text:
Chebyshev polynomials
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev,[1] are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and Chebyshev polynomials of the second kind which are denoted Un. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
The Chebyshev polynomials Tn or Un are polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence.
Chebyshev polynomials are polynomials with the largest possible leading coefficient, but subject to the condition that their absolute value on the interval [-1,1] is bounded by 1. They are also the extremal polynomials for many other properties.[2]
Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the polynomial of best approximation to a continuous function under the maximum norm. This approximation leads directly to the method of Clenshaw–Curtis quadrature.
In the study of differential equations they arise as the solution to the Chebyshev differential equations
for the polynomials of the first and second kind, respectively. These equations are special cases of the Sturm–Liouville differential equation.
- 1 Definition
- 2 Relation between Chebyshev polynomials of the first and second kinds
- 3 Explicit expressions
- 4 Properties
- 5 Examples
- 6 As a basis set
- 7 Spread polynomials
- 8 See also
- 9 Notes
- 10 References
- 11 External links
The Chebyshev polynomials of the first kind are defined by the recurrence relation
The ordinary generating function for Tn is
the exponential generating function is
The generating function relevant for 2-dimensional potential theory and multipole expansion is
The Chebyshev polynomials of the second kind are defined by the recurrence relation
The ordinary generating function for Un is
the exponential generating function is
Trigonometric definition[edit]
The Chebyshev polynomials of the first kind can be defined as the unique polynomials satisfying
or, in other words, as the unique polynomials satisfying
for n = 0, 1, 2, 3, ... which is a variant (equivalent transpose) of Schröder's equation, viz. Tn(x) is functionally conjugate to nx, codified in the nesting property below. Further compare to the spread polynomials, in the section below.
The polynomials of the second kind satisfy:
which is structurally quite similar to the Dirichlet kernel :
That cos(nx) is an nth-degree polynomial in cos(x) can be seen by observing that cos(nx) is the real part of one side of de Moivre's formula, and the real part of the other side is a polynomial in cos(x) and sin(x), in which all powers of sin(x) are even and thus replaceable through the identity cos2(x) + sin2(x) = 1.
This identity is quite useful in conjunction with the recursive generating formula, inasmuch as it enables one to calculate the cosine of any integral multiple of an angle solely in terms of the cosine of the base angle.
Evaluating the first two Chebyshev polynomials,
one can straightforwardly determine that
and so forth.
Two immediate corollaries are the composition identity (or nesting property specifying a semigroup)
and the expression of complex exponentiation in terms of Chebyshev polynomials: given z = a + bi,
Pell equation definition[edit]
The Chebyshev polynomials can also be defined as the solutions to the Pell equation
in a ring R[x].[3] Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:
Products of Chebyshev polynomials[edit]
When working with Chebyshev polynomials quite often products of two of them occur. These products can be reduced to combinations of Chebyshev polynomials with lower or higher degree and concluding statements about the product are easier to make. It shall be assumed that in the following the index m is greater than or equal to the index n and n is not negative. For Chebyshev polynomials of the first kind the product expands to
which is an analogy to the addition theorem with the identities For n=1 this results in the already known recurrence formula, just arranged differently, and with n=2 it forms the recurrence relation for all even or all odd Chebyshev polynomials (depending on the parity of the lowest m) which allows to design functions with prescribed symmetry properties. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion:
For Chebyshev polynomials of the second kind products may be written as:
By this, like above, with n=2 the recurrence formula of Chebyshev polynomials of the second kind forms for both types of symmetry to
- depending on whether m starts with 2 or 3.
Relation between Chebyshev polynomials of the first and second kinds[edit]
The Chebyshev polynomials of the first and second kind are closely related by the following equations
- , where n is odd.
- , where n is even.
The recurrence relationship of the derivative of Chebyshev polynomials can be derived from these relations
This relationship is used in the Chebyshev spectral method of solving differential equations.
Equivalently, the two sequences can also be defined from a pair of mutual recurrence equations:
These can be derived from the trigonometric formulae; for example, if , then
Note that both these equations and the trigonometric equations take a simpler form if we, like some works, follow the alternate convention of denoting our Un (the polynomial of degree n) with Un+1 instead.
Turán's inequalities for the Chebyshev polynomials are
- and
The integral relations are
where integrals are considered as principal value.
Explicit expressions[edit]
Different approaches to defining Chebyshev polynomials lead to different explicit expressions such as:
where 2F1 is a hypergeometric function.
Roots and extrema[edit]
A Chebyshev polynomial of either kind with degree n has n different simple roots, called Chebyshev roots, in the interval [−1,1]. The roots of the Chebyshev polynomial of the first kind are sometimes called Chebyshev nodes because they are used as nodes in polynomial interpolation. Using the trigonometric definition and the fact that
one can easily prove that the roots of Tn are
Similarly, the roots of Un are
The extrema of Tn on the interval −1 ≤ x ≤ 1 are located at
One unique property of the Chebyshev polynomials of the first kind is that on the interval −1 ≤ x ≤ 1 all of the extrema have values that are either −1 or 1. Thus these polynomials have only two finite critical values, the defining property of Shabat polynomials. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:
Differentiation and integration[edit]
The derivatives of the polynomials can be less than straightforward. By differentiating the polynomials in their trigonometric forms, it's easy to show that:
The last two formulas can be numerically troublesome due to the division by zero (0/0 indeterminate form, specifically) at x = 1 and x = −1. It can be shown that:
The second derivative of the Chebyshev polynomial of the first kind is
which, if evaluated as shown above, poses a problem because it is indeterminate at x = ±1. Since the function is a polynomial, (all of) the derivatives must exist for all real numbers, so the taking to limit on the expression above should yield the desired value:
where only is considered for now. Factoring the denominator:
Since the limit as a whole must exist, the limit of the numerator and denominator must independently exist, and
The denominator (still) limits to zero, which implies that the numerator must be limiting to zero, i.e. which will be useful later on. Since the numerator and denominator are both limiting to zero, L'Hôpital's rule applies:
The proof for is similar, with the fact that being important.
Indeed, the following, more general formula holds:
This latter result is of great use in the numerical solution of eigenvalue problems.
Concerning integration, the first derivative of the Tn implies that
and the recurrence relation for the first kind polynomials involving derivatives establishes that
