text:
# Edge question for 2006 (Lee Smolin's answer)

1. Jan 2, 2006

### marcus

http://www.edge.org/q2006/q06_4.html#smolin

every year Edge magazine chooses the Question of the Year and asks 100-some smart original-minded people and they get all different answers

this year the question is what do you think is a dangerous idea?
http://www.edge.org/q2006/q06_4.html#smolin

http://www.edge.org/q2006/q06_9.html#rovelli

http://www.edge.org/q2006/q06_6.html#susskind

Last edited: Jan 2, 2006
2. Jan 2, 2006

### marcus

BTW what do YOU personally think is a dangerous idea?
especially in physics, cosmology, intellectual history, mankind's relation to the universe.

I don't mean dangerous like the idea of the all-powerful State is a dangerous idea, or some other totalitarian ideal, or Racial Purity, or God, or whatever. I would call those things not dangerous ideas but dangerous fairytale bullsh*t.

What I mean is, can you come up with some idea that is more of an intellectual earthquake (but in the same ballpark) compared to the ideas of Smolin, Rovelli, Susskind?

3. Jan 2, 2006

### Kea

Hi Marcus

Wonderful short article by Smolin. But I'm afraid Susskind definitely wins the contest for having come up with the most dangerous idea. Of course, people have interpreted the word dangerous differently. Smolin has cleverly taken it to mean any truly revolutionary idea which has the potential to radically change science. But this isn't really a fair definition of the word in the current climate of ideas, where such things as the Landscape exist: namely, ideas that are dangerous in the most literal minded sense of the word.

Rovelli's article is a nice short rebuttal to ideas like the Landscape. It's a pity he wasn't a little more explicit.

Last edited: Jan 2, 2006
4. Jan 2, 2006

### marcus

I agree with you, you honeyblond category-bird, all the way.
what you say is what I think too. But I was expecting something more from you. Your own favorite interests have some potentially revolutionary ideas in them, n'est-ce-pas?
wouldnt it be a bit dangerous to transcend set theory and take a wild leap into the realm of abstract [***]sense?
this is a friendly challenge. suppose Edge had asked you? or John Baez?

Last edited: Jan 2, 2006
5. Jan 2, 2006

### Kea

Well, I could take the cop out that Smolin has done a pretty good job of standing up for Category Theory. Quoting Peirce? Personally, if I had been asked I would have said something about the Landscape, but I would have been much more blunt and bloody minded than Rovelli. The thing about this contest is: if a number of people come up with the same idea (although of course they may have different viewpoints), people get the message that there actually is a pretty amazingly dangerous idea out there right now.

6. Jan 2, 2006

### Kea

Marcus

Note that Woit is now into this game:

http://www.math.columbia.edu/~woit/blog/

Oh! I see now that quite a few people did in fact write about the Landscape. Also: the question was actually: What is your Dangerous Idea? That means, to be fair, one should write about one's own ideas. Well then, I would have to write about the Categorical Landscape, wouldn't I? But how to make this Dangerous? You see, I don't understand how radical ideas can be thought of as dangerous in the way that Smolin does. Hmmm....

From the Edge blurb on the question:
"The history of science is replete with discoveries that were considered socially, morally, or emotionally dangerous in their time; the Copernican and Darwinian revolutions are the most obvious. What is your dangerous idea? An idea you think about (not necessarily one you originated) that is dangerous not because it is assumed to be false, but because it might be true?"

Ok, then. They are defining dangerous to mean correct revolutionary ideas. Sigh. That rules out the Landscape. One should read the instructions first!

Last edited: Jan 2, 2006
7. Jan 2, 2006

### marcus

I dont want to get sidetracked worrying about the right interpretation of danger---I think it was a poor (sensationalist) choice of word by Edge;

what they really meant (I tell myself) is what is for you the most EARTHQUAKE idea, the most tectonic. radical.

tectonic ideas are both constructive and destructive, scary and hopeful

but they had to dramatize it in a single short sentence, so they said dangerous----which can also mean "attractive"

close your eyes and visualize all the ideas and tell me which one glows most brightly-----the one outlined in fire (forget whether it is good or bad)

HAH we beat Woit by 40 minutes! he posted at 4:30 PM EST

Last edited: Jan 2, 2006
8. Jan 2, 2006

### Kea

Perhaps I should clarify this a bit. Smolin talks about applying Darwinian ideas to the Laws of Physics themselves. He quotes Peirce, but I guess Peirce was heavily influenced by Hegel on this point: the evolution of ideas creates new worlds itself. Now, by new worlds one does not mean any sort of silly multiverse; one is talking about the world's way of viewing itself. At the core of this is the very simple idea that we should understand something about metaphysics, that is, the language in which laws themselves are written.

But we know how to do this with Category Theory. This kind of mathematics is hugely revolutionary. It takes Relationalism to a new level, and in some sense the history of physics is all about an ever increasing envelopment of diverse arenas by a mathematics of Relationalism. (Rovelli says this much better than I can, although apparently he doesn't like categories, which I find more and more astounding every time I think about it).

The unbelievable (for most) point of this is that it will help us calculate things, as any (mathematically minded) computer scientist will readily admit. If it can be measured, it can be computed.

Last edited: Jan 2, 2006
9. Jan 2, 2006

### Careful

It is not so astounding as you might think you know. There is a saying that if you start introducing categorical ideas into your area of research then you should have left the field already for a long time.
I think one should not wonder about what is the wildest, most daring or shocking idea. One should look for (a) the number of degrees of freedom involved (local as well as global), that is the complexity of the idea (b) the chance it has to make a new fasifiable prediction in a real physical situation (c) how it relates to well known physics and hence is not falsfified yet by known (bare) experiments. I am no expert in the landscape, but I once heard a lecture about it by a distinguished string theorist. I was simply astounded by the fact that
such an intelligent person could come up with a fairy tale which clearly defies all common sense.

Perhaps Kea could explain us why category theory takes up relationism at a higher level (that would be of some interest). However, it seems to me that it would be better to first understand at a deeper level the relationism involved in GR (it is my experience that this still defies many physicists).

10. Jan 2, 2006

### Kea

Dear me. I wouldn't have heard that one before, would I?

I will come back later and say some more.

11. Jan 2, 2006

### marcus

this is exciting

but I wish you would not express surprise that Rovelli puts things in his own (simple to him) terms instead of the language of categories. all the better if an idea can be expressed primitively and also translated into some elegant alternative form

Last edited: Jan 3, 2006
12. Jan 2, 2006

### Vast

Although Paul Davies dangerous idea what about environmental concerns, it reminded me of a short http://www.stnews.org/articles.php?category=guide&guide=The%20Multiverse&article_id=146" [Broken] he wrote earlier this year:

I wonder why he didn’t use this? Maybe he anticipated other physicists bringing up the same issue and decided to say something else instead?

It seems to me that the community of theoretical physicists is already split on this issue. I would have to agree with him.

Last edited by a moderator: May 2, 2017
13. Jan 2, 2006

### Kea

There are many ways to answer this question. Mathematically speaking, the question is moot. Category Theory is the mathematics of relationalism, by definition. What did you think all those arrows were for?

