Using mathematics in theoretical physics

Question: I'm a non-mathematician who is self-studying mathematics. Although I'm very interested in mathematics, my main purpose is to apply math in theoretical physics. The problem is that when I read a mathematics books, I can't see a clear way to apply these math in a concrete setting. I want to apply higher math in my study of theoretical physics (not mathematical physics). I'm not looking to put physics on a rigourous basis (e.g axiomatic field theory). I want to use math (e.g. category theory and algebraic geometry) in order to discover new ways of thinking about physics, generalizing concepts and to calculate stuff. I'm completely self taught in math. Should I read pure mathematics textbooks aimed at mathematicians? What's your advice on this?
Comment: @PeterTamaroff: "Mathematical physics" is a specific subfield of physics. That doesn't mean the rest of physics is "not mathematial". The statement is "not (mathematical physics)", not "(not mathematical) physics".
Comment: How is physics "not mathematical"?
Comment: @celtschk I interpret "mathematical physics" as a subset of mathematics rather than of physics, but maybe this is a mathematician's perspective. ArXiv has both math-ph under Physics and MP under Math. By the way, [ArXiv bookworm](http://arxiv.culturomics.org/) divides all of math into 2 areas: *Mathematical Physics* and *Other*. :)
Comment: @LeonidKovalev: Actually I've never really thought about whether it is considered part of mathematics or part of physics, but just assumed the latter because of the name (usually if something is named "XXX physics" it is considered part of physics). However I don't think it really matters too much; it's more of an example that our desire to put each field under exactly one term doesn't fit reality very well. Real world terms have fuzzy borders, and even the term "mathematics" isn't exempt from this.
Answer: While studying physics as a graduate student, I took a course at the University of Waterloo by Achim Kempf titled something like Advanced Mathematics for Quantum Physics. It was an extraordinary introduction to pure mathematics for physicists. For example, in that course we showed that by taking the Poisson bracket (used in Hamiltonian mechanics) and enforcing a specific type of non-commutativity on the elements, one will get Quantum Mechanics. This was Paul Dirac's discovery. After taking his course I left physics and went into graduate school in pure mathematics.
(I don't believe he published a book or lecture notes, unfortunately, though I just emailed him.)
In transitioning from physics to mathematics, I learned that the approach to mathematics is different in a pure setting than in a physics setting. Mathematicians define and prove everything. Nothing is left unsaid or stated. There is an incredible amount of clarity. Even in theoretical physics, I found there to be a lot of hand-waving and ill-defined statements and lack of rigor (which hilariously caused me a lot of anxiety). Overall, though, Mathematicians are focused on understanding and proving relationships between abstractions, whereas physicists are more interested in using these abstractions as tools. Therefore, the approach is very different: mathematicians don't care what the application is, they only want to understand the object under consideration.
Nevertheless, for a theoretical physicist looking to get a firm background in mathematics, you want to have the following core mathematical concepts, which will provide a foundation to explore any avenue:

Linear Algebra
Functional Analysis
Topology

But the real list is something like:

Set Theory
Group and Ring Theory
Linear Algebra
Real Analysis
Topology
Functional Analysis
Measure Theory
Operator Algebra

Set, Group, and Ring theory are used extensively in physics, especially in Hamiltonian mechanics (see Poisson Bracket). Real Analysis and Linear Algebra are needed as a foundation for Functional Analysis. Functional Analysis could be described as an extension or marriage of Ring Theory, Group Theory, Linear Algebra, and Real Analysis. Therefore, many concepts in functional analysis are extended or used directly from Real Analysis and Linear Algebra. Measure Theory is important for the theory of integration, which is used extensively in applied physics and mathematics, probability theory (used in quantum mechanics), condensed matter physics, statistical physics, etc.
Topology and Operator Algebras are used extensively in advanced quantum mechanics and Relativity. Specifically, Algebraic Geometry is studied extensively in String Theory, whereas Topology is used extensively in General Relativity. Operator Algebras are an important area for understanding advanced Quantum Mechanics (ever heard someone talk about a Lie Group before?)
Some canonical text-books I would recommend:

Linear Algebra: Advanced Linear Algebra by Steven Roman
Real Analysis: Real Analysis by H. L. Royden
Functional Analysis: A Course in Functional Analysis by John B. Conway
Measure Theory: Measure Theory by Donald L. Cohn

Those are some decent text-books. I would say: give yourself two years to digest that material. Don't be hasty. Remember: mathematics is about definitions and proofs. Do not expect to see "applications" in any of those books. Just understand that the concepts are needed in advanced physics.
Unfortunately, though, I don't know of any text-book that forms a direct bridge between the two. If Achim Kempf had published his lecture notes, those may have worked, as essentially, he was doing just that.
Good luck!
Comment: I've been spending my entire life thinking it was Pauli Dirac. Mainly because I just wouldn't believe that someone with the last name of Dirac had the first name Paul.
Comment: Not enough for an edit, and it's minor: but I think "Rothman" should be "Roman."
Comment: Also, it's "Paul Dirac", not "Pauli Dirac" (unless you meant "Pauli *and* Dirac", of course)
Answer: First of all I strongly recommend a great two-volume book by R.Courant and D.Hilbert Methods of Mathematical Physics, vol.1 (linear algebra, series expansions, integral equations, calculus of variations). vol.2 (partial differential equations) - the best ever written book on mathematical methods of physics,  uncomparable to the most of "modern" books on the subject. It comprises the (broad) base for further interests. Although these volumes are quite old they are still crucial if you want to apply mathematics in theoretical physics.
To point out more recent books let's mention (a bit over 30-years old but many times edited and revised ) great book by three women Yvonne Choquet-Bruhat, Cécile DeWitt-Morette, Margaret Dillard-Bleick :  Analysis, Manifolds, and Physics.  There is also the second volume.
Y.Choquet-Bruhat wrote (2009) also General Relativity and the Einstein Equations. For over 50 years she contibuted to mathematical analysis of Einstein theory, especially to Cauchy problem.
A very well written (3-volume) book mainly on geometry : Modern Geometry--methods and Applications  by B.Dubrovin, A.Fomenko, S.P. Novikov (the Fields prize 1970). The book covers main issues of geometry (tensors, manifolds, bundles, calculus of variations) with many references to physics ( Yang-Mills, Einstein equations etc.).
A very interesting book deeply discussing issues concerning calculus of variations is The Action Principle and Partial Differential Equations by Demetrios Christodoulou. It may be hard to read but it is really rewarding. It may open your eyes on the variational principles and their fundamental importance in modern theoretical physics.
These are the first words in a beautiful article On teaching mathematics by a preeminent mathematican V.I. Arnold :
Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.
The Jacobi identity (which forces the heights of a triangle to cross at one point) is an experimental fact in the same way as that the Earth is round (that is, homeomorphic to a ball). But it can be discovered with less expense.
No matter if you agree with him or not , if he is right or not.  These words say by themselves on great importance of intimate relations between physics and mathematics. It is very advisable to read  Arnold's another article on the subject Mathematics and physics: mother and daughter or sisters?
I wouldn't like to distinguish mathematical and theoretical physics, there is rather a good or bad theoretical physics. Well, we would say that Einstein, Maxwell, Boltzmann, Dirac  were  theoretical physicists,  while Hilbert, Poincare, Minkowski were mathematicians, nevertheless all they had their very important impact on the modern physics. Nowadays theoretical physics seems to be close to purely mathematical concepts, being rather far from experimental evidence.  Unfortunately it may be still far from  The Road to Reality. By the way Roger Penrose started his scientific career in algebraic geometry and next he made a great  contribution to mathematical physics, especially to general relativity.
It would be difficult to distinguish mathematics and theoretical physics talking about General Relativity, especially there is so called Mathematical Relativity concerning exact results on analysis of Einstein equations.  One would say that people in mathematical general relativity are mathematicians working in a physically interesting theory, and it's true but on the other hand they do really interesting things also for experimental physicists.  One of the main contributor to Mathematical Relativity  S. Klainerman wrote
a great article PDE as a Unified Subject  where discussing the main issues of the research in partial differential equations he included many interesting remarks on relations between mathematics and physics. There are numerous references to the subject.
To conclude I would refer to one of the greatest mathematicians of XX century - Hermann Weyl (D.Hilbert's student). At the dawn of the Einstein theory of gravity, he wrote one of the best books on the subject (not very technical) Space Time Matter.
All those books and articles are recommended and certainly helpful in a fascinating world of theoretical physics and its deep connections to mathematics.
Comment: @Artes: please do not insult other users. I've removed your offending comment.
Comment: @WillieWong No, you are wrong. The action principle has been the main methodology in theoretical physics since the Nother theorems (i.e. since 1918). Don't try to make an impression it is not. I'd agree it  deserves more attention, nevertheless all lectures of physics treat it as the first principle.
Comment: Richard Feynman also elevated least action as a cornerstone principle.
Comment: While I admit to not having read it all, the works by Feynman I had read did not contain the mathematical rigor I needed. This is really the problem: there are physics problems that are solved using very advanced mathematics, but the *reasons* for using these mathematics, and the axioms involved in the decision making, are never exposed.
Comment: @Artes: please read [the faq](http://math.stackexchange.com/faq#etiquette), from which I quote: "Civility is required at all times; rudeness will not be tolerated."
Comment: @WillieWong Imputing things which I did not say, that   is civility, isn't it ?
Comment: @weirdcanada We don't talk about Feynman, I pointed out a book by Christodoulou on the action principle. It really deserves more attention by mathematicians and physicists.
Comment: Analysis, Manifolds, and Physics, looks excellent. We need more women physicists!
Comment: @nbubis Thanks for the bounty !
Answer: There are some nice books that can help bridge the gap between mathematics and theoretical physics, though unfortunately not enough.
One book that I would highly recommend is V.I. Arnold's "Mathematical Methods of Classical Mechanics", A text that gives the reader quite a lot of intuition onto the meaning of the mathematical concepts of manifolds, lie algebras, etc. Although the book discusses Classical Mechanics, I think this is a good starting point even for more advanced topics such as QFT.
Answer: Pure math books may not be the best thing for you. Your job as a theoretical physicist is to make-up the math you need. Allowed operations in calculations follow from physical more than mathematical principles. Once the dust settles, then you look around and see if mathematicians have developed tools which fit your calculus.
That said, I recommend the text explaining supersymmetry for mathematicians by Varadarajan. The pdfs posted at the author's website below contain a fair amount of the text.
http://www.math.ucla.edu/~vsv/susy.html
(It has much more than just supersymmetry)
Beyond that, there are many good books. However, what you really need to do is pour over recent papers on the ArXiV.
Comment: "Allowed operations...follow from physical more than mathematical principles" Excellent comment. For example, the chain rule in stochastic calculus (eg, Ito calculus) is more complex than the chain rule in differential calculus of Newton and Leibniz who lived too early to consider Brownian motion. Both chain rules are mathematically valid of course, but physicists in this case blaze the trail and then the mathematicians work out the technical details.
Answer: It may be interesting to you to look at the math overflow question here: https://mathoverflow.net/questions/71909/book-on-mathematical-rigorous-string-theory/71919#71919
The answer I gave there points to a set of lecture notes that discuss a rigorous treatment of quantum field theory from the mathematical point of view. Edward Witten IS a physicist and mathematician, so his notes are very nice to read. Effort spent thinking about the basic ideas presented here and their physical meaning is, in my opinion, not wasted.
Witten's notes on Field Theory can be found here: http://www.math.ias.edu/QFT/spring/index.html
The rest of the notes also are available for free (there is a link to the notes in a comment in the above MO answer).
Comment: Since Peter Tamaroff commented: "How is physics "not mathematical?" in addition to the link on string theory, be sure to also check Woit's [*Not Even Wrong: the Failure of String Theory...*](http://www.amazon.com/Not-Even-Wrong-Failure-Physical/dp/0465092764) and Smolin's [*The Trouble With Physics*](http://www.amazon.com/The-Trouble-With-Physics-Science/dp/0618551050)
Answer: One book which in my opinion is very good to connect mathematical concepts with quantum mechanics is Geometry of Quantum States by Ingemar Bengtsson
and Karol Życzkowski. It is not a mathematical textbook (so if you want to learn the concepts with mathematical strength, you'll have to read in addition mathematical books about those concepts), but it definitely tells you about the connection of those concepts with physics (more exactly, with quantum states, as the title already says, and you'll be surprised how much mathematical structure you find in quantum states alone).
Answer: Mathematics for engineering books can be useful for real examples of mathematical application in real world situations. Had you thought about that, or is that a bit too practical for your purposes?
I was recommend "a guided tour of mathematical methods for the physical sciences" by Roel Snieder. It seems like a good book to use if you are not following a specific course but want a more complete understanding of what you are doing with maths in physics. It has examples at each point rather than at the end of the chapter, which I prefer because I usually forget if I don't try it at once, myself.
I have also heard good things about Mary Boas' book: Mathematical Methods in the Physical Sciences, but have no personal experience other than that.