Both the Tn and the Un form a sequence of orthogonal polynomials. The polynomials of the first kind are orthogonal with respect to the weight
on the interval [−1,1], i.e. we have:
This can be proven by letting x = cos (θ) and using the defining identity Tn(cos (θ)) = cos (nθ).
Similarly, the polynomials of the second kind are orthogonal with respect to the weight
on the interval [−1,1], i.e. we have:
(Note that the measure is, to within a normalizing constant, the Wigner semicircle distribution).
The Tn also satisfy a discrete orthogonality condition:
where the xk are the N Chebyshev nodes (see above) of TN(x)
For the polynomials of the second kind and with the same Chebyshev nodes xk there are similar sums:
and without the weight function:
Based on the N zeros of the Chebyshev polynomial of the second kind UN(x)
a different sum can be constructed
and again without the weight function:
Minimal ∞-norm[edit]
For any given n ≥ 1, among the polynomials of degree n with leading coefficient 1,
is the one of which the maximal absolute value on the interval [−1, 1] is minimal.
This maximal absolute value is
and |ƒ(x)| reaches this maximum exactly n + 1 times at
- Proof
Let's assume that is a polynomial of degree n with leading coefficient 1 with maximal absolute value on the interval [−1, 1] less than .
Because at extreme points of we have
From the intermediate value theorem, has at least n roots. However, this is impossible, as is a polynomial of degree n − 1, so the fundamental theorem of algebra implies it has at most n − 1 roots.
Other properties[edit]
The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials, which themselves are a special case of the Jacobi polynomials:
For every nonnegative integer n, Tn(x) and Un(x) are both polynomials of degree n. They are even or odd functions of x as n is even or odd, so when written as polynomials of x, it only has even or odd degree terms respectively. In fact,
The leading coefficient of Tn is 2n − 1 if 1 ≤ n, but 1 if 0 = n.
Tn are a special case of Lissajous curves with frequency ratio equal to n.
Several polynomial sequences like Lucas polynomials (Ln), Dickson polynomials (Dn), Fibonacci polynomials (Fn) are related to Chebyshev polynomials Tn and Un.
The Chebyshev polynomials of the first kind satisfy the relation
which is easily proved from the product-to-sum formula for the cosine. The polynomials of the second kind satisfy the similar relation
(with the convention )
Similar to the formula
we have the analogous formula
- .
For ,
- and
- ,
which follows from the fact that this holds by definition for .
- .
Then and are commuting polynomials:
- ,
as is evident in the Abelian nesting property specified above.
First kind[edit]
The first few Chebyshev polynomials of the first kind are A028297
Second kind[edit]
The first few Chebyshev polynomials of the second kind are
As a basis set[edit]
In the appropriate Sobolev space, the set of Chebyshev polynomials form a orthonormal basis, so that a function in the same space can, on −1 ≤ x ≤ 1 be expressed via the expansion:[4]
Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients an can be determined easily through the application of an inner product. This sum is called a Chebyshev series or a Chebyshev expansion.
Since a Chebyshev series is related to a Fourier cosine series through a change of variables, all of the theorems, identities, etc. that apply to Fourier series have a Chebyshev counterpart.[4] These attributes include:
- The Chebyshev polynomials form a complete orthogonal system.
- The Chebyshev series converges to ƒ(x) if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most cases — as long as there are a finite number of discontinuities in ƒ(x) and its derivatives.
- At a discontinuity, the series will converge to the average of the right and left limits.
The abundance of the theorems and identities inherited from Fourier series make the Chebyshev polynomials important tools in numeric analysis; for example they are the most popular general purpose basis functions used in the spectral method,[4] often in favor of trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still a problem).
Example 1[edit]
Consider the Chebyshev expansion of . One can express
One can find the coefficients either through the application of an inner product or by the discrete orthogonality condition. For the inner product,
which gives
Alternatively, when you cannot evaluate the inner product of the function you are trying to approximate, the discrete orthogonality condition gives
where is the Kronecker delta function and the are the N Gauss–Lobatto zeros of
This allows us to compute the coefficients very efficiently through the discrete cosine transform
Example 2[edit]
To provide another example:
Partial sums[edit]
The partial sums of
are very useful in the approximation of various functions and in the solution of differential equations (see spectral method). Two common methods for determining the coefficients an are through the use of the inner product as in Galerkin's method and through the use of collocation which is related to interpolation.
As an interpolant, the N coefficients of the (N − 1)th partial sum are usually obtained on the Chebyshev–Gauss–Lobatto[5] points (or Lobatto grid), which results in minimum error and avoids Runge's phenomenon associated with a uniform grid. This collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by:
Polynomial in Chebyshev form[edit]
An arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials of the first kind.[6] Such a polynomial p(x) is of the form
Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.
Spread polynomials[edit]
The 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.
See also[edit]
- Chebyshev filter
- Chebyshev cube root
- Dickson polynomials
- Legendre polynomials
- Hermite polynomials
- Chebyshev rational functions
- Approximation theory
- The Chebfun system
- Discrete Chebyshev transform
- ^ Chebyshev polynomials were first presented in: P. L. Chebyshev (1854) "Théorie des mécanismes connus sous le nom de parallélogrammes," Mémoires des Savants étrangers présentés à l'Académie de Saint-Pétersbourg, vol. 7, pages 539–586.
- ^ Rivlin, Theodore J. The Chebyshev polynomials. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney,1974. Chapter 2, "Extremal Properties", pp. 56--123.
- ^ Jeroen Demeyer Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields, Ph.D. theses (2007), p.70.
- ^ a b c Boyd, John P. (2001). Chebyshev and Fourier Spectral Methods (second ed.). Dover. ISBN 0-486-41183-4.
- ^ Chebyshev Interpolation: An Interactive Tour
- ^ For more information on the coefficients, see: Mason, J. C. and Handscomb, D. C. (2002). Chebyshev Polynomials. Taylor & Francis.
- Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 22", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 773, ISBN 978-0486612720, MR 0167642.
- Dette, Holger (1995), A Note on Some Peculiar Nonlinear Extremal Phenomena of the Chebyshev Polynomials, Proceedings of the Edinburgh Mathematical Society 38, 343-355
- Eremenko, A.; Lempert, L. (1994), An Extremal Problem For Polynomials, Proceedings of the American Mathematical Society, Volume 122, Number 1, 191-193
- Mason, J. C.; Handscomb, D. C. (2002). Chebyshev Polynomials. Taylor & Francis.
- Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
- Remes, Eugene, On an Extremal Property of Chebyshev Polynomials
- Suetin, P.K. (2001), "C/c021940", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
External links[edit]
- Weisstein, Eric W., "Chebyshev Polynomial of the First Kind", MathWorld.
- Module for Chebyshev Polynomials by John H. Mathews
- Chebyshev Interpolation: An Interactive Tour, includes illustrative Java applet.
- Numerical Computing with Functions: The Chebfun Project
- Is there an intuitive explanation for an extremal property of Chebyshev polynomials?