In the early days of Category Theory some of its proponents were a little sad to realise that categories appeared to be no more than an organisational tool, albeit a powerful one. They concluded, for technical reasons, that in practice one needed to select a specific category with nice properties in which to set up the structures that one wished to use. Then, in the 1960s, Gray came along and said: "All right, then. I choose Cat, the category of all categories." This no doubt appeared to be more like the behaviour of a petulant two year old than a sensible decision for a practical mathematician. However, after much hard work, Gray (and others) demonstrated that the study of relations between relations could be fruitful because it uncovered combinatorial structures that were previously unknown.

Eventually it was understood that even numbers themselves have contexts, and the contexts affect their type. Why shouldn't quantum numbers, for instance, be like that? Of course they should! Because it is a relational principle.

But this is just mathematics, and it remains to be seen whether or not it has anything to do with the real world. No one (sensible) doubts that GR is a relational theory. A few people have worried about how to think of QM more relationally. Rovelli is one. I guess Careful is another. Most of these people believe that this can be done without resorting to categories. Maybe that is true. It has occurred to me that I might even largely agree with this approach to a rigorous formulation of the SM if it wasn't for the fact that some aspects of the SM are already understood non-trivially in category theoretic terms. But we should want much, much more from a unified theory! And we won't get that without categories because there is no metaphysical mathematics without them.

14. Jan 3, 2006

### Chronos

Kea struck a chord with me there. Can you have a viable theory of most things [not to mention everything] that does submit to category theory? Does it not neatly avoid avoid paradoxes, like the ultra violet catastrophe and absurdly high vacuum energy? The problem with category theory is it's difficult to make falsifiable predictions. The question is, at least IMO, must it be falsifiable to be useful? Like the anthropic principle, it's ability to rule out the impossible is it's virtue. It is certainly preferable to the landscape, which fails to require or forbid hardly anything. If CAT is something close to a 'first principle', must it bootstrap itself into existence to be credible? IMO human logic has it's own planck wall. At some point it must be suspended to make further progress. I think this is strikingly similar to the struggles classical physicists had coming to grips with quantum theory? To paraphrase Sherlock Holmes . . . . the truth is what remains after eliminating the impossible.

15. Jan 3, 2006

### Careful

**There are many ways to answer this question. Mathematically speaking, the question is moot. Category Theory is the mathematics of relationalism, by definition. What did you think all those arrows were for? **

:rofl: :rofl: In this way I can even draw an arrow for the relation does agree with´´ between Kea an Careful, but I am afraid that will not be confirmed in practice. :tongue2:

** Gray (and others) demonstrated that the study of relations between relations could be fruitful because it uncovered combinatorial structures that were previously unknown. **

Fine, but you must realize how terribly hard it is to characterize an event in GR, and apart from the relation x is in the past of y or not´´, there is really no other canonical relation one can figure out generically. So, this relation between relations seems to far fetched for GR at least.

**
Eventually it was understood that even numbers themselves have contexts, and the contexts affect their type. **

You mean here for example that 5 does not make sense without specifying that it is a natural number (and not one of the prime field defined by 7 - say)?

**
Why shouldn't quantum numbers, for instance, be like that? Of course they should! Because it is a relational principle. **

I do not get that, the numbers in quantum theory are the complex ones, so there is no ambiguity.

**But this is just mathematics, and it remains to be seen whether or not it has anything to do with the real world. **

Here the inverse of our first relation applies

** No one (sensible) doubts that GR is a relational theory. **

I did not say that, I said that many people do not understand relationism properly yet...

**
A few people have worried about how to think of QM more relationally. Rovelli is one. I guess Careful is another.**

But Careful's and Rovelli's strategies are diametrically opposite to each other.

There is only one thing I will still add: start out more humble. We do not even know the boundaries of QM, how can we without this even make a suitable ansatz about what QG should be.

Cheers,

Careful

16. Jan 3, 2006

Staff Emeritus
Here's a dangerous idea for you. "There is no such thing as a geometrical point, or an event. Only equivalence classes of light cones."

17. Jan 3, 2006

### Careful

Eeuuh with respect to which equivalence relation ? It seems to me that what you want to imply is trivially equivalent to what we call points for classical spacetimes.

18. Jan 3, 2006

Staff Emeritus
I should have put in a smiley. I suppose we could start with diffeomorphism equivalence and go on from there depending on what we requite for quantization. Or causal sets if you prefer.

19. Jan 3, 2006

### Careful

Ah, but causal sets contain mathematical points and require more than just the lightcones in order to recover geometry: the counting measure is needed to compensate for the conformal degree of freedom. But anyway, all this is all just kinematics. You would make a really daring statement if you could propose a quantum dynamics (whatever that means ) -actually you would be the first one to do so

20. Jan 3, 2006

### marcus

Rovelli was, it turns out, not quite blunt enough, and he was misunderstood by commenters at Woit's blog. So Smolin added a remark (almost an hour ago) to make his message plainer.

-----quote from Not Even Wrong----
Lee Smolin Says:
January 3rd, 2006 at 11:58 am
HI everyone.

Carlo Rovelli did not say or imply that “Physics moved ahead too quickly in the last century, without properly understanding the mathematics of quantum mechanics.” Kasper, this comment was made by Adrian. Please read what he actually said, http://www.edge.org/q2006/q06_9.html#rovelli. Rovelli’s point was entirely different.

What Rovelli is complaining about is clear from his last sentence. It has nothing to do with the math, it is the extent to which string theory has become a study of cataloguing solutions to classical or at best semiclassical equations, ignoring both the background independence of general relativity and the challenges of understanding what a spacetime is within quantum mechanics. His point, with which I sadly have to agree, is that those who think of themselves as doing fundamental physics but have avoided wrestling with the implications of combining the principles of GR and QM have not appreciated the radical conceptual and technical innovations imposed on us just by those principles.

Best wishes to everyone for the new year, Lee

ps To avoid the inevitable misunderstanding, I do NOT imply here that the Einstein equations are fundamental, only that the principles of GR are.
----end quote---

http://www.math.columbia.edu/~woit/wordpress/?p=319#comment-7124

Last edited: Jan 3, 2006


text:
Chapter 20

Cardiac procedures are typically performed by cardiology or cardiothoracic surgery physicians and though are less common in the emergency or critical care setting, may be life-saving when a patient is in extremis. Two situations where ultrasound guidance may be invaluable for cardiac procedures is in performing pericardiocentesis in a patient with tamponade and in guiding cardiac pacing in a patient with complete heart block or unstable bradycardia unresponsive to medications.

Pericardial effusion can present in very nonspecific ways, with isolated dyspnea, tachycardia, or hypotension. Patients with significant pericardial effusion and signs or symptoms of tamponade may require urgent pericardiocentesis. Ultrasound guidance is used to increase success and decrease complications of pericardiocentesis, and should be considered standard of care when feasible.

Pericardiocentesis is indicated for cardiac tamponade. It is usually performed prior to a pericardial window procedure, as it is quicker and less invasive. Most patients with tamponade can safely go to the cardiac catheterization lab for their procedure. However, there are times when the patient is too unstable and an emergency pericardiocentesis must be performed in the critical care setting. If there is no cardiologist available at the treating facility, then the emergency or critical care physician may need to perform the procedure in an unstable patient.