Is user testing a luxury or a necessity?

Question: I understand that you should be thinking of your users all the time but given the current climate where every pound counts, is user testing just for the richer companies? Should you discount your services so that the smaller companies can benefit?
Comment: Yes, it's a necessity. And, sadly, all-to-often only a luxury.
Comment: Basically what you are saying is: "On time of crisis, forget the users, because money comes first!" - uau. :) Nice perspective. ;p
Comment: Is a healthy lifestyle a luxury or a necessity?
Answer: User testing is a necessity.
Heuristic evaluation will help narrow the gap between user requirements and the finished product, as will good user research, and good integration of that research into the design/development process (e.g. via personas and scenarios). But there is no substitute for user testing as part of a rapid prototyping cycle for edging your product closer to usability nirvana.
I'm a great believer in low-cost, high-speed, high-impact guerilla user testing. In four hours it's possible to run 3 user tests and deliver a report to the project team, armed with nothing more than a laptop and a quiet desk. 5 users is better, 8 maybe slightly better still, but the drop-off in terms of the number and seriousness of the issues you find is so steep that a 3-user run will often be enough for one phase in a sequence of rapid prototypes.
Steve Krug describes this in 'Don't Make Me Think' (Amazon .com .co.uk).
I take your point about the current economic climate, but a morning's user testing need cost no more than 3 users' expenses, and half-a-day's labour from one or two usability people. There's really no excuse for not doing it!
Comment: If I were a developer company, I'd rather employ a usability expert to do all that for all my projects. Gives you the ability to do user research so requirements will be better from the start, and test more often and quickly in development. So maybe the question is: as a consultant agency, how do you justify using your service instead?
Answer: i'm going to continue in this theme - not doing user research can easily and quickly cost you a lot more than doing user research, but at the same time there are plenty of ways to decrease the cost of research. Be creative, be lean, but don't stop doing research. (Perhaps stop calling it 'testing' tho'? that may be a whole other post!)
Answer: IMHO rather than undermine the real commercial needs of the user experience agency it's much better to offer a series of lower cost alternatives services.
Such as:

Remote users testing
Hallway user test

User testing doesn't have to be lab based to be effective.
Matt
Answer: Cost benefit often leaps when the solution to a tricky issue becomes blindingly obvious in a relatively quick and small scale user test.
That lightbulb moment with a user can save hours or days, sometimes days, of discussion both within internal dev/design teams and when communicating with clients.
Answer: I'm in agreement with Ali. The cost of not doing user testing is often much, much larger than the expense of running a small test.
Instead of dropping user testing, or dropping prices, change what you do.
Is lab based testing necessary? It depends. Personally I get much more value out of more informal hallway, guerilla style tests in many if not most situations.
Is writing up a long report necessary? It depends. Personally I find taking the conclusions directly to management and the developers and talking to them is much more effective than producing a large chunk of paperwork.
Do you have to do it all yourself? Train up other people to help out - get more people involved.
Cheers,
Adrian
Answer: I agree with Matt.
Also according to Jakob Nielsen, getting in 1 person will reveal nearly a third of usability problems and only 5 users will reveal 75% - using your network of friends or family members to come in and perform tasks on your website or prototype will give you so much insight into the way the site works and reveal so many usability issues (of course,the more targeted, the better). All you need to offer is some small incentive.
Answer: Keeping the user in mind cannot be achieved by just thinking,you MUST need the objective,truely,real user data to input.UT is just one of those ways to help u.
Even inside user is better than no user.
Answer: If your app is not ultra specialized, you can hire "mechanical turks" at Amazon.com.  They are real humans who are happy to test drive your app and provide feedback, for not very much money.
Answer: User testing is not a luxury and it certainly doesn't have to cost a fortune either.
I've just completed a study using 30 people. It cost us nothing aside from my time and some book vouchers, but what we learned validated our approach helped us refine the content way beyond what we could have achieved without the testing.
Thankfully I'm not an agency trying to sell my testing services, my organization has invested in getting internal resources capable of doing this type of work. Design agencies that work along side their clients and share their expertise though stand a much better chance in the current climate. Skills transfer is a viable way to add value to any project.
Comment: And Nathan as touch an important thing here that the OP may not have considered. When we talk about receiving money we cannot squarely think: "If it's cheaper then, IT WILL BE cheaper". That second don't necessarily follow the first. Plus, on a time of real crisis, food is important, shelter is important, sites with usability or w/out usability neither are important. :) Anyone, Nathan gets a point here because: "Add value" to a project is indeed a great benefit of using usability tests, it will make a difference between a success thing and another not so good. The first may save more money.
Answer: It depends how complicated the functionality of your website or software is.
The more complex or unfamiliar to users - the more you'll probably need to test it.
Comment: Indeed. Let's face it, a 6 pages website is not likely to need usability testing at all. Common. Some AI perhaps, but no more then that. On the other hand, however, it will not cost you that much, in a way that if you do you may be bankrupt.
Answer: Go on, someone say testing isn't worth it...!
OK, I'll try.  Formative testing where you do a few rapid iterations of test and redesign is always worth it, even if you are just re-using common design patterns.  But most people aren't going to outsource this type of testing.
It gets a bit more difficult when you are trying to justify recruitment and facilitation costs for a larger summative evaluation with a glossy report.  That's where remote unmoderated testing comes into it's own so I agree with Matt about low cost options.
But in the end I'd use my usability testing budget to get the best designer I could afford and skip formal testing altogether**.  The best tests can tell you what users are struggling with and why, but you've still got to figure out the best way to fix it...
Cheers
R
** Caveat: All our applications are for internal use so we have users on tap and can run guerilla tests very easily for the cost of a few donuts.
Answer: Fundamentally what is user testing and research there to do.  Will it make your users like the product more...probably. Will it look like you care what your customers want...yes. But when it comes down to it any customer research potentially allows the company to not make stuff which no one wants and produce products which are needed.
User research is a 'soft' technique that enables 'hard' decisions.
Answer: The basic premise of usablity testing early on is that it helps define the product design process along the requirements of the user and potentially enhances its acceptance in the market.
Hence if you are looking at a large scale product where it would be potentially expensive to make drastic changes to a product based upon user reactions on releasing it to the market,you will need to ensure that you are well aware of the needs of your user and their reactions to your product before you release it to the world and that kind of information can only come from doing usability testing .
However if you are looking at small scale project such as small websites who can be quickly altered based upon the user reaction and behavior after going live, usability testing might not be a required option.However since first impressions of a site matter and if people are not favorably disposed towards a site,there is a possibility that the site might lose those visitors once and for all even if it cleans up its act later.
First Impressions Count in Website Design
That said and done,there are a lot of good resources on how companies can be encouraged and do usability testing on a budget :
Quick and Dirty Usability Testing: Step Away from the Book
Why You Only Need to Test with 5 Users
Streamlining Usability Testing by Avoiding the Lab
Usability Testing on a Budget
Quick and Dirty Remote User Testing
Answer: User testing needn't be feared or difficult. By establishing very clear goals, the who and how of the test become clear. I have done many tests where I could ask literally anyone and the duration was exactly 5 seconds - this is the '5 second test'. Doing this often throughout the process will produce many lightbulb moments which will mitigate a lot of the need for large and 'expensive' tests, and even these needn't be large or expensive. I have done fairly elaborate tests with a statistically significant number of representative users in a few hours for a few hundred dollars. The key is you - do the research and preparation and get started.