text:
remember 2008's awkward opening monologue? — but this year there's just no time! You see, 2010's telecast will be live, or as Tom Bergeron would say, "Liiiiiiiiiiiive!" and it needs to end on time because it'll be replayed right away on the West Coast. And a Bob Hope Humanitarian Award is being given out for the first time in six years. Sorry, most popular shows on television! Awards have to be given out to a whole bunch of TV movies and miniseries that barely anybody watches, and George Clooney's gonna wear a tux. To which we say… READ FULL STORYReality TV fans are pissed. The Academy of Television Arts and Sciences announced last week its decision to banish the Outstanding Host for a Reality or Reality-Competition Program category to its Creative Arts ceremony, effectively banning it from Aug. 29's Primetime Emmys telecast. ATAS has deigned to allow the merry band of hosts on the Emmys stage before —
Tag: The Amazing Race (31-40 of 40)
'The Amazing Race' season finale: And the winner is...
Race fans, we've had a long and memorable season. After many centuries of philosophical debate, we learned empirically that Cowboys are better than Detectives. We saw a re-enactment of the trench warfare of World War I. Perhaps more terrifyingly, we saw Brent and Caite re-enact the same fight on five continents. ("You're dumb!" "You're dumb!" "You're a towel!" "No, YOU'RE a towel!") We saw just how easy it was to lose one coconut. (For want of a coconut, the race was lost…) And now, the 16th race around the world is over, and only one team reigns supreme. (SPOILERS APLENTY after the jump…)
'Amazing Race' winners talk hair gel and college loans!
Just because no one was surprised by the outcome of last night's Amazing Race finale doesn't mean it's any less impressive that couple Meghan Ricky and Cheyne Whitney reached the finish line first. After a night of drinking and a way-too-early Early Show appearance, the winning duo checked in to shed light on their victory, Meghan's voice, and Cheyne's hair.
EW: Where did you watch the finale last night?
Meghan: Most of the cast came out to New York so we all met up and got to watch it in a private screening room at a hotel. That was pretty fun to be with the cast again.
Cheyne: And then we stayed out drinking and went to bed about 5 hours ago [6 a.m.]
Meghan: No, we went to bed at 3:30 am and got up for the Early Show 3 hours later.
EW: How was it seeing everyone from the show? Tense?
Cheyne: Not at all. We're lucky enough that we're still really close with a lot of the cast. We had gone to visit Sam and Dan and Brian and Ericka. We've seen Zev and Justin, Ron and Marcy. We've hung out a lot with everybody. Our cast is really close. Everyone's really friendly and genuinely enjoys each other's company.
EW: Were drinks on you last night?
Meghan: Everyone wanted them to be but we said, "Let's wait until we get our check."
EW: And did you? READ FULL STORY
'The Amazing Race' finale: And the winner is...
Thankfully what happened in Vegas didn't stay in Vegas as Meghan & Cheyne, Sam & Dan, and Ericka & Brian raced that last leg to the $1 million prize. No spoilers here (BUT THE MESSAGE BOARDS ARE A DIFFERENT STORY). Jessica Shaw's got your full Amazing Race finale recap, but if you've landed here, tell us what you thought before you click over to get her take on the big finish. Could you stand the suspense??! Did your favorite team win?
More Amazing Race from EW:
Amazing Race: Talking with Flight Time and Big Easy
Amazing Race: Big Brother pair ready for next season?
Amazing Race: Last week's recap
'The Amazing Race': Talking with Flight Time and Big Easy
You can mock Nate "Big Easy" Lofton all you want for not being able to unscramble five letters to spell "Franz" last night. But the truth is we'll all miss him and Herb "Flight Time" Lang in next week's season finale. The fan favorite team of Harlem Globetrotters checked in with EW to defend their spelling skills and dish about those dastardly brothers.
EW: First of all, I'm not going to be able to call you Nate and Herb. You're Flight Time and Big Easy.
Big Easy: We're known as Flight Time and Big Easy all around the world. We've traveled so much. People know us by our nicknames. We used our regular names when we were talking to each other, though.
EW: How did you two get selected from the entire Globetrotters team to compete on Amazing Race?
Flight Time: Last year, in October 2008, [Globetrotters management] came to us with a list of reality shows and asked would we want to be on any of them if we had an opportunity. They pitched them about getting a couple of us on there. Luckily for us, we were the two selected to represent the team. We have a great relationship so we knew we would make a good team.
Big Easy: We went through everything that everyone else did to get on the show. We weren't just put on the show because we were Nate and Herb. We went through every step. We didn't get any extra help because of who we are. We had to work just as hard.
'Big Brother' winners Jeff and Jordan on 'The Amazing Race': Can it be true?
There are a couple post-Thanksgiving gifts that would bring a slap-happy grin to my face: stepping on the scale and realizing that I actually lost five pounds, and learning that my two favorite Big Brother contestants were about to participate in The Amazing Race. Some very instinctive fan took a picture of BB 11 winner Jordan Lloyd and her boyfriend, Jeff Schroeder, at the Los Angeles International airport recently and posted it on Twitter. Judging by their attire, the couple is either A) gearing up for a very exciting leg of the race or B) looking a little too precious in their matching T-shirts and backpacks. I'm going to assume the former, since Schroeder told EW last month that he and Lloyd wouldn't be "opposed" to appearing on another CBS competition show. Sadly, CBS wouldn't comment other than to say that officially, it won't announce the cast of the show's next season for a while. But pictures don't lie … or do they? An insider reminds me that Schroeder did, after all, swipe a Hawaii trip from Russell the Muscle this summer (though apparently, he cashed it in. Thanks, posters!). Could the couple just be heading off to vacay?
What do you think? Does it look like the BB couple is about to extend their 15 minutes and join the Race? Would you tune in to see if Jordo "pops a boob" again (as Jeff so eloquently said to her last summer)?
Photo Credit: Danielle Varey/Twitpic
Dubai polar bear: Best 'Amazing Race' clue keeper ever?
One of the upsides to Amazing Race challenges being super-lame lately is that the dramatic pans from the challenge to the "silent local character holding the next clue" have seemed much more hilarious than usual. I can't stop thinking about the native-to-Dubai polar bear who had to wait around during the Find/Build a Snowman Detour on last night's show. His startled expression contrasted so beautifully with his immobility and Mt. Tedium's near-total lack of suspense. Look at his baby blue scarf! Does anyone else have almost too-fond memories of this guy? Or which other international oddities have made an impression on you during Amazing Race? I smell a Colorful Local Characters EW.com photo gallery! (Not really.)
Latest Videos
- Sony hires real-life inspiration for Olivia Pope
- Madonna releases six tracks from 2015 album
- 'Amazing Race' season 25 winners are...
- Sony CEO: We've 'not caved, not given in'
- FBI officially ties North Korea to Sony hack
- Obama: 'A mistake' to pull 'The Interview'
- 'Colbert Report' has immortal, all-star finale
- 'Game of Thrones' refashions Arya's look
- 'White Collar' series finale: How it ended
- Kate Upton is People 'Sexiest Woman Alive'
- 31 Days of Holiday Binge: December picks
Stephen Colbert's Middle-earthen wear: 'Hobbit' character portraits for the Tolkien Nation
Mila and Ashton, Plus: Taylor Swift, Ryan Gosling, Shailene Woodley & More!
From Our Partners
TV Recaps
Sun, Dec 21 | Starz Comments +
Sat, Dec 20 | CBS Comments +
Fri, Dec 19 | Lifetime Comments +
Thu, Dec 18 | FX Comments +
Thu, Dec 18 | CBS Comments +
Thu, Dec 18 | Bravo Comments +