Tamponade occurs when cardiac filling is compromised due to extrinsic compression of the heart. Cardiac tamponade can occur with either small or large volumes of pericardial fluid. The quicker the fluid accumulates, the faster the rise in pericardial pressure compressing the heart. While true tamponade is a clinical diagnosis (hemodynamic compromise in the setting of pericardial effusion), signs of impending tamponade may be seen on a bedside echocardiography. The left side of the heart is used to higher pressures and has proportionally a larger and thicker muscular structure. Therefore, the thinner-walled right heart, with its lower pressures, will collapse first. Late diastolic collapse of the right atrium is seen first, followed by early diastolic collapse of the right ventricular free wall. A dilated inferior vena cava is typically present, indicating restricted filling. The physical findings of Beck's triad (hypotension, muffled heart sounds, and jugular venous distension) may be present with tamponade, however, they are not sensitive and do not provide much additional information when bedside ultrasound is available.

The first-line treatment of tamponade is to increase preload, typically with intravenous crystalloid. This effectively increases central venous pressure and right-sided heart pressures to compensate for the extrinsic pressure on the heart.

Pericardiocentesis removes extrinsic fluid volume from the pericardial sac, allowing the heart to fill. For example, if you withdraw 50 mL of fluid from the pericardial sac of a patient with tamponade, you have effectively allowed 50 mL more fluid into the heart for each beat. This will increase cardiac output by 3 L/min (50 cc × 100 × 0.6), assuming a heart rate of 100 beats per minute and an ejection fraction of 60%). ...

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text:
Document type:
Zeitschriftenaufsatz
Author(s):
Franz, C.; Pfleiderer, C.; Neubauer, A.; Schulz, M.; Pedersen, B.; B̈oni, P.
Title:
Magnetization of {{Pd}} {\textsubscript{ 1- {\emph{x}} }} {{Ni}} {\textsubscript{ {\emph{x}} }} near Quantum Criticality
Abstract:
Ni-doped Pd exhibits a paramagnetic to ferromagnetic quantum phase transition at xc $\approx$ 2.5%. We report the magnetic field dependence of the magnetization of Pd1-xNix for x = 2.59% at various temperatures. The nonlinear magnetization is well described by a conventional mean-field behavior at high temperatures and high magnetic fields. At low fields and low temperatures the magnetic field dependence deviates markedly from mean-field behavior suggesting a more complex magnetic state. Our stu...    »

Journal title:
Journal of Physics: Conference Series
Year:
2010
Journal volume:
200
Month:
jan
Journal issue:
Pages contribution:
012036
Language:
en
Fulltext / DOI:
Print-ISSN:
1742-6596


text:
## Results (1-50 of 78 matches)

Next
Label $\alpha$ $A$ $d$ $N$ $\chi$ $\mu$ $\nu$ $w$ prim $\epsilon$ $r$ First zero Origin
2-450-1.1-c1-0-0 $1.89$ $3.59$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$1.0 1 1 0 1.13669 Elliptic curve 450.c Modular form 450.2.a.c Modular form 450.2.a.c.1.1 2-450-1.1-c1-0-1 1.89 3.59 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $1.0$ $1$ $1$ $0$ $1.28283$ Elliptic curve 450.d Modular form 450.2.a.d Modular form 450.2.a.d.1.1
2-450-1.1-c1-0-2 $1.89$ $3.59$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$1.0 1 1 0 1.55715 Elliptic curve 450.e Modular form 450.2.a.e Modular form 450.2.a.e.1.1 2-450-1.1-c1-0-3 1.89 3.59 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $1.0$ $1$ $1$ $0$ $1.63197$ Elliptic curve 450.f Modular form 450.2.a.f Modular form 450.2.a.f.1.1
2-450-1.1-c1-0-4 $1.89$ $3.59$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$1.0 1 1 0 1.63862 Elliptic curve 450.g Modular form 450.2.a.g Modular form 450.2.a.g.1.1 2-450-1.1-c1-0-5 1.89 3.59 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $1.0$ $1$ $-1$ $1$ $2.17323$ Elliptic curve 450.a Modular form 450.2.a.a Modular form 450.2.a.a.1.1
2-450-1.1-c1-0-6 $1.89$ $3.59$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$1.0 1 -1 1 2.17693 Elliptic curve 450.b Modular form 450.2.a.b Modular form 450.2.a.b.1.1 2-450-1.1-c3-0-0 5.15 26.5 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $3.0$ $3$ $1$ $0$ $0.570561$ Modular form 450.4.a.a Modular form 450.4.a.a.1.1
2-450-1.1-c3-0-1 $5.15$ $26.5$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$3.0 3 1 0 0.704569 Modular form 450.4.a.c Modular form 450.4.a.c.1.1 2-450-1.1-c3-0-10 5.15 26.5 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $3.0$ $3$ $1$ $0$ $1.19540$ Modular form 450.4.a.s Modular form 450.4.a.s.1.1
2-450-1.1-c3-0-12 $5.15$ $26.5$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$3.0 3 1 0 1.26850 Modular form 450.4.a.t Modular form 450.4.a.t.1.1 2-450-1.1-c3-0-13 5.15 26.5 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $3.0$ $3$ $-1$ $1$ $1.48177$ Modular form 450.4.a.b Modular form 450.4.a.b.1.1
2-450-1.1-c3-0-15 $5.15$ $26.5$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$3.0 3 -1 1 1.57819 Modular form 450.4.a.f Modular form 450.4.a.f.1.1 2-450-1.1-c3-0-16 5.15 26.5 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $3.0$ $3$ $-1$ $1$ $1.61796$ Modular form 450.4.a.g Modular form 450.4.a.g.1.1
2-450-1.1-c3-0-17 $5.15$ $26.5$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$3.0 3 -1 1 1.64382 Modular form 450.4.a.h Modular form 450.4.a.h.1.1 2-450-1.1-c3-0-19 5.15 26.5 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $3.0$ $3$ $-1$ $1$ $2.02021$ Modular form 450.4.a.k Modular form 450.4.a.k.1.1
2-450-1.1-c3-0-2 $5.15$ $26.5$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$3.0 3 1 0 0.756904 Modular form 450.4.a.e Modular form 450.4.a.e.1.1 2-450-1.1-c3-0-20 5.15 26.5 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $3.0$ $3$ $-1$ $1$ $2.06732$ Modular form 450.4.a.m Modular form 450.4.a.m.1.1
2-450-1.1-c3-0-21 $5.15$ $26.5$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$3.0 3 -1 1 2.07118 Modular form 450.4.a.n Modular form 450.4.a.n.1.1 2-450-1.1-c3-0-22 5.15 26.5 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $3.0$ $3$ $-1$ $1$ $2.12234$ Modular form 450.4.a.o Modular form 450.4.a.o.1.1
2-450-1.1-c3-0-23 $5.15$ $26.5$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$3.0 3 -1 1 2.12825 Modular form 450.4.a.p Modular form 450.4.a.p.1.1 2-450-1.1-c3-0-3 5.15 26.5 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $3.0$ $3$ $1$ $0$ $0.776539$ Modular form 450.4.a.d Modular form 450.4.a.d.1.1
2-450-1.1-c3-0-4 $5.15$ $26.5$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$3.0 3 1 0 0.997962 Modular form 450.4.a.i Modular form 450.4.a.i.1.1 2-450-1.1-c3-0-5 5.15 26.5 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $3.0$ $3$ $1$ $0$ $1.01262$ Modular form 450.4.a.j Modular form 450.4.a.j.1.1
2-450-1.1-c3-0-7 $5.15$ $26.5$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$3.0 3 1 0 1.03050 Modular form 450.4.a.l Modular form 450.4.a.l.1.1 2-450-1.1-c3-0-8 5.15 26.5 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $3.0$ $3$ $1$ $0$ $1.14161$ Modular form 450.4.a.q Modular form 450.4.a.q.1.1
2-450-1.1-c3-0-9 $5.15$ $26.5$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$3.0 3 1 0 1.16312 Modular form 450.4.a.r Modular form 450.4.a.r.1.1 2-450-1.1-c5-0-0 8.49 72.1 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $5.0$ $5$ $1$ $0$ $0.207767$ Modular form 450.6.a.b Modular form 450.6.a.b.1.1
2-450-1.1-c5-0-1 $8.49$ $72.1$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$5.0 5 1 0 0.409381 Modular form 450.6.a.a Modular form 450.6.a.a.1.1 2-450-1.1-c5-0-10 8.49 72.1 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $5.0$ $5$ $1$ $0$ $0.827744$ Modular form 450.6.a.h Modular form 450.6.a.h.1.1
2-450-1.1-c5-0-12 $8.49$ $72.1$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$5.0 5 1 0 0.883626 Modular form 450.6.a.l Modular form 450.6.a.l.1.1 2-450-1.1-c5-0-13 8.49 72.1 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $5.0$ $5$ $1$ $0$ $0.894522$ Modular form 450.6.a.r Modular form 450.6.a.r.1.1
2-450-1.1-c5-0-15 $8.49$ $72.1$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$5.0 5 1 0 0.974202 Modular form 450.6.a.s Modular form 450.6.a.s.1.1 2-450-1.1-c5-0-18 8.49 72.1 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $5.0$ $5$ $1$ $0$ $1.06742$ Modular form 450.6.a.w Modular form 450.6.a.w.1.1
2-450-1.1-c5-0-19 $8.49$ $72.1$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$5.0 5 1 0 1.07399 Modular form 450.6.a.v Modular form 450.6.a.v.1.1 2-450-1.1-c5-0-21 8.49 72.1 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $5.0$ $5$ $-1$ $1$ $1.19777$ Modular form 450.6.a.c Modular form 450.6.a.c.1.1
2-450-1.1-c5-0-22 $8.49$ $72.1$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$5.0 5 1 0 1.20980 Modular form 450.6.a.x Modular form 450.6.a.x.1.1 2-450-1.1-c5-0-24 8.49 72.1 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $5.0$ $5$ $-1$ $1$ $1.24678$ Modular form 450.6.a.e Modular form 450.6.a.e.1.1
2-450-1.1-c5-0-25 $8.49$ $72.1$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$5.0 5 -1 1 1.25116 Modular form 450.6.a.d Modular form 450.6.a.d.1.1 2-450-1.1-c5-0-26 8.49 72.1 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $5.0$ $5$ $-1$ $1$ $1.32909$ Modular form 450.6.a.f Modular form 450.6.a.f.1.1
2-450-1.1-c5-0-27 $8.49$ $72.1$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$5.0 5 -1 1 1.33125 Modular form 450.6.a.i Modular form 450.6.a.i.1.1 2-450-1.1-c5-0-28 8.49 72.1 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $5.0$ $5$ $-1$ $1$ $1.40855$ Modular form 450.6.a.k Modular form 450.6.a.k.1.1
2-450-1.1-c5-0-31 $8.49$ $72.1$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$5.0 5 -1 1 1.59816 Modular form 450.6.a.m Modular form 450.6.a.m.1.1 2-450-1.1-c5-0-33 8.49 72.1 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $5.0$ $5$ $-1$ $1$ $1.66145$ Modular form 450.6.a.p Modular form 450.6.a.p.1.1
2-450-1.1-c5-0-34 $8.49$ $72.1$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$5.0 5 -1 1 1.68928 Modular form 450.6.a.q Modular form 450.6.a.q.1.1 2-450-1.1-c5-0-37 8.49 72.1 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $5.0$ $5$ $-1$ $1$ $1.76826$ Modular form 450.6.a.t Modular form 450.6.a.t.1.1
2-450-1.1-c5-0-38 $8.49$ $72.1$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$5.0 5 -1 1 1.78399 Modular form 450.6.a.u Modular form 450.6.a.u.1.1 2-450-1.1-c5-0-4 8.49 72.1 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $5.0$ $5$ $1$ $0$ $0.615187$ Modular form 450.6.a.j Modular form 450.6.a.j.1.1
2-450-1.1-c5-0-6 $8.49$ $72.1$ $2$ $2 \cdot 3^{2} \cdot 5^{2}$ 1.1 $$5.0 5 1 0 0.744657 Modular form 450.6.a.g Modular form 450.6.a.g.1.1 2-450-1.1-c5-0-7 8.49 72.1 2 2 \cdot 3^{2} \cdot 5^{2} 1.1$$ $5.0$ $5$ $1$ $0$ $0.750942$ Modular form 450.6.a.n Modular form 450.6.a.n.1.1
Next