What is the mechanism for fast scrambling of information by black holes?

Question: Sekino and Susskind have argued that black holes scramble information faster than any quantum field theory in this paper. What is the mechanism for such scrambling?
Answer: The mechanism by Charging Bull is interesting, but deeply flawed. It is true at any given moment, there are about $A/\ell_P^2$ entangled Hawking pairs across the horizon, and that the curvature of the black hole serves to pull them apart. The particle on the inside will of course hit the singularity. But also note that unless the direction of of the outgoing Hawking particle points nearly exactly outward in direction, it will only move out from the horizon a bit before following a curved trajectory in Eddington-Finkelstein coordinates causing it to fall back through the horizon and also hit the singularity, but at a different moment. Only an average of an order unity of the $A/\ell_P^2$ many entangled pairs lead to an outgoing Hawking particle escaping for good. Those which escape are mostly s-waves, with a few p-waves and d-waves delocalized over all directions. What we get instead are $A/\ell_P^2$ entangled pairs of ingoing Hawking particles hitting the singularity at different moments. Also, it's also true there are $A/\lambda^2$ entangled pairs of wavelength $\lambda$ for the vacuum, but they are really redshifted versions of some of the $A/\ell_P^2$ entangled pairs from an earlier moment due to the redshifting mechanism of Eddington-Finkelstein warping. They are not independent pairs. The other entangled pairs also show up as $\lambda$ wavelength entanglements later after redshifting, but of the nonvacuum sort, a squeezed state sort.
In fact, if we're only interested in pairs of infalling Hawking particles hitting the singularity at different moments, it's not even necessary for them to be produced across the horizon. That's only necessary for entangled pairs with one escaping for good and the other hitting the singularity. The location of pair production of both eventually infalling Hawking particles can be outside the horizon, across it, or inside it.
On the average, we only get an order of unity of outgoing Hawking qubits per unit Schwarzschild time. So, over a time period of $R\ln(R/\ell_P)$, only somelike like $c\ln(R/\ell_P)$ qubits could have escaped where $c$ is of order unity. It's possible for the spacecraft to dump a lot more information than that, and we have Holevo bounds. The problem with the Hayden-Preskill mechanism is, maybe Alice's top secret diary contains $k$ qubits of information. But that doesn't necessarily mean collecting only $ck\ln(R/\ell_P)$ qubits of outgoing radiation will suffice to snoop into her top secret confidential diary. What they overlooked is the microstates of the craft contain a lot more than k qubits worth of info. To suggest that only collecting that few qubits is enough to crack the k chosen diary qubits out of the that many more craft microstate qubits is to violate information causality.
But this suggests another very interesting mechanism. If the craft hits the singularity at time $t$, it hits simultaneously with $A/\ell_P^2$ ingoing Hawking particles. Actually, probably more than that, maybe $R^3/\ell_P^3$ because the location of pair production can be far inside the hole. Retrocausal influences still exist, but most of them relate to pairs of entangled infalling Hawking radiation, both infalling. Via these entanglements, this retrocausal effect zigzags back forward causally again to the singularity at a different moment. This zigzag mechanism can continue for many many iterations in both directions in Eddington-Finkelstein time, which really becomes space inside the black hole. A tiny fraction of the entangled Hawking pairs really have an escaping outgoing Hawking particle. These carry scrambled information about the craft. This information is spread out over a long time both far into the future and far into the past of the moment when the craft plunges into the hole. A very long zigzag of alternating causality and retrocausality carried over many entangled pairs of infalling Hawking particles hitting the singularity at different moments.
Answer: Fast scrambling presupposes black hole complementarity, which is why it's so mysterious.
An alternative explanation is retrocausality. Black hole singularities exert retrocausal influences. Suppose you have a black hole with a radius of one light-second. A spacecraft carrying some quantum information passes through the event horizon and hits the singularity. Work in Eddington-Finkelstein coordinates. Ingoing light rays take one second to go from the horizon to the singularity. The spacecraft moves slower than that, so it might take two or so seconds to hit the singularity. What hits the singularity at that moment? It is the spacecraft and the quantum information it carries. But also, pair entangled Hawking particle pairs of massless particles are constantly produced across the event horizon. One of them lies outside and escapes the black holes gravitational pull. The other falls in and hits the singularity. Infalling Hawking radiation also hits the singularity at the same instant. Retrocausal influences from the singularity shows up as a fixing of the form of entanglement between everything which will hit the singularity at the same instant, i.e. the internal microstates of the spacecraft and the infalling Hawking radiation. This can be described in the two-state formalism as postselection. However, the infalling Hawking radiation is also entangled with the outgoing Hawking radiation. So, information on the microstates of the spacecraft are transmitted retrocausally to the outgoing Hawking radiation via this mechanism. In flat spacetime, "virtual" particle pair entangled states don't increase in spatial separation. In the warped geometry of a black hole, a pair of "not-so-virtual" massless particles both travelling locally outward at the speed of light across the event horizon will see the spatial distance between them increasing with time. If their initial wavelength is $\lambda$, there will be around $A/\lambda^2$ many of such pairs at any given instant. The smallest wavelengths are cutoff around the Planck scale, and at that scale, there are about $A/\ell_P^2$ of them. Such pairs dominate the counting, but longer wavelengths also contribute. The infalling Hawking particle is travelling outward locally, but is still dragged inward by coordinate dragging. Initially, it is about $\lambda$ inward from the boundary. So, it would take about $R\ln(R/\lambda)$ to hit the singularity. There's a cutoff about the Planck wavelength, so the longest time it would take is around $R\ln(R/\ell_P)$ which is about 100 seconds
 or so. So, all you need is to collect all the outgoing Hawking radiation starting 100 seconds or so before the craft passes the horizon until one second after (two seconds minus one second) to extract the infalling quantum information. Any earlier Hawking radiation is not needed. Because infalling Hawking radiation hitting the singularity can come in from all directions, outgoing Hawking radition has to be collected from all directions.
The Hayden-Presskill mechanism is all wrong because it assumes collecting all info starting from the moment the craft hits the horizon for about 100 seconds or so after that, with the entire history of all the outgoing hawking radiation since the formation of the black hole needed as context.
If there's a possibility that once craft is just inside the horizon, it hits something which causes it to recoil locally outward near the speed of light, then it can take longer than two seconds to hit the singularity. In that case, we might still need to collect all outgoing Hawking radiation up till 100 secs later "just in case".
The black hole information paradox is only so mysterious because most people have an automatic mental block against retrocausality.


Free for commercial font like windows metro

Question: I need a "free for commercial use" font that looks like the font used in Windows Metro.
It will be used for display on website, both titles and text. With bold and 'slim' version.
Do you know of something like that?
Answer: Windows Metro uses the font Segoe.
The font Segoe bears resemblance to the font Frutiger, so in your search for Segoe alternatives you can expand your search to Frutiger alternatives.
The font M+ 2p is a free font that's an alternative to Frutiger:

It is indeed a strange name for a font, but it is desgined by a foundry called M+.
You might also want to do a search on other Frutiger, Segoe, or Myriad (another font similar to Frutiger) free alternatives.
Answer: thomasrutter's answer is excellent; thought I'd chip in a little extra.
Segoe, Myriad, and Frutiger are in a genre of sans-serif fonts called humanist sans-serifs. Just as all rhombii are not rectangles, not all humanist fonts will look like Segoe and friends. Most won't. But this might help you in your search.
One free humanist font that is in the same ballpark as Segoe is Open Sans, which is under the Apache license. Not sure of the full implications of that license in terms of attribution, but you can use it for commercial work (Anything that's hosted on Google Web Fonts has a license that allows for commercial work according to this page).


What issues are there with printing 32-bit HDR images?

Question: What issues are there with printing 32-bit HDR images?