subset: commoncrawl


text:
What Do Teen Girls and Middle-Aged Dudes Have in Common?
Compelling television.
Skip to content, or skip to search.
Skip to content, or skip to search.
Compelling television.
"Because of my dad's work — he trades oil — I have a bodyguard, which sucks."
Many say the Internet is dangerous for teen girls. It can also save them.


text:
|Wind cancels Niners tennis match
CONWAY, S.C. - The on-going battle with weather continued for the Charlotte women's tennis team, as its scheduled match Wednesday afternoon at Coastal Carolina has been cancelled. High winds in the Conway area of 25 miles per hour with gusts upwards to 37 mph have forced the two teams to nix the contest.
The 49ers have had six matches this spring either cancelled or rescheduled due to various weather issues. Charlotte has endured rain, snow, cold temperatures or high winds.
The Niners, however, own a 6-4 record despite the unstable conditions.
Charlotte will look to resume play in 10 days, March 16, when it welcomes North Carolina Central to the Halton-Wagner Tennis Complex for a 1 p.m. first serve. The match is a rescheduled date from the original Jan. 26 meeting.


text:
All Resources on Giving
Christmas Joy Vs. the Kirchensteuer
2 Corinthians 8:1-9 and 2 Corinthians 9:6-15
by John Piper
If God values joyful, loving generosity so much as to give his beloved Son to create it in his people, then we can be absolutely assured that when we are more generous, we will be more happy and more fulfilled because God is bound to work mightily for those whose behavior he values so highly.
Call upon Me in the Day of Trouble and I Will Deliver You
Psalm 50:1-15
by John Piper
Be thankful, pay your vows, call for help. And when God delivers you from the day of trouble, glorify him with all your heart.
These You Ought to Have Done Without Neglecting the Others
Luke 11:37-42
by John Piper
Don't live for comforts, live for justice and compassion.
The Kingdom of Heaven Is a Treasure
Matthew 13:44
by John Piper
The kingdom of God is so valuable that losing everything on earth, but getting the kingdom, is a happy trade-off.
Do Not Grow Weary in Well-Doing
Galatians 6:6-10
by John Piper
When you get your paycheck, do you look to the Spirit for how to use this money for God's kingdom?
Treasuring Christ Together: The North Campus Extension, Part 2
Grace Abounding for Every Good Work
2 Corinthians 9
by John Piper
He Who Sows Bountifully Will Reap Bountifully
2 Corinthians 9:6-15
by John Piper
God is the Giver, not Taker, on both sides of our giving.


