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Talk:Chebyshev polynomials

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Michael, it's not actually your browser. You can set that in your wikipedia preferences - by default it will render simple equations in normal text. And you apparently just complicated the equation by adding some white space ;-) snoyes 23:37 Mar 1, 2003 (UTC)

You're right; it works. Thanks. Why is the default the option that renders things as normal text, given that that is never what is intended when things are set with TeX? Michael Hardy 23:45 Mar 1, 2003 (UTC)

 * I don't know why that is the default option - best to ask one of the developers, maybe on the village pump. --snoyes 23:53 Mar 1, 2003 (UTC)

And BTW, there don't seem to be any other articles about special polynomial sequences on Wikipedia. Michael Hardy 23:45 Mar 1, 2003 (UTC)

//Here's the polynomial sequence on the page. Κσυπ Cyp    19:47, 25 Jan 2004 (UTC)
 * 1) include

int asdf(int, int); int choo(int, int);

int main { int n, k; for(n=0;n<=16;++n) { printf("T%d(x)=", n); for(k=0;k<=n/2;++k) { printf("%s%d", k&&asdf(n, k)>=0?"+":"", asdf(n, k)); if(n-2*k) printf("x"); if(n-2*k>1) printf("%d", n-2*k); if(k==n/2) printf(" \n"); } }  return(0); }

int tmp[1000];

int asdf(int n, int k) { int r, a; if(!n) return(1); if(!k) return(1<<(n-1)); for(a=r=0;a>(2*(k-a)); return(r); }

int choo(int n, int k) { int x, y; for(tmp[0]=y=1;y<=n;++y) for(tmp[x=y]=0;x;--x) tmp[x]+=tmp[x-1]; return(tmp[k]); }

 * If we recursively define a function (the letter ξ an arbitrary choice)
 * $$\xi(n,0)=\lceil 2^{n-1}\rceil$$
 * $$\xi(n,k)=-\sum^{k-1}_{a=0}\frac{\xi(n,a)}{4^{k-a}}{n-2a\choose k-a}$$
 * then
 * $$T_n(x)=\sum^{\lfloor n/2\rfloor}_{k=0}\xi(n,k)x^{n-2k}$$

Maxima and minima
It looks to me like the local maxima of the Chebyshev polynomials of the first kind all have value 1, and the local minima all have value -1. If so (I don't know enough to say whether it is actually true), then this is an interesting property that should be mentioned somewhere on the page. I'm guessing it probably pops right out of one of the definitions, but that's not transparent to a muggle like me. -dmmaus 04:01, 15 August 2005 (UTC)

I think that follows from the definition:
 * $$T_n(x) =

\left\{ \begin{matrix} & \cos(n\arccos(x))          & \mbox{, } \ x \in [-1,1] \\ & \cosh(n \, \mathrm{arcosh}(x))        & \mbox{, } \ x \ge 1      \\ & (-1)^n \cosh(n \, \mathrm{arcosh}(-x)) & \mbox{, } \ x \le -1    \\ \end{matrix}\right. $$

Obviously, for x in [-1, 1] since the behaviour is given by the cosine, the local maxima and minima are 1 and -1. If one can prove that there are no maxima/minima outside this interval, then you assertion would follow. Oleg Alexandrov 04:07, 15 August 2005 (UTC)

Ah, good point. I'm afraid my eyes glaze over when I see hyperbolic trig functions in the vicinity. :-) Still, I think it'd be nice to mention this property explicitly in the article, since it's the standout physical feature of the graphs when you look at them. -dmmaus 08:54, 15 August 2005 (UTC)
 * I don't know that much about these polynomials to attempt to write in the article. Let us hope somebody else will mention this fact. Oleg Alexandrov 17:04, 16 August 2005 (UTC)

 * I've mentioned it.
 * There's also an interesting paralel here, which is a bit OT. Notice that for a degree n polynomial (with fixed leading coeff) to have a minimal maximum absolute value on an interval, it has to reach this maximal absolute value the highest number of times possible: n+1 times.  (Clearly a degree n polynomial can have at most n-1 places where its derivative is 0, so it's impossible that is has more than n+1 extremal point in an interval.)  This paralels up nicely with a general theorem in numerical analysis that the polynomial p of degree n (with no restriction on the leading coeff) that approximates a continuous function f in the sense that max|p-f| on an interval is minimal, |p-f| has to take this maximum n+1 times.  I'm not sure how this general theorem is stated exactly, and I also don't know if it has any connection to this.  – b_jonas 21:45, 29 December 2005 (UTC)

 * You have cited famous approximation theorem by Chebyshev (not later than 1857). And yes, extremal properties of polynomials of first kind (minimal C-norm) follow from this theorem immediately. Mir76 (talk) 16:18, 16 August 2009 (UTC)

 * Let me say thanks to User:172.128.197.224 for his correction of the statement in the second paragraph of the article. – b_jonas 18:30, 18 February 2006 (UTC)

Optimal approximations to arbitrary functions
The theorem says that, if an nth degree polynomial P approximates a function F on a closed interval in such a way that the error function P(x)-F(x) has n+2 maxima, and those maxima have alternating signs and the same absolute value (that is, P-F has maxima +ε, -ε, +ε, ...), then that polynomial is locally optimal. That is, there is no other polynomial close to P that has maximum error less than ε. This locality situation is the same as the situation in differential calculus: elementary calculus problems involve finding the maximum value of a function. One sets the derivative to zero. That gives a local maximum (or minimum), but there might be other solutions that are far away.

I will outline the proof below. I think this theorem, and its proof, are interesting, but don't seem to be part of the standard maths curriculum. (Orthogonal polynomials in general seem to be a mathematical orphan child.) I would suggest that this proof ought to be on a separate "proof" page. WP is trying to minimize the number of such pages, because WP isn't a textbook, but this would probably be a good candidate.

The proof really requires a graph to motivate it. I'm new to WP; it should be made by someone who is comfortable making and uploading graphs, such as the graphs on the Chebyshev page.

Here's the proof: Suppose P(x)-F(x) has n+2 maxima, alternating signs, all with absolute value ε. Suppose Q is another nth degree polynomial very close to P, that is strictly better. Superimpose tha graph of Q-F on that of P-F. Since Q is close to P, they will be about the same, but at the n+2 points where P-F is maximal, Q-F must lie strictly inside P-F. This is because the maximal excursions of Q-F are δ, which is strictly less than ε. So (Q-F)-(P-F) is strictly nonzero at those n+2 points, with alternating signs. But (Q-F)-(P-F) = Q-P, an nth degree polynomial. Since it has n+2 alternating nonzero values, it must have n+1 roots. But nth degree polynomials can't do that.

Now here's an outline of why Chebyshev interpolation gets very close to this optimum polynomial: If the Taylor series for a function converges very rapidly (the way, for example, sin, cos, and exponential do), the error from chopping off the series after some finite number of terms will be close to the first term after the cutoff. That term will dominate all the later terms, and that term plus all the later ones are of course the error function. The same is true if we use a Chebyshev series instead of a Taylor series.

Now if we expand F(x) in its Chebyshev series, and chop it off after the Tn term, we have an nth degree polynomial. The error is dominated by the Tn+1 term, so it looks like Tn+1. Tn+1 is level, with n+2 maxima, with alternating signs. This means the error function is approximately level, and hence, by the theorem above, this polynomial is approximately optimal.

If one wants a truly optimal polynomial, one "tweaks" the polynomial P so that its error excursions are truly level. Remes' algorithm does this.

