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DOCUMENTA MATHEMATICA, Vol. 1 (1996), 417-447

Keiji Oguiso

Calabi-Yau Threefolds of Quasi-Product Type

According to the numerical Iitaka dimension $\nu(X,D)$ and $c_{2}(X) \cdot D$, fibered Calabi-Yau threefolds $\Phi_{|D|} : X \rightarrow W$ ($\text{\rm dim\,} W > 0$) are coarsely classified into six different classes. Among these six classes, there are two peculiar classes called of type $\text{\rm II}_{0}$ and of type $\text{\rm III}_{0}$ which are characterized respectively by $\nu(X,D) = 2$ and $c_{2}(X) \cdot D = 0$ and by $\nu(X,D) = 3$ and $c_{2}(X) \cdot D = 0$. Fibered Calabi-Yau threefolds of type $\text{\rm III}_{0}$ are intensively studied by Shepherd-Barron, Wilson and the author and now there are a satisfactory structure theorem and the complete classification. The purpose of this paper is to guarantee a complete structure theorem of fibered Calabi-Yau threefolds of type $\text{\rm II}_{0}$ to finish the classification of these two peculiar classes. In the course of proof, the log minimal model program for threefolds established by Shokurov and Kawamata will play an important role. We shall also introduce a notion of quasi-product threefolds and show their structure theorem. This is a generalization of the notion of hyperelliptic surfaces to threefolds and will have other applicability, too.

1991 Mathematics Subject Classification: Primary: 14J, secondary: 14D.

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# How do you solve and write the following in interval notation: x + 7 > 8?

Feb 8, 2017

$x > 1 , \left(1 , \infty\right)$

#### Explanation:

Isolate x by subtracting 7 from both sides of the inequality.

$x \cancel{+ 7} \cancel{- 7} > 8 - 7$

$\Rightarrow x > 1 \text{ is the solution}$

$\text{or "(1,oo)" in interval notation}$

format: plain


text:
# God AND Physics?

A lot of my physics friends talk of god in such ways that leed me to believe that they believe in him/her/it (whichever) but im not so convinced. It has always been my understanding that it is one or the other, how could one beileve in god when we ourselves understand most of what is happening around us and we are working toward the rest. I mean no disrespect or ill will to any religious poeple but i was just wondering how can you be a man of science and The Church?

I was just wanting some insight from both sides of this argument

( i wasnt quite sure if this thread should even be under this topic but i felt it was most appropriate)

Tido611 said:
A lot of my physics friends talk of god in such ways that leed me to believe that they believe in him/her/it (whichever) but im not so convinced. It has always been my understanding that it is one or the other, how could one beileve in god when we ourselves understand most of what is happening around us and we are working toward the rest. I mean no disrespect or ill will to any religious poeple but i was just wondering how can you be a man of science and The Church?
Science and religion can mix, but usually not well.

Proof of the God of Miracles comes from Saint Thomas Aquinas, Troubled by the inconsistencies in the church ideology, he, in the 13th cntury, decided to raise the level of theological debate from the vagueness of mythology to the intense rigor of logical thinking. He proposed to solve the questions of God's existance. He even put it in an easy to remember poem!

Things are in motion, hence there is a first mover.
Things are caused, hence there is a first cause
Things exist, hence there is a creator
Perfect goodness exists, hence it has a source
Things are designed, hense they serve a purpose

The first three lines are variations of what is called the cosmological proof, the forth argues on moral grounds; and the fifth is called the teleological proof. Moral proof is by far the weakest, because morality can be viewed in terms of evolving social customs.

When scientists refer to God in a non-theological way, they are refering to the the God of Order. Einstein, in writing, fondly called him "The Old Man."

Pengwuino
Gold Member
Tido611 said:
A lot of my physics friends talk of god in such ways that leed me to believe that they believe in him/her/it (whichever) but im not so convinced. It has always been my understanding that it is one or the other, how could one beileve in god when we ourselves understand most of what is happening around us and we are working toward the rest. I mean no disrespect or ill will to any religious poeple but i was just wondering how can you be a man of science and The Church?

Well I personally don't believe they are mutually exclusive. Other people certainly think they are but eh... its weird. Most sensible religious people realize that religion and faith is meant to try to explain things that is impossible for science to explain. I mean science can't really tell us what happened before the big bang (as far as i know its impossible) or if there is a divine being and if life is pre-determined. The reason there is this science vs. religion thing is because some people think that various religious books are out to teach us how the world works. For example, there was some crap about a city wanting to legislate Pi to equal 3 because there was something rather ambiguous in the bible about something being 10 cubits in diameter and 30 in circumfrence. Well some people take this to mean that pi must be 3 and they go off and say it. Some people see these people and they attack them and say God must not exist because they believe the Bible is wrong.

So this is what you have, one side taking something for what its not suppose to be (a bible as a textbook) and another side opposed to the idea of God. Add in human nature and you just have some stupid ideology war. Some people are just nuts. Even for example, the heads of the Catholic Church don't object to the idea of the big bang. Some people are just looking for fights i guess.

Like this whole evolution vs. creationism debate. I'm not really sure how the idea that DNA can change means that there is no God. Some people seem to be damn sure though! And of course... if God does exist, im pretty sure he can make the world seem like whatever he wanted it to seem. He made the laws, he can change them I suspect, he can do the impossible. If he does exist... well im pretty sure it can be beyond us to prove he exists if he wants us to not be able to. I just sit back and wait for the answer to come to me. I suppose im not really enthusiastic about dieing but I guess there is a plus side to it! You get the answer to one of the more annoying questions in life. But then again if a certain side is right, we wont really get to know we're right lol.

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Staff Emeritus
Gold Member
Dearly Missed
The question turns on monism, the belief that there is only ONE explanation/cause for anything. If monism is true then one of the following seems to result:
1) God is the cause/explanation and science isn't
2) Science tells the cause/explanation and it isn't God.
3) Science tells the cause/explanation and it is God.

In item 3, God would be a consequence of scientific analysis, which seems to deny Christian doctrine (faith in things not seen) and Islamic doctrine (God's unknowability), and possibly Jewish doctrine (God's unrepresentability?). So no Abrahamic God. What about Buddhism and Hinduism?

To answer the original question, I just do not mix religion and science together at all. One is a matter of knowledge that is proven and the other a matter of faith. There is not point to believeing if you have no faith.