text:
You must log in to access this feature. Sign in with Twellow or Facebook to get started.
By signing in to Twellow, you indicate your agreement to our terms of service.
Simply connect with Facebook and kiss those password blues away,
or go here to reset your password.
Commercial Sales Representative with Royal LePage in Ottawa, ON


text:
[My Profile] [My Settings] [
My Games
It's not all an elaborate ruse, some misguided attempt to establish for myself an online persona of dubious quality. I really am dull. If you don't find that unbearable, though, this is my blog that examines just how truly boring I can be.
When I get a cold, I really, really, REALLY get a cold...
August 01, 2009 (03:14 PM)
So, I've had most of the last few days off from work, but I haven't been working on the site much. Why? Because I've had a cold. That sounds like the sort of thing that would inspire sarcastic comments of "Boo hoo, poor baby has a cold!" and I can understand that response, but my immune system is weird. I don't often get sick with a cold or anything else, but when I do... I really do.
My typical cold lasts me anywhere from one week to a month, which makes it a good thing that they don't come along all that often. The worst of my current one seems to be over, but I'm still avoiding sweets, still guzzling immune system-boosting tea and orange juice, still eating hardly anything at all these past few days. Another day or two and I should be golden again...
So yeah, I hate getting colds. They reduce my productivity to approximately nothing. I guess I needed a bit of a rest anyway, but I would have preferred to take it while feeling mostly decent.
August 01, 2009 (04:06 PM)
That's only ever me anymore. I get a cold at least once a year
usually more, but I've been good last year I guess
, and when I do, it usually feels like a month-long affair where the first week is typical but aggressive cold symptoms like congestion, cough, and maybe a little fatigue. Then after those pass, I wind up with a dry cough that refuses to go away so I continue drinking things like orange juice and the like until it does. In fact, I found myself drikning orange juice whenever I'm at school regardless of whether I'm sick or not just to prevent me from getting ill if I can help it.
I'm just thankful I never get the flu anymore or anything else more serious.
August 07, 2009 (11:13 AM)
I have kind of an odd immune system as well, but mine works in a sort of counter-intuitive way. NORMALLY,
there are recent exceptions
when I get sick I get ALL of the symptoms at once. I get over it really fast, but it comes with the price of getting hit with everything at once. I can get over things in 1 day that take other people a week
and normally this is what happens
Your immune system seems to have a rather sadistic sense of humor. You're doing all the right things to make yourself feel better, so hopefully you'll be back at 100% in no time. Well, maybe 80%. Definitely 70%. :
2005-2012 HonestGamers
Opinions expressed in this blog represent the opinions of those expressing them and do not necessarily reflect the opinions of site staff, users and/or sponsors. Unless otherwise stated, content above belongs to its copyright holders and may not be reproduced without express written permission.


text:
No matter how well a project rolls along it's always important to get a reader's perspective from outside our perfectly structured bubbles. Unfortunately, relying on friends and family for constructive criticism isn't the best way to strengthen a weakness. Someone who likes you is probably too nice to be a critic, no matter how many times you tell them to be "honest". Remember those drawings mom put on the fridge?
In the first week of May Penguin held the soft launch of Book Country – "a place to discover and share fiction", and the reception from writers has been warm, if not hot.
So, the question is: What makes Book Country different from all the other communities out there?
Two features stand out to me: 1) a dedication to having ACTIVE community members, and 2) clear-cut ideas about genre. (Of course the best thing to do is check it out for yourself - Click here for a link to Book Country.)
It's probably the demand for members to be active that makes Book Country stand out. Anyone looking for self-promotion without engaging in the constructive criticism of others are not welcome, assuring an "earnestly interactive" community. Members must must review 3 contributions before their work becomes visible.
Prefer the "in-person" approach to writing groups? The library offers a free creative writing club program that meets once a month on Monday evenings. Next meeting is May 16! Register here.


text:
Happy blind audition hump day!
We're getting closer to the Battle Rounds after last night's new additions. Instead of stocking up solely on superstars, the judges now have to think holistically, choosing singers according to how they'll fit within their teams. "I feel like everyone on my team is different and I like that," said Adam. "This year I wanna do something different," said Blake, adding that last year he "built a country army."
Although tonight's new batch were all surprisingly terrific, there was no four-chair-salute or contestant of Tessanne-vocal-proportions (perhaps Timyra-Joi?)
Also, is anyone getting the slightest bit annoyed by Blake's same old line about being a three-peat winner? Yes, it's impressive and true, but it's quickly becoming the equivalent of Tyra Banks' robotic "You're still in the running towards becoming America's Next Top Model" spiel. Good thing he decided to switch it up and actually bring his trophies in to do the talking, much to the other judges' chagrin.
Semi-famous contestant: Briana Cuoco, sister of Kaley Cuoco (last week was E.G. Daily)
Best one-liner: "No man that wears that outfit is in a humble position," said Adam on a certain judge's Thriller-jacket.
Now the new crop...
RAY BOUDREAUX, 25, of Louisiana, is a single dad to adorable daughter Audrey (he remains friends with her mother). He's got a buttery, smooth Paolo Nutini-like voice with the looks of a Ken doll. He mows lawns on the side and flying to the blind auditions was the first time he had ever been on a plane. Without even hearing him sing, Ray gets +1,000 points for the cute-kid-peanut gallery. Backstage his daughter yelps, "C'mon daddy, you can do this!" So articulate for a toddler!
Sings: "Use me" by Bill Withers
Verdict: Blake moves first, then CeeLo. Adam is impressed by Ray's eclectic qualities. "I think this is an incredible genetic feat here, you look like Blake, but you sound like CeeLo!" CeeLo compliments his "blue-eyed soul." But Ray chooses Blake, who promises to keep his sound "swampy." Backstage, Carson was shocked he didn't go with CeeLo and I was, too.
LINA GAUDENZI, 23, of Florida, is a former model who once moved to Paris by herself, singing in cafés. She's studied classical music, opera, and has the powerhouse pipes that scream Team Christina.
Sings: "Landslide" by Fleetwood Mac
Verdict: Christina and Blake duke it out for her. Lina's falsetto finale is just gorgeous, prompting Blake to shout, "Damn, girl!" Christina asks which genre she sees herself in and Lina says blues and folk. Blake tries to interpret that as country. To which Christina zings, "I think he speaks kind of one language over there." Lina chooses Christina. (Divas unite!) "America needs change," Christina proclaims, while fanning herself.
JUHI, 16, of Tennessee, is really cutting it close for best personal style versus last week's Holly Henry. Juhi is a super-smart ("above 4.0 GPA" according to dad), blue-haired, babydoll-dress-wearing, grunge-loving chick with aspirations to major in aerospace engineering or computer science. I think I squealed a little when she said she wants to her sound to be a cross between Amy Winehouse and Kurt Cobain. CeeLo described Juhi as "someone who I would've made friends with in school." Me too, CeeLo, me too!
Sings: "Mercy" by Duffy
Verdict: Christina makes the first move, followed by CeeLo. Seeing them both turned around, Juhi had the cutest reactions, from "I'm flipping out right now!" to "This is so weird!" She chooses CeeLo.
NEXT: A little bit country, a little bit rock 'n roll