Do printing companies simply convert them to 8-bit?
Do some have printers with a wider gamut that can use additional color data?
Is there a better-than-8-bit but not 32-bit format?
Answer: It is simply impossible to print a 32bit image.
For the sake of truth it is impossible to view it on a monitor too :) ...
As far as I know there is no device being able to render 32 bit per channel, neither the human eye ... so if you'd be able to print you'll not be able to see it :)
Of course you can choose how to tone map your huge 32bit information into a smaller range, this is what is achieved by the tone mapping process.  This is a very liberal procedure, I mean that you are free to choose what you want to render, what to enhance, exasperate and so on ... HDR + tone mapping is an "artistic" tool, not a photorealistic one IMHO.
@msw: HDR tone mapping is necessary both for RGB and CMKY, none of them can represent 32 bit of information.
Comment: It seems there is a true HDR screen on the market. Dolby purchased HDRI LED technology from a company called BrightSide, which is used in new screens they are releasing. It is 16-bit floating point, so it wouldn't have the full dynamic range of a true 32-bit floating point HDR image, however it does have considerably greater contrast range than any other screen known to man at the moment. They are supposedly due out this year...however I don't know if they will be only in TV format, or if there will be a computer screen version.
Comment: I wouldn't say its "impossible" to view it on a monitor. It would depend on the monitors capabilities. Granted, I can't say that I've heard of a screen capable of full 32bit HDR color rendition, however such a thing is not *impossible*. Some of the most recent high-end LED flat panel screens are capable of at the very least 14bit display, some up to the full 16bits, which puts them well beyond the limited 8bit, 16 million colors normally available on a CRT or LCD screen (something in the area of 270 billion colors, I think, or 65k tones vs. 256.)
Comment: +1 for interesting news.  jrista, how do you keep up with all this stuff?!
Comment: I'm always digging... ;P
Comment: That monitor is super-impressive... Like looking out a window. Interestingly, it makes things look more 3D because the contrast range gives more depth-cues to our brains, or so the Brightside people told me. They also told me the estimated price :( as I saw a hand-made prototype unit.
Answer: There is simply no physical way to emulate the gamut of additive luminance (light, your monitor) with subtractive pigments (CMYK). Not even close.
The best use you could make of an HDR image for print is to compress the luminance selectively while preserving borders between light and dark regions. This is a matter of great art and you'd do well to learn of Ansel Adams' darkroom techniques as he achieved the illusion of huge tonal range on a medium that didn't support it by dodging and burning.
In some sense, you'd wind up doing the equivalent of unsharp mask, but on a regional - not pixel - level. This fools the perceiver into thinking the shadows are as detailed as the mid-tones while having equal mean luminosity.
Answer: Generally speaking, I think the format you want your images in for printing is 16bit. Most of the time, when I've taken in photos to be printed, I've always had them in 16bit per pixel (48bit) TIFF files, and used the Adobe RGB color profile. That seems to be pretty industry standard, however my experience printing with a third-party company is limited, so I can't say that definitively.
I always save a final "print" copy of my HDR images as 16bit TIFF images. Usually, Photoshop does not even give you the option to fine-tune the tone curve when Merging to HDR until you downconvert from 32bit to 16bit or 8bit anyway.
Answer: I believe that by "32 bit" you are referring to CMYK colors, that is 8 bits per channel with 4 channels. This is a color space that is usually used for offset printing, not for continous tone photographic printing.
Almost all images captured by a digital cameras or scanners are RGB colors and sometimes known as 24 bit color, 8 bits per channel with three channels. Confusingly the same images might be called 8 bit images because of the number of bits per channel. I will use the number of bits per channel terminology.
Pardon me if this all review for you, it's hard for me to be sure so I'm covering all the bases.
If you truly are referring to CMYK images I would not submit them to a service for fulfillment in this layout as almost all photographic hardcopy output devices use RGB colors (like laser printing on silver halide paper, inkjet, or thermal) converting between CMYK and RGB can be tricky and it is likely the results won't match your expectations.
Perhaps you are really asking about 16 bit images which are RGB with 16 bits per channel? I'll assume this is the case. Almost all prints made from digital images at various fulfillment services are done using 8 bit RGB images. If you submit a 16 bit image and it is not rejected the service will sample the 16 bit image to an 8 bit image. Typically (except for specialized applications) a 16 bit image allows you to have values from 0 (darkest) to 1023 (lightest). An 8 bit image is 0 (darkest) to 255 (lightest) the difference between the two is the number of intermediate steps available to be manipulated, not on the maximum lightness or darkness. The gamut of an image printed on an output device is driven by many factors: input color space, printer calibration, media, ink, minimum and maximum density achievable, etc. The number of bits in your image is a very small component.
Testing has shown that the human eye can only differentiate about 266 discrete levels in the green part of the visible light spectrum, about 200 in red and 150 in blue. A properly prepared and well-printed 8 bit image is capable of producing results indistinguishable by humans from a higher bit depth image.
High level advice for the highest gamut possible: capture your image in RAW and render it to a wide gamut color space like "Adobe RGB" or "Wide Gamut" ("sRGB" the default for most cameras encompasses the gamut of CRT monitors and is not close to the gamut available in a good hardcopy printer), carefully set white and black points in the image submitted, use a device with a large gamut (most pro level inkjet printers and laser silver halide printers do very well).
All that said it is my anecdotal opinion that most non-professional services are likely to use uncalibrated or poorly calibrated and poorly color managed devices which will override any of the careful preparation outlined above. You get what you pay for.
Comment: HDR, or High Dynamic Range, images are 32-bit floating point, rather than integer. They are not 8-bit integer, 4-channel, they are 32-bit float, 3-channel. That would mean they have a grand total of 96-bits per pixel. As floats, however, they have a considerably greater range than a 96-bit integer image would have. See: http://www.cambridgeincolour.com/tutorials/high-dynamic-range.htm


How can I seal a below-grade window permanently?

Question: I purchased my home a couple of years ago and this year I am going to replace the deck and build a fence.  The deck was in terrible condition so I completely took it down.  To my surprise I found 2 below grade windows under my deck that had been dry walled over on the inside.  In fact the previous owner even left the glass panes in the windows.  I want to fix this permanently and fill it in so it won't ever be a problem again.  What is the  best way to do this?
Comment: If it's a bedroom, they are (should be) egress windows and must be operable.
Answer: Jeff, you didn't say what material the windows were in. poured concrete foundation wall, framed wall?? Is there siding around the windows? A concrete wall could be filled with bricks and mortar. If it is a framed wall, build a framed box, shiething, then some flashing, lots of silicon caulk and match the siding. A picture would be a great help on this one.
Answer: In a similar situation, I cut a piece of flashing to the size of the window opening, then used metal screws to attach it to the (also metal) window frame.  I then sealed around the opening and the screw heads using Lexel caulk.
I thought about filling in the opening with concrete afterwards but never bothered.
Answer: If you want something relatively quick and easy, you could fill it up with a 2-part Polyurethane Foam (info).  It's really easy to apply, is airtight as long as you apply it properly and you're going to be covering it up anyway, right?  A few months ago I had a home energy audit done on my house and the technician recommended this stuff to improve temperature retention and seal the house up.  I've re-insulated a fair amount of my house with this stuff and it's great.


Good resources for Star Wars Saga Edition

Question: I was wondering if anyone knew where some good resources for SWSE (Star Wars: Saga Edition) would be. They closed down their official site besides the Dawn of Defiance campaign.
Answer: While not specific for Saga edition, the following are of general SW Gaming use:
Wookiepedia
http://starwars.wikia.com/wiki/Main_Page your best source for setting information.
http://www.starwarsmap.com/ Rest of the site is pretty lame, but it's a good map.
http://starwarsatlas.uw.hu/thelist.htm more useful, but less canonical, maps
Answer: I highly recommend www.d20radio.com and the forums in particular; the members there are very helpful and pitch in on all sorts of questions.  In addition, the associated podcasts (Order 66) are an excellent resource.
If you are a paying member of DDI, the WOTC Star Wars forums remain up and definitely active.
Answer: I actually look to Star Wars Expanded Universe as well as non-Star Wars sources for ideas for campaigns, villains, etc.
For example, the novels Death Troopers and Red Harvest provide the idea of a zombie story. Not done often in Star Wars, but can make for a creepy adventure.
TV shows like Dollhouse or other RPG systems like ShadowRun make for some interesting "cross over" ideas. You could have adventures doing different "runs" that go bad. Or where your characters get memory wiped after each adventure, etc.
Whatever you use, make sure you make it feel like Star Wars though.
That's what I've done for GM'ing and it makes for some interesting and fun adventures.
Another example of what I've done....I took inspiration from Batman and created a "Sith Jester", ie a Sith who is off his rocker a tad like the Joker. The story still feels Star Wars, but with a unique twist of a villain not traditionally seen. And so far he is a character the players love to hate. :)


What is a good place in Goa to stay with a beach and scuba diving?

Question: We are going to Goa with small children and are trying to find a nice beach resort, but with a decent dive shop, so that the adults could also do some diving. Any ideas?
Answer: The best season for enjoying water sports and scuba diving in Goa is from October to May. It's the season when the skies are cloudless and blue and the water remains placid.
SUN VILLAGE RESORT -India's only 4 star fully inclusive resort is beautifully set on the slopes of Arpora hills overlooking picturesque salt lakes and rice fields. With 135 well appointed air-conditioned suites, conference facilities, gala buffets, theme entertainment and a choice of exciting cuisines from mouth watering barbeques to Goan delicacies, this resort offers excellent facilities and a diver friendly pool. Complimentary TRY SCUBA sessions are conducted every Monday morning. Fun for all the family, join our team of Instructors to experience your first breath underwater.
CHALSTON BEACH RESORT - Overlooking the Arabian Sea along the ever popular Calangute-Baga beach, Chalston offers an ideal holiday spot for divers and families who wish to stay close to the main activity hub of Goa yet still relax in the comfort and tranquility of this friendly, family run resort. Enjoy warm hospitality, sunset cocktails and a choice of Indian or Continental cuisine at the Chalston Beach Shack. Complimentary 'TRY SCUBA' sessions are conducted every Monday afternoon. Our team of Instructors awaits you at the poolside coffee bar for scuba fun, a cappuccino and a slice of freshly baked walnut cake!
SINQUERIM: With a magnificent 17th century fort which has now been converted into a prison. It is one of the best beaches in Goa with international class facilities for water-skiing, para-sailing, fishing, scuba-diving and wind-surfing.
Comment: -1 because the *entire* answer was copy-pasted off the web. (The intro from http://www.scubadivingingoa.com/scuba-diving-vacations-goa.php and the rest from http://www.barracudadiving.com/accommodation.php) @Jaguar: You can quote relevant stuff from websites but you should clearly indicate what you wrote and what you quoted (and from where).