Lisa!
Gold Member
I don't think if I understand you well. You say your physicist friend talk in a way that they believe in God, but then you ask whether it's possible to be a scientist and church man at the same time. Well as you know some people believe in God but they aren't religious.
Anyway religious people say that if there's a God, for sure he's not left humans without giving them a plan for their lives. So if there's A God, there should be a religion.
Anyway I don't know why some people say that it is impossible to be a religious scientist. Perhaps we find some false claims in holly books. But whenever we talk to religious people about them they say we still don't know everything about the world and when we discover all the fact about nature and universe, we'll be able to understand the Bible and other holly books. On the other hand they believe in miracles and say God is able to do everything. So it's not strange for them if someone would be born of a virgin.
I don't think there's anything wrong with it, as long as they don't want to oppose new discoveries. So I really want to know what would be wrong with a religious scientist if they exist.

Tom Mattson
Staff Emeritus
Gold Member
My my, but haven't you all been busy lately?

We don't host discussions about specific religions here, either pro or con. I've just given this thread a giant enema, and I expect it to stay dogma-free from now on. I could have issued enough warnings to get a few of you banned, but I refrained. But if this thread turns into another Christian debate, I will go back and issue them.

Now don't make me turn this car around! :grumpy:

Johann
Tom Mattson said:
We don't host discussions about specific religions here, either pro or con...

While I certainly support the policy, I don't know how one can discuss religion without being specific. Perhaps it would be better not to discuss religion at all?

if this thread turns into another Christian debate

In our culture, any debate about religion is a debate about Christianity. I have never seen an American or European atheist debate with a Muslim. And of course many atheists happen to be Buddhist!

Tom Mattson
Staff Emeritus
Gold Member
Johann said:
While I certainly support the policy, I don't know how one can discuss religion without being specific. Perhaps it would be better not to discuss religion at all?

We do, in fact, have a ban on religious discussion here at PF. This thread probably should not have been allowed to proceed, but unfortunately we missed it.

However, for the sake of the precious few philosophical points (such as those made by Mk and selfAdjoint), I didn't delete the whole thread.

We do allow philosophical discussions of the general concept of god (ontological, cosmological, teleological arguments; idealist metaphysics; etc.)

So on that note, let me throw in my $0.02: I think that the question of the coexistence of god and science is trivial. If "god" is taken to be an all-powerful, creative, intelligent being, then in principle I don't see how it could be in conflict with any scientific knowledge. I mean, this god (no matter whose god it is) could simply choose at any moment to suspend natural law and cause what humans would recognize as a "miracle". I really don't see how this discussion is philosophically interesting, but hey that's just me. Lisa! Gold Member Tom Mattson said: We do, in fact, have a ban on religious discussion here at PF. This thread probably should not have been allowed to proceed, but unfortunately we missed it. So finally you saw this thread! I was surprised that nobody hadn't closed this thread and somehow happy! Becaus you even don't let we talk about religion in general, but this thread was about a specific religion! I think that the question of the coexistence of god and science is trivial. If "god" is taken to be an all-powerful, creative, intelligent being, then in principle I don't see how it could be in conflict with any scientific knowledge. I mean, this god (no matter whose god it is) could simply choose at any moment to suspend natural law and cause what humans would recognize as a "miracle". I really don't see how this discussion is philosophically interesting, but hey that's just me. I agree with that. And well I think this thread was a good experience. I always thought why you don't let us to discuss religion, although I knew it would be something like fighting instead of discussing. The problem is that I don't know why people can't let each other to believe in whatever they want. We're human and we're free to think in a way we want. And un/religious people should learn to be tolerant of each other. Tido611 said: A lot of my physics friends talk of god in such ways that leed me to believe that they believe in him/her/it (whichever) but im not so convinced. It has always been my understanding that it is one or the other, how could one beileve in god when we ourselves understand most of what is happening around us and we are working toward the rest. I mean no disrespect or ill will to any religious poeple but i was just wondering how can you be a man of science and The Church? I was just wanting some insight from both sides of this argument ( i wasnt quite sure if this thread should even be under this topic but i felt it was most appropriate) There are inherent limitation to the questions science can ever answer. I don t belief in god, but i can understand why people might find comfort in one. Tom Mattson said: We do, in fact, have a ban on religious discussion here at PF. This thread probably should not have been allowed to proceed, but unfortunately we missed it. However, for the sake of the precious few philosophical points (such as those made by Mk and selfAdjoint), I didn’t delete the whole thread. We do allow philosophical discussions of the general concept of god (ontological, cosmological, teleological arguments; idealist metaphysics; etc.) So on that note, let me throw in my$0.02:

I think that the question of the coexistence of god and science is trivial. If “god” is taken to be an all-powerful, creative, intelligent being, then in principle I don’t see how it could be in conflict with any scientific knowledge. I mean, this god (no matter whose god it is) could simply choose at any moment to suspend natural law and cause what humans would recognize as a “miracle”.

I really don’t see how this discussion is philosophically interesting, but hey that’s just me.

Strange that you do have a ban on religious discussion. Science and its methodologies are quite religious after you go back to about the 3rd or 4th definition of religious. You say that if God is all-powerful etc... then you don’t see how it could be in conflict with scientific knowledge, then Im guessing the ban is just for practicals considerations ie the typical arguments between purely first definition religious minded people and strictly science-minded types. I agree its huge waste of server space to keep these kind of one-sided arguments on file, but I guess I don’t see how either side is any worse than the other. Both mindsets that create (re-create) the arguments Im referring to are closed to large extent. One says OnlY SCIENCE such as I/WE understand it, the other says ONLY RELIGION such I/WE understand it. Personally I cant see why its difficult to acknowledge from a scientific mindset that if it is a consideration (such as religious) that it must be something that can at some point be put into scientific terms. Perhaps those specific terms haven’t yet been established,(perhaps we should have a little patience and realize that just because we haven’t explained something doesn’t mean we wont or cant) but in that notion, is the challenge. From a religious mindset I cant see how one can avoid considering that if all things that exist are the work and will of a supreme being or GOD, and that science clearly does exist at least to some extent, then all of the rest of it (science not considered valid) must in some way be valid, unless one is to believe that there exists phenomena ( from physical to mental etc) that is not valid and not created by the God. Thats a huge contradiction to the original premise and there must be a solution. Finding the solution is the challenge.
All of everything that we can consider no matter what the angle of our perspective exists in the realm of mind. The bottom line problem that I assume is responsible for the prejudiced ban on religion is a perspective, or perspectives that are somehow closed.

So isn’t creating a ban closing some point of consideration or perspective? It seems that in that somewhere are the grounds for similar problems.

If server space consumed by redundant debating is a problem perhaps selective and strict termination of such conversation based on certain established conditions should be used instead. What do you think?