text:
Services on Demand
Related links
- Similars in SciELO
Estudos Avançados
Print version ISSN 0103-4014
ROSEMBERG, Fúlvia. O branco no IBGE continua branco na ação afirmativa?. Estud. av. [online]. 2004, vol.18, n.50, pp. 61-66. ISSN 0103-4014. http://dx.doi.org/10.1590/S0103-40142004000100007.
THIS TEXT aims to present and discuss the challenges related to the use of categories such as skin color and race for affirmative action programs in education. It describes the procedures adopted by the Ford Foundation's International Graduate Scholarship Program and compares the results with data obtained from the Monthly Job Survey analyzed by Petruccelli (2001).


text:
Student administration
This section provides a guide to student administration at the University.
Student self-administration
Sydney Student is your online administration service, which you can use to update your personal details, enrolment information, manage your finances and many other candidature-related tasks. Find out more about what you can do in Sydney Student.
Log in to your student email account and Learning Management System (Blackboard).
Get more information about student IT services including student email, access labs, internet use, printing and your UniKey username and password.
Help during the University closedown
On Thursday 11 December the Student Centre will close at 12pm and re-open at 9am on Friday 12 December.
The University will be closed from 5pm Thursday 18 December 2014 to 9am Monday 5 January 2015.
For help with your enrolment and using the Sydney Student system during this time, please contact our help line on 1800 SYD UNI or +61 2 8627 1444 (from overseas), then press 2.
The help line is open from 9am to 5pm (Eastern Standard Time) on the following days:
- Monday 22 December 2014 Wednesday 24 December 2014
- Monday 29 December 2014 Wednesday 31 December 2014
- Friday 2 January 2015
The Student Centre
The Student Centre assists with matters relating to admissions, enrolments, HECS and domestic fees, class timetables, examinations, results and graduations. The Student Centre also issues documents including Academic Transcripts, letters and proof of enrolment.
If you are unable to complete your enquiry or self-administration task online, you can visit the Student Centre in person on Level 3 of the Jane Foss Russell Building or make an enquiry by phone or email.
When you visit the Student Centre in person, please ensure you have your student card, drivers licence, passport or other government issued photo identification with you.
|Student Centre Enquiries Counter and Call Centre
|Opening times
|Throughout the year
|9am - 5pm Monday to Friday
|Formal examination period (last 2 weeks of each semester)
|8.30am - 5pm Monday to Friday
|Saturday during formal exam period
|8.30am - 2.30pm
Please note: on Wednesday 11 December the Student Centre will close at 12pm. We will re-open at 9am on Thursday 12 December.
International Services
International Services provides advice and assistance relating to academic programs and visa regulations for international students, as well as administrative services including admission and fee payment. To make an enquiry in person, visit International Services on Level 4 of the Jane Foss Russell Building (open 9am-5pm, Mon-Fri).
Your faculty provides information and advice about course requirements, applying for a leave of absence, credit for previous study or special consideration; and to change your subject choices (if you cannot change them online).
Student Affairs Unit
The Student Affairs Unit can advise on areas such as: the academic progression program, student misconduct and appeals against academic decisions.


text:
The NY Mets No-Hitter Makes History - My proposal!
Proposal Date:
Proposal Location: New York
Views: 525
My fiance, Matthew, is a huge Met's fan. He listens to all the games on the radio if he cannot be at the game or by a tv. So, on a Friday night upon returning home from a friend's dinner party sans cell phones(!), Matt immedicately checks his blackberry for work & game scores. Getting into my pj's I hear a scream and lots of "whoas" as if something bad had happened - so I run into the living room asking what's wrong and is everything ok?! Matt is almost brought to tears when telling me that the Mets had their first no hitter ever (after 50 years) - and we missed it! Long story short, he gets down on one knee and says, "I want to be able to tell our kids that on this great night, although I missed the most important game in Mets history, it's also the night I proposed to their mother". So, at close to midnight on June 1, 2012, in my pajamas, I said YES! We have a brick at Citi Field with the engagement date and are excited to torture our future kids with visiting it at every gam
Comments: 0