Prove $\log_{\frac{1}{4}} \frac{8}{7}> \log_{\frac{1}{5}} \frac{5}{4}$

Question:
Prove $$\log_{\frac{1}{4}} \frac{8}{7}> \log_{\frac{1}{5}} \frac{5}{4}$$

How to prove without a computer?
Answer: Here is another argument using power-series-based computations of $\log$, but slightly different in its details to Ivan's.
$$\log_{1/4} 8/7 = -\log_4 8/7 = \log_4 7/8 =  \log_4 (1 - 1/8),$$
while
$$\log_{1/5} 5/4 = -\log_5 5/4 = \log_5 4/5 = \log_5 (1 - 1/5).$$
So the problem reduces to showing that
$$\log_4(1-1/8) >\log_5 (1- 1/5).$$
This is slightly delicate, because both numbers of which we are taking the $\log$ are close to $1$, so both sides are close to $0$, and we have to estimate how close.
Recall that $$\ln (1- x) = - x - x^2/2 - x^3/3 - \cdots$$
when $|x| < 1$,
and that $\log_a (1-x) = (1/\ln a)\ln (1-x)$.
So we have to prove
$$\ln 4 \, (1/5 + 1/50 + \cdots ) > \ln 5 \,(1/8 + 1/128 + \cdots).$$
Now clearly
$$(1/5 + 1/50 + \cdots ) > 8/5 (1/8 + 1/128 + \cdots),$$
so it suffices to show that
$$(8/5)\ln 4 > \ln 5,$$
i.e. that $$\ln 4^8 > \ln 5^5,$$
i.e. that $$65536 > 3125,$$
which is true.
Answer: My second attempt, hopefully not flawed this time. (Although I think that there must be a simpler way to do this.)
I will use $\ln$ to denote natural logarithm. (Some people use $\log$)
The inequality
$$\log_{\frac{1}{4}} \frac{8}{7}> \log_{\frac{1}{5}} \frac{5}{4}$$
can be rewritten as
$$\frac{\ln\frac87}{\ln\frac14} > \frac{\ln\frac54}{\ln\frac15}.$$
This is equivalent to the inequality
$$\ln\frac87 \ln \frac15 > \ln \frac54 \ln\frac14$$
(we have multiplied both sides of the inequality by $\ln\frac14$ and $\ln\frac15$. So we have multiplied the inequality by a negative number twice - as a final result the inequality signs remains unchanged.)
This leads to equivalent inequalities
$$-\ln 5 \ln\frac87 > -\ln 4 \ln \frac54\\
\ln 4 \ln \frac54 > \ln 5 \ln\frac87. \tag{*}$$
So the above inequality $(*)$ is what we want show.
We know that $\ln(1+x)<x$ and $\ln(1+x)>\frac{x}{1+x}$ for $x>0$.
(Both these inequalities can be verified simply by taking derivative of functions $\ln(1+x)-x$ and $\ln(1+x)-\frac{x}{1+x}$ and showing that the derivative is negative/positive for $x>0$.
See WolframAlpha: 1, 2. Of course, there can be other ways how to prove this.
Both these properties are mentioned at Wikipedia.)
Now if we use one of these inequalities for $x=\frac17$, we get $\ln\frac87 < \frac17$, which gives
$$\ln\frac87\ln 5 < \frac17\ln5.$$
For $x=\frac14$ we have $\ln\frac54 > \frac{1/4}{1+1/4} = \frac{1/4}{5/4} = \frac15$ and thus
$$\ln \frac54 \ln 4 > \frac15 \ln 4.$$
Now we may notice that the following inequalities are equivalent to each other:
$$\frac15 \ln 4 > \frac17\ln5 \tag{**}\\
7\ln 4 > 5\ln 5\\
\ln(4^7) > \ln{5^5}\\
4^7 > 5^5\\
$$
The last inequality is true, which can be seen by evaluating $4^7=16384$ and $5^5=3125$. (And this can be shown by hand, too, since
$4^7=2^{14}=16\cdot2^{10}=16\cdot1024 > 16000$ and $5^5=125\cdot 25 < 128 \cdot 25 =  32 \cdot (4\cdot 25) = 3200$.)
This proves $(**)$.
So we have shown the following chain of inequalities
$$\ln \frac54 \ln 4> \frac15 \ln 4 > \frac17\ln5 > \ln\frac87\ln 5.$$
If we compare the leftmost and the rightmost expression, this is precisely $(*)$.
Answer: $\newcommand{\Log}{\operatorname{Log}}$I will use approximations by power series. It is easy to show that the inequality is equivalent to:$\Log(3.5) \Log(5) > \Log(4)^2$. On the other hand consider the power series expansion for $\Log(x)$ around $3.5$.It is: $$\Log(3.5)+\frac{2}{7}(x-3.5)-\frac{2}{49}(x-3.5)^2+O(x-3.5)^3$$ For $x=5$ we get: $\Log(5)>33/98+\Log[3.5]$
On the other hand for $x=4$ the linear term gives: $\Log(4)<1/7+\Log(3.5)$. Now we aim to prove the following:$$
\Log(3.5) \Log(5) > \Log(3.5)*(33/98 + \Log(3.5)) > (1/7 + \Log(3.5))^2 >
 \Log(4)^2$$
The first and last are proved already by the approximations. We need the middle. It is equivalent to the positivity of:$$
\Log(3.5)*(33/98 + \Log(3.5)) - (1/7 + \Log(3.5))^2
=-(1/49) + 5/98 \Log(3.5)
=(1/49)(5/2\Log(3.5)-1)
$$
The last is positive because $\Log(3.5)>1>2/5$.
Comment: I would have thought you needed $$\frac{\log(5) - \log(4)}{\log(1) - \log(5)}<\frac{\log(8) - \log(7)}{\log(1) - \log(4)}$$
Comment: This is exactly what I prove. Reversing up the first inequality by multiplying the denominators by -1.
Comment: My denominators are not the same as your final line
Comment: yes, you are right... this proof is wrong
Comment: I have completely redone the proof, so the comments above are obsolete.
Comment: @ivan: I've changed $Log$ to $\operatorname{Log}$ (TeX: `$\operatorname{Log}$`, I think it looks better. (Of course, you can rollback the edit if you don't like it, or you can change it to $\log$ (TeX: `$\log$`.)
Comment: Thanks! I am no good at TeX.
Answer: Note that
$\log_{\frac{1}{4}}\frac{8}{7}=-\log_4\frac{8}{7}=\log_47-\log_48=\log_4 7-\frac{3}{2}$,
$\log_{\frac{1}{5}}\frac{5}{4}=-\log_5\frac{5}{4}=\log_54-\log_55=\log_54-1$.
Hence we need to prove $$\log_47-\frac{3}{2}>\log_54-1.\ (*)$$ Now, $$(*)\Leftrightarrow2\log_47-3>2\log_54-2\Leftrightarrow \log_27-3>\log_516-2\Leftrightarrow\log_27>\log_516+1\Leftrightarrow$$ $$\Leftrightarrow\log_27>\log_516+\log_55\Leftrightarrow\log_27>\log_580.$$
I claim that $7^4>2^{11}$ and that $5^{11}>80^4$. Then $$4\log_27=\log_27^4>\log_22^{11}=\log_55^{11}>\log_580^4=4\log_580,$$ proving the desired result.
Since you explicitly ask not to use a computer let me show the two claimed inequalities by hand:
$$7^4-2^{11}=(2^3-1)^4-2^{11}=2^{12}-4\cdot 2^9+6\cdot 2^6-4\cdot 2^3+1-2^{11}= $$ $$=2^{12}-2^{11}+3\cdot 2^7-2^5+1-2^{11}=3\cdot 2^7-2^5+1=3\cdot 128-32+1=353>0.$$
Finally, since $80^4=5^4\cdot 2^{16}$ we only need to show that $5^7>2^{16}$. But
$$5^7-2^{16}=(5^3)^2\cdot 5-(2^7)^2\cdot 4=125^2\cdot 5-128^2\cdot 4=125^2\cdot 5-(125+3)^2\cdot 4=$$ $$=125^2\cdot 5-125^2\cdot 4-24\cdot 125-36=125^2-24\cdot125-36=125\cdot(125-24)-36=$$ $$=125\cdot 101-36=12625-36=12589>0.$$


Self-hosted URL shorteners with built-in statistics?

Question: I'm looking for self-hosted URL shortener scripts and projects (e.g. YOURLs) that track traffic sources, locations and other statistics. I need one with a solid API and JSON support. Any suggestions?
Comment: Note that if self-hosting is not an absolute requirement, http://bit.ly provides a lot of the information you are looking for. Just create an account to track your own or add "+" to the end of any bit.ly URL to view its stats.
Answer: if you have to go self hosted take a look at YOURLS
http://yourls.org/
Did you know bitly has an api, and allows you to use your own custom domain name
FOR FREE?
why not let bitly handle all that traffic?
YOURLS has its advantages since you can simply dump table.
But bitly also gets the job done for free
Answer: I have recently been researching the same thing.
YOURLS looks like a good option although I'm not entirely sure what reporting level you can access through this.
Bit.ly offer a custom domain setting when shortening links which works great! it requires you to point your site DNS at bitly (very simple step by step instructions).
In my opinion the best option (and the one I'm going to be using) is bit.ly. It's a tried and tested platform with solid reporting and flexibility.
https://bitly.com/a/custom_domain_settings
Answer: This script is self-hosted and has a stats page. Not sure what kind of functions you're looking for in an API but hope this helps get you started
http://codecanyon.net/item/url-shortener-script-with-stats-page-/1429705


How to cook a pork sirloin roast?

