemera Technologies. &0183;&32;Properties Out Of Limits! in unincorporated areas, those outside the city limits, use the county's name or a nearby city for a mailing address. Abbreviation to define. &215; HOW I FIND K IN ANY BUSINESS IN 45 MINUTES. Flying a fictional flight SAEZ-SBGR with MD11, prepared the bird with 60000lbs of. when I go to plot and preview the drawing set on limits the drawing is way off to the left side (about half way) I have double checked the limits and where the drawing is in relation to the limits, everything is fine. Welcome to Beatport. 6 Review the identification. Listen to Out of Limitsby Dan Whaley for free. This topic is now archived and is closed to further replies. Find more ways to say limit, along with related words, antonyms and example phrases at Thesaurus. If your house or property is not within city boundaries, the city in your mailing address is used only for mailing and directional purposes. One accurate version. &0183;&32;Re: "X+ out of soft limit". In I co-founded Out of Limits, with the aim of transform it in a world wide reference in BHS - baggage handling system - software solutions. So, the defense costs outside the limits don’t erode your policy limits available to pay settlements resulting from a suit. “Out of Limits” is one of two bonus tracks only available in the vinyl release of the “Destroy All Astromen! LIMITS BEYOND B5. Guitar Tabs Universe. Or perhaps you can help us out. Posted J. Out Of Limits (1963) by The Ventures. "You Don't Own Me" by Lesley Gore was at 2; it had jumped from 13 but because of the Fab Four it never reached 1. Out Of Limits Tab by The Ventures with free online tab player. Out Of Limits by Ventures Tab Different Versions Chords, Tab, Tabs. OOL is defined as Out of Limits somewhat frequently. Out Of Limits by The Kilaueas, released 12 December 1998 supported by 4 fans who also own “The Volcanic Eruption Of Surf” Nice collection of instrumental surf rock songs. Since then I have lead and implemented several solutions in Airports and Airlines, focusing on software applications to support decision making in all steps of the baggage process. In business applications, three-sigma refers. A hit, "I Want To Hold Your Hand", moved into the 1 spot. Vote up content that is on-topic, within the rules/guidelines, and will likely Out Of Limits! stay relevant long-term. You’ll receive instant access to the video as well as our email training series on how to build a successful and profitable business! Changing MBase helps to bring that machine within the limits. Features the title track which became a US No. Ask Question Asked today. Printer friendly. Out Of Limits (1964) by Billy Vaughn. Setting Soft Limits for Your Industrial CNC Router: When you're getting started, in order to make sure that your Industrial CNC Router doesn't run into the side of the table, it can be a good idea to set up limits in your Mach 3 software for safety. 5 Based on the information gathered, determine the actions to be initiated as per Investigation report and proceed accordingly. warehouses in an effort to make sure it had enough space to store all their goods for the holidays. New search features Acronym Blog Free tools "AcronymFinder. Exciter problems are often caused by overloaded generators (Pgen or Qgen) in loadflow or wrong dynamic data. I believe Jeff Beck ripped off Tommy Tedesco's vibe for some of his Yardbirds. Out Of Limits (1964) by The Challengers. We've got 1 shorthand for Concurrency Out Of Limits &187; What is the abbreviation for Concurrency Out Of Limits? This page is about the various possible meanings of the acronym, abbreviation, shorthand or slang term: Concurrency Out Of Limits. • If the out of limit result is observed in only one sample location and rest of the system is conforming to specifications, then verify the sample location for any discrepancies in the sample/user point and the location. &0183;&32;There’s more out there (so to speak), and we’ll probably be hearing even more interpretations versions in the future. Top synonyms for out of bounds (other words for out of bounds) are restricted, forbidden and off limits. Governor problems are caused by overloaded Pgen in loadflow or wrong dynamic data. Unfortunately we don't have the lyrics for the song "Out Of Limits" yet. HELP Post by dhellew2 &187; Sat 9:00 pm Since your project is a rectangle all you need to do is put a box around it allowing enough room for the bit to finish the complete cut. Out of Limits" is a popular choice for television and film soundtracks; it can be heard in: Pulp Fiction (1994) Slayground (1983) The Outsiders (1983) Mafioso: The Father, the Son (). Barry from Sauquoit, Ny On January 26th, 1964, "Out of Limits" by the Marketts peaked at 3 for two weeks on Billboard's Hot Top 100 chart. Another word for limit. . &215; EVERYTHING You Know About Marketing Your Small Business Is WRONG! &0183;&32;Three-Sigma Limits: Three-sigma limit (3-sigma limits) is a statistical calculation that refers to data within three standard deviations from a mean. We at LetsSingIt do our best to provide all songs with lyrics. It is also available on the compilations Billboard Top Rock'n'Roll Hits: 1964, Elvira Presents Haunted Hits and Classic Rock (Time-Life Music). Find the limit by factoring. This is a 1964 pressing. 1963年の12月にポップ・チャートの3位まで昇ったザ・マーケッツの代表作、「Out Of Limits」（当初は「Outer Limits」というタイトルで発売された）がフューチャーされたサード・アルバム。アルバムの発売は1964年。アルバムを通じてのサウンドは、全体的に宇宙を連想させる「スペース・サウンド. Out of Limits on Burgess Hill Radio Review with Dan Whaleyby Dan Whaley. &0183;&32;INITIALIZED OUT OF LIMITS means that dynamic limits in the model do no match the load flow situation for the generator. And on this very same day the Beatles' first U. 0 to try and organize a weapons production operation. Recommended by The Wall Street Journal. &0183;&32;Amazon in. We have a large team of moderators working on this day and night. &0183;&32;The Marshall Islands could be wiped out by climate change – and their colonial history limits their ability to save themselves Decem 12. Viewed 9 times 0 $\begingroup$ $$\lim_x \to 0+(e^\sin x\, \cdot\, \ln x) = e^\lim_x \to 0+(\sin x\, \cdot\, \ln x)$$ Can someone give me a simple explanation of why this works (and what it's called)? This should not to be overlooked because if your company is facing a law suit, the type of coverage you’ve elected will be pivotal in. &0183;&32;supported by 4 fans who also own “Out Of Limits” The horn at the beginning of Geist convinced me to get the whole album. Factoring is the method to try when plugging in fails — especially when any part of the given function is a polynomial expression. Tagged r&b rhythm and blues soul rock 'n' roll. COLLISION COURSE A4. THE VENTURES " Out Of Limits " Instrumental Ventures In Space (1963) Out Of Limits; He Never Came Back; Moon Child; Fear; Exploration In Terror; War Of The Satellites; The Bat; Penetration; Love Goddess Of Venus; Solar Race; The Fourth Dimension; The Twilight Zone; THE VENTURES videos - Out Of Limits. edit flag offensive delete link more Comments. FSantos 0 FSantos 0 Members; 0 5 posts; Location: 5nm NW SSTO, Brasil. Autumn Bordner, University of. Simply enter your name and email below to receive instant access to the video. TWILIGHT CITY B2. It’s actually a cover of the original “Out of Limits” song, from. "Out Of Limits" lyrics - THE VENTURES. What does OOL stand for. /21/7108298e2a3c5 /558-7ae073ee04545 /bff41d74-267 /24871745 Out Of Limits! email: [email protected] - phone:(347) 357-7610 x 8612 -> Lazarus -> CARYOTYPE Sitemap 1	2021-08-05 17:41:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1750245839357376, "perplexity": 5480.558636554918}, "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-2021-31/segments/1627046156141.29/warc/CC-MAIN-20210805161906-20210805191906-00176.warc.gz"}
http://matthematics.com/sandwiches-and-the-ontology-of-definitions/	# Sandwiches and the Ontology of Definitions Is a hot dog a sandwich? More than a provocative conversation-starter at a party, this is the kind of question that invites a healthy scrutiny of our own assumptions and implicit definitions. Here’s a first-day-of-class activity aimed at challenging the sanctity of definitions, so that students can begin to take ownership of a more humanized mathematical discourse. Click here to jump to the PDF activity. Update: Phil DeOrsey made an interactive Desmos version! More than fifty years ago, Supreme Court Justice Potter Stewart famously offered from the bench a definition of “obscenity” that would be long remembered: I shall not today attempt further to define the kinds of material I understand to be embraced within that shorthand description [“hard-core pornography”], and perhaps I could never succeed in intelligibly doing so. But I know it when I see it. —Potter Stewart, 1964 We all are born with a natural cognitive instinct to classify: to, after experiencing many specific examples of a phenomenon, develop a sensibility that will permit us to recognize when other examples fit that same pattern. I suspect that not many of our classification schemes are explicit — we could not if asked come up with a precise definition in writing for them. Many more of our classifications are tacit like Stewart’s. “I know it when I see it.” And so it is with sandwiches. Americans eat a lot of sandwiches. By some accounts, a majority of us eat at least one sandwich every day. And the USDA estimates that about one-eighth of all calories we consume comes in sandwich form. Studies like those linked here, though, would be impossible to conduct without operationalizing the very concept of sandwich. What counts? What doesn’t? What definition can we all agree upon? Does such a consensus exist? Is it even possible? Sandwiches are an excellent entré to the topic of definition, ontology, and deontology precisely because we all have some tacit belief about what’s a sandwich and what’s not, and there is just enough variation in those beliefs that it leads quickly to interesting negotiations. Is a hot dog a sandwich? What about a taco? A calzone? Where does lasagna fit? How about a Pop-Tart? Do we make allowances for open-faced sandwiches? Does that put us on a slippery slope toward pizza? And if pizza and Pop-Tarts are both in, aren’t we forced into saying that a frosted cookie is a sandwich? Is everything a sandwich? Where does the madness end? Can’t we just “know a sandwich when we see it?” Math Twitter has been fairly obsessed with these questions in recent months. A trip through the hashtag #HoagieHomies turns up more good-natured controversy (and not a little good-natured trolling) than you might think this question could generate. But if there’s one group of people who can take a definitional argument too far, and who don’t have well-paying lawyer jobs that make them too busy for Twitter, it’s mathematicians. #### Ontology? Deontology? I am not a philosopher — insofar as any mathematician can avoid being one. But teaching mathematics, especially teaching it to students who themselves will be called upon to teach it one day, quickly puts one face-to-face with a wide range of beliefs about what math “is,” and its proper orientation with respect to humanity. What is our place in mathematics? Do we invent math, or merely discover it? Does math embody a supernatural form of truth, and if so, what business do humans have with that truth? Or is it in fact more ordinary than that? Our beliefs about these things matter more than we often give them credit. Students’ beliefs about the nature of math can inspire them with a taste of the divine – or discourage them with divine retribution. Teachers’ beliefs about the nature of math likewise shape our pedagogical choices, often in more ways than we are conscious of. As Bonnie Gold writes (Ch. 2), [T]here is quite a range of philosophical questions on which you take a position when you teach. By what we say in class we take positions on what mathematical objects are, the role of definitions in mathematics, the kind of logical rules that are to be followed, and how we determine truth in mathematics. What something “is” in mathematics (ontology), Gold writes, is a question mathematicians are often uninterested in addressing. Instead, our primary stance is often the opposite: it doesn’t matter what something “is” — it matters how it behaves. It matters what patterns it fits. Don’t ask me whether it’s a duck; ask me whether it walks, talks, and quacks like a duck. That’s deontology. And that’s what happens as soon as we begin to interrogate the sandwich question. Any attempt at ontology, an attempt to write down an unambiguous definition of what a sandwich “is,” quickly leads us to examples, or counterexamples, that we find unacceptable. • Definition. A sandwich is an arrangement of two parallel slices of bread, between which an edible filling has been placed. • Trouble. But then a club sandwich isn’t a sandwich, since it has three slices of bread. Or, a stack of three bread slices would count as a sandwich, the middle slice being the edible filling. Is that what we expected? • Definition. A sandwich is a means by which to surround a solid or semi-solid filling by a solid layer of starchy food whose surface area far exceeds the isoperimetric limit ( $$A \gg 3 V^{2/3} \sqrt[3]{\frac43 \pi}$$ ). • Trouble. But then a Pop-Tart, a raviolo, a pierogi, and a potsticker are all sandwiches. And depending on what we mean by “surrounded,” we might also need to rule out open-faced sandwiches and hot dogs. This exercise in deontology, of proposing, testing, and perhaps discarding and revising, definitions is as fun as it can be frustrating. And that is exactly the point I want students to grasp. Because mathematical definitions are not that different. Is the number 1 a prime? Beware any mathematician who gives you a glib answer — because the question has both a very long and rich history and a variety of answers. And the consensus that has emerged in the 20th century isn’t so much that 1 “is not” prime, but rather that it “ought not” be prime, because the consequences of its primality would make other definitions and results more troublesome. If the primality of 1 is the hot dog, those other results are the tacos, the Choco Tacos, the eclairs, and all the other delicious things that we might rather not admit are sandwiches after all. The ink is never really dry on a definition. That unsettles a lot of novice mathematicians. It’s not the message their school-mathematical curriculum has taught them. But it is an essential message I want them to learn, because it unsettles them in exactly the right ways. I hope that it helps them shake loose some of the bounds on their mathematical exploration, and help them to see their own self, agency, ownership, and humanity in their work. They can, in fact, “invent” mathematics. And test their inventions to see if they need updating. And assess others’ mathematics in the same way. #### The Activity Okay: now I’ll stop acting like a recipe-blogger, and give you what you came for. Below is my first-day activity on definitions intended both to break the ice among students, and to help set the tone for a semester of inquiry and humanized mathematics. At least, it sparked some very spirited debate in my precalculus class — and would likely do so in any group of math students. (Maybe best with an intro-to-proofs crowd!) This site uses Akismet to reduce spam. Learn how your comment data is processed.	2020-09-26 00:08:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.5156252980232239, "perplexity": 2073.2650638471223}, "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/1600400228998.45/warc/CC-MAIN-20200925213517-20200926003517-00700.warc.gz"}
http://physics.stackexchange.com/questions/82476/can-i-usefully-interpret-a-non-unital-completely-positive-cp-map-as-a-cooling?answertab=votes	# Can I usefully interpret a non-unital completely positive (CP) map as a cooling process? Non-unital completely positive (CP) maps take a maximally mixed quantum state (aka a normalized identity matrix aka an infinite temperature state) and map it to something else. This necessarily decreases its von Neumann entropy and, depending on how you define it, reduces its temperature. Can a stronger connection be made between thermodynamics and CP maps? To what extent do non-unital CP maps reliably cool states that aren't infinite temperature? -	2014-04-20 21:32:50	{"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": 0.871918797492981, "perplexity": 1355.0096147827135}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00440-ip-10-147-4-33.ec2.internal.warc.gz"}
https://www.degruyter.com/view/j/math.2017.15.issue-1/math-2017-0098/math-2017-0098.xml	Show Summary Details More options … # Open Mathematics ### formerly Central European Journal of Mathematics Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo IMPACT FACTOR 2018: 0.726 5-year IMPACT FACTOR: 0.869 CiteScore 2018: 0.90 SCImago Journal Rank (SJR) 2018: 0.323 Source Normalized Impact per Paper (SNIP) 2018: 0.821 Mathematical Citation Quotient (MCQ) 2018: 0.34 ICV 2018: 152.31 Open Access Online ISSN 2391-5455 See all formats and pricing More options … Volume 15, Issue 1 # Coloring subgraphs with restricted amounts of hues Wayne Goddard / Robert Melville Published Online: 2017-09-22 \| DOI: https://doi.org/10.1515/math-2017-0098 ## Abstract We consider vertex colorings where the number of colors given to specified subgraphs is restricted. In particular, given some fixed graph F and some fixed set A of positive integers, we consider (not necessarily proper) colorings of the vertices of a graph G such that, for every copy of F in G, the number of colors it receives is in A. This generalizes proper colorings, defective coloring, and no-rainbow coloring, inter alia. In this paper we focus on the case that A is a singleton set. In particular, we investigate the colorings where the graph F is a star or is 1-regular. Keywords: Vertex colorings; Rainbow; Monochromatic; Defective MSC 2010: 05C15 ## 1 Introduction Consider a (not necessarily proper) coloring of the vertices of a graph G. For a set S of vertices, denote by c(S) the number of colors used on the set S. Let F be some fixed graph and let A be some fixed subset of the positive integers (the allowed numbers). We consider colorings of G where for every copy of F in G the number c(V(F)) is in the set A. (Note that F is not required to be an induced subgraph.) We call this a coloring with a Restricted Amount of Subgraph Hues, or RASH for short. Below we will simply refer to it as a valid coloring. This idea has been studied in other contexts. Most obviously, proper colorings are the case that F = K2 and A = {2}. Thereafter, probably most studied is the case of coloring the vertices without creating some monochromatic subgraph, such as a star; these are often called defective colorings (see for example [14]). Defective colorings correspond to RASH colorings where A = {2,3,…,\|F\|} (where we use \|F\| to denote the order of F). More recently, at least in the setting of graphs, there is work on no-rainbow colorings [57], which correspond to RASH colorings where A = {1,2,…,\|F\|−1}, and on Worm colorings [8], which correspond to RASH colorings where A = {2,3,…,\|F\|−1}. These three types of colorings were already considered and generalized for hypergraphs; see [9, 10]. In this paper we will focus on the case that A is a singleton set {a}. That is, we consider colorings where every copy of F receives precisely a colors. And especially, we investigate the case that the subgraph F is a star or is 1-regular. We will assume throughout that the graphs are simple and have no isolates. ## 2 Preliminary remarks The main question is the existence of RASH colorings. But another question is the minimum or maximum number of colors. We will use the following notation. If there is a coloring of graph G where c(V(H)) ∈ A for all subgraphs H isomorphic to F, then we let W+(G,F,A) denote the maximum number of colors in such a coloring and let W(G,F,A) denote the minimum number of colors in such a coloring. (In hypergraphs these are the upper and lower chromatic numbers, respectively.) Note that if G has a valid coloring then so does any subgraph; thus these parameters are monotonic. Now, if A contains the integer 1, then the existence is guaranteed and the minimum number of colors in a valid coloring is 1. Similarly, if A contains the integer \|F\|, then the existence is guaranteed and the maximum number of colors is \|G\|. In particular, if A contains both 1 and \|F\|, all three questions are trivial. This yields two special cases of RASH colorings. Consider the case that A = {1}. Define an auxiliary graph HF(G) with the same vertex set as G but with two vertices adjacent if and only if they lie in a common copy of F. For example, HP3(G) is the square of G, provided G has no component of order 2. Then in a valid coloring of G, two vertices of G must have the same color if and only if they are in the same component of the auxiliary graph HF(G) . It follows that $W+(G,F,{1}) is the number of components of HF(G).$ Consider the case that A = {\|F\|}. Then, in a valid coloring of G, two vertices of G must have different colors if they are adjacent in the auxiliary graph HF(G) ; if they are not adjacent in HF(G) then they can have the same color or different. It follows that $W−(G,F,{\|F\|}) is the chromatic number of HF(G).$ The following result is straight-forward. It shows that if F is connected, then we may restrict our discussion to connected graphs G. (However, if F is not connected, then the situation is more complex.) #### Observation 2.1 Assume F is connected but G is not. Then the existence of a valid coloring of G is equivalent to the existence of such a coloring in all its components Gi. Further, the value W(G,F,A) is the maximum of the values W(Gi,F,A) over all components Gi; and the value W+(G,F,A) is the sum of the values W+(Gi,F,A) over all components Gi. It is not surprising that these parameters are often NP-hard to calculate. For example, W(G,K2,{2}) is just the ordinary chromatic number of G, while several hardness results for WORM and no-rainbow colorings were shown in [58, 11]. ## 3 Stars In this section we focus on the case that F is a star. We begin by observing that all the possibilities for F = K1,2 are trivial or have already been studied. Then we provide a few general results, after which we focus on F = K1,3 and F = K1,4. ## 3.1 The star with two leaves For P3, the star with two leaves, almost all the cases are covered by the above discussion or have been previously studied. The ones where the allowed set A does not contain both 1 and 3 are: • A = {1}. Here the minimum number of colors is 1. The maximum number is 2 for K2, and 1 in all other connected graphs. • A = {2}. This is the WORM coloring number (see for example [8]). • A = {3}. The maximum number of colors is \|G\|. The minimum number of colors is 1 for K2, but for other connected graphs it is the chromatic number of the square of G. • A = {1, 2}. This is the no-Rainbow coloring (see for example [7]). • A = {2, 3}. This is equivalent to 1-defective coloring: every vertex has at most one neighbor of its color (see for example [1, 3]). The situation for other subgraphs F with 3 vertices is similar. For example, the auxiliary graph HK3(G) is G minus all edges not in a triangle. We will look at other small stars shortly, but first some general results. ## 3.2 Arbitrary stars In [8] it was shown that, if a graph has a coloring where every P3 receives two colors, then there is such a coloring that uses only two colors. That is, if it exists, W(G,P3,{2}) ≤ 2. Now, this result does not generalize to most other F, such as K3 (see [11]). But we show next that it does generalize somewhat to other stars. #### Theorem 3.1 If graph G has minimum degree at least f and has a coloring where every copy of K1,f receives 2 colors, then G has such a coloring using only two colors. #### Proof Consider a valid coloring of G. Let EM be the set of edges that are monochromatic; that is, those edges whose two ends have the same color. Note that EM does not contain a copy of K1,f. Let H be the graph GEM. Since H has no monochromatic edge, it is properly colored. Consider two vertices u and v in H with a common neighbor w. Then since H is properly colored, neither u nor v has the same color as w. Since w has degree at least f in G, it follows that u and v have the same color. It follows that every cycle C in H has even length, since it is properly colored and every pair of vertices two apart in C have the same color. This means that H is bipartite. It follows that we can (re)color V(H) = V(G) so that H is properly colored and use only two colors. Now consider this 2-coloring in G. Since the new coloring is a proper coloring in H, every edge that is monochromatic in this new coloring must be in EM. But that means there is no monochromatic copy of K1,f. That is, every copy of K1,f receives exactly two colors. □ It is unclear what happens if one drops the condition that G have minimum degree at least f. For another general result, we consider bounds on the maximum degree of graphs that have a valid coloring. This is equivalent to asking which stars have such colorings. #### Lemma 3.2 For the star F = K1,f and A = {a} with 3 ≤ af, the maximum degree in a graph with a valid coloring is at most $(f−1)(a−1)(a−2).