Johann
Tom Mattson said:
We do, in fact, have a ban on religious discussion here at PF. This thread probably should not have been allowed to proceed, but unfortunately we missed it.

(I hope discussing forum policies on religion doesn't qualify as religious debate)

I agree with you that, from my understanding of the forum rules, this thread shouldn't be allowed. But I don't understand why you deleted most posts and kept the whole thread. I see nothing in the remaining posts that is less religion-laden than whatever else was left.

However, for the sake of the precious few philosophical points (such as those made by Mk and selfAdjoint), I didn't delete the whole thread.

This is just my personal opinion, but I don't see any post here that is not addressing "philosophical" points from a Western/Christian perspective. Where is Nirvana mentioned? Reincarnation? Karma? Enlightenment? Shouldn't a philosophical debate on religion address those religious concepts just as well?

I think that the question of the coexistence of god and science is trivial. If "god" is taken to be an all-powerful, creative, intelligent being...

In other words, the god of the bible, right? See what I'm saying?

I really don't see how this discussion is philosophically interesting

Neither do I. If God can be whatever you want, then the concept is meaningless and not worth talking about.

Johann said:
Neither do I. If God can be whatever you want, then the concept is meaningless and not worth talking about.

If there is a god then it cant be whatever anyone wants it to be, it can only be what it is. Abandoning an arguement out of frustration or of thinking its futile is understandable, but its certainly not in the direction of arriving at a solution, unless its temporary. Perhaps a tempory solution is all thats desired here.

arildno
Homework Helper
Gold Member
Dearly Missed
Tom Mattson said:
I think that the question of the coexistence of god and science is trivial. If "god" is taken to be an all-powerful, creative, intelligent being, then in principle I don't see how it could be in conflict with any scientific knowledge. I mean, this god (no matter whose god it is) could simply choose at any moment to suspend natural law and cause what humans would recognize as a "miracle".
Sure, if you start out with the promise anything goes (everything's possible, which the positing of the all-powerful God implies), then of course, nothing can ever be in "conflict" with scientific knowledge either.

In fact, crocodiles that turn blue and climb into trees to sing might exist as well.
What a wonderful world that would be!

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Tom Mattson
Staff Emeritus
Gold Member
arildno said:
Sure, if you start out with the promise anything goes (everything's possible, which the positing of the all-powerful God implies), then of course, nothing can ever be in "conflict" with scientific knowledge either.

Yes, that's my point. That's why I think this is trivial.

As for the comments from Vossistarts and Johann:

It's OK to discuss our policies, but it should be done in the Feedback and Announcements Forum. All I'll say in response to your posts is that we used to have a Religion forum (it's archived, in case you want to thumb through it). It didn't work out so well, so now we don't host that sort of thing here anymore.

Les Sleeth
Gold Member
The question turns on monism, the belief that there is only ONE explanation/cause for anything. If monism is true then one of the following seems to result:
1) God is the cause/explanation and science isn't
2) Science tells the cause/explanation and it isn't God.
3) Science tells the cause/explanation and it is God.

I see another possbility.

If the "one cause" is some uncaused, forever-existing "base substance" that possesses dynamics which might spontaneously form consciousness, and if that consciousness were to evolve for zillions of eons, then it might learn to work with its own "base substance" in creative ways. And if one of the things it learned was to generate a long series of steps which establish what we now call "physical," then science is the explanation for all the steps that lead from physicalness to the point of generation, but isn't the explanation before/after that. Even then, if it is as I suggest, God (if you want to call the evolved consciousness that) is only the cause and explanation to the point of its own origin, and then we are back to the monistic concept that something neutral possesses dynamics which can accidentally cause consciousness to form.

I think that the question of the coexistence of god and science is trivial. If "god" is taken to be an all-powerful, creative, intelligent being, then in principle I don't see how it could be in conflict with any scientific knowledge. I mean, this god (no matter whose god it is) could simply choose at any moment to suspend natural law and cause what humans would recognize as a "miracle".

Well, it gets more interesting and less trivial if God is not all powerful, but is instead a consciousness that had a beginning and has evolved over a vast period, and if the unfolding of our universe (particularly in the corridor from the Big Bang to now on Earth) reflects the same sort of evolutive dynamic which developed the evolved consciousness. Since the universe is law-bound, we'd have to assume the creationary consciousness is as well. And while we don't need the concept of a creationary consciousness to explain how physics works, we might need it to explain certain aspects which are too creatively formed to be explained by any known physical principles (like the origin of life and the emergence of consciousness in biology).

Tido611 said:
A lot of my physics friends talk of god in such ways that leed me to believe that they believe in him/her/it (whichever) but im not so convinced. It has always been my understanding that it is one or the other, how could one beileve in god when we ourselves understand most of what is happening around us and we are working toward the rest. I mean no disrespect or ill will to any religious poeple but i was just wondering how can you be a man of science and The Church?

Think about this: the "founding fathers" of modern science were deeply religious theists, not atheists. Why, for instance, would a rational investigation of nature be successful? Because a rationally orderly God created the universe. (Nature consistently operating in mathematical patterns would especially be confirmative for this belief.) Indeed, many of the founders of modern science were Christians trying to demonstrate that humanity lived in an orderly universe.

Consider what Eugene Wigner says in his oft-quoted paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences:

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.

A theist can be a physicist easily (as Newton was) because theism straightforwardly explains why we find sophisticated mathematical patterns imprinted in the universe.

Lisa!
Gold Member
For sure someone's beliefe has nothing to do with his job. But the question is whether physicists can accept what religion says about universe ?

In my opinion the logical ordered universe is more of an indication that God didn't create it than that he did. Most scientists who believe in God see the hand of an intelligent designer in its logic and order, I feel just the opposite. Most religions were created when the laws of the universe weren't known and it was believed to be a place of magic and chaos. I think that model fits a creator much better than the logical universe that we know today. Take the concept of heaven for example. It is inherently just the opposite of the ordered universe, a metaphysical place. Why would the creator create an orderly logical universe when it could create an analog of the concept of heaven for his soul forge? It's like attributing bathroom graffiti and the Mona Lisa to the same artist.

Religion has evolved to accept science over the years but as we know more and more of the universe, religion is rebelling against science because they are becoming incompatable in the fanatical (read that born again and fundamentalist) religious view. You can see that happening everywhere but worst of all for the world here in the US. The cause I think is that deep down, fanatics have doubts about their beliefs and they react by trying to stifle anything that challenges those beliefs, namely science like evolution, and try to impose their beliefs on others. Fanatical religion is a sickness, a disease and it's spreading.