Question: I picked up a pork sirloin roast yesterday, it's a little over three pounds.  I've never cooked one of these before and I am not sure what to do.  My seasoning plan is salt, pepper, thyme, and rosemary, and I might poke a couple holes for cloves of garlic as well.  My cooking plan is 8+ hours in the crock pot on low, with about an inch of water in there.  Will this turn out OK?  I'd like a tender and moist result, obviously.  Any tips would be greatly appreciated.
Comment: For what it's worth, the roast I have is short and fat.  As opposed to being long and thin, like some other "tenderloin" roasts I saw.
Comment: Some that posted replies to this are confusing pork LOIN and SIRLOIN. The LOIN has less fat than the SIRLOIN, and can dry out easily.
Answer: I don't think a pork sirloin roast will stand up to that kind of cooking very well.  That is a very lean piece of meat, which does not lend itself to long cooking times at low temperatures.  That type of cooking is best reserved for cuts of meat with lots of fat and/or connective tissue.  If you put a pork sirloin roast in your cooker for 8 hours on low, you're going to have one tough, dry piece of meat when you are through.
You would be better served by cooking it in the oven at higher heat, for a relatively short period of time, like 400F for an hour or so.  Bear in mind, you're not going to get to a fall-apart tender state with a sirloin roast.  This isn't a pork butt with which you're dealing, which you'd take up to an internal temperature of 195F or more.  Rather, you're going to want to target the safe temperature, and no higher.  You want the meat to be juicy.  Cut into thinner slices to achieve maximum tenderness.
Answer: The cooking technique you are describing, braising, is good for meat that would otherwise be tough, with a lot of connective tissue, such as a shoulder roast. When you try to braise meat which is low on connective tissue, such as the sirloin, you risk drying the meat out.
I would recommend roasting instead. If you have a roasting rack (or a metal cookie rack, anything to elevate the meat and provide circulation), place the pork on the rack with a tray underneath to catch juices. (if you don't have a rack, no big deal). Preheated to 400 degrees; this high temperature will give your meat a nicely browned exterior. Immediately turn the heat down to 300 (or lower if you have time). Every half hour, turn the meat on the rack and baste the meat with the juices. Turning it will keep the juices distributed inside.
When your meat is finished (as decided by a thermometer), bring it out of the oven to rest for at least 30 minutes. Turn it every 5-10 minutes. While the meat rests, turn your juices into a nice pan sauce.
Answer: I agree. Roasting in a conventional oven is probably your best option. Keep in mind that a roast of that size could take a couple of hours to cook. Also, you'll want to baste it quite frequently - maybe every 15 mins or so. The last time I made a pork roast I basted with butter and whole grain mustard.
Answer: One of the best things to ever happen was when the USDA finally lowered its recommended internal temp for pork chops and roasts to to a minimum of 145F. YAY! That's medium rare! So it's really unnecessary to do a low, slow and moist method for a lean cut like the sirloin. Save your crock-pot for cuts like the shoulder, which will benefit much more from long, slow cooking because of all the fat and collagen.
You can get downright quick with sirloin, and still have tender, luscious meat.
Here is a great chart with guidelines for every cut of pork:
Answer: I put the pork sirloin roast in my cast iron dutch oven with salt, pepper, garlic powder, onion powder, and paprika.  Then I cook it on 250 for about 2.5-3 hours. It is fall apart amazing, and so tasty. Better than a pork loin (more flavor and juice).
Answer: Heat the oven to 325ºF (162ºC) with a little water in bottom of the roaster (i.e. the pan used for roasting). Cover and roast, calculating roat time with f30 minutes per pound. Married 51 years...have been doing my roasts this way all my life and never had a failure!
Answer: I wash my roast in cold water, let dry a minute or two, then place it on an aluminum foil lined lasagna pan.  I do not put any rub or flavoring on the meat. If there is any visible fat, I put that side up. I loosely cover the pan with another piece of foil. I preheat my convection oven to 300.  When ready I place the pan in the oven. After 45 minutes, I turn the meat over and leave the aluminum top tented only above the meat. I cook it for another 45 minutes. My roast always comes out with just a touch of golden brown and tender.  I let it stand for about 20 minutes and then slice it into 1-inch rounds end to end. We add spices to the meat if desired at the table.  We love lemon zest, light clarified garlic butter of BBQ sauce. We serve this as a main meal on day one;  then as pulled pork sandwiches on day two. Yummy (very easy and fool proof…always good)
Answer: If you want it to be even more juicy, brine it!
Just get 8 cups warm water, 1 cup sugar, and 1 cup salt and whisk it to dissolve. Let the pork roast sit in this solution in the fridge, covered for 6-8 hours and then remove and pat dry. Then cook as you normally would, the result will be much more succulent.
Answer: We had a 10 lb sirloin. Cut in half. Injected with a mixture of apple juice, brown sugar and molasses. Then heavily coated with coarse kosher salt, lightly sprinkled with garlic powder and just a little pepper, Smoked for 5.5 hours on Apple wood chunks. At 200°F over tin foil. The fat side up over the foil keeping the meat in the juices. So awesome. Big hit! Love my little smoker.... enjoy
Answer: I think oven at 400 for 15min then turned down to 350 then add slits and baste with saucy susan garlic and even a little soy sauce but rubbed first with veg oil
Answer: I cooked a 4.5 pound pork sirloin pork roast at 400 degrees Fahrenheit for an hour, then turned the temperature down to 375 degrees Fahrenheit for two more hours. Came out amazing!
Answer: I cooked a 3lb pork sirloin roast for the first time today and I loved it. I saw the above advice not too cook it in a crock pot but I was already 3 hours into it and figured it was too late. It turned out wonderfully after 7.5 hours of cooking. I fully submerged it in liquid made of chicken broth, Worcestershire sauce, Powdered onion soup mix, and chopped (fresh) garlic and chopped (fresh) jalapeños, and olive oil. Just didn't want people to be scared to use the crock pot (I think the fully submerged part is the most important). Oh! And I let it rest in a cool ceramic dish with a lid for about15 mins before cutting and serving.
Answer: I cooked a 3.5 seasoned pork sirloin for a little over an hour in a 425 degree oven. Covered it with foil the first 15 minutes, then uncovered for the rest. It got rave reviews, yet so easy. Pared it with corn, baked potato and homemade garlic bread. Super easy and super good.
Answer: If you'd like to try something different, the Szechuan recipe for Double Cooked Pork is a particular favourite of mine.
Dowse the joint of pork in boiling water for two minutes, drain and pat dry.
Prepare a basting fluid with mushroom soy sauce, rice wine (dry sherry will work as well), star anise, and chilli oil.
Then cut the pork joint into medallions half an inch thick and brush with the basting fluid.
In a flat bottomed pan, put a little groundnut oil and bring it to 180 C.
Place the pork medallions in the pan and fry for 3 minutes each side.
Serve with steamed rice, broccoli, snow peas, and french beans.
Answer: I do mine in a rotisserie. Perfect.
Comment: This isn't much information to go on. Care to share at least the time and temperature you would use for a three pound pork sirloin?

How to cook a pork sirloin roast?


Creating a Link Directory using WordPress?

Question: I would like to create a dmoz like website, using WP.
Any suggestions on plugins or site's architecture would be welcomed.
Answer: A great starting point would be Mike's answer to the question about cloning CrunchBase.  You'll want to do something similar with custom post types for entries in your directory.
If you want to allow visitors to submit sites, you could perhaps use the TDO Mini Forms plugin to allow visitors to create a new listing, and adjust the settings so that any new listings created through that form get held for administrator approval before being published.
Comment: How can I argue that this answer? :)  Seriously great answer.
Answer: A bit of extra info.
In case you want to have you're site in Hebrew, TDO Mini Forms will not work.
The configuration part of the plugin fails once you try to change strings to hebrew.
See this post by Nitzan Brumer for a good example on how to create a custom user-generated-content page.


Should I skin the common sole for pan?

Question: I have already cooked a common sole once before, and have skinned only the upper side (the dark side). The result was quite tasty. However, some people say that there's no need to skin it. For what recipes would you pan-dress a sole by skinning it, and for what recipes would you leave the skin on, and why?
Comment: The original question asked strictly for opinion, which is off-topic for this board.  I sumbitted and edit to permit a factually-based answer.
Answer: I would say it's totally up to you.
I prefer it with the skin still on, it get's really juicy inside. However it's a bit more difficult to eat. So perhaps if you were making it for kids (although, you could just skin them after they are fried), you could skin it.
One opportunity I really think is better without skin is when you serve it with sauce. Having to fiddle around with the skin covered in sauce is not really handy nor fun.

Should I skin the common sole for pan?