$ and this is realizable for all a and f. #### Proof Since a < f+1, a vertex v can have at most a − 1 new colors among its neighbors. Say v has c neighbors of its own color. Then, the sum of the counts of the most numerous a − 2 other colors can be at most f − 1 − c. Thus, the total number of neighbors is at most f − 1+(f − 1 − c)/(a − 2). This quantity is maximized at c = 0, where it has the above value. And it is achievable by using a − 1 colors on the neighbors divided as equally as possible among them. □ More generally one can say the following: #### Lemma 3.3 For the star F = K1,f, if the allowed set A contains none of 1, 2 and f+1, then the maximum degree in graphs G that have valid colorings is bounded. #### Proof For a crude upper bound, we argue that G cannot have a vertex v of degree (f − 1)f+1 or more. For, then either v has f neighbors of the same color (yielding a copy of K1,f with c ≤ 2) or v has f neighbors of different colors each distinct from v’s color (yielding a rainbow copy of K1,f). □ Note that in contrast, if A contains 1 or f+1 then every graph has a valid coloring. Further, if A contains 2, then there are graphs with arbitrarily large degree that have a valid coloring, such as the complete bipartite graph Km,m. ## 3.3 The star with three leaves We consider next the star K1,3 with three leaves. There are six cases not covered by general results, or by the colorings described earlier. These are the two singletons A = {2} and A = {3}, and the four pairs A = {1,2}, A = {1,3}, A = {2,4}, and A = {3,4}. ## 3.3.1 Colorings where every K1,3 receives 2 colors Consider F = K1,3 and A = {2}. This is equivalent to a coloring where every vertex of degree at least 3 sees at most one color other than itself and has at most two neighbors of its color. We consider first families of planar graphs. Several authors (e.g. [1]) showed that one can partition the vertex set of an outerplanar graph into two forests of maximum degree 2. In particular, it follows that W(G,K1,3,{2}) exists and is at most 2. But for maximal (outer)planar graphs, one can go a bit further. #### Theorem 3.4 1. If G is a maximal outerplanar graph of order at least 4, then it has a valid coloring and W(G,K1,3,{2}) = W+(G,K1,3,{2}) = 2. 2. If G is a maximal planar graph of order at least 4 and G has a valid coloring, then W(G,K1,3,{2}) = W+(G,K1,3,{2}) = 2. #### Proof 1. We saw above that such a graph has a valid coloring. Since G has maximum degree at least 3, one cannot use only one color. So it remains to show that one cannot use more than two colors. If G has order 4 then this is easily checked; so assume the order is at least 5. We know that G has minimum degree 2. Consider a vertex u of degree 2 with neighbors v and w, necessarily adjacent. Then at least one of these vertices has degree at least 4, say v. Then, by induction, every valid coloring of Gu uses exactly two colors. Since vertex v has degree at least 3 in Gu, it must have a neighbor y (possibly w) of different color in Gu. It follows that u must have the color of v or y. That is, G has only two colors. 2. If G is hamiltonian, then it contains a maximal outerplanar graph as a spanning subgraph, and so the result follows from part (a). So assume it is not hamiltonian. Then it has connectivity at most 3 and so there is a cut-triangle T. Let G1 be the graph formed from G by removing the vertices inside T; let G2 be the graph formed by removing the vertices outside T. By induction, the valid coloring of G when restricted to G1 uses only two colors, and similarly when restricted to G2. Let v be a vertex of the triangle. Then, since G1 and G2 both have minimum degree at least 3, the vertex v must see both colors in G1 and both colors in G2. Since v can see at most two colors in total, it follows that G1 and G2 use the same pair of colors. That is, G has only two colors. □ For another graph family, consider cubic graphs. Such graphs always have a valid coloring since they have a coloring with two colors where every vertex has at most one neighbor of its color (a1-defective 2-coloring [12]). It might be interesting to determine the maximum number of colors in such a coloring: #### Problem 3.5 What is the maximum possible number of colors in a coloring of a connected cubic graph of order n where every K1,3 receives exactly 2 colors? Note that for regular graphs, the case of F = K1,f and A = {f − 1} also corresponds to what we called a near-injective coloring; see [13]. ## 3.3.2 Colorings where every K1,3 receives 3 colors Consider F = K1,3 and A = {3}. By Lemma 3.2, the maximum degree of a graph G with a valid coloring is at most 4. So one natural family to consider is the set of 4-regular graphs. Note that a valid coloring must be proper, and every vertex must have two neighbors of one color and two neighbors of another color. It follows that if only three colors are used in total, that each color must be used an equal number of times, and in particular, that the order of G is a multiple of 3. However, there are other orders for which such a coloring exists. For example, consider the direct product of a cycle Cn with 2K1; that is, duplicate each vertex of the cycle so that one ends up with a 4-regular graph on 2n vertices. Then a valid coloring is achieved by using n different colors, giving every pair of similar vertices the same color. Another family of interest is the set of cubic graphs. Computer search shows that all small cubic graphs have such a coloring. What about in general? We conjecture yes. #### Conjecture 3.6 Every cubic graph has a coloring where every K1,3 receives 3 colors. If the cubic graph G has a matching M none of whose edges are in a triangle, then one has a valid coloring by assigning a different color to each edge in M (and so every vertex has a neighbor of its color but no other repetition). Indeed, computer search suggests that the maximum colors in a valid coloring is always at least n/2. If true, this would be a strengthening of the conjecture from [7] that every cubic graph has a coloring with at least n/2 colors without a rainbow copy of K1,3. In contrast, there are cubic graphs that require more than 3 colors, since each color class in such a coloring would be a dominating set, and there are infinitely many cubic graphs of order n with domination number more than n/3 (see e.g. [14]). ## 3.3.3 Other restrictions on K1,3 In each of the remaining cases (where A is a doubleton), the coloring is guaranteed to exist since A contains either 1 or 4. But note that these situations are different to the associated singletons. For example, let F = K1,3. Let G be a graph that has a valid coloring for A = {3} and H be a graph that does not. Then their disjoint union has W+(GH,K1,3,{1,3}) ≥ 4. ## 3.4 The star with four leaves Finally in this section we briefly consider the star K1,4 with the allowed set A a singleton. Consider the case that A = {4}. By Lemma 3.2 above, the maximum degree is at most 4. Clearly, vertices with degree smaller than 4 pose no constraints as centers of stars. So a natural class to consider is 4-regular graphs. Let K2,2,2 denote the 4-regular graph of order 6. Computer search shows that all 4-regu1ar graphs through order 12 have such a coloring, except for K2,2,2. This raises the question: #### Problem 3.7 Does every connected 4-regular graph, apartfrom K2,2,2, have a coloring where every K1,4 receives exactly 4 colors? Consider the case that A = {3}. By Lemma 3.2 above, the maximum degree can be as high as 6. It is not true that every 6-regular graph has a valid coloring; for example through order 11 only the complete multipartite graph K3,3,3 has one. Nor is it true that every 5-regular graph has a valid coloring; for example only 3459 of the 7848 5-regular graphs of order 12 have one. But computer search shows that all 4-regu1ar graphs through order 12 have a valid coloring. This raises the question: #### Problem 3.8 Does every 4-regular graph have a coloring where every K1,4 receives exactly 3 colors? ## 4 Stripes In this section we consider colorings where F is 1-regular. We begin by studying the case where F = 2K2. ## 4.1 Colorings where every 2K2 receives 2 colors Let 𝓜 denote the set of graphs that do not contain two disjoint edges. That is, 𝓜 is the set of all stars and K3 together with isolates. #### Theorem 4.1 A graph G has a coloring where every 2K2 receives 2 colors if and only if 1. V(G) has a bipartition (R, S) such that R and S both induce graphs of 𝓜, or 2. G is the disjoint union of stars and K3s. #### Proof First observe that such a graph has the requisite coloring. In case (i), color every vertex in R red and every vertex in S sapphire. Then since there is no monochromatic 2K2, every copy of 2K2 receives both colors. In case (ii), color every component monochromatically with a different color. Suppose now that the graph G has a valid coloring. If every edge is monochromatic, then the graph is the disjoint union of stars and K3’s, with each component monochromatic, and we are in case (ii) of the characterization. So, assume there is a proper edge, say rs with r red and s sapphire. Suppose there is another color present; say vertex t is taupe. Then all edges out of t must go to {r, s}. If t has degree 2, then {r, s, t} is an isolated triangle; and indeed no other edge possible. So assume that t has degree 1; say with neighbor s. By similar argument, vertex r has degree 1 too. Indeed, any vertex x that is not sapphire must have degree 1 with s as its only neighbor. Then, if we change all non-sapphire vertices to be red, we will still have a valid coloring. That is, we may assume that the coloring uses only two colors. It follows that we are in case (i) of the characterization. □ It is straight-forward to argue that one can recognize such graphs in polynomial-time and calculate the minimum and maximum number of colors used. For a crude algorithm, simply consider all possible stars and triangles and check whether what remains after their edges are removed is bipartite. If we are in case (ii) with two or more components, then the number of colors used equals the number of components. If we are in case (i), then the minimum number of colors used is at most 2, as we argued in the above proof; we can only use more than two colors if there is a vertex s with multiple leaf neighbors and the rest of the graph has a suitable structure. We omit the details. ## 4.2 Colorings where every 2K2 receives 3 colors We consider next the case that every 2K2 has 3 colors. Like the above result, the characterization is that the graph G must be “nearly bipartite”. Define the following graphs and graph classes. Let 𝓗1 be the set of graphs that contain two adjacent vertices x and y such that every other edge is incident to x or y. Let 𝓗2 be the graphs that contain a triangle x, y, z such that every other vertex is a leaf with a neighbor in {x, y, z}. Let F1 be the graph that is obtained from the disjoint union of K4 and K2 by identifying a vertex of each. Let F2 be the graph that is obtained from K4e by picking a vertex x of degree 3 and a vertex y of degree 2 and adding a leaf adjacent only to x and one only to y. Let F3 be the graph that is obtained from the disjoint union of K4e and P3 by identifying a minimum-degree vertex of each. Let F4 be the graph that is obtained from P6 : v1, v2, …, v6 by adding edge v3v5. We draw these in Figure 1. Fig. 1 Coloring graphs so that every 2K2 receives 3 colors For a graph G, we define the reduced version of G by considering every vertex in turn and, if it has more than one leaf neighbor, discarding all but one of these neighbors. It is immediate that a graph has a valid coloring if and only if its reduced version has, because we may assume all leaves at a vertex are the same color. #### Theorem 4.2 A graph G has a coloring where every 2K2 receives 3 colors if and only if 1. G is bipartite; 2. G is formed from the disjoint union of a bipartite graph H and K3 by identifying one vertex of each; 3. G is the disjoint union of graphs H and M where H ∈ 𝓗1 ∪ 𝓗2 and M ∈ 𝓜 (the family of graphs with matching number 1); or 4. the reduced version of G is F1, F2, F3, or F4, or a subgraph thereof. #### Proof These graphs each have a valid coloring. In case (i), color all vertices in one partite set red and give all remaining vertices unique colors. In case (ii), let x be the vertex that was identified. Color it and all other vertices in its partite set of H red, color its two neighbors in the K3 sapphire, and then give all remaining vertices unique colors. In case (iii), color the subgraph H as shown in the above picture and color the subgraph M monochromatically with color 4. In case (iv), color the graphs with three colors as shown in the above picture. We turn to the proof that these are all the graphs. So assume G has a valid coloring. #### Claim 4.3 Graph G cannot have an odd cycle of length 5 or more. #### Proof We claim first that there cannot be two consecutive vertices of the same color. Consider a portion of the cycle abcdef (where possibly a = f) where c and d have the same color, say 1. Then, by considering the pair bc, de, without loss of generality b has color 2 and e has color 3. By considering the pair cd, ef, the vertex f must have a color other than 1 and 3. By considering pair bc, ef, it follows that vertex f has color 2. By similar argument, a has color 3 (and in particular af). But then pair ab, ef is a contradiction. Now suppose there are vertices at distance three of the same color. Consider a portion of the cycle abcdef (where possibly a = f) where b and e have same color, say 1. Then by considering bc, de, without loss of generality c has color 2 and d has color 3. By considering pair cd, ef, vertex f must have a color from {1, 2, 3}. By the lack of consecutives of the same color, f cannot be color 1; by considering pair bc, ef, vertex f cannot be color 2. Therefore f has color 3. Similarly, vertex a has color 2. In particular af, and so there is another vertex next to f, say g. Then by considering pair bc, fg, vertex g must have a color from {1, 2, 3}. But it can easily be checked that each choice leads to a contradiction. So we have shown that there cannot be two consecutive vertices of the same color nor two vertices at distance three. Now consider again a portion of the cycle abcdef (where possibly a = f). There must be a duplicate color at distance two within abcd. Say a and c have color 1, with b of color 2 and d of color 3. Then consider the pair bc, de. Since vertices b and d have different colors, it must be that vertices c and e have the same color. Then f must have a color different from c, d and e. Further, it cannot have the same color as b, by the pair ab, ef. By repeated application of this, it follows that every alternate vertex on the cycle has color 1. In particular the cycle has even length. □ If the graph G is bipartite, we are done. So assume there is a triangle T. #### Claim 4.4 If T is properly colored, then we are in case (iii) or have reduced graph F1 or F2. #### Proof Say triangle T has vertex a of color 1, vertex b of color 2, and vertex c of color 3. Suppose there is another color present, say vertex d has 4. Then consider an edge incident with d. If it goes to triangle T, we have a contradiction. So say it is de. By considering the pairs formed by de and each edge of T, it follows that e cannot have a new color, nor can it have color from {1, 2, 3}. Thus e has color 4. That is, the edges incident with the vertices not colored from {1, 2, 3} are monochromatically colored; and indeed must induce a component from 𝓜 colored 4. So consider the vertices with color from {1, 2, 3}. We cannot have an edge disjoint from the triangle, since if it is monochromatic one can pair it with an edge of T that includes that color, and if proper one can pair it with the edge of T that is identically colored. So all such edges have (at least) one end in the triangle. If the component containing T is in 𝓗1, then we are done. So assume the component containing T is not in 𝓗1. That is, every vertex of T has a neighbor outside the triangle. Consider the possibilities. One possibility is that the three vertices of T have a common neighbor v. By the above, v must have one of their colors, say 1. If not just K4, there is another vertex, say w. This vertex cannot be incident with either a or v, so say incident with vertex c. Then w must be color 2 and must be a leaf. One can check that all remaining edges must be similarly incident with c. Further, the edge va being monochromatic implies that there is no monochromatic edge of color 4. That is, the reduced version of G is F1 or K4. A second possibility is that a, b have a common neighbor v while c has neighbor w. Suppose that w has color 3. Then considering the pair va, cw it follows that v must be color 2, but then the pair vb, cw is not validly colored. Thus vertex w has color 1, say. By considering the pair va, cw, it follows that the vertex v has color 2. Any other neighbor x of a must have color 2 because of ax, cw but not color 2 because of vb, ax, so a has no other neighbor. Any other neighbor y of b must have color 3 because of the pair av, by. And, any other neighbor z of c must have color 1 because of the pair bv, cz. Further, the edge vb being monochromatic means that there is no monochromatic edge of color 4. That is, the reduced version of G is F2 or F2 with one/both of the leaves deleted. A third possibility is that a, b, and c have only distinct neighbors outside the triangle. That is, the component containing T is in 𝓗2. □ So we can assume that there is no properly colored triangle T. #### Claim 4.5 If triangle T contains two colors, then we are in case (ii) or have reduced graph F3 or F4. #### Proof Say T has two vertices of color 1, say a and b, and the remaining vertex c has color 2. By the conditions and the lack of properly colored triangles, there is no other common neighbor of a and c, nor of b and c. Assume first that there is no edge disjoint from T. Then the component containing T is in 𝓗2 and we are in case (iii) (with M being null) unless a and b have another common neighbor, say d. The vertex d must have a different color, say 3. Then the component containing T is in 𝓗1 (albeit with an alternative coloring) unless c has another neighbor, say e. The vertex e must have color 3. Then neither a nor b can have any other neighbors, and we are in case F2 minus the leaf incident with the vertex of degree 4. So assume second there is an edge ds disjoint from T. Since one can pair it with ab, edge ds must be properly colored and neither end is color 1. Since one can pair it with ac, one end must have color 2. Further, no isolated vertex w of GT can have color 1 or 2, since V(T) ∪ {w} contains a 2K2. In particular, GT is bipartite with bipartition (D,S), where d and every other vertex in D has color 2, while s and every other vertex in S has color distinct from {1, 2}. Note that none of {a, b, c} can have a neighbor in D. Therefore, if neither a nor b has a neighbor in S, then we are in case (ii). So assume one of them, say a, has a neighbor e in S, say of color 3. Note that possibly e = s. Then consider any vertex v other than {a, b, c, d, e}. Suppose it has a color not in {1, 2, 3}. By potential pairing with ae or ac, it follows that all neighbors of v have color 1 (so in particular vs) and then there is a contradiction from pairing with ds. That is, all vertices other than a, b have colors from {2, 3}. It follows that the nontrivial component in G − {a, b} is a star. Now, note that if both b and c have degree 2, we are in case (ii) of the theorem. So either b is adjacent to d, or c is adjacent to s. In the first case we get the reduced graph F3 or possibly with edge removed so that it is just K4eK2 (in which case G is covered by case (iii)). In the second case we get F4. And one can check that all remaining vertices are clones of existing leaves. □ Finally suppose that all triangles are monochromatic. Then there can be only one. Indeed it must be an isolated component, while the remainder of the graph is bipartite, so that we are in case (iii). This completes the proof of Theorem. □ ## 4.3 General stars Consider colorings where every copy of fK2 has a colors. We start with the case that a = 2. Theorem 4.1 showed that if G is connected and a valid coloring exists, then W(G,2K2,{2}) ≤ 2. (We do need the connectivity, since the disjoint union of copies of K3 has a valid coloring but each triangle must have a different color.) The analogue of Theorem 4.1 for more edges turns out to be simpler. #### Theorem 4.6 Let f > 2. A graph G has a coloring where every fK2 receives two colors if and only if G has a bipartition (R, S) such that both R and S induce subgraphs with matching number less than f. #### Proof If G has such a bipartition then that coloring is a valid coloring. So, consider a graph G that has a valid coloring. In particular, consider a valid coloring of G that uses the fewest total number of colors, and suppose that total is more than two. Then G has vertices of three colors, say colors 1, 2, and 3. Consider recoloring every vertex of color 2 with color 1. This cannot increase the number of colors in any copy of fK2, but by minimality this coloring is invalid. That means that in G there must be a copy F12 of fK2 with colors 1 and 2. Consider a vertex v of color 3 with neighbor w say. If w is disjoint from F12, then we can take vw, an edge of F12 containing color 1 and an edge of F12 containing color 2 and f − 3 more edges of F12, and so obtain a bad fK2. So w must be in F12. In particular, there is no edge where both ends are color 3. By similar logic, there is no monochromatic edge of any color. But that means every edge of F12 is properly colored. And so vw and f − 1 disjoint edges of F12 produces a bad fK2, a contradiction. Thus, G has a valid coloring using only two colors. □ We end with some comments about the general case. #### Theorem 4.7 Consider colorings where every fK2 has a colors. One can color mK2 for all m if and only if a ∈ {1, 2, f, f + 1, 2f}. #### Proof The colorings are as follows. For a = 1 give all vertices the same color; for a = 2 properly color each edge with red and sapphire; for a = f color all edges monochromatically but with different colors; for a = f+1 color all edges properly with one end red and the other end a unique color; and for a = 2f give all vertices different colors. Now consider a coloring of mK2 for m large. One can obtain arbitrarily large collection of edges such that are either all edges are monochromatic or all are properly colored. In the former case we can further find large collection of edges that either are all the same color or are all different colors. Thus for the coloring to be valid we need a ∈ {1, f}. So assume all the edges in the collection are properly colored. Again we can find a large collection such that either all have the same pattern or all have different patterns. In the former case it follows that a = 2. So assume the latter case. We can then find a large collection where every edge ei has (at least one) end of color i. For the other end, we can again assume that they are all the same color or all different colors. In the former case we need a = f+1. In the latter case we can again prune so that each color appears on only one edge, and so a = 2f. □ We saw in the proof of Theorem 4.2 that the 5-cycle does not have a coloring where every 2K2 receives three colors. This can be generalized. #### Lemma 4.8 Consider colorings where every fK2 has a colors. One can color all odd cycles if and only if a ∈ {1, 2, 2f}. One can color all even cycles if and only if a ∈{1, 2, f+1, 2f}. #### Proof The result is clear for a = 1 and a = 2f (color all vertices the same or all different). To do a = 2 (assuming f > 1), color every alternate vertices with red and sapphire except that possibly one pair of adjacent vertices receive the same color. To do a = f (assuming f > 2) in a large cycle, every edge must have a different monochromatic coloring, which is impossible. To do a = f+1 (assuming f > 1) in a large cycle, every edge must have the same color pattern but be properly colored. So the cycle length must be even. □ For example, for F = 3K2 the above lemma shows that the cycle length is bounded for colorings where a∈{3, 5}. If every 3K2 receives three colors, then the longest cycle colorable is the 10-cycle: color red a vertex v and the two vertices at distance two from it, color sapphire the vertex v′ opposite v and the two vertices at distance two from it, and color the remaining vertices taupe. If every 3K2 receives five colors, then the only cycle of length more than 6 that can be colored is the 8-cycle: color red a vertex u and the vertex u′ opposite u, color sapphire the two vertices adjacent to neither u nor u′, and give the remaining vertices unique colors. It can also be shown that the cycle length is bounded even if A = {3, 5}. ## 5 Conclusion We proposed RASH colorings, where every copy of a specified graph F has a restriction on the number of colors it receives. We focused on the case that F is a star or is 1-regular. It would be interesting to explore RASH colorings where the restricted subgraph is a path or cycle. Another direction is suggested by the case of F = K1,r and A = {r} in r-regular graphs. These are colorings where every closed neighborhood has precisely one repeated color; see [13]. ## References • [1] Mihók P., On vertex partition numbers of graphs, In Graphs and other combinatorial topics (Prague, 1982), volume 59 of Teubner-Texte Math., pages 183–188. 1983 Google Scholar • [2] Borowiecki M., Broere I., Frick M., Mihók P., Semanišin G., A survey of hereditary properties of graphs, Discuss. Math. Graph Theory, 17 (1997), 5–50 • [3] Cowen L., Goddard W., Jesurum C., Defective coloring revisited, J. Graph Theory, 24 (1997), 205–219 • [4] Alon N., Ding G., Oporowski B., Vertigan D., Partitioning into graphs with only small components, J. Combin. Theory Ser. B, 87 (2003), 231–243 • [5] Bujtás C., Sampathkumar E., Tuza Z., Subramanya M., Dominic C., 3-consecutive C-colorings of graphs, Discuss. Math. Graph Theory, 30 (2010), 393–405 • [6] Bujtás C., Sampathkumar E., Tuza Z., Dominic C., Pushpalatha L., Vertex coloring without large polychromatic stars, Discrete Math., 312 (2012), 2102–2108 • [7] Goddard W., Xu H., Vertex colorings without rainbow subgraphs, Discuss. Math. Graph Theory, 36 (2016), 989–1005 • [8] Goddard W., Wash K., Xu H., Worm colorings, Discuss. Math. Graph Theory, 35 (2015), 571–584 • [9] Voloshin V., On the upper chromatic number of a hypergraph, Australasian J. Comb., 11 (1995), 25–45 Google Scholar • [10] Tuza Z., Voloshin V., Problems and results on colorings of mixed hypergraphs, In Horizons of combinatorics, volume 17 of Bolyai Soc. Math. Stud., pages 235–255. 2008 Google Scholar • [11] Bujtás C., Tuza Z., K3-worm colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum, Discuss. Math. Graph Theory, 36 (2016), 759–772 • [12] Lovász L., On decomposition of graphs, Studia Sci. Math. Hungar., 1 (1966), 237–238 Google Scholar • [13] Goddard W., Melville R., Xu H., Almost injective colorings, To appear in Discuss. Math. Graph Theory Google Scholar • [14] Kostochka A., Stodolsky B., On domination in connected cubic graphs, Discrete Math., 304 (2005), 45–50 ## About the article Accepted: 2017-08-17 Published Online: 2017-09-22 Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 1171–1180, ISSN (Online) 2391-5455, Export Citation	2019-12-12 05:45:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7217105627059937, "perplexity": 546.5661110105998}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540537212.96/warc/CC-MAIN-20191212051311-20191212075311-00381.warc.gz"}