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## Algebra and Trigonometry 10th Edition

$r =13n-0.5n^2$
Revenue= number of people times the rate r = n * rate $r = n(10-0.5(n-60))=n(10-0.5n+3)=13n-0.5n^2$


text:
# What role does H2O2 have in copper (II) acetate formation?

Mixing solid copper, 5% vinegar, and hydrogen peroxide ($\ce{H2O2}$) causes copper acetate to form. The process will occur very slowly without hydrogen peroxide. Adding $\ce{H2O2}$ speeds the formation of copper acetate.

How does Hydrogen Peroxide promote the formation of Copper (II) Acetate?

The redox potentials $E$ for $\mathrm{pH} = 0$ show that $\ce{H+}$ cannot oxidize $\ce{Cu}$ to $\ce{Cu^2+}$:

\begin{alignat}{2} \ce{Cu^2+ + 2e- \;&<=> Cu}\quad &&E^\circ = +0.340\ \mathrm{V}\\ \ce{2H+ + 2e- \;&<=> H2}\quad &&E^\circ = +0.000\ \mathrm{V} \end{alignat}

Thus, non-oxidizing acids such as acetic acid cannot directly oxidize copper.

However, $\ce{Cu}$ can be oxidized by $\ce{O2}$:

$$\ce{O2 + 4H+ + 4e- <=> 2H2O}\quad E^\circ = +1.229\ \mathrm{V}$$

Therefore, copper is slowly oxidized in acetic acid in contact with air.

The oxidation can be increased by addition of oxidizing agents such as hydrogen peroxide:

$$\ce{H2O2 + 2H+ + 2e- <=> 2H2O}\quad E^\circ = +1.763\ \mathrm{V}$$

Presumably hydrogen peroxide causes copper metal to donate 2 electrons to form $\ce{Cu(II)}$ -- which is soluble in water -- and we'd say the hydrogen peroxide is acting as an oxidizing agent.


text:
SEARCH HOME
 Math Central Quandaries & Queries
 Question from Gene, a parent: what absolute number is one millionth of one percent of 90,000,000,000

Hi Gene,

one percent of 90,000,000,000 is

$\frac{90,000,000,000}{100} = 900,000,000$

and hence one millionth of this is 900,000,000 divided by one million.

Penny

Math Central is supported by the University of Regina and the Imperial Oil Foundation.


text:
## tex4ht - Bugs: bug #429, htlatex: Does not delete...