Should there be a separate WP page for this stuff, linked from Chebyshev? What should it be called? William Ackerman 18:46, 21 February 2006 (UTC)

 * Thanks for stating the theorem. – b_jonas 21:13, 21 February 2006 (UTC)

The relationship
I removed the following:

 * Starting with T0( cos u ) = 1 and T1( cos u ) = cos u, one quickly calculates that cos 2u = 2 cos^2 u - 1, cos 3u = 4 cos^2 u - 3 cos u, cos 4u = 8 cos^4 u - 8 cos^2 u + 1, etc. By setting u = 0, whence cos u = 1, one can immediately detect that both sides of each equation evaluate to unity, which is a good clue that the relationship may be correct—which, of course, it is.

What does "the relationship may be correct" mean? which relationship? Not $$T_n(\cos\theta)=\cos(n\theta)$$ because thats a definition. It may show that the trignometric identities are true, but thats not the point of the article. PAR 13:27, 11 August 2006 (UTC)

reasoning not correct
I removed the following part because it is nonsense:

From reasoning similar to that above, one can develop a closed-form generating formula for Chebyshev polynomials of the first kind:

\cos(n \theta)=\frac{e^{i n \theta}+e^{-i n \theta}}{2}=\frac{(e^{i \theta})^n+(e^{i \theta})^{-n}}{2} \,\! $$

which, combined with DeMoivre's formula: THIS IS FALSE:

\! e^{i \theta}=\cos\theta+i \sin\theta=\cos\theta+i \sqrt{1-\cos^2\theta}=\cos\theta+\sqrt{\cos^2\theta-1} \,\! $$

yields:

\cos(n \theta)=\frac{\left(\cos\theta+ \sqrt{\cos^2\theta-1}\right)^n+\left(\cos\theta+ \sqrt{\cos^2\theta-1}\,\right)^{-n}}{2} \,\! $$

which, of course, is far more expedient for determining the cosine of N times a given angle than is cranking through almost N rounds of the recursive generator calculation. Finally, if we replace $$\cos(\theta)$$ with x, we can alternatively write:

T_n(x)=\frac{\left(x+ \sqrt{x^2-1}\right)^n+\left(x+ \sqrt{x^2-1}\right)^{-n}}{2}. $$

Linear operators
Should someone write a section about these as the eigenfunctions of a second order linear differential operator (which is symmetric wrt the inner product $$(f,g)= \int_{-1}^{1} f(x)g(x)\frac{1}{\sqrt{1-x^2}}\,dx $$)? I would, but I'm not really comfortable with the theory yet... Marbini (talk) 17:54, 2 April 2008 (UTC)

Factoring
Can anything of interest be said about factorization of Chebyshev polynomials? Michael Hardy (talk) 00:04, 9 December 2008 (UTC)

 * Apparently. Googling for "factoring chebyshev" turned up this and this which seem to at least contain some pointers. Fredrik Johansson 00:15, 9 December 2008 (UTC)

So there's really not very much.....maybe. There is a sense in which the spread polynomials Sn, satisfying
 * $$ S_n(\sin^2\theta) = \sin^2(n\theta)\,$$

are trivially equivalent to the Chebyshev polynomials Tn, satisfying
 * $$ T_n(\cos\theta) = \cos(n\theta).\,$$

But the article titled spread polynomials has some material on factorization, and maybe there the story is different. Or maybe not? Michael Hardy (talk) 21:36, 10 March 2009 (UTC)

Some interest in factoring the Chebyshev T-polynomials stems from reduction of cosines to rational numbers as PDF here. - R. J. Mathar (talk) 19:35, 12 March 2014 (UTC)

A few remarks about divisibility of Chebyshev polynomials by other Chebyshev polynomials have now been added in the section Composition and divisibility properties. Perhaps more could be said, but I wanted to keep it simple, and also didn't feel qualified to say more. Will Orrick (talk) 23:24, 8 January 2022 (UTC)

Explicit formulas
Can someone provide sources for the given explicit formulas, or at least some hints on how to obtain them?

I found some other formulas in Mason, J. C. "Chebyshev polynomials", 2003. CRC Press, 2003., pp.35-36 and in Snyder, M. A. "Chebyshev methods in numerical approximation", Englewood Cliffs, NJ, USA, 1966., p. 14.

In the first book, there are some other references to Rivlin (1974) and Clenshaw (1962). Mstempin (talk) 21:23, 3 August 2009 (UTC)

 * After expanding formula
 * $$T_{n}(x)=\sum_{k=0}^{\lfloor\frac{n}{2}\rfloor}{n \choose 2k}x^{n-2k}(x^2-1)^{k}$$
 * for n=8 i made conjecture that
 * $$T_{n}(x)=\sum_{k=0}^{\lfloor\frac{n}{2}\rfloor}\sum_{m=k}^{\lfloor\frac{n}{2}\rfloor}(-1)^k {n \choose 2m}\cdot {m \choose k} x^{n-2k}$$
 * This conjecture should be proven and in my opinion induction should be enough
 * For the sum
 * $$\sum_{m=k}^{\lfloor\frac{n}{2}\rfloor} {n \choose 2m}\cdot {m \choose k}$$
 * i found with help of Wolfram Alpha only piecewise definition
 * $$\sum_{m=k}^{\lfloor\frac{n}{2}\rfloor} {n \choose 2m}\cdot {m \choose k} = \begin{cases}1 \qquad n=0 \\ \frac{n}{2}\cdot\frac{1}{n-k}\cdot {n-k \choose k}\cdot 2^{n-2k} \qquad n > 0\end{cases}$$ — Preceding unsigned comment added by 188.47.53.242 (talk) 11:56, 4 July 2023 (UTC)

History
I believe Chebyshev developed these polynomials while attempting to solve the problem of the inversor, a mechanical linkage that converts circular motion into linear motion. DonPMitchell (talk) 03:42, 13 March 2010 (UTC)

Dickson polynomial
I would like to propose merging Dickson polynomial into Chebyshev polynomials. All the properties, except the analysis over finite fields, is already here. — Arthur Rubin (talk) 00:27, 21 September 2011 (UTC)

 * Dickson polynomials are mainly studied over finite fields, when they are not equivalent to Chebyshev polynomials.
 * It appears in their application, these two polynomials groups do really differ, and I wouldn't support a merge... Gryllida (talk) 22:16, 2 January 2014 (UTC)

Proposed section to merge: Spread polynomials and some other Chebyshev-linked Polynomials
spread polynomials are in a sense equivalent to the Chebyshev polynomials of the first kind, but enable one to avoid square roots and conventional trigonometric functions in certain contexts, notably in rational trigonometry.
 * Dickson polynomials   ,Dn(x), are  equivalent to Chebyshev polynomials Tn(x), with a slight and trivial change of variable:
 * $$ D_n(2x) = 2 T_n(x) \,$$

Dickson polynomials are given by:

 * $$D_n(x)=\sum_{p=0}^{\lfloor n/2\rfloor}\frac{n}{n-p} \binom{n-p}{p} (-x^{n-2p}. $$

The first few Dickson polynomials are

 * $$ D_0(x) = 2 \,$$

 * $$ D_1(x) = x \,$$

 * $$ D_2(x) = x^2 - 2 \,$$

 * $$ D_3(x) = x^3 - 3x \,$$

 * $$ D_4(x) = x^4 - 4x^2 + 2 \,$$

Boubaker polynomials are also linked to Chebyshev polynomials. they can be defined by the recurrence relation   :

 * $$B_n(x)= \begin{cases}

1, & \mbox{if } n = 0, \\ x, & \mbox{if } n = 1, \\ x B_{n - 1}(x) - B_{n - 2}(x),& \mbox{if } n \geq 3. \end{cases},$$

Lucas polynomials Ln(x) are a polynomial sequence defined by the following Chebyshev-like recurrence relation:

 * $$L_m(x) = \begin{cases}

2, & \mbox{if } m = 0 \\ x, & \mbox{if } m = 1 \\ x L_{m - 1}(x) + L_{m - 2}(x), & \mbox{if } m \geq 2. \end{cases}$$

Section on Orthogonality and Example 1 has errors
The session on Orthogonality has a somewhat major issue, namely that the discrete orthogonality condition as written is wrong. The discrete orthogonality condition neglects to mention that the formula given is only valid for $$0\leq i,j \leq N-1$$. For other values of $$i,j$$ the aliasing condition gives alternating sign "tiled" versions of what's currently written.