Partial sums of multiplicative functions

Question: It is well known that some statements about partial sums of multiplicative functions are extremely hard. For example, the Riemann hypothesis is equivalent to the assertion that $|\mu(1)+\mu(2)+\dots+\mu(n)|$ is bounded above by $n^{1/2+\epsilon}$ for all $\epsilon > 0$. However, I want to know about lower bounds for such partial sums, valid for infinitely many n. For instance, is it known that the sums of the Möbius function must infinitely often be somewhere near $n^{1/2}$ in magnitude? Can one at least prove that they are unbounded? And what about the Liouville function $\lambda$? Are its partial sums unbounded? If so, can anyone give me a good reference or quick proof? And how about general completely multiplicative functions that take values of modulus 1?  Is anything known about those?
Comment: I heard from Andrew Granville that the following is (roughly) true: Let $f(n)$ be a completely multiplicative function, with |f(n)| = 1, such that the mean value of $f(p)$ is equal to $\alpha \neq 0, -1$. Then under some technical conditions, $\sum_{n < x} f(n)$ is asymptotic to a constant times $(\log x)^{\alpha - 1}$. Apparently this is due to Tenenbaum, based on ideas of Selberg.   Unfortunately I'm not familiar with the proof, nor do I know a reference.
Comment: Frank, you must mean a constant times $x (\log x)^{a-1}$. One reference is my paper with Finch and Sebah (Google knows it), although the best use of that hint is to find more primary references from my paper's bibliography.
Comment: Greg -- indeed, that's what I meant, thank you!
Comment: I had an idea that it might be worth a) defining exhaustively the algebra of functions which are multiplicative on the domain of positive integers and b) the modular form of the same algebra based on any two elements being considered equivalent if they yield the same Euler product, which although an ambitious enterprise should surely yield some new insight into this area. https://math.stackexchange.com/questions/2436382/
Answer: This reference contains the best result of this kind currently known for $\mu(n)$:
Tadej Kotnik and Herman te Riele The Mertens Conjecture revisited. Algorithmic number theory, 156--167, Lecture Notes in Comput. Sci., 4076, Springer, Berlin, 2006.
They prove that $\limsup_{x \rightarrow +\infty}M(x)/\sqrt{x} \geq 1.218$ and that $\liminf_{x \rightarrow +\infty}M(x)/\sqrt{x} \leq -1.229$. Here
$
M(x) = \sum_{n \leq x}\mu(n)
$
is the conventional notation for the summatory function of the Möbius function. Their proof is a mixture of analytic number theory and large scale computations. They also have a survey of what is known and what is conjectured about the size of $M(x)$.
Now on to the Liouville function $\lambda(n)$ and its summatory function $L(x)$. The latter is very closely connected with $M(x)$, for
$
\sum_{n \leq \sqrt{x}}\mu(n)L\left(\frac{x}{n^2}\right) = \sum_{n \leq \sqrt{x}}\mu(n)\sum_{m \leq x/n^2}\lambda(m) = \sum_{N \leq x}\sum_{mn^2 = N}\mu(n)\lambda(m) = \sum_{N \leq x}\mu(N) = M(x).
$
Thus
$
|M(x)| \leq \sum_{n \leq \sqrt{x}}|L\left(\frac{x}{n^2}\right)|
$
and so the assumption
$
L(x) = o\left(\frac{\sqrt{x}}{\log^{1 + \epsilon}(x)}\right)
$
for example, leads to a contradiction with the Kotnik-te Riele result (or earlier results) for any $\epsilon > 0$.
My guess is that if one looks up the old (pre-computer) results on $|M(x)|$ from below, one might prove that $\limsup_{x \rightarrow +\infty}|L(x)|/\sqrt{x} > 0$. This may even be in the literature somewhere.
Alternatively
$
\sum_{n = 1}^{\infty}\lambda(n)n^{-s} = \prod_{p}\sum_{k=0}^{\infty}\lambda(p^k)p^{-ks} = \prod_{p}\sum_{k=0}^{\infty}(-1)^kp^{-ks} = \prod_{p}(1 + p^{-s})^{-1} = \frac{\zeta(2s)}{\zeta(s)}
$
by the Euler product formula, and from here on it is more elementary than it was with $M(x)$ in the argument that David Speyer gave, because we don't need the zeros on the critical line. For $\zeta(2s)$ has a pole at $s = 1/2$ which is not canceled by a pole of $\zeta(s)$ at the same point. Thus $L(x) = O(x^{\alpha})$ is impossible with $\alpha < 1/2$ by partial summation.
For multiplicative functions of modulus $1$, the situation is much less clear. For simplicity, assume that $f(n)$ is a totally multiplicative function (multiplicative and $f(p^k) = f(p)^k$) with $|c_p| = 1$ where $c_p = f(p)$. The Liouville function is the case
$c_p \equiv -1$. Then
$
\sum_{n = 1}^{\infty}f(n)n^{-s} = \prod_{p}\frac{1}{1 - c_pp^{-s}}{\quad},{\quad}A(x) = \sum_{n \leq x}f(n)
$
by the Euler product formula. The basic principle is that if $A(x) = O(x^{\alpha})$, then the Dirichlet series on the left hand side is convergent in the half plane $\sigma > \alpha$, so the sum is holomorphic in that half plane. If we can find a singularity $s_0$ of the product on the right hand side with $\mathrm{Re}(s_0) = \sigma_0$, that tells us that
$A(x) = O(x^{\alpha})$ with $\alpha < \sigma_0$ is impossible. The bad thing now is that the product may diverge at a point without having a singularity there, because the product may diverge to zero, and a holomorphic function may be zero at a point without being singular there.
But it is straightforward to show that if $\mathrm{Re}(c_p) \geq \delta > 0$ for all $p$, then $A(x) = O(x^{\alpha})$ is impossible for any $\alpha < 1$, by showing that the series
$
\sum_{p}c_pp^{-\sigma}
$
goes to infinity as $\sigma \rightarrow 1^{+}$.
Answer: You probably know this, but: Set $s(n) = \mu(1) + \cdots + \mu(n)$. Suppose, for the sake of contradiction, that $s(n) = O(n^{1/2 - \epsilon})$. Then
$$\sum s(n) \left( n^{-s} - (n+1)^{-s} \right)$$
would converge for $Re(s) > 1/2-\epsilon$.
This would give an analytic extension of $1/\zeta(s)$ to this half plane, contradicting that $\zeta$ has zeroes on the critical line.
So we know that the partial sums of $\mu$ are NOT $O(n^{1/2 - \epsilon})$. I don't know if this can be pushed to show that they are not $o(n^{1/2})$.
Comment: By the way, it is not necessary to know that $\zeta(s)$ has any zeros on the critical line, just that it has at least one zero in the critical strip $0 \leq \sigma \leq 1$. For then it has at least two zeros in $\sigma \geq 1/2$ by the functional equation and symmetry about the real axis, and only a single zero in $\sigma \geq 1/2$ is needed.
Comment: I was fairly sure that partial sums of mu were not better than the square root of n, but I didn't in fact know this argument, so thanks for giving it. I'll think about whether it can be adapted to work for the Liouville function.
Answer: This paper shows that $L(n) > .061867\sqrt{n}$ for infinitely many $n$.
As for somewhat elementary methods (in the sense of avoiding the Riemann zeta function) to show that $L(n)$ is "usually" of order $\sqrt{n}$, one can use the Lambert series
$$\sum_{n=1}^{\infty}{\frac{\lambda(n)q^n}{1-q^n}} = \sum_{n=1}^{\infty}{q^{n^2}}.$$
As
$$\frac{q^n}{1+q^n} = \frac{q^n}{1-q^n} - 2\frac{q^{2n}}{1-q^{2n}},$$
we have
$$\sum_{n=1}^{\infty}{\frac{\lambda(n)}{q^{-n}+1}} = \sum_{n=1}^{\infty}{q^{n^2}} - 2\sum_{n=1}^{\infty}{q^{2n^2}}$$
or equivalently, letting $q = e^{-\pi/x}$ and $\psi(x) = \sum_{n=1}^{\infty}{e^{-\pi xn^2}}$, where $x$ is large,
$$\sum_{n=1}^{\infty}{\frac{\lambda(n)}{e^{n\pi/x}+1}} = \psi(1/x) - 2\psi(2/x)$$
Now $\psi(x)$ satisfies the functional equation
$$\frac{1+2\psi(x)}{1+2\psi(1/x)} = \frac{1}{\sqrt{x}},$$
and so we can rewrite this as
$$\sum_{n=1}^{\infty}{\frac{\lambda(n)}{e^{n\pi/x}+1}} = \frac{1-\sqrt{2}}{2}\sqrt{x} + \frac{1}{2} + (\psi(x)-2\psi(x/2))\sqrt{x}.$$
For large $x$, the left-hand side "looks like" $L(x)$, whereas the right-hand side is dominated by the term $\frac{1-\sqrt{2}}{2}\sqrt{x}$. This also explains why $L(n)$ is predominantly negative, as $\frac{1-\sqrt{2}}{2}$ is negative.
Answer: Here is a largely historical remark:
Littlewood showed that there exists $x$ so that $\pi(x)$ is greater than the log-integral $\mathrm{li}(x)$. In fact,  $\pi(x) - \mathrm{li}(x)$ is (more or less) a linear combination of factors $x^{1/2+it}$, where $1/2+it$ are zeroes of $\zeta$. Form the multiplicative convolution with a suitable smooth function: you can thereby make the sum over only finitely many zeroes. Now find (by Dirichlet) some $x$ for which all the numbers $i t \log(x)$ are near multiples of $2 \pi$, etc.
This shows that $\pi(x) - \mathrm{li}(x)$ is not bounded by $C \sqrt{x}$ for any $C$;  in the case of Mobius, you get some $C$ but not any $C$, because it's harder to understand the coefficients of $x^{1/2+it}$.
Finally, there are some results valid for any completely multiplicative function: See the paper by A. Granville and K. Soundararajan entitled "The spectrum of multiplicative functions."
Comment: Would you please provide reference to Littlewood's paper (or its not original explanation)? Many thanks!
Answer: This might not be exactly what you have in mind, but it is an old result of Paley that for an infinite (but very sparse) set of real Dirichlet characters the inequality
$max_{N} |\sum_{n}^{N} \chi(n)| > c \sqrt{Q} \ln\ln(Q) $
holds, where Q is the modulus of the character. Montgomery and Vaughan have shown the reverse inequality (on RH) for all Dirichlet characters, that is to say,
$max_{N} |\sum_{n}^{N} \chi(n)| < c \sqrt{Q} \ln\ln(Q)$.
Recently Paley's theorem was improved by Granville and Soundararajan in their work on Pretentious characters (among other things, they prove the lower bound holds for a larger set of characters).
references:
R.E.A.C. Paley, A theorem on characters, J. London Math. Soc 7 (1932), 28-32
H.L. Montgomery and R.C. Vaughan, Exponential sums with multiplicative coefficients, Invent.
Math 43 (1977), 69-82.
LARGE CHARACTER SUMS: PRETENTIOUS CHARACTERS
AND THE POLYA-VINOGRADOV THEOREM: http://arxiv.org/abs/math.NT/0503113.
Comment: It should be $\sqrt{Q}\log \log Q$ in the above comment. The sums are trivially bounded by $Q$ and it is not too hard to show that the inequality (Polya-Vinogradov)

 $\sum_{M\leq n \leq M+N} \chi(n) = O(\sqrt{Q} \log Q)$

holds no matter how large $N$ is.