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## bug #429: htlatex: Does not delete intermediate files

 Submitted by: Hilmar Preusse Submitted on: Tue Jul 2 16:12:16 2019 Category: None Priority: 5 - Normal Severity: 3 - Minor Status: None Privacy: Public Assigned to: None Open/Closed: Closed

Sun Jul 28 16:29:55 2019, comment #5:

OK, I accept your explanation. Feel free to close the case. Thanks!

Hilmar Preusse <hpreusse>
Fri Jul 5 13:13:13 2019, comment #4:

You can use a simple wrapper around htlatex if you don't mind the issues I described in my previous post. This version is for Unix based OSes:

#!/bin/bash
htlatex "$@" base=${1%.tex}
rm $base.dvi rm$base.idv
rm $base.lg rm$base.tmp
# following files should be kept IMHO
# the log file can be usefull for error investigation
rm $base.log # these files are reused between TeX runs, it is not a good idea to delete them rm$base.4tc
rm $base.4ct rm$base.aux
rm \$base.xref

I would say that deleting the dvi, idv, lg and tmp files should be quite safe, as they are generated every time from the scratch and the subsequent compilations don't depend on them.

Michal Hoftich <michal_h21>
Fri Jul 5 11:36:59 2019, comment #3:

I'm aware that TeX related programs are not very good in deleting intermediate files. However in this case we speak about htlatex, some kind of wrapper script. I would expect that this script cares about intermediate files, which are not needed any more.
Note that we speak about the files, which were created by tex4ht & t4ht i.e. the files generated by the post-processor.

Feel free to lower that bug to wish list and introduce a clear switch if you think it is not a good idea to delete all intermediate files by default.

Many thanks,
Hilmar

Hilmar Preusse <hpreusse>
Wed Jul 3 07:58:02 2019, comment #2:

Hi Hilmar, as Karl and Nasser said, there is no cleanup performed by htlatex or other tex4ht scripts by default. The auxilary files are in fact quite important for the correct conversion, so it is not a good idea to delete them after every compilation. For example some complex tables may need more than three compilations by LaTeX to correctly resolve the structure, so you would never get a correct result with removed aux files.

Michal Hoftich <michal_h21>
Tue Jul 2 21:25:07 2019, comment #1:

Hi Hilmar - What cleanup is attempted now? I wasn't aware that htlatex etc. removed anything.

In any case, I don't think it would be a good idea to change the behavior at this late date. Those intermediate files can be helpful for debugging and bug reports. I could imagine adding an option for such cleaning (if there isn't one already ... I haven't checked).

Thanks for the suggestion.

Karl Berry <karl>
Tue Jul 2 16:12:16 2019, original submission:

I've noticed that running

htlatex test.tex "xhtml,ooffice,enumerate+" "ooffice/! -cunihtf -utf8" "-coo"

pollutes current directory with the following intermediate files:

test.4ct
test.4tc
test.dvi
test.idv
test.tmp
test.xref

As htlatex is actually trying to do some cleanup, I think these should be removed as well. I only expect the following files to stay:

test.lg
test.odt

Test input file:
\documentclass{article}
\begin{document}
test
\end{document}

Hilmar Preusse <hpreusse>

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# Spectroscopy and thermal modelling of the first interstellar object 1I/2017 U1 'Oumuamua

Alan Fitzsimmons, Colin Snodgrass, Ben Rozitis, Bin Yang, Meabh Hyland, Tom Seccull, Michele T. Bannister, Wesley C. Fraser, Robert Jedicke, Pedro Lacerda

Research output: Contribution to journalArticlepeer-review

65 Citations (Scopus)
375 Downloads (Pure)

## Abstract

During the formation and evolution of the Solar System, significant numbers of cometary and asteroidal bodies were ejected into interstellar space$^{1,2}$. It can be reasonably expected that the same happened for planetary systems other than our own. Detection of such Inter- stellar Objects (ISOs) would allow us to probe the planetesimal formation processes around other stars, possibly together with the effects of long-term exposure to the interstellar medium. 1I/2017 U1 'Oumuamua is the first known ISO, discovered by the Pan-STARRS1 telescope in October 2017$^3$.The discovery epoch photometry implies a highly elongated body with radii of $\sim 200 \times 20$ m when a comet-like geometric albedo of 0.04 is assumed. Here we report spectroscopic characterisation of 'Oumuamua, finding it to be variable with time but similar to organically rich surfaces found in the outer Solar System. The observable ISO population is expected to be dominated by comet-like bodies in agreement with our spectra, yet the reported inactivity implies a lack of surface ice. We show this is consistent with predictions of an insulating mantle produced by long-term cosmic ray exposure. An internal icy composition cannot therefore be ruled out by the lack of activity, even though 'Oumuamua passed within 0.25 au of the Sun.
Original language English 133-137 5 Nature Astronomy 2 18 Dec 2017 https://doi.org/10.1038/s41550-017-0361-4 Published - 29 Jan 2018

### Bibliographical note

4 figures, 1 table; Accepted for publication in Nature Astronomy, data are available at https://github.com/asteroidnerd/Oumuamua-spectroscopy

• astro-ph.EP

## Fingerprint

Dive into the research topics of 'Spectroscopy and thermal modelling of the first interstellar object 1I/2017 U1 'Oumuamua'. Together they form a unique fingerprint.
• ### Revealing Refractory Materials on Trans-Neptunian Objects and Centaurs via Reflectance Spectroscopy

Author: Seccull, T., Jul 2020

Supervisor: Fitzsimmons, A. (Supervisor) & Watson, C. (Supervisor)

Student thesis: Doctoral ThesisDoctor of Philosophy


text:
# How do I find this limit: $\lim_{x \to \infty} \sqrt{x^4-3x^2-1}-x^2$ [duplicate]

$$\lim_{x \to \infty} \sqrt{x^4-3x^2-1}-x^2$$

The answer is $$\frac{-3}{2}$$ according to Wolfram alpha.

-

## marked as duplicate by Aryabhata, Micah, Amzoti, Davide Giraudo, vonbrandApr 12 '13 at 16:09

*find this limit – Jp McCarthy Apr 12 '13 at 14:59

I would consider $$\lim_{x \to \infty} \sqrt{x^4-3x^2-1}-x^2$$ as $$\lim_{x \to \infty} \dfrac{\sqrt{x^4-3x^2-1}-x^2}{1}$$

Then multiply $\sqrt{x^4-3x^2-1}+x^2$ top and bottom to get rid of the ridiculous looking square root sign on the top. Then bingo!

-

Putting $x=\frac1h$ as $x\to\infty, h\to0$

$$\lim_{x \to \infty} \left(\sqrt{x^4-3x^2-1}-x^2 \right)$$

$$=\lim_{h\to0}\left(\frac{\sqrt{1-3h^2-h^4}-1}{h^2}\right)$$

$$=\lim_{h\to0}\left(\frac{\{1-h^2(h^2+3)\}^\frac12-1}{h^2}\right)$$

$$=\lim_{h\to0}\left(\frac{1-\frac12h^2(h^2+3)+O(h^4)-1}{h^2}\right)$$

$$=\lim_{h\to0} \left(-\frac12(h^2+3)+O(h^2)\right)$$

$$=-\frac32$$

-

first, substitute : $t=x^2$, you get :

$$\lim_{t\to +\infty} \sqrt{t^2-3t-1}-t=\lim_{t\to \infty} \frac{-3t-1}{\sqrt{t^2-3t-1}+t}=\lim_{t\to \infty} \frac{-3-\frac{1}{t}}{\sqrt{1-\frac{3}{t} -\frac{1}{t^3}}+1} =\frac{-3}{2}.$$

Explanation : first we use this identity for $a\neq -b$ : $a-b =\frac{a^2-b^2}{a+b}$, then we factor $t$ from the numerator and the denomenator, the limit of the numerator is $-3$ and the limit of the numerator is $2$.