Also, in Basis Set, Example 1, the closed-form Chebyshev coefficients $$a_n$$ for Log(1+x) is given correctly. However, immediately afterward it is written that these same coefficients $$a_n$$ can alternately be computed via the discrete Gauss-Lobatto sum. This is false, as is readily verified numerically. The $$a_n$$ in the Gauss-Lobatto discretization are the aliasing-folded versions of the $$a_n$$ computed in the continuous integral case, and are not the same. To a newcomer who does not know this already, this is completely non-obvious, especially considering that the same symbol $$a_n$$ is used for both. I suggest that different symbols should be used for the two cases to emphasize the fact that the two quantities are different. — Preceding unsigned comment added by MathDoobler (talk • contribs) 04:13, 3 January 2014 (UTC)

Wrong discrete orthogonality
There is a very wrong formula in the Orthogonality section. The discrete orthogonality shouldn't have the weight that appears in the continuous orthogonality condition. See for example Eq. (3.30) of https://www.siam.org/books/ot99/OT99SampleChapter.pdf Revision of 14 March 2016 should be undone because wrong! Pliskin14 (talk) 09:21, 12 May 2016 (UTC)

Third and Fourth Kind
Is there any reason why the Chebyshev polynomials V_n(x) and W_n(x) of the third and fourth kind are not mentioned (at least: defined) in the article? R. J. Mathar (talk) 19:29, 12 March 2014 (UTC)
 * I hadn't heard of them. Got a reference which is uses them?  — Arthur Rubin  (talk) 16:27, 18 March 2014 (UTC)
 * For example

and many, many others R. J. Mathar (talk) 17:25, 1 May 2014 (UTC)

In response to my earlier edit being reverted, p_1 as listed are not orthogonal to the respective weight functions despite being consistent with the definitions. While definitions for V and W are sometimes interchanged, the ones used here are consistent with Mason and Handscomb's 2003 book (definition 1.3), but the weight functions are not (section 4.2.2). In regards to this page, I believe that the weight functions for V and W need to be switched. — Preceding unsigned comment added by Youboo4 (talk • contribs) 21:12, 21 July 2022 (UTC)

 * I now agree that the weights were incorrect and have switched them. Do you agree that everything else is consistent now? Will Orrick (talk) 22:15, 21 July 2022 (UTC)
 * Yes, I think that was the only issue. Youboo4 (talk) 23:45, 21 July 2022 (UTC)

Regarding the 24 May 2024 edits and reversions: it seems to have escaped the participants' notice that Chebyshev polynomials of the third and fourth kind were already in the article. My suggestion would be to roll the article back to its state prior to 24 May 2024 and to have a discussion before changing the article in such a drastic way. (As it is now, the article is only partially rolled back.) Will Orrick (talk) 06:50, 26 May 2024 (UTC)

 * I rolled back all the recent edits that introduced polynomials of the third and fourth kind early in the article. I'm willing to entertain contrary opinions, but my feeling is that 99% of readers coming to this article are here to learn about polynomials of the first or second kind, which have a history going back to Viète and are ubiquitous in mathematics and science. Polynomials of the third and fourth kinds, while implicit in the work of Jacobi, and having been studied in the aerospace community for more than half a century, have only started to be called by that name in the past few decades and have a much smaller literature. I feel that introducing them early in the article gives them undue weight.

 * Furthermore, it is untenable to have an article that opens by stating that there are two sets of polynomials, then immediately introduces four of them, but then goes back mostly to talking as if there are only two sets, only to reintroduce the other two sets at the very end of the article as if they hadn't been mentioned before. This is what we had prior to the rollback. If treating Chebyshev polynomials as four sets of polynomials is the right thing to do from the outset, then the article needs to maintain that point of view throughout.

 * I'm not saying that such a change would need to take place in a single edit—that would be impractical—but anyone proposing to make that change needs to provide a rationale that achieves broad support from other editors and needs to outline a plan for carrying it out. Will Orrick (talk) 16:09, 26 May 2024 (UTC)
 * I think it's possibly worth mentioning in the lead section, but perhaps not worth elaborating too much up front. My impression is that these are mostly used in the context of fast transforms, where a polynomial of type X is broken into sub-transforms of types X, Y, Z (precisely which ones depends on X and on the degree). –jacobolus (t) 07:24, 27 May 2024 (UTC)

 * The questions of whether and where to mention the third and fourth kinds comes down to how widespread usage of those terms is. I was the person who added mention of them to the article back in January 2022. I did that because R. J. Mathar's comment on this Talk page persuaded me that usage of the terms was common enough that we couldn't ignore it. On the other hand, my judgement was that this usage was restricted to certain specialties and was relatively new, so I restricted the mention to a late section of the article, "Families of polynomials related to Chebyshev polynomials".

 * Whenever terminology with longstanding use is adopted for a new purpose, the new thing is piggybacking on the fame of the older one. Sometimes the new usage catches on; sometimes it doesn't; sometimes it catches on in some quarters but not in others. I find it hard to judge whether the applications you mention are important enough to mention in the lead of this article. Will Orrick (talk) 11:46, 28 May 2024 (UTC)
 * Sure, I agree. I might even focus on the "first-kind" T polynomials, which are what people mean when they say "Chebyshev polynomials" without qualification, and seem by far the most important to discuss, and then lump the second, third, and fourth-kind spin-offs into a later section. –jacobolus (t) 15:59, 28 May 2024 (UTC)

 * This is valuable input, and shows that there is a wide variety of different views. It's hard to get many editors to weigh in on issues like this, but I hope we will hear from others.

 * I think I learned this topic from volume II of Bateman (1953), or maybe from Abramowitz and Stegun (1964), both of which use the language "first kind", "second kind" and deal with two sets of polynomials from the start. It's also interesting that polynomials essentially equivalent to $T(x)$ and $U(x)$ already appear together in Viète's book (1646). But it seems you may be right that this "first kind", "second kind" vocabulary is of relatively recent vintage. I find very little evidence of this language prior to Bateman. Regarding terminology, Bateman (p. 183) has this to say,

 * "Sometimes (especially in the French literature) orthogonal polynomials in general are called Tchebichef polynomials. There are also several special systems of orthogonal polynomials called Tchebichef polynomials. In this chapter we shall reserve the name Tchebichef polynomials of the first and second kind for the suitably standardized orthogonal polynomials associated with
 * $$a=-1,\quad b=1,\quad w(x)=(1-x^2)^{\mp1/2},\quad X=1-x^2$$."