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Les Sleeth
Gold Member
Psi 5 said:
Most scientists who believe in God see the hand of an intelligent designer in its logic and order. . . Fanatical religion is a sickness, a disease and it's spreading.

What makes you think religion has anything to do with God?

Les Sleeth said:
What makes you think religion has anything to do with God?

What makes you think I do?

ranger
Gold Member
how can you be a man of science and The Church?

"Science without religion is lame. Religion without science is blind." - Albert Einstein

Les Sleeth
Gold Member
Psi 5 said:
What makes you think I do?

This . . . "In my opinion the logical ordered universe is more of an indication that God didn't create it than that he did. Most scientists who believe in God see the hand of an intelligent designer in its logic and order, I feel just the opposite. Most religions were created when the laws of the universe weren't known and it was believed to be a place of magic and chaos. I think that model fits a creator much better than the logical universe that we know today. Take the concept of heaven for example. It is inherently just the opposite of the ordered universe, a metaphysical place. Why would the creator create an orderly logical universe when it could create an analog of the concept of heaven for his soul forge?"

You clearly link the beliefs about God to religion, and then reason in relation to religious concepts about God.

You ask why a creator would "create an orderly logical universe"? Well, maybe the creator isn't supernatural, but rather ultra-natural. That is, a consciousness that developed out of that chaotic situation you spoke of, and then continued to evolve. This creator can't do anything it pleases, it can only work within the potentials and limitations of the existence principles that gave it its beginning and allowed it to evolve. It is "ultra-natural" because (compared to the current state of our consciousness) it has learned highly advance ways to manipulate natural conditions (apparently, according to this model, so that new consciousnesses, us, will emerge out of those natural conditions).

You also say (excepted), ". . . religion is rebelling against science because they are becoming incompatable in the fanatical (read that born again and fundamentalist) religious view . . . they react by trying to stifle anything that challenges those beliefs, namely science like evolution, and try to impose their beliefs on others. Fanatical religion is a sickness, a disease and it's spreading."

I dislike fundamentalist religion myself (actually, all religion), so I am not defending it. But I also don't see much difference between that and the growing fanaticism with the secular religion of scientism. You cite evolution as an example, well I wish I didn't have to admit I agree with the anti-evolutionists since it makes people think I am an IDer or creationist (of the Biblical sort) which isn't so.

But I don't think what we know, as fact, about evolution allows us to leap to the conclusion it alone has done anything but alter rather superficial aspects of an organism (like bird beak size, or moth color). Yet we have the scientific community forcing schools (via Supreme Court decisions) to teach that evolution has evolved entire organs and organisms. What is scientific about pushing a theory as "truth" when the evidence isn't there (yet) to support it?

If pressed, the evolutionist may admit the evidence isn't there, but then will say "what better theory do you have?" or "it's the only reasonable theory." Why do they say that? Well, because they are committed to a scientific (aka, physicalist) explanation a priori, and so nothing can possibly be considered which isn't purely physical.

You are worried about religious fanaticism, but I am more worried about scientism fanaticism, which is so determined to explain the world as only physcial they do things like misrepresent the strength of evidence they have to courts in order to push their physicalist agenda. If you ask me, that is growing faster than religious fundamentalism and is to be more feared.


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# W. Kosek 1 , A . Rzeszótko 1 , W . Popiński 2 PowerPoint PPT Presentation

Contribution of wide-band oscillations excited by the fluid excitation functions to the prediction errors of the pole coordinates data. W. Kosek 1 , A . Rzeszótko 1 , W . Popiński 2 1 Space Research Centre, Polish Academy of Sciences, Warsaw, Poland

W. Kosek 1 , A . Rzeszótko 1 , W . Popiński 2

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## Contribution of wide-band oscillations excited by the fluid excitation functions to the prediction errors of the pole coordinates data

W. Kosek1, A. Rzeszótko1, W. Popiński2

1Space Research Centre, Polish Academy of Sciences, Warsaw, Poland

2Central Statistical Office, Warsaw, Poland

Journées "Systèmes de référence spatio-temporels"and X. Lohrmann-Kolloquium 22, 23, 24 September 2008 - Dresden, Germany

### DATA

• x, y pole coordinates data from theIERS: EOPC04_IAU2000.62-now (1962.0 - 2008.6), Δt = 1 day, http://hpiers.obspm.fr/iers/eop/eopc04_05/,

• Equatorial components of atmospheric angular momentum from NCEP/NCAR, aam.ncep.reanalysis.* (1948-2008.6) Δt=0.25 day, ftp://ftp.aer.com/pub/anon_collaborations/sba/,

• Equatorial components of ocean angular momentum (mass + motion): 1) c20010701.oam (gross03.oam) (Jan. 1980 - Mar. 2002) Δt = 1 day, 2) ECCO_kf049f.oam(Mar. 2002 - Mar. 2006), Δt = 1 day,http://euler.jpl.nasa.gov/sbo/sbo_data.html,

• Equatorial components of effective angular momentum function of the hydrology obtained by numerical integration of water storage data from NCEP: water_ncep_1979.dat, water_ncep_1980.dat, …, water_ncep_2004.dat, Δt = 1 day,ftp://ftp.csr.utexas.edu/pub/ggfc/water/NCEP.

### x, y pole coordinates model data computed from fluid excitation functions

Differential equation of polar motion:

- pole coordinates,

• equatorial excitation functions corresponding to AAM, OAM

• and HAM,

• complex-valued Chandler frequency,

• where and

Approximate solution of this equation in discrete time moments can be obtained using the trapezoidal rule of numerical integration:

### THE MORLET WAVELET TRANSFORMCOHERENCE

The WT coefficients of complex-valued signal are defined as:

where are dilation and translation parameters, respectively,

is the CFT ofcomplex-valued Morlet wavelet function:

and is the CFT of

Spectro-temporal coherence between and time series is defined as:

where M is a positive integer and Δt is the sampling interval.