Further more, for any reals $a,b$ : $$\lim_{x\to+ \infty } \sqrt{x^2+ax+b}-x=\frac{a}{2}.$$

-

Hint: Can you relate the expression under the root sign to $x^2$ in some way? That might make it easier to see where the value goes as $x\to\infty$.

-


text:
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Astronomy 2e

17.4Using Spectra to Measure Stellar Radius, Composition, and Motion

Astronomy 2e17.4 Using Spectra to Measure Stellar Radius, Composition, and Motion

Menu

Learning Objectives

By the end of this section, you will be able to:

• Understand how astronomers can learn about a star’s radius and composition by studying its spectrum
• Explain how astronomers can measure the motion and rotation of a star using the Doppler effect
• Describe the proper motion of a star and how it relates to a star’s space velocity

Analyzing the spectrum of a star can teach us all kinds of things in addition to its temperature. We can measure its detailed chemical composition as well as the pressure in its atmosphere. From the pressure, we get clues about its size. We can also measure its motion toward or away from us and estimate its rotation.

Clues to the Size of a Star

As we shall see in The Stars: A Celestial Census, stars come in a wide variety of sizes. At some periods in their lives, stars can expand to enormous dimensions. Stars of such exaggerated size are called giants. Luckily for the astronomer, stellar spectra can be used to distinguish giants from run-of-the-mill stars (such as our Sun).

Suppose you want to determine whether a star is a giant. A giant star has a large, extended photosphere. Because it is so large, a giant star’s atoms are spread over a great volume, which means that the density of particles in the star’s photosphere is low. As a result, the pressure in a giant star’s photosphere is also low. This low pressure affects the spectrum in two ways. First, a star with a lower-pressure photosphere shows narrower spectral lines than a star of the same temperature with a higher-pressure photosphere (Figure 17.9). The difference is large enough that careful study of spectra can tell which of two stars at the same temperature has a higher pressure (and is thus more compressed) and which has a lower pressure (and thus must be extended). This effect is due to collisions between particles in the star’s photosphere—more collisions lead to broader spectral lines. Collisions will, of course, be more frequent in a higher-density environment. Think about it like traffic—collisions are much more likely during rush hour, when the density of cars is high.

Second, more atoms are ionized in a giant star than in a star like the Sun with the same temperature. The ionization of atoms in a star’s outer layers is caused mainly by photons, and the amount of energy carried by photons is determined by temperature. But how long atoms stay ionized depends in part on pressure. Compared with what happens in the Sun (with its relatively dense photosphere), ionized atoms in a giant star’s photosphere are less likely to pass close enough to electrons to interact and combine with one or more of them, thereby becoming neutral again. Ionized atoms, as we discussed earlier, have different spectra from atoms that are neutral.

Figure 17.9 Spectral Lines. This figure illustrates one difference in the spectral lines from stars of the same temperature but different pressures. A giant star with a very-low-pressure photosphere shows very narrow spectral lines (bottom), whereas a smaller star with a higher-pressure photosphere shows much broader spectral lines (top). (credit: modification of work by NASA, ESA, A. Field, and J. Kalirai (STScI))

Abundances of the Elements

Absorption lines of a majority of the known chemical elements have now been identified in the spectra of the Sun and stars. If we see lines of iron in a star’s spectrum, for example, then we know immediately that the star must contain iron.

Note that the absence of an element’s spectral lines does not necessarily mean that the element itself is absent. As we saw, the temperature and pressure in a star’s atmosphere will determine what types of atoms are able to produce absorption lines. Only if the physical conditions in a star’s photosphere are such that lines of an element should (according to calculations) be there can we conclude that the absence of observable spectral lines implies low abundance of the element.

Suppose two stars have identical temperatures and pressures, but the lines of, say, sodium are stronger in one than in the other. Stronger lines mean that there are more atoms in the stellar photosphere absorbing light. Therefore, we know immediately that the star with stronger sodium lines contains more sodium. Complex calculations are required to determine exactly how much more, but those calculations can be done for any element observed in any star with any temperature and pressure.

Of course, astronomy textbooks such as ours always make these things sound a bit easier than they really are. If you look at the stellar spectra such as those in Figure 17.6, you may get some feeling for how hard it is to decode all of the information contained in the thousands of absorption lines. First of all, it has taken many years of careful laboratory work on Earth to determine the precise wavelengths at which hot gases of each element have their spectral lines. Long books and computer databases have been compiled to show the lines of each element that can be seen at each temperature. Second, stellar spectra usually have many lines from a number of elements, and we must be careful to sort them out correctly. Sometimes nature is unhelpful, and lines of different elements have identical wavelengths, thereby adding to the confusion. And third, as we saw in the chapter on Radiation and Spectra, the motion of the star can change the observed wavelength of each of the lines. So, the observed wavelengths may not match laboratory measurements exactly. In practice, analyzing stellar spectra is a demanding, sometimes frustrating task that requires both training and skill.

Studies of stellar spectra have shown that hydrogen makes up about three-quarters of the mass of most stars. Helium is the second-most abundant element, making up almost a quarter of a star’s mass. Together, hydrogen and helium make up from 96 to 99% of the mass; in some stars, they amount to more than 99.9%. Among the 4% or less of “heavy elements,” oxygen, carbon, neon, iron, nitrogen, silicon, magnesium, and sulfur are among the most abundant. Generally, but not invariably, the elements of lower atomic weight are more abundant than those of higher atomic weight.

Take a careful look at the list of elements in the preceding paragraph. Two of the most abundant are hydrogen and oxygen (which make up water); add carbon and nitrogen and you are starting to write the prescription for the chemistry of an astronomy student. We are made of elements that are common in the universe—just mixed together in a far more sophisticated form (and a much cooler environment) than in a star.

As we mentioned in The Spectra of Stars (and Brown Dwarfs) section, astronomers use the term “metals” to refer to all elements heavier than hydrogen and helium. The fraction of a star’s mass that is composed of these elements is referred to as the star’s metallicity. The metallicity of the Sun, for example, is 0.02, since 2% of the Sun’s mass is made of elements heavier than helium.

Appendix K lists how common each element is in the universe (compared to hydrogen); these estimates are based primarily on investigation of the Sun, which is a typical star. Some very rare elements, however, have not been detected in the Sun. Estimates of the amounts of these elements in the universe are based on laboratory measurements of their abundance in primitive meteorites, which are considered representative of unaltered material condensed from the solar nebula (see the Cosmic Samples and the Origin of the Solar System chapter).

Radial Velocity

When we measure the spectrum of a star, we determine the wavelength of each of its lines. If the star is not moving with respect to the Sun, then the wavelength corresponding to each element will be the same as those we measure in a laboratory here on Earth. But if stars are moving toward or away from us, we must consider the Doppler effect (see The Doppler Effect section). We should see all the spectral lines of moving stars shifted toward the red end of the spectrum if the star is moving away from us, or toward the blue (violet) end if it is moving toward us (Figure 17.10). The greater the shift, the faster the star is moving. Such motion, along the line of sight between the star and the observer, is called radial velocity and is usually measured in kilometers per second.