 * A quick search yielded articles on Chebyshev polynomials of the fifth, sixth, seventh, and eighth kinds, so it looks like this is never going to end. Will Orrick (talk) 13:19, 29 May 2024 (UTC)
 * It's probably worth going with whatever is used in widely cited dedicated sources, e.g. Rivlin (1974) or Mason & Handscomb (2002). I don't feel particularly motivated to do a survey right now though. –jacobolus (t) 14:57, 29 May 2024 (UTC)
 * Just my perspective, but I've read through most of this page multiple times in the capacity of my research (computer science / applied math), and not yet encountered a reason to use Chebyshev polynomials of the third & fourth kind. My impression is that the first kind is by far the most important, and expressions involving the second kind come up often when analyzing the first kind, so it is important by proxy. I haven't seen any use for these other polynomials in the areas I work in, and to my knowledge it's not even common to find it in approximation theory textbooks. So, keeping weight off of the third and fourth kind makes sense to me. Fawly (talk) 20:26, 29 May 2024 (UTC)

 * So I guess the Mason of Mason and Handscomb is the Mason that wrote all those articles on 3rd and 4th kind polynomials mentioned by R. J. Mathar in the opening comment of this thread. It's therefore perhaps not surprising that the blurb for the book says that it treats "all four kinds" of Chebyshev polynomials, and, indeed, the table of contents bears that out.
 * Rivlin had a second edition in 1990, which was just about the time that the 3rd kind/4th kind terminology was being introduced, so one wouldn't expect to find it in that book. From what I can see of the first chapter, "the" Chebyshev polynomials are the $T(x)$. But Rivlin does introduce the second kind polynomials as early as page 7.
 * The NASA document by Desmarais and Bland, cited in the bibliography discusses the original motivation for looking at the 3rd and 4th kind polynomials.
 * I don't work in approximation theory, so I don't have much of an opinion. Chebyshev polynomials, of both of the first two kinds, come up from time to time in my work in mathematical physics, combinatorics, and number theory, but that's about it. Will Orrick (talk) 13:18, 30 May 2024 (UTC)
 * Fox & Parker (1968) also seems to focus on the T polynomials. –jacobolus (t) 02:30, 31 May 2024 (UTC)
 * Boyd (1989) Chebyshev and Fourier Spectral Methods focuses on the T polynomials, but also has two appendices covering the "Chebyshev polynomials: $T_n(x)$" and the "Chebyshev polynomials of the second kind: $U_n(x)$", respectively. Davis (1963) Interpolation and Approximation also focuses on the T for Tschebyscheff polynomials, and when he later introduces the U polynomials, describes the two types as "Tschebyscheff polynomials (of the first kind)" and "Tschebyscheff polynomials of the second kind". –jacobolus (t) 02:47, 31 May 2024 (UTC)
 * Trefethen (2013) Approximation Theory and Approximation Practice focuses on the T polynomials. –jacobolus (t) 02:50, 31 May 2024 (UTC)
 * Cheney (1966) Introduction to Approximation Theory focuses on the "Tchebycheff polynomials" T, but introduces the "Tchebycheff polynomials of the second kind" U in an exercise, and then uses them a couple more times toward the end of the book. –jacobolus (t) 03:22, 31 May 2024 (UTC)

Series coefficients found using Gauss–Lobatto zeros are approximate
I edited the section "As a basis set," Example 1, to indicate that the coefficients found by exploiting the discrete orthogonality property and the Gauss–Lobatto zeros give approximate results. This is discussed in Numerical Recipes (which is about my level of expertise here).

Also, I believe that the "exact" results in the same example for n > 0 are wrong. I don't have a correct form but I have a short Mathematica notebook which compares these exact coefficients with the approximate ones using a large N and they aren't even close. — Preceding unsigned comment added by 97.117.195.123 (talk) 10:33, 22 September 2015 (UTC)

Oh--sorry. The comment not far above this, "Section on Orthogonality and Example 1 has errors," deals with the same subject. — Preceding unsigned comment added by 97.117.195.123 (talk) 10:35, 22 September 2015 (UTC)

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Proposed remove mention of spread polynomials
They seem to be part of the research program of one author and to have not caught on more widely. Googling "spread polynomials" and "Chebyshev polynomials" gives me 1.5k vs 364k hits. Spread polynomials are introduced in a book which is not widely cited apart from by its author. The work may be good but that shouldn't be wikipedia's job to evaluate. This mathoverflow thread suggests that they are basically equivalent, and that it's enough to just describe shifted Chebyshev polynomials. I propose removing all mention of spread polynomials from the article. The only reason to keep them, I think, would be if we find the approach to geometry in the book "Divine Proportions..." to be an important enough application of Chebyshev polynomials to mention in the main article. But Chebyshev polynomials have many many important applications I'd mention first, like their role in filter design, linear algebra algorithms, etc. Aram.harrow (talk) 19:17, 23 March 2021 (UTC)

update: I've edited the article to remove them. Aram.harrow (talk) 20:02, 28 March 2021 (UTC)

Generalized Chebyshev polynomials
The term "generalized Chebyshev polynomials" appears to have a variety of uses: among others, it is used to refer to Shabat polynomials, to certain two-variable or multi-variable generalizations, and to generalizations to unions of disjoint intervals. As used in the section "Generalized Chebyshev polynomials", however, it appears to refer to objects that are not actually polynomials most of the time. I have not been able locate any sources attesting to this use of the term. Would anyone be able to help? I have placed a "dubious" tag on the section. Will Orrick (talk) 05:41, 9 January 2022 (UTC)

 * I have removed the section. It seems unlikely that this is a common use of the term "generalized Chebyshev polynomials" if extensive searching doesn't turn up any examples of the term being used in this way. Will Orrick (talk) 03:40, 15 January 2022 (UTC)

Hyperbolic trigonometric functions; romanization of "Chebyshev"
Your argumet of cosh is purely imaginary assuming both x,t in R in your exponential generating function of T_{n} Would be better to use trig cos instead By the way this guy was named Chebyshov because over last e there is diaeresis which Russians usually dont write Russians usually dont write it but diareresis is still here — Preceding unsigned comment added by 188.47.27.31 (talk) 17:57, 29 May 2023 (UTC)

 * Wikipedia's biographical article on Chebyshev mentions the romanization issue. Unfortunately, the current spelling is well established in the mathematics literature, and Wikipedia needs to reflect what's in the majority of authoritative sources. Will Orrick (talk) 01:31, 30 May 2023 (UTC)
 * Anyone interested in Chebyshev and romanization might enjoy Philip Davis's The Thread. –jacobolus (t) 16:09, 28 May 2024 (UTC)

Yes but current transcription leads to misreading this name Many people misread this name — Preceding unsigned comment added by 188.47.53.242 (talk) 23:38, 7 July 2023 (UTC)

Incorrect integration formula for n=0
In the integration "The last formula can be further manipulated to express the integral of Tn as a function of Chebyshev polynomials of the first kind only:" the derivation is incorrect for n=0. It uses T_1 T_n = 1/2 (T_{n+1} + T_{n-1}). But according to the below section "Products of Chebyshev polynomials" the T_{n-1} should be T_{|n-1|}. So the formula is incorrect for n=0. 100.7.38.140 (talk) 19:41, 26 August 2023 (UTC)

title: Talk:Chebyshev polynomials


text:
Talk:Bennington (disambiguation)

There may be a Bennington, Vermont that is not Old Bennington, Vermont or Bennington County, Vermont. Check http://www.bennington for references. It may be the "5th-largest community in Vermont".

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Bennington (disambiguation)


text:
Acetyl-CoA

Acetyl-CoA (acetyl coenzyme A) is a molecule that participates in many biochemical reactions in protein, carbohydrate and lipid metabolism. Its main function is to deliver the acetyl group to the citric acid cycle (Krebs cycle) to be oxidized for energy production.

Coenzyme A (CoASH or CoA) consists of a β-mercaptoethylamine group linked to pantothenic acid (vitamin B5) through an amide linkage and 3'-phosphorylated ADP. The acetyl group (indicated in blue in the structural diagram on the right) of acetyl-CoA is linked to the sulfhydryl substituent of the β-mercaptoethylamine group. This thioester linkage is a "high energy" bond, which is particularly reactive. Hydrolysis of the thioester bond is exergonic (−31.5 kJ/mol).