text:
Fundraiser helps family devastated by fatal I-65 crash
By Christin Nance Lazerus firstname.lastname@example.com August 18, 2013 7:54PM
Updated: September 20, 2013 6:31AM
Friends, aquaintances and even complete strangers are banding together to help the Williams family pay for funeral arrangements in the wake of a horrific Thursday night crash on Interstate 65 in Jasper County that claimed the lives of seven family members.
Merrillville resident Lindsey Williams, 27, was killed, alongside her daughters, 5-year-old Yazmin Goldman and Arielle Goldman, 3, aunt Yvette Williams of Atlanta, Yvette's two children Jamin, 5, and Jasmin, 7, and uncle Amado Mangual, 49, of Merrillville, when a semi rear-ended their Jeep Cherokee and ignited a fire which engulfed the SUV.
A preliminary investigation showed that the Jeep was stopped behind a semi for work zone traffic at 9:52 p.m., when they were hit by a 2007 Volvo semi truck driven by Howard Stratton, 55, of Grand Rapids, Mich.
Family friend Jeff Julkes of Schererville established an online fundraiser on Friday and it's generated more than $7,600 in donations through Sunday afternoon. On Facebook, Julkes said he is in the process of transferring the money to Gerald Williams, who is the brother of Yvette and Lindsey Williams.
Funeral arrangements and a memorial service are in the works, but have not been finalized, according to the family's Facebook page.
The Indiana State Police are investigating the crash, using accident reconstruction of the scene and evidence and witness statements from the crash. Police also will try to determine if the Volvo truck was in operating condition.
The Jasper County Prosecutor's Office will review information from the investigation once it is complete.


text:
Welcome to CfaN, the evangelistic ministry of Reinhard Bonnke and Daniel Kolenda. . .
This year we are celebrating our 40 year anniversary as a ministry. Our hearts are full as we recount and revisit some of the memories of what God has done over the last 4 decades. However, we are just as aware now as ever before that Africa, and the world, needs transformation that only God can bring. We are pouring our lives and efforts into working with the Lord of the Harvest to bring in the full end-time harvest. You can be part of this too!
You will rejoice with us as we celebrate over 1 MILLION completed decision cards for Christ in our Gospel Campaigns so far in 2014! Please take some time to browse through our reports from recent campaigns or read our posts about current and relevant things to pray for, particularly in Africa.
As you browse you will see our passion and purpose summarised as - 'Winning Millions to Christ - One Soul at a Time', each and every person who attends a CfaN Gospel Campaign is an individual who is important to God and to us. Your support helps to touch individuals not just the masses. Please consider becoming a ONE SOUL PARTNER with CfaN today and help us to continue to change individual lives. Your gifts of support and prayers are valued and needed. We don't do this alone, but with YOU!
We are approaching the final In His Presence worship encounter for 2014 at Bethel Convention Centre on Saturday 15 November. You would not want to miss this! See our EVENTS PAGE for more information and to reserve your seats.
Finally, make sure you've signed up for our email updates (bottom of page). Also, please visit and 'Like' our FACEBOOK PAGE to keep up to date on what is happening here in the UK and in Africa and for more material and resources that will be a blessing to you.
Mark My Word with Reinhard Bonnke
Family Planning
For we do not have a High Priest who cannot sympathize with our weaknesses, but was in all points tempted as we are, yet without sin. (Hebrews 4:15)
Jesus Christ was born because He wanted to be born.