https://www.esaral.com/q/solve-this-following-25422/	Solve this following Question: Compound $\mathrm{A}\left(\mathrm{C}_{9} \mathrm{H}_{10} \mathrm{O}\right)$ shows positive iodoform test. Oxidation of $\mathrm{A}$ with $\mathrm{KMnO}_{4} / \mathrm{KOH}$ gives acid $\mathrm{B}\left(\mathrm{C}_{8} \mathrm{H}_{6} \mathrm{O}_{4}\right)$. Anhydride of $\mathrm{B}$ is used for the preparation of phenolphthalein. Compound $\mathrm{A}$ is :- Correct Option: 1 Solution:	2022-06-29 00:27:37	{"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": 0.8681351542472839, "perplexity": 7043.388731214367}, "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/1656103619185.32/warc/CC-MAIN-20220628233925-20220629023925-00530.warc.gz"}
https://optimization-online.org/tag/constrained-weighted-ell_r-ell_1-minimization/	## A new sufficient condition for non-convex sparse recovery via weighted $\ell_r\!-\!\ell_1$ minimization In this letter, we discuss the reconstruction of sparse signals from undersampled data, which belongs to the core content of compressed sensing. A new sufficient condition in terms of the restricted isometry constant (RIC) and restricted orthogonality constant (ROC) is first established for the performance guarantee of recently proposed non-convex weighted $\ell_r-\ell_1$ minimization in recovering … Read more	2023-03-27 14:24:45	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.835136353969574, "perplexity": 427.74037469756263}, "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/1679296948632.20/warc/CC-MAIN-20230327123514-20230327153514-00290.warc.gz"}
http://math.stackexchange.com/questions/232436/how-to-solve-lim-x-rightarrow-infty-sqrtx-ax-b-x	# How to solve $\lim_{x\rightarrow +\infty} \sqrt{(x-a)(x-b)}-x$? How do I solve? I've tried to multiply and divide by the conjugate cannot advance. $$\lim_{x\rightarrow +\infty} \sqrt{(x-a)(x-b)}-x$$ - Multiplying and dividing by the conjugate works fine. Let $x$ be positive and larger than $a$ and $b$. We quickly obtain $$\frac{-ax-bx+ab}{\sqrt{(x-a)(x-b)}+x}.$$ Divide top and bottom by $x$. (That is another commonly useful kind of move.) We get $$\frac{-a-b+\frac{ab}{x}}{\sqrt{\left(1-\frac{a}{x}\right)\left(1-\frac{b}{x}\right)}+1}.$$ Now finding the limit is straightforward. I was answering the OP, who knew about the conjugate. Actually, it is not quite the conjugate, but I went along with it. Multiply top and bottom (which is $1$) by $\sqrt{(x-a)(x-b)}+x$. On top we get $(x-a)(x-b)-x^2$, which simplifies to $-ax-bx+ab$. Multiplying by a conjugate is a widely useful trick. When a problem involves $a+b\sqrt{d}$, it is often useful to get $a-b\sqrt{d}$ involved. You may have seen this with complex numbers. There, when you see $a+bi$, the number $a-bi$ is often helpful. – André Nicolas Nov 9 '12 at 6:54	2013-12-20 02:27:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9534128308296204, "perplexity": 185.90014533828744}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345768787/warc/CC-MAIN-20131218054928-00071-ip-10-33-133-15.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/290209/diff-eq-transformation-polar-coordinates/290220	# Diff eq. transformation polar coordinates I have $(x',y')=(x-y-x(x^2+y^2)+\frac{xy}{\sqrt{x^2+y^2}},x+y-y(x^2+y^2)-\frac{x^2}{\sqrt{x^2+y^2}} )$ Now I want to use polar coordinates $(x,y)=(r\cos(t),r\sin(t))$ to get $(r',t')=(r(1-r^2),2\sin(\frac{t}{2})^2)$ I do not see this relation. When I put $x=\cos t$, $y=\sin t$ into the system of differential equations, I only get $(r\cos(t)-r\sin(t)-r^3\cos(t)+r\cos(t)\sin(t),r\cos(t)+r\sin(t)-r^3\cos(t)-r\cos(t)^2)$. - Two things. Your notation is obscure, as $t$ is function of the variable of integration, i.e. $\frac{d t}{d \xi} \equiv {t}'$, hence you are using the chain rule wrong; also, the cubic term on the second component of your substitution should read $-r^3 \sin t$. See my answer for details. – Pragabhava Jan 30 '13 at 1:25 Using $x=r \cos{t}$, $y=r \sin{t}$: $$x'=(\cos{t}) r' - r (\sin{t}) \, t'$$ $$y'=(\sin{t}) r' + r (\cos{t}) \, t'$$ So we get $$\left ( \begin{array}\\ \cos{t} & -r \sin{t} \\ \sin{t} & r \cos{t} \end{array} \right ) \left ( \begin{array}\\ r' \\ t' \end{array} \right ) = \left ( \begin{array}\\ r \cos{t} - r \sin{t} - r^3 \cos{t} + \sin{t} \cos{t} \\ r \cos{t} + r \sin{t} - r^3 \sin{t} - \cos^2{t} \end{array} \right )$$ Multiply both sides by the matrix inverse to get $$\left ( \begin{array}\\ r' \\ t' \end{array} \right ) = \frac{1}{r} \left ( \begin{array}\\ r\cos{t} & r\sin{t} \\ - \sin{t} & \cos{t} \end{array} \right ) \left ( \begin{array}\\ r \cos{t} - r \sin{t} - r^3 \cos{t} + \sin{t} \cos{t} \\ r \cos{t} + r \sin{t} - r^3 \sin{t} - \cos^2{t} \end{array} \right )$$ Just do out the multiplication. It is messy, but there is a lot of cancellation and we get $$\left ( \begin{array}\\ r' \\ t' \end{array} \right ) = \left ( \begin{array}\\ r-r^3 \\ 1 - \cos{t} \end{array} \right )$$ - Your notation is obscure and is getting in the way of the problem. What you have is \begin{align} x\color{red}{(t)} &= r\color{red}{(t)} \cos \theta \color{red}{(t)}\\ y\color{red}{(t)} &= r\color{red}{(t)} \sin \theta \color{red}{(t)} \end{align} so \begin{align} x'(t) &= r' \cos \theta - r \theta' \sin \theta \\ y'(t) &= r' \sin \theta + r \theta' \cos \theta \\ \end{align} In vectorial form $$\vec{X}(t) = r(t) \hat{r}(t), \mbox{ where } \hat{r} = \pmatrix{\cos \theta\\ \sin \theta}$$ and $$\vec{X}'(t) = r'\hat{r} + r \theta' \hat{\theta}$$ Then $$r'\hat{r} + r \theta' \hat{\theta} = r \hat{r} + r \hat{\theta} - r^3\hat{r} - r \cos\theta \,\hat{\theta}$$ Due orthogonality \begin{align} r' &= r(1-r^2) \\ \theta' &= 1 - \cos \theta = 2 \sin^2 \left(\frac{\theta}{2}\right) \end{align} -	2014-12-22 00:23:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 3, "equation": 3, "x-ck12": 0, "texerror": 0, "math_score": 1.0000100135803223, "perplexity": 772.7027127937594}, "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/1418802772757.23/warc/CC-MAIN-20141217075252-00016-ip-10-231-17-201.ec2.internal.warc.gz"}
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=141&t=28159	## 14.91 Electrolyte $\Delta G^{\circ} = -nFE_{cell}^{\circ}$ Emily Glaser 1F Posts: 156 Joined: Thu Jul 27, 2017 3:01 am ### 14.91 Electrolyte What does it mean conceptually when it says "A negatively charged electrolyte flows from the cathode to the anode," which completes the circuit. Is this the salt bridge? Chew 2H Posts: 34 Joined: Fri Sep 29, 2017 7:05 am ### Re: 14.91 Electrolyte i think this is referring to the external wire circuit. From what i know, salt bridges don't exist in Electrolytic cells. Nancy Dinh 2J Posts: 59 Joined: Fri Sep 29, 2017 7:07 am ### Re: 14.91 Electrolyte Chew 2H wrote:i think this is referring to the external wire circuit. From what i know, salt bridges don't exist in Electrolytic cells. It mentioned the external wire circuit in the problem so I don't think it's that. The problem wants to know how the "current is carried through the cell itself." Emily Glaser 1F wrote:What does it mean conceptually when it says "A negatively charged electrolyte flows from the cathode to the anode," which completes the circuit. Is this the salt bridge? The solution manual may be referring to a porous disk. On page 570, the book says, "To prevent this charge buildup [in the solution], which would quickly stop the flow of electrons, the two solutions are in contact through a porous wall; ions provided by the electrolyte solutions move between the two compartments and completes he electrical circuit." Since a salt bridge and a porous disk function in the same way, you could use the salt bridge as an answer.	2020-07-05 20:22:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.41794049739837646, "perplexity": 2080.5216246531268}, "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/1593655888561.21/warc/CC-MAIN-20200705184325-20200705214325-00542.warc.gz"}
https://forum.math.toronto.edu/index.php?PHPSESSID=1u0iucfle60t5k23oom8iace62&topic=385.0	### Author Topic: Writing the quiz with another section (Read 1496 times) #### Victor Ivrii • Administrator • Elder Member • Posts: 2563 • Karma: 0 ##### Writing the quiz with another section « on: September 15, 2014, 10:57:17 AM » Students of one evening section are allowed to write Quiz with another evening section but not with a day section. Students of the day section are allowed to write a Quiz with one of the evening sections.	2021-09-28 01:42:50	{"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": 0.9002169370651245, "perplexity": 13565.537339092305}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780058589.72/warc/CC-MAIN-20210928002254-20210928032254-00582.warc.gz"}
https://www.gradesaver.com/textbooks/math/algebra/elementary-linear-algebra-7th-edition/chapter-3-determinants-3-4-applications-of-determinants-3-4-exercises-page-136/4	Elementary Linear Algebra 7th Edition Published by Cengage Learning Chapter 3 - Determinants - 3.4 Applications of Determinants - 3.4 Exercises - Page 136: 4 Answer $$\operatorname{adj}(A)=\left[ \begin {array}{ccc} 4&2&-5\\ -2&-4&1 \\ -2&2&1\end {array} \right] .$$ $$A^{-1}= -\frac{1}{6}\left[ \begin {array}{ccc} 4&2&-5\\ -2&-4&1 \\ -2&2&1\end {array} \right].$$ Work Step by Step The matrix is given by $A=\left[ \begin {array}{ccc} 1&2&3\\ 0&1&-1 \\ 2&2&2\end {array} \right] .$ To find $\operatorname{adj}(A)$, we calculate first the cofactor matrix of $A$ as follows $$\left[ \begin {array}{ccc} 4&-2&-2\\ 2&-4&2 \\ -5&1&1\end {array} \right] .$$ Now, the adjoint of $A$ is $$\operatorname{adj}(A)=\left[ \begin {array}{ccc} 4&2&-5\\ -2&-4&1 \\ -2&2&1\end {array} \right] .$$ To find $A^{-1}$, we have to calculate $\det(A)$ which is given by $$\det(A)=-6.$$ Finally, we have $$A^{-1}=\frac{1}{\det(A)}\operatorname{adj}(A)=-\frac{1}{6}\left[ \begin {array}{ccc} 4&2&-5\\ -2&-4&1 \\ -2&2&1\end {array} \right].$$ After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	2019-12-09 05:08:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8634157180786133, "perplexity": 506.9897239726492}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540517557.43/warc/CC-MAIN-20191209041847-20191209065847-00015.warc.gz"}
http://havenwoodbaptist.org/poacvdkd/9fsuld.php?ada95d=examples-of-nuclear-fusion	# examples of nuclear fusion Nuclear fusion is the Holy Grail of energy production. Nuclear fusion is the breakdown of a heavy nucleus into two lighter nuclei due to bombardment of neutron. Nuclear fission power plants generate unstable nuclei; some of these are radioactive for millions of years. In addition, the wastes will not be of weapons-grade nuclear materials as is the case in fission reactors. The specific type of fusion that occurs inside of the Sun is known as proton-proton fusion. \nonumber\], The Sun’s mass decreases by $$0.0276 \, u = 4.58 \times 10^{-29}kg$$ per fusion reaction, so the rate at which its mass decreases is (9.26 \times 10^{37} reaction/s)(4.58 \times 10^{-29} kg/reaction) = 4.24 \times 10^9 kg/s. Tritium is radioactive (a beta emitter) but its half-life is short. The main difference between these two processes is that fission is the splitting of an atom into two or more smaller ones while fusion is the fusing of two or more smaller atoms into a larger one. A fission reaction at a nuclear power plant provides enough energy to give electricity to large cities. Eventually, much of the material lost by stars is pulled together through the gravitational force, and it condenses into a new generation of stars and accompanying planets. Both nuclear fusion and fission produce a massive amount of energy. The release of this energy produces an outward thermal gas pressure that prevents the Sun from gravitational collapse. 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Nuclear Fusion Example. Have questions or comments? Scientist now believe that many heavy elements found on Earth and throughout the universe were originally synthesized by fusion within the hot cores of the stars. During this event, the flood of energetic neutrons reacts with iron and the other nuclei to produce elements heavier than iron. Most of the energy radiated from the surface of the sun is produced by the fusion of protons to form helium atoms within its core. Hydrogen burning does very little to change the mass of the Sun. Two light atoms combined have a lot more energy than one heavier atom, so when the two fuse together the excess energy is released. Expanding shock waves generated within the star due to the collapse cause the star to quickly explode. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Requires a lot of heat and pressure for the process to happen. Nuclear fusion is when two or more lightweight atoms join together to form one heavier nucleus, with any energy released due to the conversion being converted into nuclear energy. Oct 30, 2020Jul 27, 2015 by Editor in Chief Nuclear fusion is considered the most basic form of energy used today. Supernovae and the formation of planetary nebulas together play a major role in the dispersal of chemical elements into space. Astrophysicists find that hydrogen fusion supplies the energy stars require to maintain energy balance over most of a star's life span. \end{align}, Thus, a stable helium nucleus is formed from the fusion of the nuclei of the hydrogen atom. By the end of this section, you will be able to: The process of combining lighter nuclei to make heavier nuclei is called nuclear fusion. Comparing the binding energy per nucleon for oxygen, carbon, and helium, the oxygen nucleus is much more tightly bound than the carbon and helium nuclei, indicating that the reaction produces a drop in the energy of the system. 3. This energy is released in the form of gamma radiation. A great deal of work still has to be done on fusion reactor technology, but many scientists predict that fusion energy will power the world’s cities by the end of the twentieth century. It is the reaction in which 2 atoms of hydrogen combine, or fuse together, to form an atom of helium. Legal. Where does the energy from the Sun originate? nuclear fusion synonyms, nuclear fusion pronunciation, nuclear fusion translation, English dictionary definition of nuclear fusion. Now, iron has the peculiar property that any fusion or fission reaction involving the iron nucleus is endothermic, meaning that energy is absorbed rather than produced. Brings two or more small atoms together to form one large atom. Nuclear Fusion: Nuclear Fusion is a reaction that occurs when two atoms combine together to form one or more different atomic nuclei and subatomic particles like protons and neutrons. It is only used in low amounts so; unlike long-lived radioactive nuclei, it cannot produce any serious danger. One way to explain this phenomenon is to assume that hydrogen nuclei in the core of stars fuse with each other to form the nuclei of helium atoms. Your email address will not be published. This would change as star formation began and produced more elements through the process of nuclear fusion. In 1938, Hans Bethe proposed that the Sun produces energy when hydrogen nuclei ($$\ce{^1H}$$) fuse into stable helium nuclei ($$\ce{^{4}He}$$) in the Sun’s core (Figure $$\PageIndex{1}$$). The basic differences between Nuclear Fission and Nuclear Fusion are: We are still at an experimental stage as far as nuclear fusion reactions are concerned. The reaction between deuterium and tritium, both isotopes of hydrogen, is given by, $\ce{_1^2H + _1^3H \rightarrow _2^4He + _0^1n} + 17.6 \, MeV.$, Deuterium is relatively abundant in ocean water, but tritium is scarce. The new generation of stars begins the nucleosynthesis process anew, with a higher percentage of heavier elements. The total energy output per second is given in the problem statement. This chain, like the proton-proton chain, produces energy without any radioactive by-product. Sustained nuclear fusion is the holy grail of the power industry. In about five billion years, the central core of the Sun will be depleted of hydrogen. After five billion years, the Sun is very nearly the same mass as it is now. Inside the sun, hydrogen nuclei fuse together to form helium, creating heat energy that warms the Earth. What is the rate at which the mass of the Sun decreases? Q1. Breaks heavy atom into two or smaller ones. Example of Nuclear Fission Energy is required in order for fission to occur. Less nuclear waste - Fusion reactors will not produce high-level nuclear wastes like their fission counterparts, so disposal will be less of a problem. Multiplying this rate by five billion years gives the total mass lost by the Sun. For example, in lighter stars, hydrogen combines to form helium through the proton-proton chain. He was also the first to create table-top nuclear fusion. Uranium 235 is a fissile isotope and its fission cross-section for thermal neutrons is about 585 barns (for 0.0253 eV neutron). Nuclear fusion is a reaction in which two or more atomic nuclei are combined to form one or more different atomic nuclei and subatomic particles (neutrons or protons). Nuclear fusion is when two small, light nuclei join together to make one heavy nucleus. The first hydrogen bomb was detonated in 1952 on the remote island of Eniwetok in the Marshall Islands. The factual basis for such beliefs is that stars consist primarily of hydrogen gas. to produce the desired tritium. The environmental impacts of nuclear power results from the nuclear power cycle, its operation, and the effects of nuclear accidents. It also produces and consumes tritium within the plant in a closed circuit. English examples for "nuclear fusion" - Research into nuclear fusion started in the early part of the 20th century. Fusion results in a release of energy because the mass of the new nucleus is less than the sum of the original masses. [ "article:topic", "authorname:openstax", "nuclear fusion", "nuclear fusion reactor", "nucleosynthesis", "proton-proton chain", "license:ccby", "showtoc:no", "Q value", "program:openstax" ], 10.8: Medical Applications and Biological Effects of Nuclear Radiation, Creative Commons Attribution License (by 4.0), Describe the process of nuclear fusion in terms of its product and reactants, Calculate the energies of particles produced by a fusion reaction, Explain the fission concept in the context of fusion bombs, the production of energy by the Sun, and nucleosynthesis. Once the hydrogen fuel is exhausted, the star enters the next stage of its life and fuses helium. 3. It is incredibly inexpensive to create. In the method, some of the mass of the hydrogen is converted into energy. Nuclear fusion, on the other hand, rather than splitting an atom, collides two lighter atoms (typically hydrogen) until they fuse together into one heavier atom (helium). Nuclear fission generates a lot of radioactive particles. \nonumber\], In $$5 \times 10^9 \, y = 1.6 \times 10^{17}s$$, the Sun’s mass will therefore decrease by \begin{align} \Delta M &= (4.24 \times 10^9 kg/s)(1.6 \times 10^{17}s) \\[4pt] &= 6.8 \times 10^{26}kg.\end{align} The current mass of the Sun is about $$2.0 \times 10^{30} kg$$, so the percentage decrease in its mass when its hydrogen fuel is depleted will be $\left(\frac{6.8 \times 10^{26}kg}{2.0 \times 10^{30}kg}\right) \times 100\% = 0.034\%. Like fission, nuclear fusion can also transmute one element into another. If utilised properly, nuclear fusion is the answer to the world’s power crisis problem. This energy is released in the form of gamma radiation. In nuclear fusion, atoms are fused or combined together to create energy. Fission is induced by neutrons. These elements, along with much of the star, are ejected into space by the explosion. This value is divided by the original mass of the Sun to determine the percentage of the Sun’s mass that has been lost when the hydrogen fuel is depleted. Although nuclear power plants do not emit carbon dioxide, high amounts of carbon dioxide are emitted during operation and activities that are related to building and running the plant. For example, hydrogen nuclei fuse in stars to form the element helium. In 1942, Robert Oppenheimer suggested that the extremely high temperature of an atomic bomb could be used to trigger a fusion reaction between deuterium and tritium, thus producing a fusion (or hydrogen) bomb. Fusion is also used to force together atomic nuclei to form the newest elements on the periodic table. Lacking an outward pressure from fusion reactions, the star begins to contract due to gravity. The universe is full of instances of nuclear fusion reactions. This temperature together with incredibly high pressure, two isotopes of Hydrogen, Deuterium, and Tritium combine for forming Helium and releases the enormous amount of energy in the form of heat. This calculation assumes that only the proton-proton decay change is responsible for the power output of the Sun. Plasma must be kept at very high temperatures with the support of external heating systems and confined by an external magnetic field. Some representative reactions are, \[\ce{_6^{12}C + _6^{12}C \rightarrow _{11}^{23}Na + _1^1H,}$, $\ce{_6^{12}C + _6^{12}C \rightarrow _{12}^{24}Mg + \gamma,}$, $\ce{_6^{12}C + _8^{16}O \rightarrow _{14}^{28}Si + \gamma.}$. Fusion reactions have been duplicated in … Helium melting temperature, at -272 ° C. Melting temperature of hydrogen, at -259 ° C. Fusion of ice in liquid water, when the temperature is 0 ° C. This process, called the proton-proton chain, is summarized by three reactions: \begin{align} \ce{_1^1H + _1^1H} & \rightarrow \ce{_1^2H + _1^0e + \nu + Q,} \\[4pt] \ce{_1^1H + _1^2H} &\rightarrow \ce{_2^3He + \gamma + Q,} \\[4pt] \ce{_2^3He + _2^3He} &\rightarrow \ce{_2^4He + _1^1H + _1^1H + Q.} Ans:Â The balanced nuclear reaction is given as: Stay tuned with BYJU’S to learn more about nuclear fusion, energy, and much more. Fusion, on the other hand, does not create any long-lived radioactive nuclear waste. For example, the so-called hydrogen bomb (or H bomb) is actually a deuterium–tritium bomb (a D–T bomb), which uses a nuclear fission reaction to create the very high temperatures needed to initiate fusion of solid lithium deuteride (6 LiD), which releases neutrons that then react with 6 Li, producing tritium. Q1. The fuel for fusion, Deuterium, and Tritium, are also readily available in nature. Hence, nuclear energy cannot be generated in an iron-rich core. An example of nuclear fusion is the process of four hydrogens coming together to form helium. Based on the principle of mass-energy equivalence, this mass difference means that some mass that was "lost" has been converted into energy. The excess vastly exceeds the energy produced during fission. However, there is a very difficult problem that must be overcome before fusion can be used to produce significant amounts of energy: Extremely high temperatures $$(\approx 10^7 \, K)$$ are needed to drive the fusion process. Nuclear fusion and nuclear fission are two different types of energy-releasing reactions in which energy is released from high-powered atomic bonds between the particles within the nucleus. Sometimes shortened to: fusion Compare nuclear fission See also... Nuclear fusion - definition of nuclear fusion by The Free Dictionary. This energy is transmitted outward by the processes of convection and radiation. If the mass loss per fusion reaction is known, the mass loss rate is known. As with fission reactions, fusion reactions are exothermic—they release energy. Nuclear fusion is a reaction through which two or more light nuclei collide into each other to form a heavier nucleus. Nuclear fusion, process by which nuclear reactions between light elements form heavier elements (up to iron). By what percentage will the mass of the Sun have decreased from its present value when the core is depleted of hydrogen? The deuterium-tritium reaction releases energy explosively. The sun is one of the best examples of nuclear fusion. Example: Fusion occurs in the sun where the atoms of (isotopes of hydrogen, Hydrogen-3, … These two are the major nuclear reactions that take place. Carbon and oxygen nuclei produced in such processes eventually reach the star’s surface by convection. A fusion reactor produces helium, which is an inert gas. \nonumber Thus, to supply $$3.8 \times 10^{26} J/s = 2.38 \times 10^{39} MeV/s$$, there must be $\frac{2.38 \times 10^{39} MeV/s}{25.7 \, MeV/reaction} = 9.26 \times 10^{37} \, reaction/s. Modern hydrogen bombs are approximately 1000 times more powerful than the fission bombs dropped on Hiroshima and Nagasaki in World War II. What are the environmental effects of nuclear power? The net Q value is about 26 MeV. ΔE = -1.697 × 10 9 kJ.mol-1. However, tritium can be generated in a nuclear reactor through a reaction involving lithium. Suppose that we fuse a carbon and helium nuclei to produce oxygen: \[\ce{_6^{12}C + _2^4He \rightarrow _8^{16}O + \gamma.}$. Near the end of its lifetime, the star loses its outer layers into space, thus enriching the interstellar medium with the nuclei of heavier elements (Figure $$\PageIndex{2}$$). Most of this energy is produced in the Sun’s core by the proton-proton chain. Coupled with very high pressure, two isotopes of Hydrogen, Deuterium, and Tritium, fuse to form Helium and releases the massive amount of energy in the form of heat. Over time, however, hydrogen gas is used up in stars, and helium gas is produced. No, because fusion energy production is not based on chain reaction as nuclear fission. Example $$\PageIndex{1}$$: Energy of the Sun. Nucleosynthesis continues until the core is primarily iron-nickel metal. The difference in mass between the reactants and products is manifested as either the release or the absorption of energy. The decrease in mass for the fusion reaction is \[\begin{align*} \Delta m &= 4m (_1^1H) - m(_2^4He) - 2m(_1^0e) \\[4pt] &= 4(1.007825 \, u) - 4.002603 \, u = 2(0.000549 \, u) \\[4pt] &= 0.0276 \, u. An important example of nuclear fusion in nature is the production of energy in the Sun. The energy changes in this reaction can be understood using a graph of binding energy per nucleon. It is produced by a nuclear reaction, where two atoms of similar lightweight elements (usually a hydrogen isotope) combine into one molecule of helium to release energy in the form of photons, which are visible as light. An example is the Joint European Torus (JET) shown in Figure $$\PageIndex{4}$$. The energy from the Sun - both heat and light energy - originates from a nuclear fusion process that is occurring inside the core of the Sun. This process heats the core to a temperature on the order of $$5 \times 10^9K$$. This process is known as nucleosynthesis. Every shift or change of the working configuration in the reactor causes the cooling of plasma or the loss of its containment; in such a case, the reactor would automatically come to a halt within a few seconds, since the process of energy production is arrested, with no effects taking place on the outside. Considered to be inherently safe occurrence of a novel phenomenon of electrochemically induced nuclear if. 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Posted in Uncategorized Commentary	2021-08-05 08:39:17	{"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": 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": 0.606514573097229, "perplexity": 1229.8993795836557}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046155458.35/warc/CC-MAIN-20210805063730-20210805093730-00005.warc.gz"}