CoA is acetylated to acetyl-CoA by the breakdown of carbohydrates through glycolysis and by the breakdown of fatty acids through β-oxidation. Acetyl-CoA then enters the citric acid cycle, where the acetyl group is oxidized to carbon dioxide and water, and the energy released is captured in the form of 11 ATP and one GTP per acetyl group.

Konrad Bloch and Feodor Lynen were awarded the 1964 Nobel Prize in Physiology or Medicine for their discoveries linking acetyl-CoA and fatty acid metabolism. Fritz Lipmann won the Nobel Prize in 1953 for his discovery of the cofactor coenzyme A.

Role
Acetyl-CoA is a metabolic intermediate that is involved in many metabolic pathways in an organism. It is produced during the breakdown of glucose, fatty acids, and amino acids, and is used in the synthesis of many other biomolecules, including cholesterol, fatty acids, and ketone bodies. Acetyl-CoA is also a key molecule in the citric acid cycle, which is a series of chemical reactions that occur in the mitochondria of cells and is responsible for generating energy in the form of ATP.

In addition, acetyl-CoA is a precursor for the biosynthesis of various acetyl-chemicals, acting as an intermediate to transfer an acetyl group during the biosynthesis of those acetyl-chemicals. Acetyl-CoA is also involved in the regulation of various cellular mechanisms by providing acetyl groups to target amino acid residues for post-translational acetylation reactions of proteins.

Biosynthesis
The acetylation of CoA is determined by the carbon sources.

Extramitochondrial

 * At high glucose levels, glycolysis takes place rapidly, thus increasing the amount of citrate produced from the citric acid cycle. This citrate is then exported to other organelles outside the mitochondria to be broken into acetyl-CoA and oxaloacetate by the enzyme ATP citrate lyase (ACL). This principal reaction is coupled with the hydrolysis of ATP.
 * At low glucose levels:
 * CoA is acetylated using acetate by acetyl-CoA synthetase (ACS), also coupled with ATP hydrolysis.
 * Ethanol also serves as a carbon source for acetylation of CoA utilizing the enzyme alcohol dehydrogenase.
 * Degradation of branched-chain ketogenic amino acids such as valine, leucine, and isoleucine occurs. These amino acids are converted to α-ketoacids by transamination and eventually to isovaleryl-CoA through oxidative decarboxylation by an α-ketoacid dehydrogenase complex. Isovaleryl-CoA undergoes dehydrogenation, carboxylation and hydration to form another CoA-derivative intermediate before it is cleaved into acetyl-CoA and acetoacetate.

Intramitochondrial

 * At high glucose levels, acetyl-CoA is produced through glycolysis. Pyruvate undergoes oxidative decarboxylation in which it loses its carboxyl group (as carbon dioxide) to form acetyl-CoA, giving off 33.5 kJ/mol of energy. The oxidative conversion of pyruvate into acetyl-CoA is referred to as the pyruvate dehydrogenase reaction. It is catalyzed by the pyruvate dehydrogenase complex. Other conversions between pyruvate and acetyl-CoA are possible.  For example, pyruvate formate lyase disproportionates pyruvate into acetyl-CoA and formic acid.
 * At low glucose levels, the production of acetyl-CoA is linked to β-oxidation of fatty acids. Fatty acids are first converted to acyl-CoA. Acyl-CoA is then degraded in a four-step cycle of oxidation, hydration, oxidation and thiolysis catalyzed by four respective enzymes, namely acyl-CoA dehydrogenase, enoyl-CoA hydratase, 3-hydroxyacyl-CoA dehydrogenase, and thiolase. The cycle produces a new fatty acid chain with two fewer carbons and acetyl-CoA as a byproduct.

Intermediates in various pathways

 * In cellular respiration
 * Citric acid cycle:
 * Through a series of chemical reactions, stored energy is released through the oxidation of acetyl-CoA derived from carbohydrates, fats, and proteins into adenosine triphosphate (ATP) and carbon dioxide.
 * Fatty acid metabolism
 * Acetyl-CoA is produced by the breakdown of both carbohydrates (by glycolysis) and lipids (by β-oxidation). It then enters the citric acid cycle in the mitochondrion by combining with oxaloacetate to form citrate.
 * Two acetyl-CoA molecules condense to form acetoacetyl-CoA, which gives rise to the formation of acetoacetate and β-hydroxybutyrate. Acetoacetate, β-hydroxybutyrate, and their spontaneous breakdown product acetone are frequently, but confusingly, known as ketone bodies (as they are not "bodies" at all, but water-soluble chemical substances). The ketone bodies are released by the liver into the blood. All cells with mitochondria can take ketone bodies up from the blood and reconvert them into acetyl-CoA, which can then be used as fuel in their citric acid cycles, as no other tissue can divert its oxaloacetate into the gluconeogenic pathway in the way that the liver does. Unlike free fatty acids, ketone bodies can cross the blood–brain barrier and are therefore available as fuel for the cells of the central nervous system, acting as a substitute for glucose, on which these cells normally survive. The occurrence of high levels of ketone bodies in the blood during starvation, a low-carbohydrate diet, prolonged heavy exercise, and uncontrolled type-1 diabetes mellitus is known as ketosis, and in its extreme form in out-of-control type-1 diabetes mellitus, as ketoacidosis.
 * On the other hand, when the insulin concentration in the blood is high, and that of glucagon is low (i.e. after meals), the acetyl-CoA produced by glycolysis condenses as normal with oxaloacetate to form citrate in the mitochondrion. However, instead of continuing through the citric acid cycle to be converted to carbon dioxide and water, the citrate is removed from the mitochondrion into the cytoplasm. There it is cleaved by ATP citrate lyase into acetyl-CoA and oxaloacetate. The oxaloacetate is returned to the mitochondrion as malate (and then converted back into oxaloacetate to transfer more acetyl-CoA out of the mitochondrion). This cytosolic acetyl-CoA can then be used to synthesize fatty acids through carboxylation by acetyl-CoA carboxylase into malonyl CoA, the first committed step in the synthesis of fatty acids. This conversion occurs primarily in the liver, adipose tissue and lactating mammary glands, where the fatty acids are combined with glycerol to form triglycerides, the major fuel reservoir of most animals. Fatty acids are also components of the phospholipids that make up the bulk of the lipid bilayers of all cellular membranes.
 * In plants, de novo fatty acid synthesis occurs in the plastids. Many seeds accumulate large reservoirs of seed oils to support germination and early growth of the seedling before it is a net photosynthetic organism.
 * The cytosolic acetyl-CoA can also condense with acetoacetyl-CoA to form 3-hydroxy-3-methylglutaryl-CoA (HMG-CoA) which is the rate-limiting step controlling the synthesis of cholesterol. Cholesterol can be used as is, as a structural component of cellular membranes, or it can be used to synthesize steroid hormones, bile salts, and vitamin D.
 * Acetyl-CoA can be carboxylated in the cytosol by acetyl-CoA carboxylase, giving rise to malonyl-CoA, a substrate required for synthesis of flavonoids and related polyketides, for elongation of fatty acids to produce waxes, cuticle, and seed oils in members of the Brassica family, and for malonation of proteins and other phytochemicals. In plants, these include sesquiterpenes, brassinosteroids (hormones), and membrane sterols.
 * Steroid synthesis:
 * Acetyl-CoA participates in the mevalonate pathway by partaking in the synthesis of hydroxymethyl glutaryl-CoA.
 * Acetylcholine synthesis:
 * Acetyl-CoA is also an important component in the biogenic synthesis of the neurotransmitter acetylcholine. Choline, in combination with acetyl-CoA, is catalyzed by the enzyme choline acetyltransferase to produce acetylcholine and coenzyme A as a byproduct.
 * Melatonin synthesis
 * Acetylation
 * Acetyl-CoA is also the source of the acetyl group incorporated onto certain lysine residues of histone and nonhistone proteins in the posttranslational modification acetylation. This acetylation is catalyzed by acetyltransferases. This acetylation affects cell growth, mitosis, and apoptosis.
 * Allosteric regulator
 * Acetyl-CoA serves as an allosteric regulator of pyruvate dehydrogenase kinase (PDK). It regulates through the ratio of acetyl-CoA versus CoA. Increased concentration of acetyl-CoA activates PDK.
 * Acetyl-CoA is also an allosteric activator of pyruvate carboxylase.