The MWT spectro-temporal coherence between IERS x, y pole coordinates data and x, y pole coordinates model data computed from AAM, OAM and HAM excitation functions

The MWT spectro-temporal coherence between IERS x, y pole coordinates data and x, y pole coordinates model data computed from AAM, AAM+OAM and AAM+OAM+HAM excitation functions

### Prediction of x, y pole coordinates data by the LS+AR method

x, y LS model

(Chandler circle + annual and semiannual ellipses + linear trend)

x, y

LSresiduals

x, y

LS extrapolation

AR prediction

Prediction of

x, y

LSresiduals

Prediction of

x, y

x, y

LS extrapolation

LS+AR prediction errors of IERS x, y pole coordinates data and of x, y pole coordinates model data computed from AAM, OAM and HAM excitation functions

The mean LS+AR prediction errors of IERS x, y pole coordinates data (black), and of x, y pole coordinates model data computed from AAM (orange), OAM (blue) and HAM (green) excitation functions

The mean LS+AR prediction errors of IERS x, y pole coordinates data (black), and of x, y pole coordinates model data computed from AAM+OAM (red) and AAM+OAM+HAM (purple) excitation functions

### DISCRETE WAVELET TRANSFORM BAND PASS FILTER

The DWT j-th frequency component of the complex valued signal x(t) is given by:

Signal reconstruction:

- the DWT coefficients,

- discrete Shannon wavelets.

For fixed lowest frequency indexand time index

For higher frequency index and time index

The DWT frequency components of xpole coordinate data

longer period

Chandler + Annual

Semiannual

shorter period

The mean LS+AR prediction errors of IERS x, y pole coordinates data (black), and of x, y pole coordinates model data computed by summing the chosen DWTBPF components

The mean LS+AR prediction errors of IERS x, y pole coordinates data (black), and of x, y pole coordinates model data computed from AAM+OAM (red) excitation functions as well as by summing the DWTBPF components corresponding to Chandler, annual and shorter period oscillations (green)

### CONCLUSIONS

• The contributions of atmospheric or ocean angular momentum excitation functions to the mean prediction errors of x, y pole coordinates data from 1 to about 100 days in the future is similar and of the order of 60% of the total prediction error.

• The contribution of ocean angular momentum excitation function to the mean prediction errors of x, y pole coordinates data for prediction lengths greater than 100 days becomes greater than the contribution of the atmospheric excitation function.

• The contribution of the joint atmosphere and ocean angular momentum excitation to the mean prediction errors of x, y pole coordinates data is almost equal to the contribution of the sum of Chandler + annual and shorter period frequency components. Both contributions explain about 80÷90% of the total prediction error.

• Big prediction errors of IERS x, y pole coordinates data in 1981-1982 and in 2006-2007 are mostly caused by wide-band ocean and atmospheric excitation, respectively.

• The contribution of the hydrologic angular momentum excitation to the mean prediction errors of x, y pole coordinates data is negligible.

### Acknowledgements

This paper was supported by the Polish Ministry of Education andScience, project No 8T12E 039 29 under the leadership of Dr. W. Kosek. The authors of this poster are also supported by the Organizers of Journées "Systemes de référence spatio-temporels" and X. Lohrmann-Kolloquium.

poster available: http://www.cbk.waw.pl/~kosek


text:
Optimal polygonal L_{1} linearization andfast interpolation of nonlinear systems

# Optimal polygonal L1 linearization and fast interpolation of nonlinear systems

Guillermo Gallego, Daniel Berjón and Narciso García This work has been partially supported by the Ministerio de Economía y Competitividad of the Spanish Government under project TEC2010-20412 (Enhanced 3DTV). G. Gallego is supported by the Marie Curie - COFUND Programme of the EU (Seventh Framework Programme). G. Gallego, D. Berjón and N. García are with Grupo de Tratamiento de Imágenes (GTI), ETSI Telecomunicación, Universidad Politécnica de Madrid, Madrid, Spain, e-mail: {ggb,dbd,narciso}@gti.ssr.upm.es. Copyright (c) 2014 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE. DOI: 10.1109/TCSI.2014.2327313
###### Abstract

The analysis of complex nonlinear systems is often carried out using simpler piecewise linear representations of them. A principled and practical technique is proposed to linearize and evaluate arbitrary continuous nonlinear functions using polygonal (continuous piecewise linear) models under the L1 norm. A thorough error analysis is developed to guide an optimal design of two kinds of polygonal approximations in the asymptotic case of a large budget of evaluation subintervals N. The method allows the user to obtain the level of linearization (N) for a target approximation error and vice versa. It is suitable for, but not limited to, an efficient implementation in modern Graphics Processing Units (GPUs), allowing real-time performance of computationally demanding applications. The quality and efficiency of the technique has been measured in detail on two nonlinear functions that are widely used in many areas of scientific computing and are expensive to evaluate.

Piecewise linearization, numerical approximation and analysis, least-first-power, optimization.

## I Introduction

The approximation of complex nonlinear systems by simpler piecewise linear representations is a recurrent and attractive task in many applications since the resulting simplified models have lower complexity, fit into well established tools for linear systems and are capable of representing arbitrary nonlinear mappings. Examples include, among others, complexity reduction for finding the inverse of nonlinear functions [1, 2], distortion mitigation techniques such as predistorters for power amplifier linearization [3, 4], the approximation of nonlinear vector fields obtained from state equations [5], the obtainment of approximate solutions in simulations with complex nonlinear systems [6], or the search for canonical piecewise linear representations in one and multiple dimensions [7].

In the last decades, the main efforts in piecewise linearization have been devoted both to find approximations of multidimensional functions from a mathematical standpoint and to define circuit architectures implementing them (see, for example, [8] and references therein). In the one-dimensional setting, a simple and common linearization strategy consists in building a linear interpolant between samples of the nonlinear function over a uniform partition of its domain. Such a polygonal (i.e., continuous piecewise linear) interpolant may be further optimized by choosing a better partition of the domain according to the minimization of some error measure. This is a sensible strategy in problems where there is a constraint on the budget of samples allowed in the partition.

Hence, in spite of the multiple benefits derived from modeling with piecewise linear representations, a proper selection of the interval partitions and/or predefining the number of partitions is paramount for a satisfactory performance. Some researchers [2] use cross-validation based approaches to select such a number of pieces within a partition. In other applications, the budget of pieces may be constrained by an internal design requirement (speed, memory or target error) of the approximation algorithm or by some external condition.

Simplified models may be built using descent methods [9], dynamic programming [10] or heuristics such as genetic [1] and/or clustering [11] algorithms to optimize some target approximation error. In some cases, however, the resulting piecewise representation may fail to preserve desirable properties of the original nonlinear system such as continuity [1].

We consider the simplified model representation given by the least-first-power or best approximation of a continuous nonlinear function by some polygonal function. The generic topic of least-first-power approximation has been previously considered in several references, e.g., [12, 13, 14], over a span of many years and it is a recurrent topic and source of insightful results.