https://crazyproject.wordpress.com/2011/01/01/show-that-a-given-polynomial-is-irreducible-over-the-rationals-with-%E2%88%9A-2-adjoined/	## Show that a given polynomial is irreducible over the rationals with √(-2) adjoined Prove that $p(x) = x^4 - 4x^2 + 8x + 2$ is irreducible over the quadratic field $\mathbb{Q}(\sqrt{-2})$. We saw previously that $\mathbb{Z}[\sqrt{-2}]$ is a Euclidean domain with field of fractions $\mathbb{Q}(\sqrt{-2})$. Thus it is a unique factorization domain, and thus by Gauss’ lemma, $p(x)$ is irreducible over $\mathbb{Q}(\sqrt{-2})$ if and only if it is irreducible over $\mathbb{Z}[\sqrt{-2}]$. Suppose $p(x)$ is reducible in $\mathbb{Z}[\sqrt{-2}][x]$. Suppose $p(x)$ has a linear factor; say $x - a$. Then we must have that $a$ divides $2$ in $\mathbb{Z}[\sqrt{-2}]$. Recall that the Euclidean norm in this ring is given by $N(a+b\sqrt{-2}) = a^2 + 2b^2$; thus $N(a)$ divides 4, so that $N(a) \in \{ \pm 1, \pm 2, \pm \sqrt{-2}\}$. Evidently, however, we have $p(1) = 7$, $p(-1) = -9$, $p(2) = 18$, $p(-2) = -14$, $p(\sqrt{-2}) = 14 + 8\sqrt{-2}$, and $p(-\sqrt{-2}) = 14 - 8\sqrt{-2}$. So $p(x)$ does not have a linear factor. Suppose instead that $p(x)$ has a quadratic factor, say $x^2 + ax + b$. Using the long division algorithm for polynomials, we have $p(x) = (x^2 + ax + b)(x^2 - ax + (a^2 - b - 4)) + (8 - a^3 + 2ab +4a)x + (2 - a^2b + b^2 + 4b)$. Since quotients and remainders in the Euclidean algorithm in $F[x]$ are unique, we have $8 - a^3 + 2ab + 4a = 0$ and $2 - a^2b + b^2 + 4b = 0$. Moreover, we have $b \in \{ \pm 1, \pm 2, \pm \sqrt{-2} \}$ as in the linear factor case, and similarly $a^2 - b - 4 \in \{ \pm 1, \pm 2, \pm \sqrt{-2}$. Note that $8 = a(a^2 - 2b - 4)$; computing norms, we see that $N(a)$ divides 64. If $b = 1$, then the second equation gives $2a^2 = -5$. Computing norms, we have that 4 divides 25, a contradiction. If $b = 1$, then we have $-2a^2 = 3$, which is similarly a contradiction. If $b = 2$, then we have $a^2 = -5$. Then $N(a)$ is either 1, 5, or 25; since $N(a)$ divides 64, $N(a) = 1$ and so $a = \pm 1$. Then $a^2 = 1$, a contradiction. If $b = -2$, then we have $a^2 = -1$, a contradiction since no such element exists in $\mathbb{Z}[\sqrt{-2}]$. If $b = \sqrt{-2}$, then the second equation yields $\sqrt{-2}(a^2 + 2) = 1$, a contradiction since the only units in $\mathbb{Z}[\sqrt{-2}]$ are 1 and -1. If $b = -\sqrt{-2}$, then we likewise have $\sqrt{-2}(a^2 + 2) = -1$, again a contradiction. Thus $p(x)$ does not have a quadratic factor. Thus $p(x)$ is irreducible over $\mathbb{Z}[\sqrt{-2}]$, and thus over $\mathbb{Q}(\sqrt{-2})$.	2017-01-22 05:53:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 57, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9878255128860474, "perplexity": 48.001083556345264}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281353.56/warc/CC-MAIN-20170116095121-00030-ip-10-171-10-70.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/2325600/eigenvalues-of-matrix	# Eigenvalues of matrix Let $\mathbb{C}^n$ be given. Consider a matrix $A$ on it with one simple eigenvalue zero and all other eigenvalues having strictly negative real part. Now, let $v$ be the eigenvector to eigenvalue $0$ and $V$ be a subspace of $\mathbb{C}^n$ such that $v \notin V$ and $AV \subset V.$ Does this imply that on $V$ all eigenvalues have strictly negative real part? If anything is unclear, please let me know. Yes. Since $AV \subset V, \tag{1}$ $A$ may be considered a linear operator on $V$, and since $V \subset \Bbb C^n$ is indeed a subspace, it is a complex vector space in its own right; thus, $A$ has its own eigenstructure on $V$. If $w \in V$ is an eigenvector of $A$, we have $Aw = \lambda w \tag{2}$ for some $\lambda \in \Bbb C^n$. But now we have $w \in V \subset \Bbb C^n, \tag{3}$ thus $w \in \Bbb C^n$ is also an eigenvector of $A$ acting on $\Bbb C^n$. Since $\lambda \ne 0$, it follows from our hypothesis that $\Re(\lambda) < 0$.	2019-06-18 12:37:28	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9542552828788757, "perplexity": 63.674143091968595}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998724.57/warc/CC-MAIN-20190618123355-20190618145355-00226.warc.gz"}
https://wizedu.com/questions/25996/suppose-your-firm-uses-2-inputs-to-produce-its	In: Economics # Suppose your firm uses 2 inputs to produce its output: K (capital) and L (labor). the... Suppose your firm uses 2 inputs to produce its output: K (capital) and L (labor). the production function is q = 50K^(1/2)L^(1/2). prices of capital and labor are given as r = 2 and w = 8 a) does the production function display increasing, constant, or decreasing returns to scale? how do you know and what does this mean? b) draw the isoquants for your firms production function using L for the x axis and K for y. how are the factors K and L? c) derive the expansion path equation. represent it graphically. how does the expansion path change when r = 1 and w = 8? d) find the total cost function as a function of quantity e) represent the firms cost minimizing choice of factors to produce a given quantity q in a diagram. if q = 1000, calculate K and L ## Related Solutions ##### There is a firm who manufacturers and uses capital (K) and labor (L) to product output... There is a firm who manufacturers and uses capital (K) and labor (L) to product output Q such that Q=10KL. The unit price for K and L are w = $15 and r =$5, respectively. 1).Does the firm’s production exhibit decreasing, constant, or increasing returns to scale? 2)What is the optimal input bundle (K, L) to produce 480 unit of output? 3)Derive the long run cost function. ##### A competitive firm uses two inputs, capital (?) and labour (?), to produce one output, (?).... A competitive firm uses two inputs, capital (?) and labour (?), to produce one output, (?). The price of capital, ??, is $1 per unit and the price of labor, ?? , is$1 per unit. The firm operates in competitive markets for outputs and inputs, so takes the prices as given. The production function is ?(?, ?) = 3? 0.25? 0.25. The maximum amount of output produced for a given amount of inputs is ? = ?(?, ?) units.... ##### A firm produces output using capital (K) and labor (L). Capital and labor are perfect complements... A firm produces output using capital (K) and labor (L). Capital and labor are perfect complements and 1 unit of capital is used with 2 units of labor to produce 1 unit of output. Draw an example of an isoquant. If wages and rent are $2 and$3, respectively, what is the Average Total Cost? A firm has a production function given by Q=4KL where K, L and Q denote capital, labor, and output, respectively. The firm wants to produce... ##### A firm uses two inputs, labor and capital, to produce a good. To keep up with... A firm uses two inputs, labor and capital, to produce a good. To keep up with the story, let zℓ ≥ 0 denote the units of labor and zk ≥ 0 the units of capital. The firm’s technology is expressed as a production function f(zℓ, zk) = 20 z1/5 ℓ z3/5 k . Let w > 0 and r > 0 be the cost of hiring a unit of labor and a unit of capital, respectively. (a) Find the technical... ##### . A firm uses the inputs of fertilizer (sacks), labor and hothouses to produce roses. Suppose... . A firm uses the inputs of fertilizer (sacks), labor and hothouses to produce roses. Suppose that when the quantity of labor and hothouses is fixed, the relationship between the quantity of fertilizer (F) used and the number of roses (TP) produced is given by the following table. F TP APF MPF 0 0 10 1,100 20 2,200 30 4,800 40 7,600 50 9,800 60 11,600 70 12,200 80 11,800... ##### Consider an economy that uses two factors of production, capital (K) and labor (L), to produce... Consider an economy that uses two factors of production, capital (K) and labor (L), to produce two goods, good X and good Y. In the good X sector, the production function is X = 4KX0.5 + 6LX0.5, so that in this sector the marginal productivity of capital is MPKX = 2KX-0.5 and the marginal productivity of labor is MPLX = 3LX-0.5. In the good Y sector, the production function is Y = 2KY0.5 + 4LY0.5, so that in this sector... ##### A firm discovers that when it uses K units of capital and L units of labor... A firm discovers that when it uses K units of capital and L units of labor it is able to produce q=4K^1/4 L^3/4 units of output. Continue to assume that capital and labor can be hired at $40 per unit for labor and$10 for capital. In the long run if the firm produces 600 units of output, how much labor and capital will be used and what is the LR Total cost of production? ##### A firm discovers that when it uses K units of capital and L units of labor... A firm discovers that when it uses K units of capital and L units of labor it is able to produce q=4K^1/4 L^3/4 units of output. a) Calculate the MPL, MPK and MRTS b) Does the production function (q=4K^1/4 L^3/4) exhibit constant, increasing or decreasing returns to scale and why? c) Suppose that capital costs $10 per unit and labor can each be hired at$40 per unit and the firm uses 225 units of capital in the short run.... ##### A firm produces output y using two factors of production (inputs), labour L and capital K.... A firm produces output y using two factors of production (inputs), labour L and capital K. The firm’s production function is ?(?,?)=√?+√?=?12+?12. The wage rate w = 6 and the rental price of capital r = 2 are taken as parameters (fixed) by the firm. a. Show whether this firm’s technology exhibits decreasing, constant, or increasing returns to scale. b. Solve the firm’s long run cost minimization problem (minimize long run costs subject to the output constraint) to derive this... ##### Suppose output, Q, is produced by labor, L, and capital, K, according to the following function:... Suppose output, Q, is produced by labor, L, and capital, K, according to the following function: Q = K ½ L½.. Suppose the firm sells each unit of output in a competitive market for a price P = $100. Suppose the firm hires each unit of labor in a competitive market for a wage W =$25. Suppose the firm has to make do for now with a stock of capital K = 49; moreover, suppose each unit of capital...	2023-03-25 00:48:33	{"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": 0.5442512631416321, "perplexity": 3421.2469519418187}, "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/1679296945292.83/warc/CC-MAIN-20230325002113-20230325032113-00656.warc.gz"}
http://math.stackexchange.com/questions/236939/limit-of-a-function-at-infinity	# Limit of a function at infinity I have a question that finding the limit : $\text{lim}_{x\rightarrow \infty}x(\sqrt{x^2+1}-x)$. My strategy is follows : $\text{lim}_{x\rightarrow \infty}x(\sqrt{x^2+1}-x)=\text{lim}_{x\rightarrow \infty}\dfrac{x}{\sqrt{x^2+1}+x}$ From this if I divide both the denominator and the numerator by $x$, then it wil depend whether $x\rightarrow +\infty$ or $x\rightarrow -\infty$ to conclude and two case wil give two answer $1$ and $-1$. So, am I wrong any where ? How can I solve it ? - The question asked for $\lim_{x\to\infty}$. Why are you worried about what happens when $x\to-\infty$? – Gerry Myerson Nov 14 '12 at 3:39 I think that $\infty$ can be $+\infty$ or $-\infty$ – knot Nov 14 '12 at 3:40 @knot: But what matters is what the person who asked the question thinks. – André Nicolas Nov 14 '12 at 3:48 When someone says $\lim_{x\rightarrow \infty} f(x) = L$, they mean the following: $\forall \epsilon > 0, \exists \delta >0$ such that $x \in (\delta,\infty) \Rightarrow \|f(x)-L\|< \epsilon$. Thus $+\infty$ and $-\infty$ are two different things. – Gautam Shenoy Nov 14 '12 at 4:46 Corrected: Presumably you got $$x\left(\sqrt{x^2+1}-x\right)=\frac{x}{\sqrt{x^2+1}+x}$$ by some version of the trick of multiplying by $1$ in a carefully chosen disguise. To continue, do it again, but this time with the disguise $1=\dfrac{1/x}{1/x}$, using the fact that $\sqrt{x^2+1}=\sqrt{1+\frac1{x^2}}$ for positive $x$: \begin{align} \frac{x}{\sqrt{x^2+1}+x}&=\frac{x}{\sqrt{x^2+1}+x}\cdot\frac{1/x}{1/x}\\\\ &=\frac1{\sqrt{1+\frac1{x^2}}+1} \end{align} for $x>0$. (Since we’re going to take the limit as $x\to\infty$, we care only about $x>0$.) Now go ahead and take the limit as $x\to\infty$. - Sorry, but how did you get $\sqrt{x^2+1}=x\sqrt{1+\dfrac{1}{x^2}}$ ? If we take $x$ out of $\sqrt{x^2+1}$, I think we have to care about the sign of $x$. – knot Nov 14 '12 at 3:51 @knot: Yes, I should really have written $\|x\|$, but we’re interested in the limit as $x\to\infty$, so we’re interested only in positive $x$. I’ll revise it slightly to make that clear. – Brian M. Scott Nov 14 '12 at 3:54 @Gerry: Ouch. Indeed. – Brian M. Scott Nov 14 '12 at 4:51 @knot: My apologies for casting aspersions on your algebra, which was fine: I misread one of the signs. – Brian M. Scott Nov 14 '12 at 4:54 $\displaystyle \lim_{x \rightarrow \infty} \left[ x \left( \sqrt{x^2 + 1} - x \right) \right] = \lim_{x \rightarrow \infty} \left[ x \left( x\sqrt{1 + \frac 1{x^2}} - x \right) \right] = \lim_{x \rightarrow \infty} \left[ x^2 \left( \sqrt{1 + \frac 1{x^2}} - 1 \right) \right] = \lim_{x \rightarrow \infty} \left[ x^2 \left( 1 + \frac 1{2x^2} - 1 \right) \right] = \frac 12$ - It'd be great if you explain how you pass from the LHS to the middle term, justifying the step of course. – DonAntonio Nov 14 '12 at 13:21 It's a power series expansion of square root function after taking $x^2$ out of the root as $x$, with assumption $x>0$ due to the $x \rightarrow \infty$ of course. Anyway, edited my answer with a bit more details. – Kaster Nov 14 '12 at 19:11	2015-08-01 16:30:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9600480794906616, "perplexity": 378.6157007843917}, "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-2015-32/segments/1438042988840.31/warc/CC-MAIN-20150728002308-00111-ip-10-236-191-2.ec2.internal.warc.gz"}
https://www.math.temple.edu/events/seminars/algebra/2019/	Algebra Seminar 2019 Current contacts: Vasily Dolgushev, Ed Letzter or Martin Lorenz. The Seminar usually takes place on Mondays at 1:30 PM in Room 617 on the sixth floor of Wachman Hall. Click on title for abstract. • Monday January 28, 2019 at 13:30, Wachman 617 Paschke Categories, K-homology and the Riemann-Roch Transformation Khashayar Sartipi, University of Illinois at Chicago For a separable C^-algebra A, we introduce an exact C^-category called the Paschke Category of A, which is completely functorial in A, and show that its K-theory groups are isomorphic to the topological K-homology groups of the C^*-algebra A. Then we use the Dolbeault complex and ideas from the classical methods in Kasparov K-theory to construct an acyclic chain complex in this category, which in turn, induces a Riemann-Roch transformation in the homotopy category of spectra, from the algebraic K-theory spectrum of a complex manifold X, to its topological K-homology spectrum. This talk is based on the preprint https://arxiv.org/abs/1810.11951 • Monday February 4, 2019 at 13:30, Wachman 617 Detecting free objects in associative algebras: A survey Edward Letzter, Temple University In the 1970s, Lichtman asked whether or not the multiplicative group of units of a noncommutative division algebra contains a free subgroup and Makar-Limanov asked whether or not a finitely generated infinite dimensional noncommutative division algebra must contain a free subalgebra. These questions are still open in general, even if many important special cases have been resolved, and have recently received renewed attention. (These questions can be considered in analogy to the Tits Alternative for linear groups as well as Gromov's Theorem on groups with polynomial growth.) My talks will survey both older and newer results. • Monday February 11, 2019 at 13:30, Wachman 617 Detecting free objects in associative algebras, II Edward Letzter, Temple University A discussion of more recent results, and still-open questions, on free subalgebras and free multiplicative subsemigroups of associative algebras. • Monday February 18, 2019 at 13:30, Wachman 617 On the Replacement Property for $PSL(2,p)$ Aidan Lorenz, Temple University The replacement property (or Steinitz Exchange Lemma) for vector spaces has a natural analog for finite groups and their generating sets. For the special case of the groups $PSL(2,p)$, where $p$ is a prime larger than 5, first partial results concerning the replacement property were published by Benjamin Nachman. The main goal of this talk is to outline the methods involved in providing a complete answer for $PSL(2,p)$ (which was accomplished during the Summer of 2018). This talk is based on a paper in preparation joint with Baran Zadeoglu and David Cueto Noval. • Monday February 25, 2019 at 13:30, Wachman 617 Prime Torsion of the Brauer Group of an Elliptic Curve Charlotte Ure, Michigan State University The Brauer group of an elliptic curve $E$ is an important invariant with intimate connections to cohomology and rational points. Elements of this group can be described as Morita equivalence classes of central simple algebras over the function field. The Merkurjev-Suslin theorem implies that these classes can be written as tensor product of symbol (or cyclic) algebras. In this talk, I will describe an algorithm to calculate generators and relations of the $q$-torsion ($q$ a prime) of the Brauer group of $E$ in terms of these tensor products over any field of characteristic different from $2$,$3$, and $q$, containing a primitive $q$-th root of unity. This is work in progress. • Monday March 11, 2019 at 13:30, Wachman 617 Catalan Functions and k-Schur functions Anna Pun, Drexel University Li-Chung Chen and Mark Haiman studied a family of symmetric functions called Catalan (symmetric) functions which are indexed by pairs consisting of a partition contained in the staircase (n-1, ..., 1,0) (of which there are Catalan many) and a composition weight of length n. They include the Schur functions, the Hall-Littlewood polynomials and their parabolic generalizations. They can be defined by a Demazure-operator formula, and are equal to GL-equivariant Euler characteristics of vector bundles on the flag variety by the Borel-Weil-Bott theorem. We have discovered various properties of Catalan functions, providing a new insight on the existing theorems and conjectures inspired by Macdonald positivity conjecture. A key discovery in our work is an elegant set of ideals of roots that the associated Catalan functions are k-Schur functions and proved that graded k-Schur functions are G-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We exposed a new shift invariance property of the graded k-Schur functions and resolved the Schur positivity and k-branching conjectures by providing direct combinatorial formulas using strong marked tableaux. We conjectured that Catalan functions with a partition weight are k-Schur positive which strengthens the Schur positivity of Catalan function conjecture by Chen-Haiman and resolved the conjecture with positive combinatorial formulas in cases which capture and refine a variety of problems. This is joint work with Jonah Blasiak, Jennifer Morse and Daniel Summers. Here are the links to the papers on ArXiv: https://arxiv.org/abs/1804.03701, https://arxiv.org/abs/1811.02490 • Monday March 18, 2019 at 13:30, Wachman 617 Classifying Actions of $T_n \otimes T_n$ on Path Algebras of Quivers Delaney Aydel, Temple University Let $T_n$ denote the $n$th Taft algebra. We fully classify inner-faithful actions of $T_n \otimes T_n$ on four-vertex Schurian quivers as extensions of the actions of $\mathbb{Z}_n \times \mathbb{Z}_n$. One example will be presented in full, with the remaining results briefly given. • Monday March 25, 2019 at 13:30, Wachman 617 Topics in Galois Theory, I Martin Lorenz, Temple University This is the first lecture in a minicourse (probably three lectures) surveying some topics in Galois Theory that are not typically covered in the graduate algebra course (Math 8011/12): inverse Galois theory, Noether's rationality problem, the Chebotarev density theorem,... The Galois Theory portion of Math 8011/12 will be sufficient background for the material presented in this minicourse; so it will be accessible to all students in my current Math 8012 class. No proofs will be given; the goal is to describe some research directions that are of current interest. • Monday April 1, 2019 at 13:30, Wachman 617 Topics in Galois Theory, II Martin Lorenz, Temple University The second talk in this series will be devoted to the behavior of Galois groups under reduction mod primes. More specifically, given an polynomial $f \in \mathbf{Z}[x]$, I will discuss the question what the (cyclic!) Galois groups of the reductions of $f$ mod various primes tell us about the Galois group of $f$. • Monday April 8, 2019 at 13:30, Wachman 617 Topics in Galois Theory, III Martin Lorenz, Temple University First, I will finish (after some reminders) the proof of the reduction-mod-primes recipe for Galois groups from last time. Then I will address the following deficiency of the reduction method: while the full symmetric group is easily detected in this way, small Galois groups require further tools. I will explain a probabilistic method that is based on the Tchebotarov Density Theorem. • Monday April 15, 2019 at 13:30, Wachman 617 What is so algebraic about the algebraic fundamental group? Vasily Dolgushev, Temple University A careful definition of the fundamental group in the realm of algebraic geometry requires a lot of effort. In Chapter 4 of his book "Galois groups and fundamental groups", Tamas Szamuely gives a gentle introduction to this topic for algebraic curves. In my two lectures, I will follow Tamas's presentation from this Chapter. Most of proofs will be omitted but I will try give examples. My lectures are partially inspired by Martin Lorenz's recent mini-course. • Monday April 22, 2019 at 13:30, Wachman 617 Kazhdan-Lusztig polynomials of matroids Jacob Matherne, IAS, Princeton Kazhdan-Lusztig (KL) polynomials for Coxeter groups were introduced in the 1970s, providing deep relationships among representation theory, geometry, and combinatorics. In 2016, Elias, Proudfoot, and Wakefield defined analogous polynomials in the setting of matroids. In this talk, I will compare and contrast KL theory for Coxeter groups with KL theory for matroids. I will also associate to any matroid a certain ring whose Hodge theory can conjecturally be used to establish the positivity of the KL polynomials of matroids as well as the "top-heavy conjecture" of Dowling and Wilson from 1974 (a statement on the shape of the poset which plays an analogous role to the Bruhat poset). Examples involving the geometry of hyperplane arrangements will be given throughout. This is joint work with Tom Braden, June Huh, Nick Proudfoot, and Botong Wang. • Monday April 29, 2019 at 13:30, Wachman 617 What is so algebraic about the algebraic fundamental group? Part 2 Vasily Dolgushev, Temple University This is the second lecture devoted to Chapter 4 of Tamas Szamuely's book "Galois Groups and Fundamental Group". I will define the algebraic fundamental group of a curve and talk about the outer Galois action on the algebraic fundamental group. Examples will be given. • Monday September 9, 2019 at 13:30, Wachman 617 A brief introduction to operads I Vasily Dolgushev, Temple University The operad PaB is closely related to the Grothendieck-Teichmueller group GT introduced by Vladimir Drinfeld in 1990. This is the first talk in the mini-course devoted to the operad PaB, GT-shadows and their action on child's drawings. In this talk, I will introduce operads and give various examples. This mini-course