Interactive pathway map
''Click on genes, proteins and metabolites below to visit Gene Wiki pages and related Wikipedia articles. The pathway can be downloaded and edited at WikiPathways.''

title: Acetyl-CoA


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Traffic calming

Traffic calming uses physical design and other measures to improve safety for motorists, car drivers, pedestrians and cyclists. It has become a tool to combat speeding and other unsafe behaviours of drivers. It aims to encourage safer, more responsible driving and potentially reduce traffic flow. Urban planners and traffic engineers have many strategies for traffic calming, including narrowed roads and speed humps. Such measures are common in Australia and Europe (especially Northern Europe), but less so in North America. Traffic calming is a calque (literal translation) of the German word Verkehrsberuhigung – the term's first published use in English was in 1985 by Carmen Hass-Klau.

History
In its early development in the UK in the 1930s, traffic calming was based on the idea that residential areas should be protected from through-traffic. Subsequently, it became valued for its ability to improve pedestrian safety and reduce noise and air pollution from traffic.

For much of the 20th century, streets were designed by engineers who were charged only with ensuring smooth motor vehicular traffic flow and not with fostering the other functions of streets. Traffic calming initiatives have grown to consider other design functions as well. For example, it has been shown that car traffic severely impairs the social and recreational functions of public streets. The Livable Streets study by Donald Appleyard (1981) found that residents of streets with light traffic had, on average, three more friends and twice as many acquaintances as the people on streets with heavy traffic which were otherwise similar in dimensions, income, etc.

Measures
Traffic engineers refer to three "E's" when discussing traffic calming: engineering, (community) education, and (police) enforcement. Because neighborhood traffic management studies have shown that residents often contribute to the perceived speeding problem within their neighborhoods, instructions on traffic calming (for example in Hass-Klau et al., 1992 ) stress that the most effective traffic calming plans entail all three components—that engineering measures alone will not produce satisfactory results.

Engineering measures
Engineering measures involve physically altering the road layout or appearance to actively or passively reduce traffic speeds by one of several means: Measures include speed humps, chicanes, curb extensions, modal filters, and living street and shared space type schemes. The town of Hilden in Germany has achieved a rate of 24% of trips being on two wheels, mainly via traffic calming and the use of 30 km/h or 20 mph zones. In 1999, the Netherlands had over 6000 woonerven where cyclists and pedestrians have legal priority over cars and where a motorised "walking speed" limit applies. However, some UK and Irish "traffic calming" schemes, particularly involving road narrowings, are viewed as extremely hostile and have been implicated directly in death and injury to cyclists and pedestrians.
 * increasing the cognitive load of driving (making driving more difficult)
 * increasing the chance that an obstruction in the road will slow or momentarily stop motorists (such as replacing a bus pullout with a bus bulb, causing transit vehicles to stop for passenger loading and unloading in the travel lane, rather than pulling off and letting cars pass)
 * increasing the chance of passenger discomfort or even physical damage to a vehicle if speed limits are not observed (such as speed humps).

A number of visual changes to roads are being made to encourage more attentive driving, reduced speed, reduced crashes, and a greater tendency to yield to pedestrians. Visual traffic calming includes lane narrowings (9 –), road diets (reduction in lanes), use of trees next to streets, on-street parking, and buildings placed in urban fashion close to streets. Physical devices include speed humps, speed cushions and speed tables, sized for the desired speed. Such measures normally slow cars to between 10 and 25 mph. Most devices are made of asphalt or concrete but rubber traffic calming products are emerging as an effective alternative with several advantages.

Physical traffic calming can include the following engineering measures, grouped by similarity of method: Quite often residents have used a variety of homemade devices ranging from faux enforcement camera signs and even faux speed cameras and including dummy police. Some Canadian communities erect flexible bollards in the middle of the street in school zones. The bollards have a sign affixed indicating a 40 km/h speed limit.
 * Narrowing: Narrowing traffic lanes makes slower speeds seem more natural to drivers and are less intrusive than other treatments that limit speed or restrict route choice. Narrowing measures include:
 * Lane narrowings can be created by extending sidewalks, adding bollards or planters, or adding a bike lane or on-street parking.
 * Curb extensions (also called bulbouts) narrow the width of the roadway at pedestrian crossings
 * Chokers are curb extensions that narrow roadways to a single lane at certain points
 * Road diets remove a lane from the street. For example, allowing parking on one or both sides of a street to reduce the number of driving lanes.
 * Pedestrian refuges or small islands in the middle of the street can help reduce lane widths.
 * Converting one-way streets into two-way streets forces opposing traffic into close proximity, which requires more careful driving.
 * Vertical deflection: Raising a portion of a road surface can create discomfort for drivers travelling at high speeds. Both the height of the deflection and the steepness affect the severity of vehicle displacement. Vertical deflection measures include:
 * Speed bumps, sometimes split or offset in the middle to avoid delaying emergency vehicles
 * Speed humps, parabolic devices that are less aggressive than speed bumps.
 * Speed cushions, two or three small speed humps sitting in a line across the road that slow cars down but allows wider emergency vehicles to straddle them so as not to slow emergency response time.
 * Speed tables, long flat-topped speed humps that slow cars more gradually than humps.
 * Raised pedestrian crossings, which act as speed tables, often situated at junctions.
 * Raised intersections, like raised pedestrian crossings, act as speed tables.
 * Speed dips, sunken instead of raised (often seen as double dips in cycleways in The Netherlands).
 * Changing the surface material or texture (for example, the selective use of brick, cobblestone, or polymer cement overlay). Changes in texture may also include changes in color to highlight to drivers that they are in a pedestrian-centric zone.
 * Rumble strips, when placed perpendicular to traffic in the travel lane (rather than placed between lanes parallel to traffic) act as speed bumps as they produce unpleasant sounds and vibration when crossed at higher speeds.
 * Horizontal deflection, i.e. make the vehicle swerve slightly. These include:
 * Chicanes, which create a horizontal deflection that causes vehicles to slow as they would for a curve.
 * Pedestrian refuges again can provide horizontal deflection, as can kerb extensions and chokers.
 * Roundabouts and traffic circles provide another source of horizontal deflection especially when integrated within intersecting roads
 * Block or restrict access. Such traffic calming means include:
 * Median diverters to prevent left turns or through movements into a residential area.
 * Converting an intersection into a cul-de-sac or dead end.
 * Boom barrier, restricting through traffic to authorised vehicles only.
 * Closing of streets to create pedestrian zones.
 * Other means

Implementation strategies
There are primarily two implementation options for the creation of traffic calming measure: capital reconstruction versus operational changes.
 * Capital reconstruction is rebuilding a street, which may involve moving infrastructure (such as sewers, water lines, electrical equipment) and adjusting the location, size and path of the roadway. Typically, capital reconstruction involves years of planning and design, and millions of dollars of investment.
 * Operational changes to a road have to do with more short-term changes that typically implement minor adjustments within the existing curbline or that do not require rebuilding the road. These treatments range from a very temporary use of safety cones or other safety-approved traffic management devices, such as flexible bollards, that test a traffic-calming initiative for a few days to longer-term operational changes that may include restriping a street, which is best to plan around any routine maintenance resurfacing.