We develop a fast and practical method to compute a suboptimal partition of the interval where the polygonal interpolant and the best polygonal approximation to a nonlinear function are to be computed. This technique allows to further optimize the polygonal approximation to a function among all possible partitions having the same number of segments, or conversely, allow to achieve a target approximation error while minimizing the budget of segments used to represent the nonlinear function. The resulting polygonal approximation is useful in applications where the evaluation of continuous mathematical functions constitutes a significant computational burden, such as computer vision [15, 16] or signal processing [17, 18, 19].

Our work may be generalized to the linearization of multidimensional functions [7] and the incorporation of constraints, thus opening new perspectives also in the context of designing circuit architectures for such piecewise linear approximations, as in [8]. However, these interesting generalizations will be the topic of future work.

The paper is organized as follows: two polygonal approximations of real-valued univariate functions (interpolant and best approximation) are presented in Section II. The mathematical foundation and algorithmic procedure to compute a suboptimal partition for the polygonal approximations are developed in Section III. The implementation of the numerical evaluation of polygonal approximations is discussed in Section IV. Experimental results of the developed technique on nonlinear functions (Gaussian, chirp) are given in Section V, both in terms of quality and computational times. Finally, some conclusions are drawn in Section VI.

## Ii Piecewise Linearization

In general, a piecewise function over an interval is specified by two elements: a set of control or nodal points , also called knots, that determine a partition of into a set of (disjoint) subintervals , and a collection of functions (so called “pieces”), one for each subinterval . In particular, a polygonal or continuous piecewise linear (CPWL) function satisfies additional constraints: all “pieces” are (continuous) linear segments and there are no jumps across pieces, i.e., continuity is also enforced at subinterval boundaries, . Fig. 1 shows, for a given partition the two polygonal functions that we use throughout the paper to approximate a real-valued function : the interpolant and best approximation . Polygonal functions of a given partition generate a vector space since the addition of such functions and/or multiplication by a scalar yields another polygonal function defined over the same partition.

A useful basis for vector space is formed by the set of nodal basis or hat functions , where , displayed in Fig. 2, is the piecewise linear function in whose value is 1 at and zero at all other control points , , i.e.,

 φi(x)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩x−xi−1xi−xi−1if x∈[xi−1,xi],xi+1−xxi+1−xiif x∈[xi,xi+1],0otherwise.

Functions and associated to boundary points and are only half hats. These basis functions are convenient since they can represent any function in terms of the values of at the control points, , in the form

 v(x)=N∑i=0viφi(x). (1)

From an approximation point of view this basis is frequently used in the Finite Element Method since the hat function (simplex in arbitrary dimensions) is flexible, economic and in some way a natural geometrical element into which to decompose an arbitrary geometric object.

The polygonal interpolant of a continuous function (possibly not in ) over the interval linearly interpolates the samples of at the control points, thus using in (1),

 πTf(x)=N∑i=0f(xi)φi(x).

This polygonal approximation is trivial to construct and might be good enough in some applications (e.g., power amplifier predistorters [4], the trapezoidal rule for integration), but for us it is useful to analyze other possible polygonal approximations, such as the best one in the sense, as we discuss next.

Now, consider the problem of approximating a continuous function by some polygonal function in  using the norm to measure distances. We address natural questions such as the existence and uniqueness of such a best approximation, methods to determine it and the derivation of estimates for the minimal distance.

Let us answer the question about the existence of a best approximation, i.e., the existence of whose distance from is least. Recall that the space of continuous functions in a given closed interval , together with the norm

 ∥u∥L1(I)\coloneqq∫I|u(x)|dx (2)

is a normed linear vector space (NLVS) . Since is a finite dimensional linear subspace (with basis given by the nodal functions ) of the normed space , then for every there exists a best approximation to in [20, Cor. 15.10] [21, Thm. I.1].

The uniqueness of the best approximation is guaranteed for strictly convex subspaces of NLVSs [20, Thm. 15.19] [21, Thm. I.3], i.e., those whose unit balls are strictly convex sets. Linear vector spaces with the 1 or norms are not strictly convex, therefore (a priori) the solution might be unique, but it is not guaranteed. In these cases, the uniqueness question requires special consideration. Further insights about this topic are given in [22, ch. 4][21, ch. 3], which are general references for approximation (using polynomials or other functions) and in [23], which is a comprehensive and advanced reference about nonlinear approximation.

Next, we show how to compute such a best approximation, and later we will carry out an error analysis. As is well known [24, p. 130], generically, the analytic approach to optimization problems using the norm involves derivatives of the absolute value, which makes the search for an analytical solution significantly more difficult than other problems (e.g., those using the norm).

As already seen, a function can be written as (1). Let us explicitly note the dependence of with respect to the coefficients by . The least-first-power or best approximation to is a function that minimizes . Since , it admits the expansion in terms of the basis functions of , i.e., letting we may write

 ^f(x)≡^f(x;y)=N∑i=0yiφi(x). (3)

By definition, the coefficients minimize the cost function

 cost(v)\coloneqq∥f(x)−^f(x;v)∥L1(I). (4)

Hence, they solve the necessary optimality conditions given by the non-linear system of equations

 g(v)\coloneqq∂∂vcost(v)=0, (5)

where is the gradient of (4), with entries .

Due to the partition of the interval into disjoint subintervals , we may write (4) as

 cost(v) =N∑i=1∥f(x)−^f(x;v)∥L1(Ii) =N∑i=1∫Ii|f(x)−(vi−1φi−1(x)+viφi(x))|dx,

therefore, if , each gradient component is,

 gj(v)= −∫Ijsign(f(x)−j∑n=j−1vnφn(x))φj(x)dx −∫Ij+1sign(f(x)−j+1∑n=jvnφn(x))φj(x)dx.

Observe that solely depends on (except at extreme cases ) due to the locality and adjacency of the basis functions . In the case of the norm, the optimality conditions are linear and the previous observation leads to a tridiagonal (linear) system of equations.

A closed form solution of (5) may not be available, and so, to solve the system we use standard numerical iterative algorithms of the form to find an approximate local solution . Specifically, we use step in the Newton-Raphson iteration given by the solution of the linear system , where . Near the solution , this iteration has quadratic convergence rate. This is also the step given by Newton’s optimization method when approximating (4) by its quadratic Taylor model. Due to the locality of the basis functions , cost function (4) has the advantage that its Hessian (H) is a tridiagonal matrix, so is faster to compute than in the case of a full Hessian matrix.

The search for the optimal coefficients may be initialized by setting equal to the values of the function at the nodal points, i.e., with . A more sensible initialization to improve convergence toward the optimal coefficients is given by the ordinates of the best approximation (i.e., orthogonal projection of onto ) , which are easily obtained by solving a linear system of equations using the Thomas algorithm, with tridiagonal Gramian matrix , , and inner product .