should be accessible to first year graduate students. • Monday September 16, 2019 at 13:30, Wachman 617 A brief introduction to operads II Vasily Dolgushev, Temple University In the second talk of this series, I will give more examples of operads. I will also talk about one of the central objects of this series, the operad of parenthesized braids PaB. This is an operad in the category of groupoid and it is "assembled from" Artin's braid groups. This operad was introduced by Dmitry Tamarkin in 1998 and a very similar object was introduced by Dror Bar-Natan in 1996. • Monday October 7, 2019 at 13:30, Wachman 617 The operad PaB of parenthesized braids I Vasily Dolgushev, Temple University After a brief review of the operad PaB, I will talk about compatible equivalence relations on the truncation of PaB. A large supply of such equivalence relations come from finite index normal subgroups of $B_4$ which are contained in $PB_4$. • Monday October 14, 2019 at 13:30, Wachman 617 Highest-weight representations and global Weyl modules: from classical Lie algebras to Yangians Prasad Senesi, The Catholic University of America Highest-weight representations play a prominent role in the representation theory of Lie algebras and quantum groups. Particular examples of highest-weight representations of certain infinite-dimensional Lie algebras called the Weyl modules (for loop and quantum algebras) were introduced by Chari and Pressley in 2000. In this introductory talk, we proceed by example from the classical structure and representation theory of the special linear algebra in dimensions 2 and 3, to that of the corresponding loop algebras and quantum groups. Along the way, the utility of highest-weight representations, and of the (local and global) Weyl Modules, in all of these settings will be described. We will conclude with a discussion of the Yangian, its relation to the quantum loop algebra, and some recent work concerning its global Weyl modules. This is joint work with Bach Nguyen (Temple University) and Matt Lee (University of Illinois at Chicago). • Monday October 21, 2019 at 13:30, Wachman 617 An approach toward supersymmetric cluster algebras Ashish K. Srivastava, Saint Louis University In this talk we will propose the notion of cluster superalgebra which is a supersymmetric version of the classical cluster algebra introduced by Fomin and Zelevinsky. We show that the symplectic-orthogonal supergroup $SpO(2\|1)$ admits a cluster superalgebra structure and as a consequence of this, we deduce that the supercommutative superalgebra generated by all the entries of a superfrieze is a cluster superalgebra. We also show that the coordinate superalgebra of the super Grassmannian $G(2\|0; 4\|1)$ of chiral conformal superspace (that is, $(2\|0)$ planes inside the superspace $\mathbb C^{4\|1}$) is a quotient of a cluster superalgebra. • Monday October 28, 2019 at 13:30, Wachman 617 The operad PaB of parenthesized braids II Vasily Dolgushev, Temple University After a brief reminder of the operad PaB, I will talk about the compatible equivalence relations coming from finite index normal subgroups N in $B_4$ which are contained in the pure braid group $PB_4$ on 4 strands. If time will permit, I will introduce GT-shadows. • Monday November 4, 2019 at 13:30, Wachman 617 $k$-Schur and Catalan functions Jennifer Morse, University of Virginia We will discuss the inception, subsequent developments, and resolution of a symmetric function conjecture from the 1990's. The $k$-Schur functions arose via computer experimentation with symmetric functions called Macdonald polynomials; they are symmetric functions with coefficients involving a single $t$-parameter. Conjectures that they satisfy many strong and beautiful positivity properties compelled further study. In the special case when $t=1$, it was unexpectedly discovered that $k$-Schur functions are geometrically significant in an area called affine Schubert calculus and for computing Gromov-Witten invariants. However, the intricate combinatorics behind $k$-Schur functions involving the Bruhat order on the affine symmetric group made progress with generic $t$ extremely hard to come by. We recently discovered a new approach to the study of $k$-Schur functions; they are a subclass of Catalan functions, $G$-equivariant Euler characteristics of vector bundles on the flag variety defined by raising operators and indexed by Dyck paths. This perspective led us to settle decades old conjectures, providing tableaux enumeration formulas to do so. Joint work with Blasiak, Pun, and Summers. • Monday November 11, 2019 at 13:30, Wachman 617 GT-shadows and their action on child's drawings I Vasily Dolgushev, Temple University Last time, I introduced GT-shadows and showed that GT-shadows are morphisms in a groupoid whose objects are compatible equivalence relations on PaB. This time, I will talk about child's drawings subordinate to a given compatible equivalence relation on PaB. I will explain in what sense GT-shadows act on child's drawings. • Monday December 2, 2019 at 13:30, Wachman 617 GT-shadows and their action on child's drawings II Vasily Dolgushev, Temple University I will introduce child's drawings which are subordinate to a given compatible equivalence relation on PaB. I will introduce the action of GT-shadows on child's drawings as a (co)functor from the groupoid of GT-shadows to the category of finite sets. If time will permit, I will talk about the inverse subordination problem for child's drawings and infinite chains in the poset $NFI_{PB_4}(B_4)$. • Monday December 9, 2019 at 13:30, Wachman 617 Brill-Noether theory and degeneracy loci Linda Chen, Swarthmore College Degeneracy loci of morphisms between vector bundles have been used in a wide range of situations, including classical approaches to the Brill-Noether theory of special divisors on curves. I will give an introduction to these connections and describe recent developments, including new K-theoretic formulas for degeneracy loci and their applications to Brill-Noether loci. These recover the formulas of Eisenbud-Harris, Pirola, and Chan-Lopez-Pflueger-Teixidor for Brill-Noether curves. This is joint work with Dave Anderson and Nicola Tarasca.	2020-02-26 22:23:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.6651450991630554, "perplexity": 1083.0331732146315}, "config": {"markdown_headings": false, "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-2020-10/segments/1581875146562.94/warc/CC-MAIN-20200226211749-20200227001749-00368.warc.gz"}
https://socratic.org/questions/how-do-you-find-the-slope-perpendicular-to-2x-3-4y	# How do you find the slope perpendicular to 2x + 3 = 4y? Jan 20, 2016 Perpendicular slope has value $- 2$ #### Explanation: Since the given equation is a simple linear equation, it's not difficult to find the slope perpendicular to the given equation. Now, we know that the product of the value of the slopes of 2 perpendicular lines is equal to negative of unity, that is ${m}_{1} \cdot {m}_{2} = - 1$ The above equation should first be made into a general slope equation first, so it becomes $2 \left(x + \frac{3}{2}\right) = 4 y \setminus \implies y = \frac{1}{2} \left(x + \frac{3}{2}\right)$ So the slope of the above given equation is taken as ${m}_{1} = \frac{1}{2}$ Now, substituting this in the given equation, we get ${m}_{2} \cdot \frac{1}{2} = - 1 \setminus \implies {m}_{2} = - 2$	2019-11-20 12:59:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 5, "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": 0.9258013367652893, "perplexity": 192.51620524361294}, "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/1573496670558.91/warc/CC-MAIN-20191120111249-20191120135249-00553.warc.gz"}
https://artofproblemsolving.com/wiki/index.php/Nichomauss%27_Theorem	# Nichomauss' Theorem ## Nichomauss' Theorem Nichomauss' Theorem states that $n^3$ can be written as the sum of $n$ consecutive integers, thus giving us $1^3+2^3+...+n^3=(1+2+...+n)^2$. ## A Visual Proof Imagine a cuboid with a height of $1$, and length and width of $n$. Divide it into unit cubes. Now, starting from the bottom right of the cuboid (when it's flat on the ground), imagine the first cube to have $l=w=h=1$ (call this $a_1$). Now, extend its length and width by $2$ to get another cuboid with $l=w=3$ and $h=1$ (call this $a_2$). Color $a_1$ yellow and $a_2$ blue. Now, it's easy to see that $a_2$ has $(3 \cdot 3 \cdot 1)-(1 \cdot 1 \cdot 1)=8$ unit cubes and $a_1$ has $1 \cdot 1 \cdot 1=1$ unit cube. We can then rearrange the $8$ unit cubes into a $2 \cdot 2 \cdot 2$ cube. Thus, we clearly have $(1+2)^2=1^3+2^3$. And we can continue this process to $n$ unit cubes and $n$ cuboids with $l=w=n$, $h=1$, which gets us to $(1+2+...+n)^2=1^3+2^3+...+n^3$.	2020-12-04 17:20:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 25, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9618394374847412, "perplexity": 184.58118279998646}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141740670.93/warc/CC-MAIN-20201204162500-20201204192500-00232.warc.gz"}
http://uncyclopedia.wikia.com/wiki/User_talk:Boothman?oldid=1360134	# User talk:Boothman (diff) ← Older revision \| Latest revision (diff) \| Newer revision → (diff) ## editRebutal • You don't seem to have seen my rebutal of sorts on the UotM page. I'm not asking for a vote at all, but I would atleast like a reason why you have deemed my "non-notable". Thanks in advance (for a responce). --ZB 22:55, 17 May 2006 (UTC) ## editUntitled Folder • Hi, man. I read your comment on Zombiebaron's page where you ask why Garth Crooks is tagged as ugly. Yeah I think the article is well formatted, and quite funny. The reason it was tagged was bacause there were no wikilinks. I've taken the liberty of adding some for you and removing the tag. Welcome to Uncyclopedia, by the way! --Hindleyite Talk 10:36, 7 May 2006 (UTC) • Yes welcome! If you look at my talk page you will see my and Hindleyites disscussion on the topic (after your comment) and I agree that Garth Crooks was funny, just lacking in links. Keep it up, and don't make a jackass of yourself (not that you have, but I hate to see n00bs banned for doing stuff they might not have knowen was a bannable offense) -- 00:49, 9 May 2006 (UTC) ## editYou want an award? You got one. Albeit a not very significant one. For the mintedness of your Garth Crooks and Robbiesavage articles. Enjoy your pie. --Hindleyite Talk 18:09, 8 May 2006 (UTC) ## edit A second award for you Seriously, thanks for the vote. It means a lot to me. I hope your Uncyclopedia exploits are all uphill (after the whole UGotM buisness is over :)... oddly enough, previous winners of that award often become useful contributors to Uncyclopedia. I hope this means you're on your way).-- 02:51, 4 June 2006 (UTC) I personally think it's a little silly to nominate someone for UGOTM because you didn't like the way they voted on another page, which is what happened here. This is definately a case of taking the awards way too seriously, which is why I'm glad that Benson came along to give us a chance to make fun of the awards a little bit Forum:BENSON_NEEDS_SUPPORT_TO_BE_N00B_OF_THE_MONTH. --Hrodulf 00:16, 7 June 2006 (UTC) ## edit Ban From Voting Hi, as you probably know by now, you suck at voting. Not only do you issue senseless against votes, but you have also been known to nominate things as a joke on pages that other users take seriously. It is those reasons that you are now being issued a ban from voting and/or nominating on any of the several official Uncyclopedia major award pages (NotM, UotM, WotM, UGotM.) Any votes made on these pages will be instantly invalidated, and if you continue to ignore the rule, you will be banned. Anywho, This ban on voting may be lifted within 2 months, but there are no guarantees. (See Also: Special Rules) 06:47, 11 June 2006 (UTC) • Cosign. --KATIE!! 06:48, 11 June 2006 (UTC) • Cotangent --Splaka 06:50, 11 June 2006 (UTC) • ln(sec + tan) --PantsMacKenzie 07:15, 11 June 2006 (UTC) • $Cos^n$ -- Sir Codeine K·H·P·B·M·N·C·U·Bu. · (Harangue) 08:48, 11 June 2006 (UTC) ## edit Trebor Extra Strong Man Hi there. The {{construction}} tag on your article, Trebor Extra Strong Man, expired. So I moved it to a subpage for you to work on it until it's ready. It is now at Trebor Extra Strong Man. Spang 22:19, 19 June 2006 (UTC) You're lucky I don't ban you... VERY lucky. The rules say you get a warning first, but I've been known to break the rules, and I could easily argue that you addition to "PENIS PENIS PENIS" was vandalism. However, I have a heart, and you were on the right track before now. So, I won't ban you today, but if I happen to be in a bad mood tomorrow, or the next day, or a week from now, you'll get the honor of being my stress cushion. 22:33, 25 August 2006 (UTC) I see above that your two months have expired...does this mean they get refreshed? I mean, I appreciated the vote tremendously, but if one has a history of messing around with votes, such a thing isn't wise at all. So anyway, since Tompkins rolled back your changes, I made sure that your vote stands. I mean, even if I have to do it myself. It's that important to me. But you'd better lay off the voting pages before Tompkins goes all INFINITE BAN on your ass. Because that's the sort of shit I appreciate more than a creative vote. 23:02, 25 August 2006 (UTC) ## edit Merry Christmas Hindleyite was throwing away last year's Christmas cards, and realised they had purposefully forgotten about you.This user doesn't care about Multi-culturalism, and DEMANDS you have a Merry Christmas... NOW! Well, dunno if you hang about here amy more, but an 'appy Chrimbo to thee lad. -- Hindleyite Converse 11:45, 17 December 2006 (UTC) Bradaphraser was throwing away last year's Christmas cards, and realised they had purposefully forgotten about you.This user is completely thoughtless, doesn't care about Multi-culturalism, and therefore DEMANDS you have yourself a Merry little Christmas... NOW! Failure to comply with result in disciplinary action up to and including excommunication from the Capitalist Church May you focus on your successes and forget your failures here at the end of the year. Never forget how we all improve one another's lives. Season's Greetings.-- 17:27, 17 December 2006 (UTC)	2014-10-01 19:26:42	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3876298666000366, "perplexity": 4155.845830837998}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1412037663551.47/warc/CC-MAIN-20140930004103-00130-ip-10-234-18-248.ec2.internal.warc.gz"}
https://en.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-6/v/both-bounds-being-a-function-of-x	If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains .kastatic.org and .kasandbox.org are unblocked. Finding derivative with fundamental theorem of calculus: x is on both bounds AP.CALC: FUN‑6 (EU) , FUN‑6.A (LO) , FUN‑6.A.1 (EK) , FUN‑6.A.2 (EK) How do you apply the fundamental theorem of calculus when both integral bounds are a function of x. Created by Sal Khan. Want to join the conversation? • can someone explain why he added the times 2x at the back of F'(x)? I can't seem to understand that. I did however understand why he had to in the previous video. but in this case. i don't see the need. • Because our upper bound was x², we have to use the chain rule to complete our conversion of the original derivative to match the upper bound. The derivative of x² is 2x, and the chain rule says we need to multiply that factor by the rest of the derivative. • What would happen if you have intervals of numbers rather than x and x^2? For example if you have over the interval of 0 to 1 of the function cos(t)/t. • Then, F(x) would be a constant since the input x is not used in the expression. If you take the derivative of a constant, the result is 0, meaning F'(x) would be 0. I hope this helps! • I don't really get how c is constant if its value depends on x. If by constant he means for a particular value of x, then wouldn't the bounds of the original integral also be constant and so wouldn't there be no point in splitting the integral into two parts in the first place? • This is one of those confusing contraptions where the underlying function -- the one portrayed in the graph -- is a function of t, not a function of x. The constant c is a t-value, not an x-value. It doesn't depend on x. If you aren't getting how F(x) can be a function of x when we're doing an integral of a function of t, it may help to go back over the earlier videos in this section on the fundamental theorem. • What about when the lower and upper limits of the integral both contain a variable for instance the integral from 3x to x^2 of 1/(2+e^t) ? How would you solve that problem? • You simply do the integral in the normal way, and then substitute in the limits which are functions of x. You end up with an expression which is a function of x. This is quite reasonable, if you think about it -- a definite integral gives you the area below the curve between the two specified limits. If the limits depend on x, then the area is not going to be constant, but will also depend on x. In your example we have integral_(3x)^(x^2) 1/(2+e^t) dt = [-ln(2+e^t) / 2 + t/2]_(3x)^(x^2) = -ln(2+e^(x^2)) / 2 + (x^2)/2 + ln(2+e^(3x)) / 2 - 3x/2. • How to calculate the indefinite integral ∫cos(x)/x dx? • Instead of using c, would you be able to take the integral from 0 to x^2 - x ? • I just solved it using 0 instead of c and it gave me the same answer, so I think it's acceptable. • At Sal cancelled one x out of x^2 in the denominator with the only x in the numerator and wrote simply x but why can't x be negative?So i think it's reasonable to write ABSOLUTE VALUE OF X instead of simply x.Please help! (1 vote) • Remember x is a variable, a placeholder for any value. Now . . . . Simplify x²/x. The answer is x. In no way have we limited x to be non negative. In fact, by placing absolute values around the x, as you suggest, you are actually saying, "I don't care if the value x takes on is negative, make the result positive." Example: let x = -5, then x²/x = (-5)²/-5 = (-5)(-5)/-5 = -5. If we did it your way we would be saying that (-5)²/-5 = 5, and that is not correct. • At , why does Sal take the derivative of x^2? Didn't we already apply the Fundamental Theorem of Calculus, or does the fundamental theorem also state that we must take the derivative of the upper-bound? • What do you mean by secant square of x • Instead of thinking of a constant c between x and x^2, couldn't we also choose a constant, say k, whose value is lesser than x? The solution would then be: (derivative of integral from k to x^2)-(derivative of integral from k to x). The results are the same, but then we don't need to switch the bounds. • That depends. If the function is defined outside the interval given, (in this case x to x^2), which it should be, then yes, and you are correct in that the results would be the same. However, because the interval is given as x to x^2, it is considered "more correct" to use a value in between. If you have to show your work, I would do it this way, but otherwise, use whichever method works best for you. Video transcript So let's see if we can take the derivative of this expression right over here, if we can find capital F prime of x. And once again, it looks like you might be able to use the fundamental theorem of calculus. A big giveaway is that you're taking the derivative of a definite integral that gives you a function of x. But here I have x on both the upper and the lower boundary, and the fundamental theorem of calculus, is at least from what we've seen, is when we have x's only on the upper boundary. And then, of course, it's an x squared, but we've seen examples of that already when we used the chain rule to do it. But how can we break this up and put this in a form that's a little bit closer to what we're familiar with when we apply the fundamental theorem of calculus? And to realize that, we really just have to attempt to graph what this is representing. So let's say that this is our lowercase f of x, or I should say f of t. So let's call this lowercase f of t. And let's graph it over the interval between x and x squared. So let's say this is my y-axis. This is my t-axis. And let's say that this right over here is y is equal to f of t. I'm drawing it generally. I don't know what this exactly looks like. And we're going to talk about the interval between x and x squared. So if we're going to talk about the interval between x, which is right over here, it's the lower bound, so x and x squared. It's the lower bound, at least for this definite integral. We don't know for sure. It depends on what x you choose on which one is actually smaller. But let's just say that for the sake of visualizing, we'll draw x right over here, and we will draw x squared right over here. So this whole expression, this entire definite integral, is essentially asking for, is essentially representing this entire area, the entire area under the curve. But what we could do is introduce a constant that's someplace in between x and x squared. Let's say that constant is c, and break this area into two different areas with c as the divider. So that same exact whole area we can now write it as two separate integrals. So one integral that represents this area right over here, and then another integral that represents this area right over there, and where we just say c is some constant between x and x squared. Well, how can we denote this area in purple? Well, that's going to be-- So this thing is going to be equal to the sum of these two areas. The purple area we can show is the definite integral from x to c of our function of t, cosine t over t dt. And then to that we're going to add the green area. And then we'll get the original area. So for the green area, our lower bound of integration is now our constant c, and our upper bound of integration is x squared, and it's going to be of cosine t over t dt. And this is a form where, if we know how to apply the chain rule, we can apply the fundamental theorem of calculus. And this is almost in a form. We're used to seeing it where the x is the upper bound. And, well, we already know what happens. We can swap these two bounds, but it'll just be the negative of that integral. So this is going to be equal to-- let me rewrite it-- the negative of the definite integral from c to x of cosine t over t dt. And then we have plus the definite integral that goes from c to x squared of cosine t over t dt. So all we've done is we've rewritten this thing in a way that we're used to applying the fundamental theorem of calculus. So if we want to find F prime of x, well, applying the derivative operator over here, we're going to have a negative out front. It's going to be equal to negative cosine x over x. Once again, just the fundamental theorem of calculus. And then plus-- we're first going to take the derivative of this thing with respect to x squared, and that's going to give you cosine of x squared over x squared. Wherever you saw t, you replace it with an x squared. And then you're going to multiply that times the derivative of x squared with respect to x. So that's just going to be-- derivative of x squared with respect to x is just 2x. And we're done. We just need to simplify this thing. So all of this is going to be equal to negative cosine x over x plus-- well, this is going to cancel out with just one of those-- plus 2 cosine x squared over x. And I guess we could simplify it even more as being equal to-- and we can swap these-- everything over x 2 cosine of x squared minus cosine of x. And we are done.	2023-03-21 09:05:22	{"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": 0.865973949432373, "perplexity": 274.2467569019692}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2023-14/segments/1679296943637.3/warc/CC-MAIN-20230321064400-20230321094400-00322.warc.gz"}