Enforcement and education measures
Enforcement and education measures for traffic calming include:
 * Reducing speed limits near institutions such as schools and hospitals (see below)
 * Vehicle activated signs and radar speed feedback signs, signs which react with a message if they detect a vehicle exceeding a pre-determined speed.
 * Embedded pavement flashing-light systems which react to pedestrian presence at crossings to signal drivers and increase awareness.
 * Watchman, traffic calming system

Speed limits
Speed reduction has traditionally been attempted by the introduction of statutory speed limits. Traffic speeds of 30 km/h (20 mph) and lower are said to be more desirable on urban roads with mixed traffic. The Austrian city of Graz, which has achieved steady growth in cycling, has applied 30 km/h limits to 75% its streets since 1994. Zones where speeds are set at 30 km/h (or 20 mph) are preferred by some as they are found to be effective at reducing crashes and increasing community cohesion. Speed limits which are set below the speed that most motorists perceive to be reasonable for the given road require additional measures to improve compliance. Attempts to improve speed limit observance are usually by either education, enforcement or road engineering. "Education" can mean publicity campaigns or targeted road user training.

Speed limit enforcement techniques include: direct police action, automated systems such as speed cameras or vehicle activated signs or traffic lights triggered by traffic exceeding a preset speed threshold. One cycling expert argues for placing direct restrictions on motor-vehicle speed and acceleration performance. An EU report on promoting walking and cycling specifies as one of its top measures comprehensive camera-based speed control using mainly movable equipment at unexpected spots. The Netherlands has an estimated 1,500 speed/red-light camera installations and has set a target for 30 km/h limits on 70% of urban roads. The UK has more than 6,000 speed cameras, which took more than £100 million in fines in 2006/07.

Europe
Traffic calming has been successfully used for decades in cities across Europe. For example, a living street (sometimes known as home zones or by the Dutch word woonerf, as the concept originated in the Netherlands) towards the end of the 1960s, initially in Delft, is a street in which the needs of car drivers are secondary to the needs of other road users; traffic calming principles are integrated into their design. From the Netherlands, the concept spread rapidly to Germany, starting in North Rhine-Westphalia in 1976, and had become very widespread by the early 1980s. The ideas and techniques also spread to the UK towards the end of the 1980s, and practice there was advocated by academics such as Tim Pharaoh and Carmen Hass-Klau. The guidelines published by Devon County Council (of which Tim Pharaoh was the principal author) in 1991 were particularly well received.

In the United Kingdom, Low Traffic Neighbourhoods incorporate traffic calming and filtered permeability. In 2020, some LTNs were introduced with emergency funding from the government, in response to the COVID-19 pandemic. A study found that people living in LTNs in Waltham Forest became less likely to own a car and were more likely to walk or cycle.

School Streets are another UK scheme which involves part-time restrictions on motor vehicles during school pick up and drop off times.

In Ireland, traffic calming schemes have been implemented on national roads since 1993, typically on those with a hard shoulder, on the approach to towns and villages where the speed limit is reduced from 100 km/h to 50 or 60 km/h. This is done by reducing the width of the hard shoulder and the carriageway, various landscaping and installation of 'gateways' in order to reduce the driver's field of view and thus their speed. A gateway marks the transition from high-speed to low-speed road and may feature a pavement, cycle lane, central island (where the road is sufficiently wide enough) or all three, and is accompanied with town/village entrance and speed limit signs as well as bollards and a lamppost in the island.

An evaluation of 91 traffic calming schemes implemented between 1997 and 2002 showed that they were successful in reducing road collisions, the number of which decreased by 13%. The number of fatal collisions was reduced by 52%.

North America
By 2017, San Francisco's Vision Zero program, which heavily features traffic calming, has reduced fatalities by 33%.

A 2018 study found that traffic calming measures in Portland, Oregon reduced excessive speeds, reduced daily traffic volume by 16% and increased home prices by 1%.

Japan
Various forms of traffic calming are used in Japanese cities, particularly in large cities like Tokyo and Yokohama. Tokyo's narrow streets force automobiles and pedestrians to be close to one another; a common traffic calming technique in Tokyo is to change the surface material and/or texture of the shoulder of narrow roads, which helps define the boundary between cars and pedestrians, while allowing cars to use those spaces to pass oncoming traffic.

Reception and evaluation
A Cochrane Review of studies found that there is evidence to demonstrate the efficacy of traffic calming measures in reducing traffic-related injuries and may even reduce deaths. However, the review found that more evidence is needed to demonstrate its efficacy in low income countries.

title: Traffic calming


text:
Joel Garreau

Joel Garreau (born 1948) is an American journalist, scholar, and author.

In 1981, Garreau published The Nine Nations of North America. In 1991, he published Edge City: Life on the New Frontier. In 2005, he published Radical Evolution: The Promise and Peril of Enhancing Our Minds, Our Bodies—and What It Means to Be Human. He has served as a fellow at Cambridge University, a Bernard L. Schwartz Fellow at New America Foundation, the University of California at Berkeley and George Mason University. Previously, he was a reporter and editor at The Washington Post. He is a senior fellow at the School of Public Policy at George Mason University, leading two groups, one studying the future of universities and the other examining which global gateway city regions will be the winners and losers in the year 2020.

title: Joel Garreau


text:
Paul D'Amour

Paul D'Amour (born May 12, 1967) is an American musician and was the first bassist for Tool. His bass sound is recognized by the aggressive picked tone he developed with his Chris Squire Signature Rickenbacker 4001CS, which can be heard on Tool's first full-length album, Undertow. Since March 2019, he has been the bassist for industrial metal band Ministry.

Biography
D'Amour was born in Spokane, Washington. Originally a guitarist, D'Amour became Tool's bassist after being introduced to the band by Adam Jones. Like Jones, D'Amour was in Los Angeles because of his wish to enter the film industry. D'Amour built movie sets and worked in an art department on music videos and commercials.

D'Amour notably appeared on Tool's debut release, Opiate, in 1992 and also Undertow the following year. The band achieved mainstream success due to Undertow, fueled by the charting singles "Sober" and "Prison Sex". In late 1995, D'Amour left Tool during the early recording stages of the Ænima album. He was ultimately credited as a co-writer on five songs. According to drummer Danny Carey, D'Amour left the band because he wanted to play guitar rather than bass. D'Amour corroborated this in 2020 saying, "I always wanted to do other things, and it felt like I was too much in a box with that band ... I'm not just a bass player; I'm a creator, I wanted to have a bigger role, and it just wasn't happening in that situation."

After his departure from Tool, he formed the psychedelic pop band Lusk with Brad Laner, Chris Pitman (future member of Guns N' Roses), and Greg Edwards of Failure and Autolux. In 1997, they released their only album, entitled Free Mars. In addition, D'Amour played guitar in a group named Replicants, a cover band that included Ken Andrews and Greg Edwards from Failure, as well as Chris Pitman. They released one self-titled album in 1995, with a guest appearance from former Tool bandmate Maynard James Keenan.