From a numerical point of view, it is also a reasonable choice to replace by some smooth approximation, for example, , with parameter controlling the width of the transition around .

In summary, the coefficients that specify the best approximation (3) on a given partition are computed numerically via iterative local optimization techniques starting from an initial guess .

## Iii Optimizing the partition

Given a vector space , we are endowed with a procedure to compute the least-first-power approximation of a function and the corresponding error, . However, the approximation error depends on the choice of , which is specified by the partition . Hence, the next problem that naturally arises is the optimization of the partition for a given budget of control points, i.e., the search for the best vector space to approximate for a given partition size. This is a challenging non-linear optimization problem, even in the simpler case (less degrees of freedom) of substituting by the polygonal interpolant . Fortunately, a good approximation of the optimal partition can be easily found using an asymptotic analysis.

Next, we carry out a detailed error analysis for the polygonal interpolant and the polygonal least-first-power approximation . This will help us derive an approximation to the optimal partition that is valid for both and , because, as it will be shown, their approximation errors are roughly proportional if a sufficiently large budget of control points, i.e., large number of subintervals, is available.

### Iii-a Error in a single interval: linear interpolant

First, let us analyze the error generated when approximating a function , twice continuously differentiable, by its polygonal interpolant in a single interval , of length . To this end, recall the following theorem on interpolation errors [25, sec. 4.2]: Let be a function in , with and closed, and let be a polynomial of degree or less that interpolates at distinct points . Then, for each there exists a point for which

 f(x)−p(x)=1(n+1)!f(n+1)(ξx)n∏i=0(x−xi). (6)

In the subinterval , letting , the polygonal interpolant is written as

 πTf(x)=f(xi−1)(1−δi(x))+f(xi)δi(x). (7)

Since interpolates the function at the endpoints of , we can apply theorem (6) (with ); hence, the approximation error only depends on and , but not on or :

 f(x)−πTf(x)=−12f′′(ξx)(x−xi−1)(xi−x). (8)

Let us compute the error over the subinterval by integrating the magnitude of (8), according to (2):

 ∥f−πTf∥L1(Ii) =∫Ii∣∣∣−12f′′(ξx)(x−xi−1)(xi−x)∣∣∣dx =12∫Ii∣∣f′′(ξx)∣∣|(x−xi−1)(xi−x)|dx.

Next, recall the first mean value theorem for integration, which states that if is a continuous function and is an integrable function that does not change sign on , then there exists a number such that

 ∫BAu(x)v(x)dx=u(ξ)∫BAv(x)dx. (9)

Applying (9) to the error and noting that gives

 ∥f−πTf∥L1(Ii) 12∣∣f′′(η)∣∣∫Ii(x−xi−1)(xi−x)dx =h3i12∣∣f′′(η)∣∣, (10)

for . Finally, if , a direct derivation of the error bound yields

 ∥f−πTf∥L1(Ii)≤112|f′′i|maxh3i. (11)

Formula (11) states that the deviation of from being linear between endpoints of is bounded by the maximum concavity/convexity of the function in (e.g., limits the amount of bending) and the cubic power of the interval size , also known as the local density of control points.

### Iii-B Error in a single interval: best L1 linear approximation

To analyze the error due to the least-first-power approximation and see how much it improves over that of the interpolant , let us first characterize the error incurred when approximating a function by a linear segment not necessarily passing through the endpoints of ,

 linei(x;Δyi−1,Δyi) =(f(xi−1)+Δyi−1)(1−δi(x)) +(f(xi)+Δyi)δi(x), (12)

where and are extra parameters with respect to that allow the linear segment to better fit the function in . Letting by analogy to (7), the corresponding error is

 ϵ =f(x)−linei(x;Δyi−1,Δyi), (13) =f(x)−πTf(x)−(πTΔy)(x;Δyi−1,Δyi), −12f′′(ξx)(x−xi−1)(xi−x) −(πTΔy)(x;Δyi−1,Δyi). (14)

#### Iii-B1 Characterization of the optimal line segment

To find the line segment that minimizes the distance

 ∥ϵ∥L1(Ii)=∥f−linei∥L1(Ii)=∫Ii|ϵ(x;Δyi−1,Δyi)|dx,

i.e., to specify the values of the optimal in (12), we solve the necessary optimality conditions given by the non-linear system of equations

 0=∂∥ϵ∥L1(Ii)∂Δyi−1 =∫Iisign(ϵ(x;Δyi−1,Δyi))(1−δi(x))dx, (15) 0=∂∥ϵ∥L1(Ii)∂Δyi =∫Iisign(ϵ(x;Δyi−1,Δyi))δi(x)dx,

where we used that, for a function ,

 ∂∂x|g(x)|=∂∂x√g2(x)=sign(g(x))∂∂xg(x).

Adding both optimality equations in (15) gives

 ∫Iisign(ϵ(x;Δyi−1,Δyi))dx=0,

which implies that must be positive in half of the interval and negative in the other half.

In fact, [26][21, Cor. 3.1.1] state that if has a finite number of zeros (at which changes sign) in , then is a best approximation to if and only if (15) is satisfied. To answer the uniqueness question, [27][28][21, Thm 3.2] state that a continuous function on has a unique best approximation out of the set of polynomials of degree . Hence, the solution of (15) provides the best linear approximation.

Let us discuss the solution of (15). If changes sign only at one abscissa , e.g., , and , the non-linear system of equations (15) cannot be satisfied since the first equation gives while the second equation gives . However, in the next simplest case where changes sign at two abscissas , the non-linear system (15) does admit a solution. This is also intuitive to justify since it corresponds to the simplified case constant in , where the sign change occurs if , i.e., according to (14), which is a quadratic equation in . It is also intuitive by looking at a plot of a candidate small error, as in Fig. 3, right.

Next, we further analyze the aforementioned case of changing sign at , with . Assume that for and in the other half of . If we apply the change of variables , and let for , then (15) becomes

 t22−t21−2(t2−t1)+12 =0, t21−t22+12 =0.

Adding both equations gives, as we already mentioned, , i.e., , stating that in half of the interval. This equation can be used to simplify the second equation, , yielding . Therefore (15) is equivalent to the linear system whose solution is , , i.e. , .