https://www.emerald.com/insight/content/doi/10.1108/OMJ-02-2022-1482/full/html	# Harnessing virtual reality for management training: a longitudinal study Julita Haber (Strategy and Statistics Area, Gabelli School of Business, Fordham University, New York, New York, USA) Heng Xu (Leading People and Organizations Area, Gabelli School of Business, Fordham University, New York, New York, USA, and) Kanu Priya (Department of Management, College of Business, Missouri State University, Springfield, Missouri, USA) ISSN: 2753-8567 Article publication date: 29 December 2022 91 ## Abstract ### Purpose Virtual reality (VR) technologies have been gaining popularity in training and development in many fields to promote embodied training. However, its adoption in management has been slow and rigorous empirical research to understand its impact on learning and retention is scarce. Thus, this paper aims to examine the benefits of VR technologies for management training. ### Design/methodology/approach Through a longitudinal experiment comparing VR platforms and a traditional video platform, this study examines two as yet unexplored benefits of VR technologies vis-à-vis management training – the cognitive outcome and affective reaction of the training experience over time. ### Findings This study finds that, for cognitive outcomes, immediate gains are similar across video and VR platforms, but subsequent knowledge retention is significantly higher for VR platforms. In terms of affective reaction, VR platforms generate significantly more enjoyment, which carries over to two weeks later, and is partially associated with higher knowledge retention. ### Practical implications This study has implications for management and human resource trainers and system designers interested in integrating VR for training and development purposes. ### Originality/value This study makes a unique contribution by unpacking the long-term benefits of an embodied training system, as well as identify a possible link between cognitive outcomes and affective reaction. ## Citation Haber, J., Xu, H. and Priya, K. (2022), "Harnessing virtual reality for management training: a longitudinal study", Organization Management Journal , Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/OMJ-02-2022-1482 ## Publisher : Emerald Publishing Limited ## Introduction In recent years, there has been an exploding interest in immersive virtual reality (VR) technologies. Technology giants like Meta, Google, Apple and Sony have invested heavily in moving from desktop VR to immersive VR with head-mount displays, fundamentally altering the way people interact with content. The research on VR usage in management and human resource (HR) training has been incommensurate with the increasing popularity of such systems (Schmid Mast, Kleinlogel, Tur, & Bachmann, 2018). Other disciplines, such as medicine, have come to recognize the tremendous promise VR holds for the future. As an example, a randomized double-blinded study found that VR-trained residents performed surgery almost 30% faster and six times less likely to make surgical errors than non-VR-trained residents (Seymour et al., 2002). Similarly, VR usage has been increasing in variety of education and training programs, such as environmental education (Markowitz, Laha, Perone, Pea, & Bailenson, 2018), science and engineering (Potkonjak et al., 2016), mindfulness training for increased decentring (Chandrasiri, Collett, Fassbender, & Foe, 2020), emergency respondents training (Carlson & Caporusso, 2019), virtual chemistry laboratory (Su & Cheng, 2019) and a systematic analysis of VR studies and their impact on learning outcomes in higher education (Radianti, Majchrzak, Fromm, & Wohlgenannt, 2020). The field of business and management, in comparison, has lagged behind in VR application. Although there have been some efforts in combining VR systems into HR training, they are few and far between (Khandelwal & Upadhyay, 2021). Rigorous empirical research to understand its impact on learning is even more scarce. This is despite the fact that management practitioners have already recognized the usefulness and importance of VR technology in areas such as leadership skill development (Chirino-Klevans, 2017) or diversity training in the boardroom (Hinchliffe, 2019). This paper seeks to fill the gap. The main objective of this study is to assess the impact of immersive technologies on cognitive outcomes, specifically to examine twofold usefulness of VR technology. First, we conduct an experiment to compare how well participants learn concepts through a VR platform versus a traditional video platform (which has been the norm in management training). Second, we investigate the affective reaction from engaging in a virtual learning experience to understand if it offers any superior affective benefits. ## Virtual learning through virtual reality technology VR technology provides a visual representation of a virtual world. The immersive VR technology that we used in this study uses a head-mounted display with head motion tracking and a handset and a computer system control that includes PC-tethered or stand-alone VR headsets. This offers a more immersive experience than 360-degree video viewing. More specifically, immersion, interaction and simulation are the three key features of the VR system examined here. The experience is immersive because the fully enclosed space is devoid of the external physical world environment. Through the head-mounted gear combined with a headphone, users can have 360-degree view of the virtual world integrated with audio content. The VR experience is interactive as the sensors can detect user movement and provide corresponding feedback. Users can move inside the virtual world and see a setting from different perspectives. Through a hand console, users can touch and move virtual objects, or even talk to avatars and get responses. Simulated presence is the third important feature of VR. New technological advances enable designers to make the virtual world in striking resemblance to the real world and can trigger emotional responses in users similar to how they would feel in the real world. For instance, users may feel scared that they would fall off a virtual cliff even though they are fully aware that there is no cliff outside of the VR system. Immersive and simulated VR’s adoption and research in management training and development is scarce. A recent study demonstrated using an immersive VR application in the field of HR training (Ventura, Cardenas, Miragall, Riva, & Baños, 2021). It examined the 360-degree video-based experience in sexual harassment training and found it to be far superior in increasing empathy and perspective taking toward a victim of sexual harassment. However, more research is needed to understand the learning and long-term retention benefits of this new technology in management. ### Theory of embodied learning Embodiment is an emerging cognitive sciences theory that emphasizes the connection between knowledge and the activity of our bodies. The unifying framework captures cognitive linguistics, perceptual symbol theory, action-based theories and emotions in the social psychology literature (Glenberg, 2010). In particular, we draw from the action-based theory that highlights the role of sensorimotor activity in human mental processing. Cognition is not only a result of our neural systems but is also affected by the bodily states such as movement, shape and scale (Barsalou, 2008). Evidence shows that movement has beneficial effects on memory and hippocampal neurogenesis, as well as executive function, the prefrontal cortex and anterior cingulate cortex (Madan & Singhal, 2012). An interesting brain magnetic resonance imaging (MRI) study demonstrated how cognition is embedded in the action of the body (Hauk, Johnsrude, & Pulvermüller, 2004). When participants heard words relating to various body parts (e.g. pick, lick, kick), their brain’s sensorimotor areas corresponding to those movements were found to be activated. Another MRI study observed similar concept-body integration (Boulenger, Hauk, & Pulvermüller, 2009). When participants silently read abstract statements containing action words such as “John grasped the idea” or “Pablo kicked the habit,” it activated areas of the brain associated with the arm or leg along with the area for language processing. These studies suggest that our higher-order conceptual knowledge is deeply grounded in the action-perception system. Many areas of our knowledge, abstract concepts included, have close connections with body movements. It can be argued that our knowledge is inherently embodied. Similarly, Alibali and Nathan (2012) argue that abstract contents such as mathematics are based on actions and grounded in the physical environment. Experiments have shown that gesturing can facilitate the acquisition of new knowledge. For instance, children who were taught to produce hand gestures while solving a math problem learned better than those who were not taught to make any gestures (Goldin-Meadow, Cook, & Mitchell, 2009). In recent years, theories of embodiment have received increasing attention as new virtual learning technologies tap into the body movement and perception (Lindgren & Johnson-Glenberg, 2013). Immersive experiences allow users to be inside a setting and align the body movement with the environment. While embodied learning can occur without advanced technologies, VR platforms are well suited to promote embodied learning as they combine the physical movement and a virtual environment in new ways. ### Hypothesis development Learning activities can be classified into three categories: cognitive, affective and metacognitive activities (Vermunt, 1996). Cognitive activities include comprehension, analysis, memory and application of knowledge. Affective activities include learners’ subjective perception of the learning experience, emotion, satisfaction and attitude. Metacognitive activities refer to planning, evaluation and reflection. Similarly, learning outcomes can be cognitive or skills based (Kraiger, Ford, & Salas, 1993). This study focuses on cognitive learning outcomes as well as affective reaction and also attempts to explore the relationship between the two. ### Cognitive outcome: learning gain There are two opposing perspectives on the potential of learning gain in a virtual environment. Theories of embodied learning suggest that bodily movements congruent with the learning materials is beneficial for knowledge acquisition and recall. Researchers of gestures, in particular, have found evidence that motor representation of the body plays an important role in retrieving existing knowledge and comprehending new information (Saltz & Donnenwerth-Nolan, 1981). When people gesture and perform actions along with a speech, the information encoded in that speech is more memorable than without gesture (Cook, Mitchell, & Goldin-Meadow, 2008; `Stevanoni & Salmon, 2005). Different theories seem to offer competing predictions about whether immersive technologies would aid or hinder learning. At the time of VR exposure, embodied learning could lead to learning advantage, but that may be outweighed by extraneous processing of vivid details and subsequent cognitive overload (Makransky et al., 2019). After examining the influence of three levels of audiovisual immersive technology using two-week long intersession intervals, Pollard et al. (2020) found performance on directional bearings as a U-shaped relationship with level of immersion, suggesting that higher levels of immersion may not always improve learning. In an experiment examining how embodied systems affect learning of physics knowledge, Johnson-Glenberg, Megowan-Romanowicz, Birchfield, and Savio-Ramos (2016) found that although immediate learning appeared to be the same across different systems, users of high embodiment systems were able to retain more knowledge after a week and better apply it to new problems, thus exhibiting better transfer of training. Similarly, studies on children found that gesturing during the instruction helped them to retain knowledge better, suggesting that movement “makes learning last” (Cook et al., 2008). Therefore, we hypothesize that immediate cognitive learning outcomes will be similar for low- and high-embodied conditions. However, over the long run, the extraneous processing ceases, whereas the benefits of embodied learning may endure leading to VR participants to remember more over time. H1a. The immediate learning gains after the instruction will be similar in both low- and high-embodied platform. H1b. The learning retention at a follow-up test will be higher in a high-embodied platform than in a low-embodied platform. ### Affective reaction: enjoyment Much of the literature on embodied learning focuses on cognitive outcomes, but not enough attention has been paid to affective reaction. Existing studies have shown that users of a VR instructional platform find it entertaining and fun (Beltrán Sierra, Gutiérrez, & Garzón-Castro, 2012), and that this positively influences their motivation to continue using the tool (Gallego, Bueno, & Noyes, 2016). Thus, in this study, we focus on enjoyment as an element of affective reaction. Three key factors make immersive VR experience an enjoyable one. First is enhanced simulation. Empirical studies have exhibited increased interest, engagement and perceived effectiveness from incorporating simulation into management teaching (Keys & Wolfe, 1990; Lu, Hallinger, & Showanasai, 2014). As a new simulation technology, immersive VR not only offers the same value but also provides additional benefits because of its enhanced technical features. Second, immersive VR has been consistently demonstrated to induce greater sense of presence among users than traditional desktop environment (Bailey, Bailenson, Won, Flora, & Armel, 2012; Makransky et al., 2019; Moreno & Mayer, 2002). Presence (place presence, social presence and copresence) is the psychological experience of “being there,” and it leads to higher satisfaction of the learning experience (Bulu, 2012; Vrellis, Avouris, & Mikropoulos, 2016). Third, the novelty of this emerging technology adds to the appeal for users, though some argue that with widespread use it is likely to fade over time (Kavanagh, Luxton-Reilly, Wuensche, & Plimmer, 2017). A combination of enhanced simulation, presence and novelty can contribute to a more enjoyable and preferable experience in an immersive VR environment. We therefore hypothesize more positive experience with the VR platform in self-reports of enjoyment. H2. The experience of a high-embodied platform will be more enjoyable than the experience of a low-embodied platform. ### Enjoyment and learning retention Both theory and empirical evidences suggest that positive emotions enhance learning through the mediating effect of increased motivation. In addition to advances in motivational theory (Renninger & Hidi, 2016; Wentzel & Miele, 2009), several emerging theoretical perspectives also point to the important role of positive affect on learning. Some examples include the control value theory of achievement emotions (Pekrun, 2000), cognitive affective theory of learning with media (Moreno & Mayer, 2007) and affective computing (Picard, 1997). They argue that positive emotions are likely to increase the “interest and motivation to learn” (Pekrun, 2006, p. 326). And motivation has been found to have positive effects on virtual learning (Benbunan-Fich & Hiltz, 2003; Piccoli, Ahmad, & Ives, 2001; Salzman, Dede, Loftin, & Chen, 1999). In some recent empirical studies, structural equation models have shed light on an “affective path” where VR features increased presence and motivation, which in turn improves learning results (Ai-Lim Lee, Wong, & Fung, 2010; Makransky & Petersen, 2019). The impact of positive affect on learning can potentially occur at two different junctures in the learning process: the initial knowledge acquisition, subsequent knowledge retention or both. An experiment demonstrated that positive emotions at the time of initial learning had not improved subsequent recall (Isen, Shalker, Clark, & Karp, 1978). During the study, participants were induced with either positive or negative affect before being asked to learn a list of words. They then were induced with positive or negative affect again with a different set of stimuli before a recall test. The difference in recall was attributable to the affective state at the time of recall, regardless of the affective state at the time of knowledge acquisition. This study suggested a need to examine affect beyond the initial learning stage, and that long-term retention could be related to learners’ emotional state at the time of recall, not just at the time of knowledge acquisition. We hypothesize that enjoyment has a positive impact on immediate learning and subsequent retention. When people feel good about the learning experience, they are more likely to remember the content over a longer period. H3a. Participant enjoyment at the time of learning is positively correlated with the immediate learning gain. H3b. Participant enjoyment at the time of follow-up is positively correlated with the learning retention. ## Methods ### Experimental design and virtual reality apparatus We conducted an experiment to assess how well participants learned statistical concepts, as a representation of an example of abstract conceptual learning that can be found in management, via different instructional platforms. There were three conditions varying by the level of embodiment of the platform. The low-embodied condition (Video) used a video with Excel demonstration viewed from a laptop. The medium-embodied condition SVR (Static VR) utilized a VR demonstration viewed from a head-mount display, and the high-embodied condition EVR (Exploratory VR) used the same equipment as the medium-embodied condition but with the addition of handheld controls to manipulate freely the VR dataset at the end of the instruction. Plate 1 shows participants in the SVR and EVR conditions during the experiment. Instructions in all three conditions were recorded by the same speaker, following the same script, using the same data set and covering the same length in time. The VR system used in this experiment was the Oculus Rift S, a PC-tethered head-mounted display. The Rift S allows users to move with six degrees of freedom, which refers to tracking display movement in three-dimensional space with forward/backward, up/down and left/right moves around three perpendicular axes. Accordingly, the user’s position changes in virtual space reflect their movement in the physical world. The Rift S is navigated with two controllers, which the user can see in VR. In this experiment, the EVR group interacted with a virtual laser pointer that they could use to select various objects and menus in the virtual space. ### Participants, procedure and measures Seventy-five undergraduate sophomore business students enrolled from a private university in the USA participated in the experiment. In total, 38 were male (51%) and 37 were female (49%). Students received information about a study on technology platforms and learning outcomes for basic statistics and were asked to sign up. All participants were each given a $5 Starbucks coupon and entered into a lottery with five winners each receiving$100. Participants were randomly assigned to one of the three conditions, with 25 participants in each condition. First, the participants were given a pretest to assess their self-reported familiarity with statistics and with the technology of their experimental platform (e.g. VR or Excel). They were also given three statistical questions to objectively assess their baseline statistics knowledge. Participants in the SVR condition watched a recorded 8-min instruction on statistical concepts using a real estate price data set. Afterward, they were given 12 multiple choices questions to assess their understanding of concepts explained in the session. They were then asked about how much they enjoyed the session and how much they enjoyed the data presentation. The only difference for participants in the EVR condition was that they were given 5 min after the instruction to use their handheld controls to freely manipulate the data set in the VR environment. In the Video condition, the participants watched an 8-min recording on the computer. The video covered the same content as the VR video with Excel illustration. Participants in all conditions received the same pretest and posttest. They were also given an online follow-up survey 15 days later. It contained the same statistics and subjective questions as the posttest. In total, 62 of the 75 subjects (83%) participated in the follow-up test (21 for Video, 22 for SVR, 19 for EVR). ## Results Participants’ pretest responses were compared to validate initial equivalence between conditions. ANOVA showed no significant difference across conditions on either self-reported statistics proficiency or the objective test from three statistical questions (see Table 1). Participants reported significantly higher proficiency with Excel than with VR, but the two VR groups had similar level of familiarity with the technology (t(48) = 0.77, p > 0.10). To test H1a, a measure of immediate learning gain was constructed by the difference of the participant score on the three pretest statistics questions from the scores on the same questions at posttest (MEVR = 0.80, SDEVR = 2.08, MSVR = 1.04, SDSVR = 1.93, MVideo = 1.28, SDVideo = 1.51). ANOVA results showed no significant difference among three groups on immediate learning gain, F(2, 72) = 0.42, p > 0.10. H1a was supported as the immediate learning gain was similar in high-, medium- and low-embodied platforms. To test H1b on learning retention, we constructed a measure of learning retention by the difference between posttest and follow-up test on 12 statistical questions. There was a general learning loss across all conditions, as negative retention score indicates loss in knowledge (MEVR −1.58, MSVR = −0.82, MVideo −3.52, MTotal −1.97). ANOVA analysis showed a significant difference between groups (F(2, 59) = 3.24, p < 0.05). Eta squared was 0.099, indicating a medium to large effect size. Post hoc tests suggested significant difference between SVR and Video conditions (see Table 2). Participants in the Video group experienced more than four times the learning loss as those in the SVR group. There was no significant difference between SVR and EVR in retention. Figure 1 shows the comparison of immediate learning gain and retention by condition. To understand affective reaction to the learning experience, we asked participants to rate how much they enjoyed the session and the data presentation. The same assessments were taken at posttest and follow-up. Responses to the two questions were highly correlated (r = 0.72 at posttest, r = 0.78 at follow-up). Paired t-test showed that reported enjoyment for the three groups all decreased significantly over two weeks (session enjoyment Mpost-test = 5.15, Mfollowup = 4.50, t(61) = 3.75, p < 0.001; data presentation enjoyment Mpost-test = 5.44, Mfollowup = 5.00, t(61) = 3.13, p < 0.01). ANOVAs showed that both measures of enjoyment at both times differed significantly across conditions (see Table 3). Post hoc Tukey tests showed that at posttest, participants in VR conditions enjoyed the session much more than those in the Video condition with no significant difference between SVR and EVR. Enjoyment of the data presentation yielded the same result: Video participants enjoyed it much less than those in SVR and EVR conditions, and the two VR conditions had similar ratings. The same results were found at follow-up test, that is, participants in the Video condition enjoyed the session and the data presentation significantly less than both VR conditions, and there was no significant difference between SVR and EVR participants. H2 was partially supported in that low-embodied Video platform was experienced as significantly less enjoyable, but there was no difference between medium- and high-embodied conditions. To test the relationship between affective reaction and cognitive learning outcomes in H3a, Pearson correlation was calculated between enjoyment at the time of learning (i.e. posttest) and immediate learning gain. The correlation was not significant (r = −0.15, p > 0.10 for session enjoyment, r = −0.75, p > 0.10 for data presentation enjoyment). H3a was not supported as there was no significant relationship between enjoyment at the time of learning and immediate learning gain. For H3b, similar Pearson correlation was calculated between enjoyment and learning retention score. At the time of recall, enjoyment of the data presentation was significantly correlated with retention (r = 0.27, p < 0.05), whereas enjoyment of the session had no significant correlation (r = 0.20, p > 0.10). H3b was partially supported. At the time of recall, participants who enjoyed the data presentation more were able to retain more of the initial learning. ## Discussion Through a longitudinal experimental design, the study compared three instructional platforms with varying degrees of embodiment. Several key findings have emerged. First, the study showed that an immersive VR platform produced longer lasting learning benefits than traditional video platforms. This is extremely important to note because traditional HR training has been relying on video platforms as the main medium of training. All platforms achieved similar level of immediate learning gain post instruction. However, participants in the Static VR condition experienced significantly higher retention two weeks afterward (see Figure 2). Second, subjective experience of VR instructions was significantly more enjoyable than the traditional video format, and the enjoyment advantage persisted after two weeks. Moreover, there was evidence suggesting a relationship between enjoyment and learning retention. Correlation tests showed participants expressing higher enjoyment at the time of recall were also able to better retain their initial learning. Consistent with findings from previous research (Isen et al., 1978), enjoyment at the time of learning had no relationship with learning gain. It was positive emotions at the time of recall that were associated with better retention. Lastly, we hypothesized that cognitive learning outcomes and affective reaction would be different between medium-embodied SVR and high-embodied EVR conditions. Results suggested that neither immediate learning gain nor retention was significantly different. Assessment of enjoyment did not differ significantly either. Based on the cognitive theory of multimedia learning theory (Mayer, 2005, 2009), VR platforms may cause cognitive overload through extraneous processing and hinder learning as a result. It can be argued that the potential benefits of high-embodied EVR setup may have been offset by the additional mental resources required when participants were manipulating the virtual environment with the controllers after the instruction. Another explanation could point to the research design where physical movement of handheld controls was only available to the participants after instruction in the EVR condition. This hand movement was open to individuals for voluntary exploration of the VR data set without any monitoring. As a result, the movement may have not been sufficient in an overall magnitude across all participants to show a different impact on learning from the other conditions. It could also be plausible that the participants may have been distracted by playing with a novel handheld device at the end and failed to remember principles of statistics taught earlier. The impact of recency bias could have been at play here and a worthy pursuit for further exploration. ### Theoretical and practical implications This study makes several contributions both to the embodied learning theory and its applications. This is the first study that investigates the superiority of VR platforms for measurable retention of abstract concepts in addition to subjective experiences. Another theoretical contribution comes from the longitudinal design for exploring learning outcomes over time. Although we did not observe an immediate learning advantage, we did find longer lasting learning outcomes for the VR platform (Pollard et al., 2020). This study also contributes to the research on affect and learning. Most models of virtual learning primarily focus on cognitive outcomes. While VR platforms did not improve cognitive outcomes immediately after instruction, they produced more enjoyable experience, which was positively associated with better retention over time. Future research is needed to understand how to elicit certain emotions in the virtual learning environment, how to manage the evolution of the emotional reaction over time and how both the immediate emotional reaction and longitudinal change in emotional response can facilitate learning. This study demonstrates that VR learning platforms can be a valuable tool for management and business training. VR system can not only improve affective reaction to the learning experience but can also keep learners motivated to retain knowledge. The study can enlighten educators to use carefully design VR application for other than statistics content such as sustainable operations and decision-making and can even be useful for other management skills and training such as giving feedback, sexual harassment or ethics training or even new employee orientations. Educators can create technical features and cognitive contents to make remote learning more effective and affective. Certain elements of the system can be designed to elicit lasting emotions to facilitate learning (Plass & Kaplan, 2016). ## Limitations This study seeks to shed light on how VR platforms impact cognitive learning outcomes and affective reactions with several limitations. First, the study participants were college students. People of different age groups may interact and learn differently in embodied systems. Second, enjoyment from VR could be associated with the novelty of the technology. Majority of the participants not familiar with VR may have been more likely to experience excitement. Whether enjoyment would diminish as a result is a question for further study. Finally, the longitudinal approach of this study has offered some interesting insights about knowledge retention, and even a longer time frame could reveal different dynamics. ## Figures ### Figure 1. Immediate learning gain and learning retention by condition ### Figure 2. Learning gain and learning retention per question over time ### Plate 1. Participants in EVR and SVR conditions ## Table 1. ANOVAs to validate initial equivalence between conditions Variable Mean (SD) Low-embodied (Video) Medium-embodied (SVR) High-embodied (EVR) F-test p-value Statistics: self-reported 4.28 (1.40) 4.76 (1.33) 4.56 (1.19) 0.85 0.434 Statistics: objective 3.76 (1.45) 3.68 (1.38) 3.92 (1.68) 0.16 0.849 Platform familiarity 4.76 (1.14) 2.60 (1.61) 3.00 (2.06) 11.71*** 0.000 Notes: p < 0.05, p < 0.01, *p < 0.001. N = 75 ## Table 2. Descriptive statistics and ANOVA for learning retention score at follow-up Condition N Mean SD EVR 19 −1.58 4.30 SVR 22 −0.82 3.25 Video 21 −3.52 3.16 Total 62 −1.97 3.70 Sum of squares df Mean square F Sig. Between groups 82.79 2 41.40 3.24 0.046 Within groups 753.14 59 12.77 Total 835.94 61 (I) Group (J) Group Mean difference (I − J) Sig. EVR SVR −0.76 0.776 Video 1.94 0.207 SVR EVR 0.76 0.776 Video 2.71* 0.042 Video EVR −1.94 0.207 SVR −2.71* 0.042 Notes: p < 0.05, p < 0.01, p < 0.001 ## Table 3. 