In early 2005, D'Amour wrote and performed under the name Feersum Ennjin. The name is inspired by the science fiction novel Feersum Endjinn by Iain Banks, an author whose novel The Wasp Factory was conceptual inspiration for Lusk as well. The project released a self-titled EP on Silent Uproar Records. In 2011, a self-titled LP was released on Dissociated Press, featuring some songs that had been released previously and some new ones. On the first track of the LP, "The Fourth", former Tool band-mate Danny Carey plays drums.

D'Amour also played bass in the band Lesser Key. The group consists of Andrew Zamudio (vocals), Brett Fanger (guitar), and Justin Hanson (drums). The band "represents an exploration into personal and artistic freedom." On July 26, 2013, the band released a video of their debut single "Intercession." Their debut EP was produced by former Tool producer Sylvia Massy and released on April 1, 2014 on Sumerian Records.

As of late March 2019, D'Amour joined Ministry as the band's newest bass player, replacing Tony Campos.

Selected discography

 * With Tool
 * Opiate (1992)
 * Undertow (1993)

 * With Replicants
 * Replicants (1995)

 * With Lusk
 * Free Mars (1997)

 * With Feersum Ennjin
 * Feersum Ennjin (2011)

 * With Lesser Key
 * Lesser Key (2014)

 * With Ministry
 * Moral Hygiene (2021)
 * Hopiumforthemasses (2024)

title: Paul D'Amour


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Admiralty Islands

The Admiralty Islands are an archipelago group of 40 islands in the Bismarck Archipelago, to the north of New Guinea in the South Pacific Ocean. These are also sometimes called the Manus Islands, after the largest island.

These rainforest-covered islands constitute Manus Province, the smallest and least-populous province of Papua New Guinea, in its Islands Region. The total area is 2100 km2. The province had a population of 60,485 at the 2011 Census. Many of the smaller Admiralty Islands are atolls and uninhabited.

Islands
The larger islands in the center of the group are Manus Island and Los Negros Island. The other larger islands are Tong Island, Pak Island, Rambutyo Island, Lou Island, and Baluan Island to the east, Mbuke Island to the south and Bipi Island to the west of Manus Island. Other islands that have been noted as significant places in the history of Manus include Ndrova Island, Pitylu Island and Ponam Island.

Geography
The temperature of the Admiralty Islands varies little throughout the year, reaching daily highs of 30 - 32 C and 20 - 24 C at night. Average annual rainfall is 3382 mm and is somewhat seasonal, with June–August being the wettest months.

Manus reaches an elevation of 700 m and is volcanic in origin and probably broke through the ocean's surface in the late Miocene, 8–10 million years ago. The substrate of the island is either directly volcanic or from uplifted coral limestone.

The main town in the islands is Lorengau on Manus, connected by road to an airport on nearby Los Negros Island, otherwise transport around the islands is by boat. There is little tourism, although the seas are attractive to divers, including Jean-Michel Cousteau who spent time on nearby Wuvulu Island in the 1970s.

Ecology
Due to the isolated location, the rain forests of the Admiralty Islands are home to rare and endemic species of birds, bats and other animals, and are considered a separate ecoregion, the Admiralty Islands lowland rain forests. The majority of the forests on Manus still remain, but some of the smaller islands have been cleared for coconut farming. The typical tree species are various Calophyllum and Sararanga species.

The 58.5 km2 Ndrolowa Wildlife Management Area was created in March 1985 south of Lorengau on Manus Island and contains both terrestrial and marine regions. A further 240 km2 protected area has been established around the highest mountain on Manus, Mount Dremsel, but the level of protection remains undetermined in the UNEP World Database on Protected Areas.

Three of the bird species endemic to Admiralty Islands have been listed as vulnerable in the IUCN Red List: Manus fantail (Rhipidura semirubra), superb pitta (Pitta superba) and Manus masked owl (Tyto manusi). Three other birds are endemic to Admiralty Islands but are classified as non-threatened or least concern: white-naped friarbird (Philemon albitorques), Manus monarch (Monarcha infelix) and Manus hawk owl (Ninox meeki). Birds found mainly but not exclusively on the Admiralty Islands include Melanesian megapode (Megapodius eremita), yellow-bibbed fruit-dove (Ptilinopus solomonensis), yellowish imperial-pigeon (Ducula subflavescens), pied cuckoo-dove (Reinwardtoena browni), Meek's pygmy parrot (Micropsitta meeki), black-headed white-eye (Zosterops hypoxanthus) and ebony myzomela (Myzomela pammelaena).

Mammals found only here or on nearby island groups include the large fruit bats, Admiralty flying-fox (Pteropus admiralitatum), Andersen's naked-backed fruit bat (Dobsonia anderseni) and Seri's sheathtail-bat (Emballonura serii) while the two pure-endemics are Admiralty Island cuscus (Spilocuscus kraemeri) and a local mosaic-tailed rat (Melomys matambuai). The islands are home to two endemic Cornufer frogs (Cornufer admiraltiensis and Cornufer latro) and four lizards, while the emerald green snail of Manus was the first terrestrial snail to be listed as vulnerable by the IUCN.

Prehistory
Along with New Guinea, the Bismarck Archipelago and the Solomon Islands Archipelago, the Admiralty Islands were first inhabited approximately 40,000 years ago, in the initial wave of migration out of Southeast Asia that also populated Australia. This early society appears to have cultivated taro, and to have deliberately introduced wild animals from New Guinea such as bandicoots and large rats. Obsidian was gathered and traded throughout the Admiralty Islands archipelago.

The Lapita culture arose around 3,500 years ago, and its extent ranged from the Admiralty Islands to Tonga and Samoa. Its origins are contested, but it may well have been a product of another wave of migration from Southeast Asia. Lapita society featured renowned pottery, stilt houses, the introduction of domestic animals such as pigs, dogs, and chickens, and substantial developments in agriculture and boat technology, allowing long-distance trade to develop. Lapita society, as a distinct culture and extended trade network, collapsed around 2,000 years ago.

European and Japanese periods
The first European to visit the islands was the Spanish navigator Álvaro de Saavedra when trying to return from Tidore to New Spain in the summer of 1528. Saavedra charted Manus as Urays la Grande. Its visit was also reported in 1616 by the Dutch navigator Willem Schouten. The name 'Admiralty Islands' was devised by Captain Philip Carteret of the British Royal Navy in 1767.

Between 1884 and 1914 the area was administered as a German colony. In November 1914, the islands were occupied by troops of the Australian Naval and Military Expeditionary Force landed from the SS Siar. A few shots fired from a machine gun on Siar over the heads of the tiny German garrison at Lorengau were the last shots fired in the battle. After the war, the islands were governed by the Commonwealth of Australia under a League of Nations mandate.

Japanese troops landed on Manus Island on 7 April 1942. In 1944, Japanese forces occupying the islands were attacked and defeated by Allied forces in Operation Brewer. Subsequently, a large American airbase and Manus Naval Base was built at Lombrum near Lorengau.

Independence
Following Papuan independence in 1975, the Admiralty Islands became part of Papua New Guinea.

They became the Manus Province of the Islands Region in Papua New Guinea.

title: Admiralty Islands


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Sibelius (disambiguation)

Sibelius may refer to:
 * Jean Sibelius (1865–1957), Finnish composer of classical music, or his namesakes:
 * Sibelius Academy, a music university in Finland
 * Sibelius (film), a 2003 film about the composer
 * Sibelius Hall, a concert hall
 * Sibelius (scorewriter), a scorewriter program developed by Sibelius Software Limited, now part of Avid
 * Sibelius monument, a monument to the composer
 * Sibelius (train), a train that operated between Helsinki and St. Petersburg on the Riihimäki–Saint Petersburg Railway
 * 1405 Sibelius, an asteroid

title: Sibelius (disambiguation)