text:
CLIMAX IN CAPE TOWN: A global wave of women leaders crashes ashore in South Africa to strengthen movements of women
After 30 hours in flight, at last I am home. Yet, honestly, at AWID, in a sea of 2,200 women from every corner of the earth, I felt more at home than I have in a long time.
At every turn, I found an oasis of sparking minds wrapped with headscarves, turbans, beaded caps, pink scarves, and horn-rimmed glasses framing bright eyes. The conference halls and passageways rang with Swahili, French, Portuguese, and the warm tongue "clicks" of Xhosa and Zulu. There were bursts of shouting and laughter at long last reunions, hugs and ready grins that spoke a sisterhood above any language.
I even turned to Leah Okeyo, my beloved Kenyan colleague and said, "It's too early to go home, I wish I could live on an island for years with these women – to deepen our understanding and creativity." To which she laughed and replied, "Ah yes, but until we all begin to wreak havoc, because you know we can!"
And so havoc was wreaked at "The Power of Movements", the first AWID conference in 3 years. But it was the most uplifting and transformative kind. The kind of havoc that turns the old, destructive power structures upside-down, and unfurls something new and palpable.
There were many climaxes.
The first climax was the pre-summit Feminist Tech Exchange (FTX) with over 90 women leaders who gathered to "take back the tech". The summit kicked off in a fury of record winds – the fiercest wind Cape Town had experienced in 40 years. Throughout the AWID conference FTX women were everywhere, trading blog links, video and audio recording key moments - and grasping each other's hands with the special human bonds that had formed over their 3 day intensive. Bonds that will surely strengthen across the "wires" long after.
Although I could barely pull myself away from the World Pulse booth, where we oriented and registered hundreds of women and men to tell their stories on PulseWire – I at last broke away to a main plenary where they were announcing "a special surprise."
The surprise turned out to be a roof-raising tribute to the recently-passed legend, "Mama Africa" Miriam Makeba. We all had lumps in our throats and cameras flashed as we danced together. Each of us knew that there could not be a more fitting way to pay homage to her legacy – surrounded by a movement of women in her homeland of South Africa.
There were decadent desserts everywhere at every lunch, dinner and tea break, and I am now going through withdrawal from the cascading tortes, custards, and fruit mousses. The message was clear: hard-working women need some indulgence!
Over 800 women took to the streets in a massive march against violence against women and to call for greater state accountability in the prevention of and response to gender based violence. The march was organized by our awesome booth neighbor - the One in Nine Campaign, a South African advocacy coalition of 26 organisations. The name stands for the one in nine women who actually report rape in South Africa.
But it was the final plenary crescendo that was the most electrifying to me. It featured young leaders looking ahead and started off with a BANG.
Sindi Blose, a community educator from Durban, took the mike and said, "I am going to speak for 5 minutes and demonstrate for 5 minutes." She said, "In the midst of all this pain, in all the violation of our rights – I feel great. And don't you all feel good about yourselves?" As the cheers rose, she jumped up and began to sing a local song. She said, "Do you recognize this song?"
The hall filled with boos and hisses from our South African sisters who called out "I hate that song!" "You hate that song?" she asked? "Well, I hate that song too. " She explained that she has been taught, "You must face your devil. You must take that devil and turn it upside down – turn it on its head. So I am going to sing this song and we are going to transform it together."
Slowly at first, the clapping began, and soon women were on their feet as we sang together reclaiming a song that obviously was designed to denigrate women. In her concluding remarks she unveiled a new platform of unity that had been developed at the conference by the women of South Africa denouncing the former Mbeke regime, and the incoming Zuma reign. She announced that, since women's vision for the country had been sidelined for too long, that for the 2009 elections "We are forming a new political party for women!"
Srilatha Batliwala, an Indian scholar and women's movement leader, took the stage to highlight the currents and undercurrents of the conference. She said, that here at AWID, "We are "celebrating the embracing of our own power as if it were a lover." She offered, we are learning that, "Building movements is like a relay – we must hold the stick tightly when it is our turn, knowing when and to whom to pass it on."
In that vein she underlined a main theme of the conference: embracing the intergenerationality of the movement, which was "here to stay." The vital voices of young women, who comprised 20% of attendees had prominent placement on all the plenaries. Pink scarves had been handed out as symbols of intergenerational movement-building, and adorned many throughout the crowds. She proudly announced that a new Young Feminists Fund had been launched at the conference.
She highlighted the growing inclusiveness of our movements. "We have worked very hard to open spaces, and to make it safe. To include those who are often on the fringe, such as the LGBT and women with disabilities, who have now found their sessions filled almost to bursting. The best ideas are coming from those that have been previously marginalized – from the young women and coming from the most oppressive domains imaginable.
"We are creating a movement that is making it alright to play - enabling us to thrive in oppression situations," she said. " These past few days we have danced, and we have wept to hear of the violence faced by our sisters. We have opened discussions of self-care. Because for too long we have subsidized the social justice movement with our bodies and with our minds. We are transforming the politics of funding, away from tradition donor/grantee relationships to true partnership."
With a nod to the global economic cloud, Batliwala acknowledged that much more emphasis was needed on the "implosion of the neoliberal economic climate. We need to address questions about what this means for women's funding which is needed more than ever."
The main complaint I heard was the agony of having to choose between 13 concurrent panel sessions that ran three times a day. I shared in this suffering :).
There were sessions on Iranian Women's One Million-Signature Campaign, Blogs Can Move the World, Disabled Women Speak Out, Funding the Future of Women's Rights, Sports for Social Change, the Coming Community Radio Movement, Ecuadorian Indigenous Organizing, Dancing the Revolution, Men as Allies, and Cyber-Quilting.
I attended a emotionally-charged session on one of today's hot spots: " The Peace Process and Political Transition In The Great Lakes Region of Africa" with women leaders from Rwanda, DRC, Uganda and Burundi.
There was an audible gasp from attendees as a panelist from the DRC announced that, in one province alone there were reports of "450 cases of rape in one day." (this fact needs to be confirmed as this was relayed by conference translator, however reports by IRIN are stating that rape is up 60% in North Kivu province in the past month.)
Tension was high as women pounded panelists, with questions of what was being done-especially on the part of Rwanda (where women have achieved 55% majority in parliament) to improve the situation.
Finally a young woman from Brazzaville, Congo stood up choking back tears and said, "Even though I am not in the affected area, my family lives there. It is not useful for us to be pointing fingers and saying, 'what are you doing? What are you doing?' We are always acting as victims. We need to stop it. We need to join our efforts and stop and think together about how to work together."
The panelist from the DRC said that a positive action that is happening for women in the DRC is the creating of space for them to speak out and come together. A solidarity statement for the women of the DRC circulated the conference.
There were many more climaxes and rallying cries .. But I will pass the baton on to the many women of AWID to share their highlights who are now on PulseWire!
I so wish I could have been
I so wish I could have been there! It sounds like an amazing experience.
Thank you Jensine This report
Thank you Jensine
This report is succint and coherent, you have conveyed in a short report the feeling and intensity of the Conference. I immediately sought out more information about AWID and will be finding my way to the next possible conference that I can. The momentum in that kind of a group can only fuel me and inspire me.
Clare Muireann Murphy
Hi Jensine!
We became friends on PulseWire - it is only right that I write a note to ask how you are getting on. I wish you well, and blessings.
Thank You
Hi Jensine,
Thank you so much for your warm greetings. I'm so happy to be a part of this network and connect with women around the world. It was so nice to meet you in Cambodia.
I would love to know more about graduate programmes and scholarships in the US. Because I've never been to US, I don't know much about where I would like to be. I am very open to suggestions if you have any ideas. I thank you so much for your offer to write a letter of recommendation. I hope someday that i can achieve my dream of studying in the US.
Thank you once again and all the best wishes from me.
"I would like to see humankind live together freely, honestly and have respect for one another"-me
great report
Thank you for sharing with us your experience in South Africa.
I enjoyed reading it because I was able to follow you in the booth and in the difference places where you took me.
Specially hearing the women voices and seeing the colors, and the hall ways full of people, the voices of the young women.
Thank you so much,
Remember to take the essentials with you today!


text: The MMC laboratory has state-of-the-art equipment and is accredited by The Joint Commission and certified by the Centers for Medicare & Medicaid Services (CMS), in accordance with Clinical Laboratory Improvement Amendments (CLIA). Physicians are board-certified clinical and anatomic pathologists. Our staff members are include certified Clinical Laboratory Scientists, Medical Technologists and Histo-Technologists. The clinical laboratory offers a full range of services including: Clinical Chemistry, Immunology, Urinalysis, Hematology, Coagulation, Microbilogy and Infectious Disease, Allergy testing, Transfusion Medicine and Anatomic and Clinical Pathology. Laboratory services are offered at the main hospital 6:00 am – 3:30 pm, Monday through Friday, and from 7:00 am – Noon on Saturday. Services are available at our HealthPlex location from 7:00 am – 1:00 pm, Monday through Friday.