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Journal of Educational Psychology, 106(1), 86104. doi: 10.1037/a0034008. ## Acknowledgements We would like to acknowledge William Miller and Akshay Pai from Fordham MBA program, and Andy Maggio from D6 and the Glimpse Group for their involvement in the study design and the experiment. We would also like to thank Dr. Yuliya Komarova for enabling partial financial support from Fordham University behavior research lab. Funding: Partial funding for this research was provided by Fordham University behavior research lab. ## Corresponding author Kanu Priya can be contacted at: [email protected]	2023-02-01 06:01:13	{"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": 0.33225855231285095, "perplexity": 6921.630264498132}, "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/1674764499911.86/warc/CC-MAIN-20230201045500-20230201075500-00163.warc.gz"}
https://byjus.com/question-answer/a-man-sold-two-articles-at-375-each-on-the-first-article-he-gains-25/	Question # A man sold two articles at $$375$$ each. On the first article, he gains $$25$$% and on the other, he loses $$25$$%. How much does he gain or lose in the whole transaction? Also, find the gain or loss percent in the whole transaction. Solution ## First article$$SP =375$$Gain$$\% =25\%$$CP $$=\cfrac{ 100 \times 375}{125} = 300$$Second article$$SP =375$$Loss $$\% =25\%$$CP $$=\cfrac{ 100 \times 375}{75} = 500$$Total CP $$= 500+300 = 800$$Total SP $$= 375+375=700$$Loss $$=$$ CP $$-$$ SP Loss $$= 800 - 750 = 50$$Loss $$\%= \cfrac{50 \times 100}{800} = 6.25$$%Mathematics Suggest Corrections 0 Similar questions View More People also searched for View More	2022-01-29 01:47:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21889102458953857, "perplexity": 10185.39589293157}, "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/1642320299894.32/warc/CC-MAIN-20220129002459-20220129032459-00673.warc.gz"}
https://math.stackexchange.com/questions/474495/field-structure-on-mathbbr2/474559	# Field structure on $\mathbb{R}^2$ I have the following question: Is there a simple way to see that if we put a multiplication $$ on $\mathbb{R}^2$ (considered as a vector space over $\mathbb{R}$) such that with usual addition and this multiplication $\mathbb{R}^2$ becomes a field, then there exists a nonzero $(x,y)$ such that $(x,y)(x,y)=-1$? Remark: 1. What I mean by "a simple way to see" is that I really don't want to refer to Frobenius's Theorem on real finite dimensional division algebras. 2. I haven't said this in the problem but I'm also assuming that with this multiplication $\mathbb{R}^2$ becomes an algebra meaning $x(\alpha y)=\alpha(xy).$ • It's not true, even if we demand $(1,0)\ast (x,y) = (x,y)$. You can let $(0,1)$ correspond to any $z \in \mathbb{C}\setminus\mathbb{R}$ and the corresponding multiplication makes $\mathbb{R}^2$ a field, but $(0,1)\ast(0,1) = -1$ only if you choose $z = \pm i$. – Daniel Fischer Aug 23 '13 at 17:57 • I see your point. Now I edited my question. – Abelvikram Aug 23 '13 at 18:05 • No. Assuming the axiom of choice, you can define multiplication to make $\mathbb{R}^2$ isomorphic to $\mathbb{R}$ (with the usual multiplication). Just consider dimensions as vector spaces over $\mathbb{Q}$. – George Lowther Aug 23 '13 at 18:27 • I'm considering $\mathbb{R}^2$ as a vector space over reals. – Abelvikram Aug 23 '13 at 18:30 • In that case, the answer is yes. As its an extension of degree 2, it is quadratic, hence obtained by appending a single square root of some non-square (hence negative) element of $\mathbb{R}$. As the square root of a negative number is a square root of $-1$ multiplied by a real, the extension is generated by appending the square root of $-1$. – George Lowther Aug 23 '13 at 18:47 Sorry if I make it too elementary: If $1\in\mathbb R^2$ denotes $1$ of your field, and if $x\in\mathbb R^2$ is not its real multiple: $1,x,x^2$ are linearly dependent (over $\mathbb R$), i.e. $ax^2+bx+c=0$ for some $a,b,c$, and $a\neq 0$ (as $x$ is not a multiple of $1$), so we can suppose it's 1. If we complete squares, we get $(x+p)^2+q=0$ for some $p,q\in \mathbb R$. Now $q$ must be positive - otherwise $(x+p+\sqrt{-q})(x+p-\sqrt{-q})=0$, so you don't have a field (we found divisors of $0$). So finally $(x+p)/\sqrt{q}$ is the element you want. • Thanks for the answer. It's really nice. – Abelvikram Aug 24 '13 at 4:46 • +1, if only for the first sentence of the post, which apologizes for making it too elementary... – Did Aug 26 '13 at 18:36 The usual addition forces $\Bbb R$ to be a two dimensional subfield of your field $F$ (Consider $\Bbb R1_F)$. If you assume the fundamental theorem of algebra, and have some background in field theory, then it is relatively straightforward. Assume no root of $x^2+1=0$, then $$F(i)=F[x]/(x^2+1)\ \text{is degree 2 over F}$$ so $F(i)$ is a degree four finite (therefore algebraic) extension of $\Bbb R$ as $$[F(i):F][F:\Bbb R]=[F(i):\Bbb R]$$ But this cannot happen as any finite extension of $\Bbb R$ is contained inside a field isomorphic to $\Bbb C$. (since $\Bbb C$ is the algebraic closure of $\Bbb R$ and is of degree two over $\Bbb R$). • sorry, but the point of the question is to find a simple proof. If you see the proof of Frobenius Theorem, this itself uses only the basic linear algebra. I was wondering that in case of $\mathbb{R}^2$ may be it's more simple. But anyway thanks for your answer. – Abelvikram Aug 23 '13 at 18:34 • The other point being that I wanted to make understand this to undergrad. students who have done only a basic course in linear algebra so that's why the question. – Abelvikram Aug 23 '13 at 18:42 You are looking at a field extension of $\Bbb R$ of degree two, but by the fundamental theorem of algebra, that field sitting above $\Bbb R$ is unique up to isomorphism and is isomorphic to $\Bbb C$. • But I fear now the question will change to "and without referring to the FTA..." :) – rschwieb Aug 23 '13 at 18:18	2020-08-07 01:10:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9593521952629089, "perplexity": 186.95155913598668}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737050.56/warc/CC-MAIN-20200807000315-20200807030315-00157.warc.gz"}
http://absolutewrite.com/forums/showthread.php?333364-Blinding-the-inner-editor&p=10319145	I also highlight font with another color when I know that work is needed on a section. It allows me to keep writing without dwelling on the issue needing more research , fleshing out, or editing. I just get the idea down and keep going. But, I do also get stuck reading and re-reading what I've written. I like the idea of a font that I can't quite read. Although I cringe to think about how many typos I'd find after changing the color back! Flubby fingers are what get me stuck the most. I stop to fix something silly and forget where I was going.	2018-08-21 09:41:49	{"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": 0.9142418503761292, "perplexity": 911.6639843828328}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221218101.95/warc/CC-MAIN-20180821092915-20180821112915-00530.warc.gz"}
https://socratic.org/questions/how-do-you-find-the-exact-value-of-the-six-trigonometric-functions-of-the-angle--6#420773	# How do you find the exact value of the six trigonometric functions of the angle whose terminal side passes through (x, 4x)? May 10, 2017 $\sin \theta = \frac{4}{\sqrt{17}}$, $\cos \theta = \frac{1}{\sqrt{17}}$, $\tan \theta = 4$, $\cot \theta = \frac{1}{4}$, $\sec \theta = \sqrt{17}$, $\csc \theta = \frac{\sqrt{17}}{4}$ #### Explanation: As the terminal side passes through $\left(x , 4 x\right)$, we have $y = 4 x$ and hence its distance from origin is $\sqrt{{x}^{2} + {\left(4 x\right)}^{2}} = \sqrt{{x}^{2} + 16 {x}^{2}} = x \sqrt{17}$ Now consider the diagram below for a typical $\theta$, whose six trigonometrical ratios are $\sin \theta = \frac{y}{r}$, $\cos \theta = \frac{x}{r}$, $\tan \theta = \frac{y}{x}$ and $\cot \theta = \frac{x}{y}$, $\sec \theta = \frac{r}{x}$, $\csc \theta = \frac{r}{y}$ As we have $y = 4 x$ and $r = x \sqrt{17}$, the six trigonometric ratios are $\sin \theta = \frac{4 x}{x \sqrt{17}} = \frac{4}{\sqrt{17}}$, $\cos \theta = \frac{x}{x \sqrt{17}} = \frac{1}{\sqrt{17}}$, $\tan \theta = \frac{4 x}{x} = 4$, $\cot \theta = \frac{x}{4 x} = \frac{1}{4}$, $\sec \theta = \frac{x \sqrt{17}}{x} = \sqrt{17}$, $\csc \theta = \frac{x \sqrt{17}}{4 x} = \frac{\sqrt{17}}{4}$	2022-06-29 09:36:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 24, "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": 0.9755913019180298, "perplexity": 345.02685077691183}, "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/1656103626162.35/warc/CC-MAIN-20220629084939-20220629114939-00653.warc.gz"}
https://codereview.stackexchange.com/questions/58714/java-sort-by-selection-which-is-better-in-regards-to-readability-and-efficiency/58715	# Java sort by selection: which is better in regards to readability and efficiency? Which algorithm do you think is better in regards to readability and efficiency? selection(): public static int[] selection(int[] tab) { int minor; int aux; for (int i=0; i<tab.length-1; i++) { minor = tab[i]; for (int j=i+1; j<=tab.length-1; j++) { if (tab[j] < minor) { aux = tab[j]; tab[j] = minor; minor = aux; } } tab[i] = minor; } return tab; } otherSelection(): public static int[] otherSelection(int A[]) { int i, j, minor, pos, tmp; for (i = 0; i < A.length - 1; i++) { minor = A[i]; pos = i; for (j = i + 1; j < A.length; j++){ if (A[j] < minor) { minor = A[j]; pos = j; } } if (pos != i){ tmp = A[i]; A[i] = A[pos]; A[pos] = tmp; } } return A; } • I think you'll have better luck with this question Programmers Exchange – MadProgrammer Aug 1 '14 at 2:02 • Better in what sense? Readability? Efficiency? Maintainability? etc – But I'm Not A Wrapper Class Aug 1 '14 at 2:03 • Mainly in readibility and efficency. Which one do you like the most? Thank you. – Víctor Aug 1 '14 at 2:05 • May I introduce a feature that's around since Java 1.5 (or nearly a decade): enhanced for each loops. And please learn about Collections (introduced in Java 1.2). And please read Clean Code. – Martin Schröder Aug 1 '14 at 12:54 The first one is far more readable because it can be translated at so: selection(tab) for each value of tab: i = (0->size-1) minor = tab[i] for each value of tab: j = (i->size-1) if (tab[j] < minor) swap(minor, tab[j]) tab[i] = minor; return tab; This is very simple and symmetric. The otherSelection is a bit more difficult to read because it tracks a pos. This is a uncommon way to do this sort (not necessarily bad). Then it does the swap post inner loop. This was personally more difficult for me to read. The first wins here. Time Efficiency: Both run $O(n^2)$. However, otherSelection does only swap once per n-iteration. This actually is more efficient to do. Space Efficiency: selection only needs 4 int variables where as otherSelection needs 5. It's a bit more space efficient to use selection. As user @200_success mentioned, the space efficiency is the same at worse case since both are constant space $O(1)$. This isn't counting the input space (which you shouldn't really anyways). Final verdict: selection is a better option than otherSelection as far as readability. However, otherSelection is better for efficiency. This is a better variant of otherSelection which is just as good when it comes to space efficiency: public static void selectionSort2(int[] x) { for (int i=0; i<x.length-1; i++) { int minIndex = i; // Index of smallest remaining value. for (int j=i+1; j<x.length; j++) { if (x[minIndex] > x[j]) { minIndex = j; // Remember index of new minimum } } if (minIndex != i) { //... Exchange current element with smallest remaining. int temp = x[i]; x[i] = x[minIndex]; x[minIndex] = temp; } } } If you care for efficiency, this is the better selection sort option. You could also do a bit-swap for when swapping the variables. Final notes: Since this is Java, you don't need to do the return statement at the end. • For all practical purposes, space efficiency is a tie. – 200_success Aug 1 '14 at 3:48 • @200_success Yes. In both cases, the worse case space efficiency is constant space. I was just nitpicking at that point. – But I'm Not A Wrapper Class Aug 1 '14 at 3:55 In terms of readability, I would go for the first one selection. You would like to do everything cleanly inside 2 nested for clarity. Both are $O \left( n^{2} \right)$. First one is definitely better. More variables you have, the harder it is to follow them. That said, I'd suggest applying some rules: 1. A <= comparison is very unusual and triggers an unnecessary attention 2. Declare a variable in the scope it it used in 3. Recognize important algorithms and factor them out 4. A loop usually represent an important algorithm Let's apply the first rule: public static int[] selection(int[] tab) { int minor; int aux; for (int i=0; i<tab.length-1; i++) { minor = tab[i]; for (int j=i+1; j<tab.length; j++) { if (tab[j] < minor) { aux = tab[j]; tab[j] = minor; minor = aux; } } tab[i] = minor; } return tab; } Then the second: public static int[] selection(int[] tab) { for (int i=0; i<tab.length-1; i++) { int minor = tab[i]; for (int j=i+1; j<tab.length; j++) { if (tab[j] < minor) { int aux = tab[j]; tab[j] = minor; minor = aux; } } tab[i] = minor; } return tab; } The code in if clause is one of the most fundamental algorithms, namely swap. I am afraid Java doesn't support swapping the way C++ programmers use; we just can't pass an integer to a function to have it modified. In this particular case Java "deficiency" pushes us to the right direction: what we actually want is to swap values in array - no need for minor at all! Here is an application of a third rule (with swap provided separately): public static int[] selection(int[] tab) { for (int i=0; i<tab.length-1; i++) { for (int j=i+1; j<tab.length; j++) { if (tab[j] < tab[i]) { swap(tab, i, j); } } } return tab; } Now we may focus on the fourth rule: what does the inner loop represent? A swap which finally settle the value of tab[i] is the smallest tab[j] in the range. This would be my final code (index_of_min provided separately): public static int[] selection(int[] tab) { for (int i=0; i<tab.length-1; i++) { int index_of_min = index_of_minimum(tab, i + 1, tab.length); if (tab[index_of_min] < tab[i]) { swap(tab, i, index_of_min) } } return tab; } BTW, now an opportunistic optimization is obvious (I am not sure I'd go for it but anyway): public static int[] selection(int[] tab) { for (int i=0; i<tab.length-1; i++) { int index_of_min = index_of_minimum(tab, i + 1, tab.length); if (tab[index_of_min] >= tab[i]) { ++i; } swap(tab, i, index_of_min); } return tab; }	2021-05-06 19:21:41	{"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": 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": 0.21942582726478577, "perplexity": 8160.097984171636}, "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-2021-21/segments/1620243988759.29/warc/CC-MAIN-20210506175146-20210506205146-00587.warc.gz"}
https://www.encyclopediaofmath.org/index.php/Kronecker%E2%80%93Capelli_theorem	Kronecker-Capelli theorem (Redirected from Kronecker–Capelli theorem) compatibility criterion for a system of linear equations A system of linear equations $$\begin{array}{ccc} a_{11} x_1 + \cdots + a_{1n}x_n &=& b_1 \\ \vdots & \vdots & \vdots \\ a_{n1} x_1 + \cdots + a_{nn}x_n &=& b_n \end{array}$$ is compatible if and only if the rank of the coefficient matrix $A = (a_{ij})$ is equal to that of the augmented matrix $\bar A$ obtained from $A$ by adding the column of free terms $b_i$. Kronecker's version of this theorem is contained in his lectures read at the University of Berlin in 1883–1891 (see [1]). A. Capelli was apparently the first to state the theorem in the above form, using the term "rank of a matrix" (see [2]). References [1] L. Kronecker, "Vorlesungen über die Theorie der Determinanten" , Leipzig (1903) [2] A. Capelli, "Sopra la compatibilitá o incompatibilitá di più equazioni di primo grado fra picì incognite" Revista di Matematica , 2 (1892) pp. 54–58 [3] A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) How to Cite This Entry: Kronecker–Capelli theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Kronecker%E2%80%93Capelli_theorem&oldid=22670	2019-11-14 01:44:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8338552117347717, "perplexity": 1796.4749149757426}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496667767.6/warc/CC-MAIN-20191114002636-20191114030636-00045.warc.gz"}
https://physics.stackexchange.com/questions/599280/what-happens-when-a-light-is-shone-across-a-spaceship-in-its-direction-of-motion	# What happens when a light is shone across a spaceship in its direction of motion? [closed] Imagine a space ship in S frame moving with speed v (in the x direction) relative to S'. An astronaut on board shines a light from the the rear of the spacecraft to the front (along the x-direction). The astronaut measures the time taken for the light to reach the other end to be T. The proper length of the spacecraft is L. The astronaut would then determine the speed of light to be: $$c = L/T\tag{1}$$ From the S' frame, a stationary observer would measure the length of the spacecraft to be: $$l' = L/\gamma$$ Then the total distance travelled by the light ray would be: $$d = vt' + l' = vT\gamma + L/\gamma$$ This is because the light ray would have to traverse the contracted length of the spacecraft ($$l'$$), but also cover the additional distance that the craft has moved from the starting point ($$vt'$$) as seen from the $$S'$$ frame. and $$d$$ would also be: $$d = ct' = cT\gamma$$ which leads to $$cT\gamma = vT\gamma + L/\gamma$$ from which follows $$c = (1/\gamma^2)(L/T) + vT\gamma\tag{2}$$ Now, if we compare equations $$(1)$$ and $$(2)$$ we would find that the two observers would disagree on the speed of light. So, my question is where has this argument gone wrong? My first thoughts would be that it is incorrect to just simply add on the extra distance that the spacecraft travels to the total distance measured from $$S'$$, but I can't seem to understand why that would be the case. The mistake is in your formula for $$d$$, which involves a $$t'$$. I don't think you've been careful about what $$t'$$ represents. So let's start over: 1. In the frame of the spacecraft, the light leaves location $$x=0$$ at time $$t=0$$ and arrives at location $$x=L$$ at time $$t=L$$. 2. Lorentz transform this to the earth-frame (which is moving at velocity $$-v$$ with respect to the ship). You'll find that the light leaves location $$x'=0$$ at time $$t'=0$$ and ultimately arrives at location $$x'=\beta L$$ at time $$t'=\beta L$$ where $$\beta=\sqrt{(1+v)/(1-v)}$$. 3. Along the way (calculating in the earth frame), the light passes the original location of the front of the ship, which is to say $$x'=L'$$. This must happen at time $$t'=L'$$. 4. So: in the earth frame, we can break the light's journey up into two parts. PART ONE: The light travels to where the front of the ship used to be --- distance $$L'$$. PART TWO: The light travels from there to its ultimate destination. According to paragraph 2), the entire journey has length $$\beta L$$. So PART TWO has length $$\beta L - L'$$. 5. You've got the length of PART TWO as $$vt'$$, but you haven't told us what $$t'$$ is. Apparently you mean for it to represent the time taken (in the earth frame) for PART TWO of the journey. If so, $$t'=\beta L-L'$$, but you seem to have assumed otherwise. Edited to add: I haven't worked through the details of your calculations, but I'm pretty sure I know what led you astray: 1. The length of a rod (in your frame) is the distance between two events that happen at the same time (in your frame). Those events are, for example, "left end of rod at noon" and "right end of rod at noon". 2. The distance between the completions of PARTS ONE and TWO of the journey is the distance between two events that happen at different times (in your frame). Those events are the arrival of the light where, according to you, the front of the ship used to be and the arrival of the light at the actual front of the ship. 3. You have a formula for computing the length of a rod in your frame, given the length in some other frame. You are (I think, though I haven't fully worked through it) trying to apply that formula to the two events in point 2) above. But those events are not the sort of events to which the formula applies. 4. One way to avoid such mistakes is to be very careful to memorize all the conditions in which you can use your formula. A better way is to forget the formula entirely and work things out from first principles and Lorentz transformations, as I've done above.	2022-05-25 01:13:12	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 34, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8491506576538086, "perplexity": 207.4512430344261}, "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/1652662577757.82/warc/CC-MAIN-20220524233716-20220525023716-00696.warc.gz"}
https://mathoverflow.net/questions/360488/lefschetz-type-theorems-for-linear-sections	# Lefschetz type theorems for linear sections Let $$X\subset\mathbb{P}^n$$ be e normal variety, $$L\subset\mathbb{P}^n$$ a linear subspace, and $$Y = X\cap L$$ a linear section. Assume that $$Y$$ is also normal. In particular, we have that $$Sing(X)$$ has codimension at least two in $$X$$, and $$Sing(Y)$$ has codimension at least two in $$Y$$. Does there exist any generalization of the Lefschetz hyperplane theorem to higher codimension linear sections of singular varieties ensuring that the restriction map $$Cl(X)\rightarrow Cl(Y)$$ is an isomorphism or at least surjective? • You would need some assumptions. For instance, a rank 5 quadric $X$ in $\mathbb{P}^n$ has $\operatorname{Cl}(X)= \mathbb{Z}$, but it admits a hyperplane section $Y$ of rank 4, hence $\operatorname{Cl}(Y)=\mathbb{Z}^2$. – abx May 16 at 8:02 • In your example we have $Sing(X) = Sing(Y)$ if I got it right. In the case I am interested in I know that $Sing(X) = Sing(Y)\cap L$ but $Sing(X)\neq Sing(Y)$. – Jessica_90 May 16 at 8:16 • Look for Grothendieck-Lefschetz theorey, for instance this paper: arxiv.org/pdf/1601.05846.pdf – Hailong Dao May 16 at 16:28 • In Theorem 1 (v) here dima.unige.it/~badescu/attivita%20scientifica/… they say that if $Y\subset \mathbb{P}^n$ is normal of dimension at least three and $Y$ can be set-theoretically defined by at most $n-3$ equations then $Pic(\mathbb{P}^n)\rightarrow Pic(Y)$ is an isomorphism. Is not this in contradiction with abx example? We could take for instace a quadric cone with vertex a point in $\mathbb{P}^4$. – Jessica_90 May 17 at 6:27 • No contradiction. $\operatorname{Pic}(Y)$ is not the same thing as $\operatorname{Cl}(Y)$. – abx May 17 at 14:28	2020-07-09 20:18:00	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 9, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7908794283866882, "perplexity": 256.61989914288466}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655901509.58/warc/CC-MAIN-20200709193741-20200709223741-00130.warc.gz"}