This agrees with the particularization of a more general result [21, Cor.3.4.1]: if is adjoined to the set of (linear, ) polynomials in , , meaning that and has at most distinct zeros in for every , its best approximation out of is the unique which satisfies

 ℓ∗(¯xj)=f(¯xj)

for , . The cosine term comes from the zeros of the Chebyshev polynomial of the second kind.

In other words, the best approximation is constructed by interpolating at the canonical points (the points of sign change of in (15)), as expressed by [13][22] in a nonlinear context. Hence, the values of that satisfy (15) are chosen so that zero crossings of occur at canonical points and length of the interval , yielding the linear system of equations

 ϵ(¯x1;Δyi−1,Δyi)=0ϵ(¯x2;Δyi−1,Δyi)=0}

whose solution is, after substituting in (14),

 Δyi−1 =3h2i64(f′′(ξ¯x2)−3f′′(ξ¯x1)), (16) Δyi =3h2i64(−3f′′(ξ¯x2)+f′′(ξ¯x1)).

The previous solution implies that the sum of the displacements has opposite sign to the convexity/concavity of the function :

 Δyi−1+Δyi=−3h2i16f′′(η), (17)

where from (16) lies between the least and greatest values of on , and by the intermediate value theorem it is for some . This agrees with the intuition/graphical interpretation (see Fig. 3, right).

#### Iii-B2 Minimum error of the optimal line segment

Now that the optimal have been specified, we may compute the minimum error. Let for , then, since , we may expand

 min∥ϵ∥L1(Ii)=(−∫¯x1xi−1ϵdx+∫¯x2¯x1ϵdx−∫xi¯x2ϵdx)s. (18)

Next, since for all , use the first mean value theorem for integration (9) to simplify

 ∫qpϵdx (???)(???)eq:FirstMVTIntegration= −12f′′(ηpq)∫qp(x−xi−1)(xi−x)dx −Δyi−1∫qp(1−δi(x))dx−Δyi∫qpδi(x)dx = −12f′′(ηpq)h3i[δ2i(x)2−δ3i(x)3]qp −Δyi−1hi[δi(x)−δ2i(x)2]qp−Δyi[δ2i(x)2]qp,

for some . In particular, using the previous formula for each term in (18) gives

 −∫¯x1xi−1ϵdx =12f′′(η1)5h3i192+Δyi−17hi32+Δyihi32, ∫¯x2¯x1ϵdx =−12f′′(η2)22h3i192−Δyi−18hi32−Δyi8hi32, −∫xi¯x2ϵdx =12f′′(η3)5h3i192+Δyi−1hi32+Δyi7hi32,

for some . Hence, (18) becomes

 min∥ϵ∥L1(Ii) =sh3i384(5f′′(η1)−22f′′(η2)+5f′′(η3)).

The segments in the best polygonal approximation may not strictly satisfy this because has additional continuity constraints across segments. The jump discontinuity at between adjacent independently-optimized pieces is

 ∣∣Δy−i−Δy+i∣∣ ≤3h2i64∣∣−3f′′(ξ¯x2,i)+f′′(ξ¯x1,i)∣∣ +3h2i+164∣∣f′′(ξ¯x2,i+1)−3f′′(ξ¯x1,i+1)∣∣,

where and are displacements with respect to of the optimized segments (16) at each side of , and evaluation points and lie in . In case of twice continuously differentiable functions in a closed interval, the extreme value theorem states that the absolute value terms in the previous equation are bounded. Accordingly, if and decrease due to a finer partition of the interval (i.e., a larger number of segments in ), the discontinuity jumps at the control points of the partition decrease, too. Therefore, the approximation is valid for large . Finally, if is sufficiently small so that is approximately constant within it, say , then

 min∥f−linei∥L1(Ii)≈h3isf′′Ii(−12)384=h3i32∣∣f′′Ii∣∣. (19)

In the last step we substituted , which can be proven by evaluation at the midpoint of interval :

 s = sign(ϵ(xi−1+xi2;Δyi−1,Δyi)) sign(−f′′Iih2i4−(Δyi−1+Δyi)) sign(−4h2i16f′′Ii+3h2i16f′′Ii) = −sign(f′′Ii).

In the same asymptotic situation, the error of the linear interpolant (10) becomes

 ∥f−πTf∥L1(Ii)≈h3i12∣∣f′′Ii∣∣, (20)

which is larger than the best approximation error (19) by a factor of .

### Iii-C Approximation to the optimal partition

Once analyzed the errors of both interpolant and least-first-order approximation on a subinterval , let us use such results to propose a suboptimal partition of the interval in the asymptotic case of a large number of subintervals.

A suboptimal partition for a given budget of control points is one in which every subinterval has approximately equal contribution to the total approximation error [29, 30]. Since such an error depends on the function being approximated, it is clear that such a dependence will be transferred to the suboptimal partition, i.e., the suboptimal partition is tailored to . Specifically, because the error is proportional to the local amount of convexity/concavity of the function, a suboptimal partition places more controls points in regions of with larger convexity than in other regions so that error equalization is achieved. Assuming is large enough so that is approximately constant in each subinterval and therefore the bound (11) is tight, we have

 |f′′i|maxh3i≈C, (21)

for some constant , and the control points should be chosen so that the local knot spacing [30] is , i.e., smaller intervals as increases. Hence, the local knot distribution or density is

 lkd(x)∝|f′′(x)|1/3, (22)

meaning, as already announced, that more knots of the partition are placed in the regions with larger magnitude of the second derivative.

The error equalization criterion leads to the following suboptimal partition : , , and take knots given by

 F(xi)=i/N, (23)

where the monotonically increasing function is

 F(x)=∫xa∣∣f′′(t)∣∣1/3dt/∫ba∣∣f′′(t)∣∣1/3dt. (24)

This procedure divides the range of into contiguous equal length sub-ranges, and chooses the control points given by the preimages of the endpoints of the sub-ranges. It is graphically illustrated in Fig. 4. The suboptimal partition is related to the theory of optimum quantization [31], particularly in the asymptotic or high-resolution quantization case [32], where a “companding” function such as enables non-uniform subinterval spacing within a partition.

This partition allows us to estimate the error bound in the entire interval starting from that of the subintervals. For any partition , the total error is the sum of the errors over all subintervals and, by (11),

 ∥f−πTf∥L1(I) ≤N∑i=1112|f′′i|maxh3i, (25)

which, under the error equalization condition (21), becomes

 ∥f−πT(∗)f∥L1(I)≤N∑i=1112C=112CN. (26)

To determine , sum over all subintervals and approximate the result using the Riemann integral:

 C1/3N ≈N∑i=1|f′′i|1/3maxhi≈∫ba∣∣f′′(t)∣