@Pete: they are quite similar from the point of view of L-functions. The key word here is "conductor", which puts "t" and "q" in the same footing. See the beginning of chapter 5 of the book by Iwaniec-Kowalski for detail.
Comment: Since Dirichlet characters are periodic, their partial sums are uniformly bounded.  Thus one seeks bounds for families of Dirichlet characters in terms of the modulus, which seems rather different from the situation of $\mu$ and $\lambda$.  (Disclaimer: I am not an analytic number theorist.  Maybe there's more of a similarity here than is immediately apparent.)


Need to remove germ from garlic or onion?

Question: My mate always tells me to remove the germ (the center) from garlic and onions, especially if it's turning green. What are the pros and cons of the germs of these plants?
Comment: i find that for garlic it imparts a bitter flavor so I always remove it. I've never had that problem with onions since I don't let them sit around long enough to sprout.
Answer: So... I was always taught that you remove it because it's bitter. I generally remove it if it's big enough.
However, there seems to be some contention as to whether this is true (and to what degree it's true). See:
http://www.examiner.com/article/remove-the-garlic-germ-few-do-this-anymore
and
http://ruhlman.com/2011/02/garlic-germ/
In short, there are mixed teachings as to whether you should remove it. But essentially the germ DOES affect the taste of garlic. Some call it bitter, some call it "more garlicky". Ruhlman suggests that if you're hitting it with heat immediately, then there isn't a point in removing it. If he's doing something that will have the garlic sitting around, then he doesn't.
From this what I'd suggest (and what I'm going to try doing from now on) is to ignore it and if you notice a difference in taste, or more importantly find the result objectionable. then take it out next time. Otherwise don't bother as you won't know the difference.
Comment: "mixed teachings" - the philosophy and divinity of an onion.
Answer: I always remove mine as I too find it affects the taste of the garlic. However, since I started keeping my garlic in the refrigerator, I've noticed that the garlic keeps much longer and rarely develops green germs.


Could quarks and leptons mix if they carried flavor charges?

Question: If quarks and leptons carried flavor charges that differed across generations (as they do in some theories), then could mixing take place?
Answer: It can happen if this flavor charge is broken at high enough energies. The symmetry has to be broken, otherwise we would either see long-range gauge forces different between different families, or else a confinement of the family charge at some accessible energies.
The breaking of family symmetry will allow the families to mix with each other by interacting with the Higgs field breaking the symmetry. However, this would mean that to mix different families using the Higgs would require both the standard model Higgs and the new Higgs, and so it would raise the dimension of the term. This would suppress the family mixing naturally by the ratio of the GUT scale to the new Higgs scale, as compared to the Higgs interaction within a family. This can be a natural explanation of the mysterious fact that the off diagonal CKM matrix elements are much smaller than they could be, naturally.
The easiest way to make this happen is to have a U(1) charge which is different for the different families. Assign one family charge 0 and the other two charge -1 and 1. Then assuming there is a Higgs field $G$ for this U(1), with mass scale $M_G$, around the GUT scale, the interaction of the standard model Higgs field H with fields in the standard model is suppressed when it requires using G by the ratio $M_g/M_{pl}$. The term in the low energy theory giving mass to the quarks would be
$$ H G Q_i q_j {M_G\over M_{pl}} $$
Where $Q_i$ is a quark SU(2) doublet and $q_j$ is a quark SU(2) singlet in a different family differing by one unit of charge. For families differing by two units of charge, you would need two G fields, and another power of ${M_G\over M_{pl}}$. The interactions within a given family would be unchanged (if you gave all the members of the family the appropriate cancelling charges).
This mechanism is like a low-energy effective theory caricature of string models proposed about a decade ago by Vafa, to explain the strange suppression of inter-family mixing. Vafa's models are based on gauge theories from branes, where the breaking can allow supression of certain off diagonal mixings based on the distances of the branes. But in the low-energy theory, this is equivalent to a Higgs mechanism on the gauge theory of the branes, and the suppressions are essentially just by the mechanism you suggest.
It might be worthwhile to see if there is a simple charge assignment which can reproduce qualitative patterns in the standard model from an idea of this sort. The biggest clue is that unlike the quark mixing, the inter-family neutrino mixing doesn't seem to be suppressed at all.
Comment: Dear Ron,thanx for the answer.


Should I perform the final color space conversion before or after final sharpening of an image?

Question: When preparing an image for the web (or other media), the last two steps is usually to resize the image to the desired resolution and sharpen it at that resolution. However, when working in a larger/different color space than the final result is going to be exported in, when should I convert to the target color space relative to these two operations?
Answer: My last steps, in order, are resize, sharpen, convert color space from ProRGB to sRGB, convert to 8-bit.
The only really critical orders are sharpening after resizing and changing to 8-bit after changing the color space.
In fact, I was curious, so I just tried.  One copy, I did my above steps.  On the other copy, I converted color space, converted to 8-bit and then resized and sharpend.  I put the images over each other and changed the blending mode to difference.
There was absolutely no pixel level difference between the two.  I expect to see a few minor differences at 100% that would never show up on the print, but in the case of this one particular photo, there were absolutely no differences at all.
Answer: It doesn't really matter to be honest.
Whether you sharpen in Adobe RGB or sRGB for example will give you pretty much the same output.
Although as a general rule, I would recommend doing a colour space conversion as the last step in the editing process because this generally involves going to a smaller colour space which risks a loss of highlights and shadows as well as gradients in the image.


Does inseam of pants convert to stand over height of a bike?

Question: If I know my jeans size, for instance 32", 32", can I convert this into an estimate for standover height on a bike?
Answer: This might not be a very good place to start from when sizing a bicycle.  Your inseam measurement is from the top of the crotch (top inseam point) to the BOTTOM SIDE OF THE ANKLE (lowest inseam point).  Your feet go all the way to the floor.
Inseam is specifically to determine what pants you wear.  I might point out that pants and bicycles are made and sized quite differently.
As @darkcanuck suggested, you're better served to stop by a local bike shop (LBS) and have them size you.  Most LBS sales people will be more than happy to put you on a couple of bikes and let you take a test ride, even if you say, "I don't plan to buy anything today, I'm just starting to shop around."  It won't cost you anything and you'll start to get a feel for how different styles of bikes feel in different sizes.
Note also that standover is not the only measure you need to size your bike.  The TOP TUBE length will also come into play--depending on how long your arms and torso are.  Different manufacturers build different bikes with different ratios of top-tube-to-stand-over.
Depending on the bike, you will also want to consider handle-bar width, crank-arm length, etc.  Don't plan to buy a bike off of the internet because it has the right stand-over!  At least, don't be surprised if you do so and then it doesn't feel right.
Try out several actual bikes--even if they aren't ones you plan to actually buy--so you know how those different geometries feel when you're riding.
AND IF YOU'RE LOOKING AT A USED BIKE... it still pays to try out some new ones and know what geometry suits you.  When you are negotiating with a seller of a used bike, show up with tape measure in hand!
Comment: @"Don't plan to buy a bike off of the internet" I just saw a posting on Craigslist that ~2 from my inseam and I wanted to estimate whether driving 45 minutes was at least worth it based on the sketchy info I have so far. Thanks for also bringing up the tube ratio, that is actually much more important to me (my last bike was an unrideable fit).
Answer: It's close enough for a very rough guess, but in most cases (especially if using a sizing formula) you'll want to measure the distance from the ground right up to your pubic bone.  Unless your pants drag along the ground, you're inseam size will be a bit shorter: somewhere in the range of 1-3" (as a guess).
You can get this quickly measured at many road bike shops as part of the shopping process (i.e. it should be free).
Answer: The short answer is no.  For information on frame sizes and all things bikes, consult the late Sheldon Brown's website: http://www.sheldonbrown.com/frame-sizing.html
It's an incredible resource!
Comment: That's good.  I was also impressed with http://www.peterwhitecycles.com/fitting.htm


How do I restore autoconf after Mountain Lion update?

Question: I upgraded my macbook to Mountain Lion today and found that I no longer have autoconf installed, is there a specific way to restore that or will I need to manually compile it etc? I have XCode 4.4 installed with Command Line Utilities downloaded and installed as well. Thanks for the information!
Answer: It sounds like you're a developer. You should install Mac Ports. It's REALLY helpful and useful to have. You can use it to get command line tools such as autoconf. It's always useful to have if you work with command line a lot.
Just install the .pkg from http://www.macports.org/install.php to install MacPorts.
After so, in terminal, run:
sudo port install autoconf
More info:
http://www.macports.org/ports.php?by=library&substr=autoconf
Comment: Heh, yeah I need it to install xdebug for PECL lol.
Comment: Better use homebrew (http://mxcl.github.com/homebrew/)
Answer: If you already have the latest XCode installed go to it's Preferences -> Downloads pane.
You can install the Command Line Tools there. You may need to reinstall Xcode and then install the Command Line Tools in the same way if that doesn't work.
Either way, to solve this specific issue you don't 'need' Mac Ports as it is a known issue that can be solved as above.
Comment: I had already done that, I reinstalled XCode 4.4.1