https://stats.stackexchange.com/questions/488176/multivariate-jensen-shannon-divergence	# Multivariate Jensen-Shannon divergence This paper says multivariate Jensen-Shannon divergence is $$JS(\mathbf{p}_1,\dots,\mathbf{p}_K) = \frac{1}{m} \sum KL(\mathbf{p}_i \|\| \bar{\mathbf{p}})$$ with $$KL$$ being the KL-divergence of the multiple probability distributions. Is it accurate compared to bivariate JS-divergence? Would a matrix of bivariate JS-divergences feasible (like the correlation matrix), and what would its diagonal consist of? Besides these, if t-distributed Stochastic Neighbor Embedding (t-SNE) in sklearn.manifold.tsne can be used to form a representation of multivariate KL-divergence for minimization (multivariate rather than one pair at a time), can the same or another algorithm do the same for a multivariate version of Jensen-Shannon divergence?	2021-08-02 15:49:42	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6122899651527405, "perplexity": 1647.5308847879478}, "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/1627046154321.31/warc/CC-MAIN-20210802141221-20210802171221-00261.warc.gz"}
https://web2.0calc.com/questions/i-need-help-pleaseeee	+0 0 115 2 +64 The five circles making up this archery target have diameters of length and. What is the total red area? Simplify your answer as much as you can. You can use pi in your answer if necessary (for example, if the answer were, you could enter "3pi" or "3*pi" or "$3\pi$"). Apr 7, 2019 #1 +102417 0 You didn't specify the diameters but.....if memory serves from answering this problem before....I believe that the diameters were 2, 4, 6, 8 and 10 So....the radiuses are 1, 2, 3, 4 and 5 The inner red area is easy = pi (1)^2 = 1pi The next red area = area of 3rd circle - area of second circle = pi (3^2 - 2^2) = pi (9 - 4) = 5pi The outer red area is = area of 5th circle - area of 4th circle = pi (5^2 - 4^2) = pi (25 - 16) = 9pi So....the total of the red areas = [ 1 + 5 + 9 ] = 15 pi [units^2 ] Apr 7, 2019 #2 +64 0 thanks! donkey Apr 13, 2019	2019-08-25 02:04:21	{"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": 0.8780478239059448, "perplexity": 6064.895326839017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027322160.92/warc/CC-MAIN-20190825000550-20190825022550-00521.warc.gz"}
https://dataspace.princeton.edu/handle/88435/dsp01s4655k203?mode=full	Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01s4655k203 DC FieldValueLanguage dc.contributor.authorKuenne, Peter- dc.date.accessioned2017-07-18T18:28:19Z- dc.date.available2017-07-18T18:28:19Z- dc.date.created2017-04-11- dc.date.issued2017-4-11- dc.identifier.urihttp://arks.princeton.edu/ark:/88435/dsp01s4655k203- dc.description.abstractThis paper expands on the framework of the M&A literature on firm knowledge bases to include indicators of both innovative and economic performance. Using a Poisson regression analysis on a dataset of high-tech M&As, the effects of the absolute size of an acquired knowledge base, the relative size of the acquired knowledge base, and the level of knowledge similarity are studied against a set of performance measures including innovation output, return on assets, and stock returns. The results suggest that the size of the acquired knowledge base has a positive effect on innovative output and a potential negative effect on long term returns. The relative size of the acquired knowledge base has a negative effect on innovation output and an insignificant effect on both economic indicators. The level of knowledge similarity has a positive effect on innovation output but a negative effect on short term returns. Taken together, the re- sults suggest while the acquired knowledge has a positive effect on innovation output, it has an insignificant or in some cases negative impact on economic performanceen_US dc.language.isoen_USen_US dc.titleM&A Knowledge Base Effectsen_US dc.typePrinceton University Senior Theses- pu.date.classyear2017en_US pu.departmentEconomicsen_US pu.pdf.coverpageSeniorThesisCoverPage- pu.contributor.authorid960862230-	2021-05-18 20:14:29	{"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": 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": 0.18041357398033142, "perplexity": 1171.6415349426068}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991514.63/warc/CC-MAIN-20210518191530-20210518221530-00008.warc.gz"}
http://www.chegg.com/homework-help/questions-and-answers/drawing-shows-three-square-surfaces-one-lying-xyplane-one-xz-plane-one-yz-plane-sides-ofth-q219517	The drawing below shows three square surfaces, one lying in the xyplane, one in the xz plane, and one in the yz plane. The sides ofthe square each have length 2.810-2 m. A uniform magnetic field exists inthis region, and its components are: Bx = 0.53 T, By = 0.88 T, and Bz = 0.29 T. Determine the magnetic flux that passesthrough the surface that lies in: (a) the xy plane xy = wb (b) the xz plane xz = wb (c) the yz plane yz = wb ### Get this answer with Chegg Study Practice with similar questions Q: The drawing below shows three square surfaces, one lying in the xyplane, one in the xz plane, and one in the yz plane. The sides ofthe square each have length 2.810-2 m. A uniform magnetic field exists inthis region, and its components are: Bx = 0.40 T, By = 0.74 T, and Bz = 0.23 T. Determine the magnetic flux that passesthrough the surface that lies in:(a) the xy planexy = wb(c) the yz planeyz =	2016-07-23 11:22:14	{"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": 0.9179928302764893, "perplexity": 2122.9293915870144}, "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/1469257822172.7/warc/CC-MAIN-20160723071022-00248-ip-10-185-27-174.ec2.internal.warc.gz"}
http://www.webelements.com/nexus/	Chemistry news, articles, and more ## Spectroscopy of element 115 decay chains The full document is not yet published but a paper accepted 7 Aug 2013 entitled Spectroscopy of element 115 decay chains by D. Rudolph et al. provides additional evidence for element 115: A high-resolution $\alpha$, $X$-ray and $\gamma$-ray coincidence spectroscopy experiment was conducted at the GSI Helmholtzzentrum f\"ur Schwerionenforschung. Thirty correlated $\alpha$-decay chains were detected following the fusion-evaporation reaction $^{48}$Ca~+~$^{243}$Am. The observations are consistent with previous assignments of similar decay chains to originate from element $Z=115$. For the first time, precise spectroscopy allows the derivation of excitation schemes of isotopes along the decay chains starting with elements $Z>112$. Comprehensive Monte-Carlo simulations accompany the data analysis. Nuclear structure models provide a first level interpretation. ## Search for element 113 concluded at last? Press release from RIKEN Nishina Center for Accelerator-Based Science The most unambiguous data to date on the elusive 113th atomic element has been obtained by researchers at the RIKEN Nishina Center for Accelerator-based Science (RNC). A chain of six consecutive alpha decays, produced in experiments at the RIKEN Radioisotope Beam Factory (RIBF), conclusively identifies the element through connections to well-known daughter nuclides. The groundbreaking result, reported in the Journal of Physical Society of Japan, sets the stage for Japan to claim naming rights for the element. Steps in chain of decays from element 113 to mendelevium-254 The search for superheavy elements is a difficult and painstaking process. Such elements do not occur in nature and must be produced through experiments involving nuclear reactors or particle accelerators, via processes of nuclear fusion or neutron absorption. Since the first such element was discovered in 1940, the United States, Russia and Germany have competed to synthesize more of them. Elements 93 to 103 were discovered by the Americans, elements 104 to 106 by the Russians and the Americans, elements 107 to 112 by the Germans, and the two most recently named elements, 114 and 116, by cooperative work of the Russians and Americans. With their latest findings, associate chief scientist Kosuke Morita and his team at the RNC are set follow in these footsteps and make Japan the first country in Asia to name an atomic element. For many years Morita's team has conducted experiments at the RIKEN Linear Accelerator Facility in Wako, near Tokyo, in search of the element, using a custom-built gas-filled recoil ion separator (GARIS) coupled to a position-sensitive semiconductor detector to identify reaction products. On August 12th those experiments bore fruit: zinc ions travelling at 10% the speed of light collided with a thin bismuth layer to produce a very heavy ion followed by a chain of six consecutive alpha decays identified as products of an isotope of the 113th element (Figure 1). While the team also detected element 113 in experiments conducted in 2004 and 2005, earlier results identified only four decay events followed by the spontaneous fission of dubnium-262 (element 105). In addition to spontaneous fission, the isotope dubnium-262 is known to also decay via alpha decay, but this was not observed, and naming rights were not granted since the final products were not well known nuclides at the time. The decay chain detected in the latest experiments, however, takes the alternative alpha decay route, with data indicating that Dubnium decayed into lawrencium-258 (element 103) and finally into mendelevium-254 (element 101). The decay of dubnium-262 to lawrencium-258 is well known and provides unambiguous proof that element 113 is the origin of the chain. Combined with their earlier experimental results, the team's groundbreaking discovery of the six-step alpha decay chain promises to clinch their claim to naming rights for the 113th element. "For over 9 years, we have been searching for data conclusively identifying element 113, and now that at last we have it, it feels like a great weight has been lifted from our shoulders," Morita said. "I would like to thank all the researchers and staff involved in this momentous result, who persevered with the belief that one day, 113 would be ours. For our next challenge, we look to the uncharted territory of element 119 and beyond." ## Element number 114: flerovium (symbol Fl) and element number 116: livermorium (symbol Lv) The International Union of Pure and Applied Chemistry (IUPAC) has recommended names for elements 114 and 116. Scientists from the Lawrence Livermore National Laboratory (LLNL) and at Dubna proposed the names as Flerovium for element 114 and Livermorium for element 116. Flerovium (atomic symbol Fl) was chosen to honor Flerov Laboratory of Nuclear Reactions, where superheavy elements, including element 114, were synthesized. Georgiy N. Flerov (1913-1990) was a renowned physicist who discovered the spontaneous fission of uranium and was a pioneer in heavy-ion physics. He is the founder of the Joint Institute for Nuclear Research. In 1991, the laboratory was named after Flerov - Flerov Laboratory of Nuclear Reactions (FLNR). Livermorium (atomic symbol Lv) was chosen to honor Lawrence Livermore National Laboratory (LLNL) and the city of Livermore, Calif. A group of researchers from the Laboratory, along with scientists at the Flerov Laboratory of Nuclear Reactions, participated in the work carried out in Dubna on the synthesis of superheavy elements, including element 116. (Lawrencium -- Element 103 -- was already named for LLNL's founder E.O. Lawrence.) In 1989, Flerov and Ken Hulet (1926-2010) of LLNL established collaboration between scientists at LLNL and scientists at FLNR; one of the results of this long-standing collaboration was the synthesis of elements 114 and 116. The creation of elements 116 and 114 involved smashing calcium ions (with 20 protons each) into a curium target (96 protons) to create element 116. Element 116 decayed almost immediately into element 114. The scientists also created element 114 separately by replacing curium with a plutonium target (94 protons). The creation of elements 114 and 116 generate hope that the team is on its way to the "island of stability," an area of the periodic table in which new heavy elements would be stable or last long enough for applications to be found. The new names were submitted to the IUPAC in late October. The new names will not be official until about five months from now when the public comment period is over. ## Printable periodic table: QR-coded Attached find a printable QR-coded periodic table with links to online periodic table data. QR codes are 2-dimensional bar codes readable by, for instance, some Apps on iPhones and others. Print on a big a piece of paper as possible, otherwise your QR reader may pick up an element you didn't intend. Periodic Table QR-coded: Periodic Table QR-coded Version history 1.1: 15 September 2011 1.0: 17 July 2011 ## Discovery of the Elements with Atomic Number 114 and 116 A news reports from IUPAC indicates the confirmation of the discoveries of elements 114 and 116. Proposals for the names of the two elements will follow in due course: ## News: Discovery of the Elements with Atomic Number 114 and 116 Priority for the discovery of the elements with atomic number 114 and 116 has been assigned, in accordance with the agreed criteria, to collaborative work between scientists from the Joint Institute for Nuclear Research in Dubna, Russia and from Lawrence Livermore, California, USA (the Dubna-Livermore collaborations). The discovery evidences were recently reviewed and recognized by a IUPAC/IUPAP joint working party. IUPAC confirmed the recognition of the elements in a letter to the leaders of the collaboration. The IUPAC/IUPAP Joint Working Party (JWP) on the priority of claims to the discovery of new elements has reviewed the relevant literature pertaining to several claims. In accordance with the criteria for the discovery of elements previously established by the 1992 IUPAC/IUPAP Transfermium Working Group, and reiterated by the 1999 and 2003 IUPAC/IUPAP JWPs, it was concluded that “the establishment of the identity of the isotope 283Cn by a large number of decaying chains, originating from a variety of production pathways essentially triangulating its A,Z character enables that nuclide’s use in unequivocally recognizing higher-Z isotopes that are observed to decay through it.” From 2004 Dubna-Livermore collaborations the JWP notes: (i) the internal redundancy and extended decay chain sequence for identification of Z = 287114 from 48Ca + 242Pu fusion (Oganessian et al. Eur. Phys. J. A 19, 3 (2004) and Phys. Rev. C 70, 064609 (2004)); and (ii) that the report of the production of 291116 from the fusion of 48Ca with 245Cm is supported by extended decay chains that include, again, 283Cn and descendants (Oganessian et al. Phys. Rev. C 69, 054607 (2004)). It recommends that the Dubna-Livermore collaborations be credited with discovery of these two new elements. A full synopsis of the relevant experiments and related efforts is presented in a technical report published online in Pure and Applied Chemistry on 1 June 2011. With the priority for the discovery established, the scientists from the Dubna-Livermore collaborations are invited to propose a name for the two super-heavy elements, elements 114 and 116. The suggested names will then go through a review process before adoption by the IUPAC Council. Review of the claims associated with elements 113, 115, and 118 are at this time not conclusive and evidences have not met the criteria for discovery. ## Synthesis of a new element with atomic number Z=117 A paper has just been accepted (5 April 2010) for publication in Physical Review Letters.1 ### International team discovers element 117 A new chemical element has been added to the Periodic Table: A paper on the discovery of element 117 has been accepted for publication in Physical Review Letters. Oak Ridge National Laboratory is part of a team that includes the Joint Institute of Nuclear Research (Dubna, Russia), the Research Institute for Advanced Reactors (Dimitrovgrad), Lawrence Livermore National Laboratory, Vanderbilt University and the University of Nevada Las Vegas. ORNL's role included production of the berkelium-249 isotope necessary for the target, which was subjected to an extended, months-long run at the heavy ion accelerator facility at Dubna, Russia. "Without the berkelium target, there could have been no experiment," says ORNL Director of Strategic Capabilities Jim Roberto, who is a principal author on the PRL paper and who helped initiate the experiment. The berkelium was produced at the High Flux Isotope Reactor and processed at the adjoining Radiochemical Engineering & Development Laboratory as part of the most recent campaign to produce californium-252, a radioisotope widely used in industry and medicine. "Russia had proposed this experiment in 2004, but since we had no californium production at the time, we couldn't supply the berkelium. With the initiation of californium production in 2008, we were able to implement a collaboration," Roberto says. Professor Joe Hamilton of Vanderbilt University (who helped establish the Joint Institute for Heavy Ion Research at ORNL) introduced Roberto to Yuri Oganessian of Russia's JINR. Five months of the Dubna JINR U400 accelerator's calcium-48 beam - one of the world's most powerful - was dedicated to the project. The massive effort identified a total of six atoms of element 117 and the critical reams of data that substantiate their existence. The two-year experimental campaign began with a 250-day irradiation in HFIR, producing 22 milligrams of berkelium-249, which has a 320-day half-life. The irradiation was followed by 90 days of processing at REDC to separate and purify the berkelium. The Bk-249 target was prepared at Dimitrovgrad and then bombarded for 150 days at the Dubna facility. Lawrence Livermore, which now has been involved in the discovery of six elements with Dubna (113, 114, 115, 116, 117, and 118), contributed data analysis, and the entire team was involved in the assessment of the results. This is the second element that ORNL has had a role in discovering, joining element 61, promethium, which was discovered at the Graphite Reactor during the Manhattan project and reported in 1946. ORNL, by way of its production of radioisotopes for research, has contributed to the discovery of a total of seven new elements. Members of the ORNL team include the Physics Division's Krzysztof Rykaczewsi, Porter Bailey of the Nonreactor Nuclear Facilities Division, and Dennis Benker, Julie Ezold, Curtis Porter and Frank Riley of the Nuclear S&T Division. Roberto says the success of the element-117 campaign underscores the value of international collaborations in science. "This use of ORNL isotopes and Russian accelerators is a tremendous example of the value of working together," he says. "The 117 experiment paired one of the world's leading research reactors--capable of producing the berkelium target material--with the exceptional heavy ion accelerator and detection capabilities at Dubna." #### Islands of Stability Roberto also says the experiment, in addition to discovering a new chemical element, has pushed the Periodic Table further into the neutron-rich regime for heaviest elements. "New isotopes observed in these experiments continue a trend toward higher lifetimes for increased neutron numbers, providing evidence for the proposed "island of stability" for super-heavy nuclei," he says. "Because the half-lives are getting longer, there is potential to study the chemistry of these nuclei," Roberto says. "These experiments and discoveries essentially open new frontiers of chemistry." —Bill Cabage The news about the claim was announced in a press release from the Oak Ridge National Laboratory. ## Tantalising news about element 117 Notes from the 31st meeting of PAC for Nuclear Physics seems to suggest that a claim for element 117 (at the base of the halogen column) may come in the coming weeks and months. It's not very clear which isotopes may have been formed so watch this space. IV. Experiments on the synthesis of element 117 The PAC heard with great interest the report on the results of the experiment dedicated to the synthesis of element 117 in the 48Ca + 249Bk reaction. The PAC congratulates the staff of the Flerov Laboratory on the discovery of element 117 and new isotopes of elements 115, 113, 111, 109, 107, and 105. The discovery of chains of two neighboring isotopes emphasizes the importance of the odd-even and odd-odd effect for such heavy nuclei. It is in fact especially interesting that the odd-odd chain (3n channel) neighboring to the odd-even chain (4n channel) is twice longer (6 α particles). ## Copernicium confirmed as name of element 112 IUPAC has officially approved the name copernicium, with symbol Cn, for the element of atomic number 112. Priority for the discovery of this element was assigned, in accordance with the agreed criteria, to the Gesellschaft für Schwerionenforschung (GSI) (Center for Heavy Ion Research) in Darmstadt, Germany. The team at GSI proposed the name copernicium which has now been approved by IUPAC. Sigurd Hofmann , leader of the GSI team stated that the intent was to "salute an influential scientist who didn't receive any accolades in his own lifetime, and highlight the link between astronomy and the field of nuclear chemistry." The name proposed by the Gesselschaft für Schwerionenforschung (GSI) lies within the long tradition of naming elements to honor famous scientists. Nicolaus Copernicus was born on 19 February 1473, in Torún, Poland and died on 24 May 1543, in Frombork/Frauenburg also in Poland. His work has been of exceptional influence on the philosophical and political thinking of mankind and on the rise of modern science based on experimental results. During his time as a canon of the Cathedral in Frauenburg, Copernicus spent many years developing a conclusive model for complex astronomical observations of the movements of the sun, moon, planets and stars. His work published as “De revolutionibus orbium coelestium, liber sixtus” in 1543 had very far reaching consequences. Indeed the Copernican model demanded major changes in the view of the world related to astronomy and physical forces and well as having theological and political consequences. The planetary system introduced by Copernicus has been applied to other analogous systems in which objects move under the influence of a force directed towards a common centre. Notably, on a microscopic scale this is the Bohr model of the atom with its nucleus and orbiting electrons. The Recommendations will be published in the March issue of the IUPAC journal Pure and Applied Chemistry and is available online at Pure Appl. Chem., 2010, Vol. 82, No. 3, pp. pp 753-755 (doi: 10.1351/PAC-REC-09-08-20) ## Approaches to element 120 (unbinilium) Attempts have been made at GSI to make element 120 (unbinilium). Several new elements have been made at GSI in the last few years. However after 120 days no decay chain of element 120 was found. With the total number of 2.6 × 1019 projectiles which impinged upon the target, it deduced that the stability in the region around Z=120, N=184 is not exceptionally high with respect to the neighbouring regions. Currently it is not clear what proton number defines the location of the "island of stability". Various theoretical models suggest numbers of Z=114, 120 or 126. Workers at GSI investigated the element Z=120 (element 120, containing 120 protons within the nucleus). Three different projectile-target combinations all lead to the same compound nucleus 302120 or 302Ubn • 64Ni + 238U • 58Fe + 244Pu, and • 54Cr + 248Cm The neutron number of the compound nucleus 302120 is N=182. This is only 2 neutrons below N=184 where the neutron shell closure is expected. Therefore, 302120 or 302Ubn is closer to the N=184 shell than any other so far produced compound nucleus with lower Z. The largest production rate for Z=120 is predicted for the most mass asymmetric projectile/target combination 54Cr + 248Cm. However, at SHIP this experiment was not possible so the reaction 64Ni + 238U was studied. If the proton shell closure is at Z=120 then an enhanced production rate and half-live of the element 120 would be expected. Depending on the magnitude of the stabilization due to the closed shell, one could expect up to a few events per week for the isotopes 299120 and 298120 produced in 64Ni + 238U reactions. The half-lives are expected to be of the order of some 10 μs, but in the end no luck, this time at least. ## WebElements chemistry nexus OK - started reorganisation of one WebElements area, the "chemistry nexus". The nexus will now bring together news, articles, the WebElements blog, and eventually the Webelements bibliography, and more. It will also contain chemistry content related to WebElements, for instance, sections on the chemistry of the various groups or periods in the periodic table, articles on periodicity, expanded disucssion of the chemistry of specific elements, and so on. All that will take years to add of course. The existing forums site will be located in a dedicated area at the WebElements chemistry forum. WebElements: the periodic table on the WWW [http://www.webelements.com/]	2015-03-02 03:29:24	{"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": 0.2878827750682831, "perplexity": 2801.5929877424273}, "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-2015-11/segments/1424936462700.28/warc/CC-MAIN-20150226074102-00029-ip-10-28-5-156.ec2.internal.warc.gz"}

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