Figure 17.10 Doppler-Shifted Stars. When the spectral lines of a moving star shift toward the red end of the spectrum, we know that the star is moving away from us. If they shift toward the blue end, the star is moving toward us.

William Huggins, pioneering yet again, in 1868 made the first radial velocity determination of a star. He observed the Doppler shift in one of the hydrogen lines in the spectrum of Sirius and found that this star is moving toward the solar system. (Better measurements now indicate that Sirius is actually moving away, but Huggins is still celebrated as a pioneer in making such measurements.) Today, radial velocity can be measured for any star bright enough for its spectrum to be observed. As we will see in The Stars: A Celestial Census, radial velocity measurements of double stars are crucial in deriving stellar masses.

Proper Motion

There is another type of motion stars can have that cannot be detected with stellar spectra. Unlike radial motion, which is along our line of sight (i.e., toward or away from Earth), this motion, called proper motion, is transverse: that is, across our line of sight. We see it as a change in the relative positions of the stars on the celestial sphere (Figure 17.11). These changes are very slow. Even the star with the largest proper motion takes 200 years to change its position in the sky by an amount equal to the width of the full Moon, and the motions of other stars are smaller yet.

Figure 17.11 Large Proper Motion. Three photographs of Barnard’s star, the star with the largest known proper motion, show how this faint star has moved over a period of 20 years. (modification of work by Steve Quirk)

For this reason, with our naked eyes, we do not notice any change in the positions of the bright stars during the course of a human lifetime. If we could live long enough, however, the changes would become obvious. For example, some 50,000 years from now, terrestrial observers will find the handle of the Big Dipper unmistakably more bent than it is now (Figure 17.12).

Figure 17.12 Changes in the Big Dipper. This figure shows changes in the appearance of the Big Dipper due to proper motion of the stars over 100,000 years.

We measure the proper motion of a star in arcseconds (1/3600 of a degree) per year. That is, the measurement of proper motion tells us only by how much of an angle a star has changed its position on the celestial sphere. If two stars at different distances are moving at the same velocity perpendicular to our line of sight, the closer one will show a larger shift in its position on the celestial sphere in a year’s time. As an analogy, imagine you are standing at the side of a freeway. Cars will appear to whiz past you. If you then watch the traffic from a vantage point half a mile away, the cars will move much more slowly across your field of vision. In order to convert this angular motion to a velocity, we need to know how far away the star is.

To know the true space velocity of a star—that is, its total speed and the direction in which it is moving through space relative to the Sun—we must know its radial velocity, proper motion, and distance (Figure 17.13). A star’s space velocity can also, over time, cause its distance from the Sun to change significantly. Over several hundred thousand years, these changes can be large enough to affect the apparent brightnesses of nearby stars. Today, Sirius, in the constellation Canis Major (the Big Dog) is the brightest star in the sky, but 100,000 years ago, the star Canopus in the constellation Carina (the Keel) was the brightest one. A little over 200,000 years from now, Sirius will have moved away and faded somewhat, and Vega, the bright blue star in Lyra, will take over its place of honor as the brightest star in Earth’s skies.

Figure 17.13 Space Velocity and Proper Motion. This figure shows the true space velocity of a star. The radial velocity is the component of the space velocity projected along the line of sight from the Sun to a star. The transverse velocity is a component of the space velocity projected on the sky. What astronomers measure is proper motion (μ), which is the change in the apparent direction on the sky measured in fractions of a degree. To convert this change in direction to a speed in, say, kilometers per second, it is necessary to also know the distance (d) from the Sun to the star.

Rotation

We can also use the Doppler effect to measure how fast a star rotates. If an object is rotating, then one of its sides is approaching us while the other is receding (unless its axis of rotation happens to be pointed exactly toward us). This is clearly the case for the Sun or a planet; we can observe the light from either the approaching or receding edge of these nearby objects and directly measure the Doppler shifts that arise from the rotation.

Stars, however, are so far away that they all appear as unresolved points. The best we can do is to analyze the light from the entire star at once. Due to the Doppler effect, the lines in the light that come from the side of the star rotating toward us are shifted to shorter wavelengths and the lines in the light from the opposite edge of the star are shifted to longer wavelengths. You can think of each spectral line that we observe as the sum or composite of spectral lines originating from different speeds with respect to us. Each point on the star has its own Doppler shift, so the absorption line we see from the whole star is actually much wider than it would be if the star were not rotating. If a star is rotating rapidly, there will be a greater spread of Doppler shifts and all its spectral lines should be quite broad. In fact, astronomers call this effect line broadening, and the amount of broadening can tell us the speed at which the star rotates (Figure 17.14).

Figure 17.14 Using a Spectrum to Determine Stellar Rotation. A rotating star will show broader spectral lines than a nonrotating star.

Measurements of the widths of spectral lines show that many stars rotate faster than the Sun, some with periods of less than a day! These rapid rotators spin so fast that their shapes are “flattened” into what we call oblate spheroids. An example of this is the star Vega, which rotates once every 12.5 hours. Vega’s rotation flattens its shape so much that its diameter at the equator is 23% wider than its diameter at the poles (Figure 17.15). The Sun, with its rotation period of about a month, rotates rather slowly. Studies have shown that stars decrease their rotational speed as they age. Young stars rotate very quickly, with rotational periods of days or less. Very old stars can have rotation periods of several months.

Figure 17.15 Comparison of Rotating Stars. This illustration compares the more rapidly rotating star Altair to the slower rotating Sun.

As you can see, spectroscopy is an extremely powerful technique that helps us learn all kinds of information about stars that we simply could not gather any other way. We will see in later chapters that these same techniques can also teach us about galaxies, which are the most distant objects that can we observe. Without spectroscopy, we would know next to nothing about the universe beyond the solar system.

Making Connections

Astronomy and Philanthropy

Throughout the history of astronomy, contributions from wealthy patrons of the science have made an enormous difference in building new instruments and carrying out long-term research projects. Edward Pickering’s stellar classification project, which was to stretch over several decades, was made possible by major donations from Anna Draper. She was the widow of Henry Draper, a physician who was one of the most accomplished amateur astronomers of the nineteenth century and the first person to successfully photograph the spectrum of a star. Anna Draper gave several hundred thousand dollars to Harvard Observatory. As a result, the great spectroscopic survey is still known as the Henry Draper Memorial, and many stars are still referred to by their “HD” numbers in that catalog (such as HD 209458).

In the 1870s, the eccentric piano builder and real estate magnate James Lick (Figure 17.16) decided to leave some of his fortune to build the world’s largest telescope. When, in 1887, the pier to house the telescope was finished, Lick’s body was entombed in it. Atop the foundation rose a 36-inch refractor, which for many years was the main instrument at the Lick Observatory near San Jose.

Figure 17.16 Henry Draper (1837–1882) and James Lick (1796–1876). (a) Draper stands next to a telescope used for photography. After his death, his widow funded further astronomy work in his name. (b) Lick was a philanthropist who provided funds to build a 36-inch refractor not only as a memorial to himself but also to aid in further astronomical research.

The Lick telescope remained the largest in the world until 1897, when George Ellery Hale persuaded railroad millionaire Charles Yerkes to finance the construction of a 40-inch telescope near Chicago. More recently, Howard Keck, whose family made its fortune in the oil industry, gave $70 million from his family foundation to the California Institute of Technology to help build the world’s largest telescope atop the 14,000-foot peak of Maunakea in Hawaii (see the chapter on Astronomical Instruments to learn more about these telescopes). The Keck Foundation was so pleased with what is now called the Keck telescope that they gave$74 million more to build Keck II, another 10-meter reflector on the same volcanic peak.

Now, if any of you become millionaires or billionaires, and astronomy has sparked your interest, do keep an astronomical instrument or project in mind as you plan your estate. But frankly, private philanthropy could not possibly support the full enterprise of scientific research in astronomy. Much of our exploration of the universe is financed by federal agencies such as the National Science Foundation and NASA in the United States, and by similar government agencies in the other countries. In this way, all of us, through a very small share of our tax dollars, are philanthropists for astronomy.

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# Question

In which of the following binomial distributions does the normal distribution provide a reasonable approximation? Use computer commands to generate a graph of the distribution and compare the results to the “rule of thumb.” State your conclusions.
a. n = 10, p = 0.3
b. n = 100, p = 0.005
c. n = 500, p = 0.1
d. n = 50, p = 0.2
MINITAB
Insert the specific n and p as needed in the procedure below. Use the Make Patterned Data commands in Exercise 6.71, replacing the first value with 0, the last value with n, and the steps with 1.
Use the Binomial Probability Distribution commands on page 251, using C2 as optional storage.
Use the Scatterplot Simple commands for the data in C1 and C2. Select Data View, Data Display, Project Lines to complete the graph.
Excel
Insert the specific n and p as needed in the procedure below. Use the RANDOM NUMBER GENERATION Patterned Distribution commands in Exercise 6.71, replacing the first value with 0, the last value with n, the steps with 1, and the output range with A1. Activate cell B1; then use the Binomial Probability Distribution commands on pages 251-252.
Use the Insert > commands for the data in columns A and B Choosing the Select Data subcommand remove Series 1.

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## Multiple Stab Wounds May be Harmful to Monkeys

A couple of items of interest with some nonzero potential. A grad student in our group presented today about a new imaging technique that’s been in the literature lately called compressed sensing. Essentially a full-resolution nearly lossless image compression technique much like the ones with which we are familiar. But that’s not what makes it special. Like its name implies, compressed sensing occurs at the image acquisition stage, meaning only as much data is taken as necessary to resolve the pixels most essential to the integrity of the image. As a result this technique works best with pictures with good contrast, some structure and complexity.

Back to the beginning. Why compressed sensing? If you’re like me with a great 10 megapixel digital camera and a lousy SD card, image compression I can imagine is a perennial problem for you. As it is for me. There are a lot of pretty standard ways of saving space by doing some basic compression right in the camera before data storage. Some of these methods are even on-chip methods, involving perhaps summing over nxn pixel arrays during readout and thereby reducing the resolution of the image. 2×2 binning is common, for instance. But now, instead of 10 million pixels, you have 2.5. Oh well. Other methods might involve some kind of averaging function which in essence smudge out the details of your image on a per pixel level, so you’ll still have 10 million pixels, but lots of the neighboring ones are the same, removing those great, crisp lines that can make a photograph look so lifelike. This isn’t really much better. Anyways, if data acquisition time and data storage room is not a problem for you, then just take the full image. Great, you have no need for this technique. But, if you could have lossless compression without losing resolution, why wouldn’t you? Let me describe it in a general way (the only way I know how).

The CCD sensor of a digital camera is a grid of photo-sensitive pixels. An image is focused onto the CCD lighting up some areas of the sensor and leaving other areas dark. This information is recorded. The grid of CCD pixels outputs a matrix of values corresponding to how many photons each pixel received during the time of exposure. These values are stored on a memory device and interpreted by software as an image.

Now imagine, instead of exposing the full grid of CCD pixels, we expose only a randomly chosen subset of those pixels at a time, and sum their values. We do this $m$ times and record the result in a vector $\vec{y}$. If you think we’re losing information by doing this, well, clearly, we are, otherwise this would be a pointless exercise. But the point of compression is to recover the most essential bits and smooth out the rest. It’s quite amazing, the end result is often indistinguishable from the original. But first, some math. Because I just realized that wordpress supports latex. And that, is huge.

We can label the N total pixels in order from 1 to N and seek a one-dimensional array of values. In this basis, we’ll call the original vector, with all the raw pixel-by-pixel data, $\vec{x}$. This we will try to reconstruct in as few steps as possible. The operation of choosing a random subset of pixels and adding up their values can be expressed by the equation $\vec{q_i} \cdot \vec{x} = y_i$ for $i = 1, 2, m$. This is just a simple inner product between two vectors. Here, $\vec{q_i}$ are binary valued vectors of length N (same length as $\vec{x}$) which represent the randomly chosen open apertures. Instead of working with $m$ equations each with index $i$, we can consolidate the statement into $\widetilde{Q} \cdot \vec{x} = \vec{y}$. Where $\widetilde{Q}$ is a matrix of 0’s and 1’s and each of the $m$ rows of $\widetilde{Q}$ is one of the randomly determined configurations $\vec{q_i}$. $\vec{y}$ is equally well-determined on account of it being what is actually measured. $\vec{x}$ is the only unknown and it is what we seek. Now, if we were to set $m=N$, and take as many samples as there are pixels in the camera. This problem would be perfectly constrained, and we would be able to solve for each pixel exactly. That would be a horrible waste of time, however, as we’ve merely made it very difficult for ourselves to have in the end accomplished no kind of compression at all. The point is to use as small an $m$ as possible, or as few samples as possible, to still reach an extremely good approximation of the original image in areas of detail, while filling in zero or near-zero values for areas of void.

It may be obvious to you that with $m < N$, there will be multiple ways to choose $\vec{x}$ and still satisfy the tensor equation above. The real insight here is offered by one Terence Tao. It is a mathematical method of selecting for the lowest noise configuration, that is, the sparsest, least complicated state which still satisfies the above constraint. This is pretty ridiculous. Think about it, the effect of undersampling the space ordinarily leaves us with no information about the exact value of any pixel. But counterintuitively, by randomizing our samples, and applying a rather straightforward minimization algorithm, the solution to the problem can be approximated with great precision. For a slightly more detailed explanation, see this document. Among other things it has a neat little comparison between conventional compression methods and compressive sensing (CS) methods. CS comes out pretty strong (but it has the advantage of some information from the other technique… you’ll see), but that’s not what interests me about the image that they used to test their algorithms:

That background… looks oddly familiar…

The camera man is apparently ubiquitous in the image processing world as a test shot. I tried to find information about its origins on google but, again, to no avail…

But, right, compressive sensing, aside from being pretty counterintuitive has one additional advantage, however. And this is quite interesting. One can imagine a single-pixel camera which spits out not an array of data, but a single data point, the sum of photon counts, with each exposure. Not super useful yet, but add a fine grid which can selectively block out light, and you have on your hands on-camera loss-less compression. Certainly, because not all the pixels are active at all times, this would increase the length of exposure by how much, I don’t know, but I can only encourage this kind of thinking which could eventually make my life just that much better.

I’ll leave you with an interesting study in the news about primates with possibly far-reaching ramifications.