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# Some suggested time-pass (including ideas for Python scripts involving vectors and tensors)

Actually, I am busy writing down some notes on scalars, vectors and tensors, which I will share once they are complete. No, nothing great or very systematic; these are just a few notings here and there taken down mainly for myself. More like a formulae cheat-sheet, but the topic is complicated enough that it was necessary that I have them in one place. Once ready, I will share them. (They may get distributed as extra material on my upcoming FDP (faculty development program) on CFD, too.)

While I remain busy in this activity, and thus stay away from blogging, you can do a few things:

1.

Think about it: You can always build a unique tensor field from any given vector field, say by taking its gradient. (Or, you can build yet another unique tensor field, by taking the Kronecker product of the vector field variable with itself. Or, yet another one by taking the Kronecker product with some other vector field, even just the position field!). And, of course, as you know, you can always build a unique vector field from any scalar field, say by taking its gradient.

So, you can write a Python script to load a B&W image file (or load a color .PNG/.BMP/even .JPEG, and convert it into a gray-scale image). You can then interpret the gray-scale intensities of the individual pixels as the local scalar field values existing at the centers of cells of a structured (squares) mesh, and numerically compute the corresponding gradient vector and tensor fields.

Alternatively, you can also interpret the RGB (or HSL/HSV) values of a color image as the x-, y-, and z-components of a vector field, and then proceed to calculate the corresponding gradient tensor field.

Write the output in XML format.

2.

Think about it: You can always build a unique vector field from a given tensor field, say by taking its divergence. Similarly, you can always build a unique scalar field from a vector field, say by taking its divergence.

So, you can write a Python script to load a color image, and interpret the RGB (or HSL/HSV) values now as the $xx$-, $xy$-, and $yy$-components of a symmetrical 2D tensor, and go on to write the code to produce the corresponding vector and scalar fields.

Yes, as my resume shows, I was going to write a paper on a simple, interactive, pedagogical, software tool called “ToyDNS” (from Toy + Displacements, Strains, Stresses). I had written an extended abstract, and it had even got accepted in a renowned international conference. However, at that time, I was in an industrial job, and didn’t get the time to write the software or the paper. Even later on, the matter kept slipping.

I now plan to surely take this up on priority, as soon as I am done with (i) the notes currently in progress, and immediately thereafter, (ii) my upcoming stress-definition paper (see my last couple of posts here and the related discussion at iMechanica).

Anyway, the ideas in the points 1. and 2. above were, originally, a part of my planned “ToyDNS” paper.

3.

You can induce a “zen-like” state in you, or if not that, then at least a “TV-watching” state (actually, something better than that), simply by pursuing this URL [^], and pouring in all your valuable hours into it. … Or who knows, you might also turn into a closet meteorologist, just like me. [And don’t tell anyone, but what they show here is actually a vector field.]

4.

You can listen to this song in the next section…. It’s one of those flowy things which have come to us from that great old Grand-Master, viz., SD Burman himself! … Other songs falling in this same sub-sub-genre include, “yeh kisine geet chheDaa,” and “ThanDi hawaaein,” both of which I have run before. So, now, you go enjoy yet another one of the same kind—and quality. …

A Song I Like:

[It’s impossible to figure out whose contribution is greater here: SD’s, Sahir’s, or Lata’s. So, this is one of those happy circumstances in which the order of the listing of the credits is purely incidental … Also recommended is the video of this song. Mona Singh (aka Kalpana Kartik (i.e. Dev Anand’s wife, for the new generation)) is sooooo magical here, simply because she is so… natural here…]

(Hindi) “phailee huyi hai sapanon ki baahen”
Music: S. D. Burman
Lyrics: Sahir
Singer: Lata Mangeshkar

But don’t forget to write those Python scripts….

Take care, and bye for now…

# My answers to the Edge questions

I was supposed to come back and write a better version of the QM-related comment which I had made at Scott Aaronson’s blog (see my last post). However, I am not going to do that right away because of two reasons: (i) I think that leaving aside a typo or two, the comment isn’t in all that bad a condition, and (ii) I have begun thinking about some of the issues involved, and so would like to come back only after getting them straightened out (with a better order of presentation).

In the meanwhile, I decided to check out my blog-ideas folder (yes, I do “maintain” one), and found that I already had some material for a post.

It had so happened that while going through Scott’s post on the Edge question, especially the beginning of that post, an idea had struck me:

why not write my own one-line answers to all the Edge questions thus far?

So, I had gone to the Edge home page, and jotted down my answers in a .txt file. That was on 4th January 2016, between 4:17 PM–5:43 PM, IST (which means, one day before I wrote my comment on Scott’s blog). The answers were jotted down very, very rapidly—average time between reading a question and starting typing answer could be barely 10–15 seconds. That’s because we were very busy with the accreditation related work, remember? Though I don’t remember that particular day, I must have finished the day’s work and only then done this thing. Here I am going to mostly copy-paste its contents.

A couple of notes before we go over to my answers:

1. I just copy-pasted the Edge questions in the order I saw them on their home page. The order of answers was, thus, determined more or less by their Web page layout. So, the answers are not in the chronological sequence of the years over which they were raised.
2. I am reproducing the contents of a very whimsically written file (which was written, as I said, near the end of a long work-day), and without effecting any editing at all. Today, right within a week of writing the answers, I find that I want to change the answers to some of the questions. In any case, I would certainly want to make the answers more streamlined. But, today’s editing is minimal; whatever whimsicality that had crept in, I have decided to keep its original flavor as is.

So, here we go, with a copy-paste from that file:

The front velocity in the linear diffusion is a finite quantity (and in the first-order analysis, it is a constant).

2003 : WHAT ARE THE PRESSING SCIENTIFIC ISSUES FOR THE NATION AND THE WORLD, AND WHAT IS YOUR ADVICE ON HOW I CAN BEGIN TO DEAL WITH THEM? – GWB

The nation (i.e. USA): I don’t care a hoot about. The position was reciprocally derived.
The world: More or less the same. More or less the same.
My advise (to any President of the USA): Embrace 100% pure capitalism.

2002 : WHAT IS YOUR QUESTION? … WHY?

In his evolution, exactly for how long has the modern man been in the junglee-like state? (Rough estimates go like: lakhs of years of the jungle state vs. 5700 years of civilization, if you want to believe in the Western/modern science.)

Why it’s important? From many viewpoints: What led to the dominance of reason implicit in civilization? To what extent are the biological structural features really important in the mind-body connection? In what subtle ways must we still be junglee-like?

2001 : WHAT NOW?

Revising my viewpoint concerning non-relativistic quantum mechanics.
Reaching a conclusion concerning the issue of “stress or strain—which one is more fundamental?”
Writing some toy CFD code

2001 : WHAT QUESTIONS HAVE DISAPPEARED?

Whether the following phenomena are real or not: (i) rebirth, (ii) tele-many things such as telepathy, mind-reading, etc. (iii) psychic attacks, (iv) the actual American depravity in practice. Also the question of where Osama was hiding. Answer: In a cantonment town in a mostly army-ruled country that is a friend of USA.

2000 : WHAT IS TODAY’S MOST IMPORTANT UNREPORTED STORY?

How telecommunications actually changed the attitudes of rural people in India

1999 : WHAT IS THE MOST IMPORTANT INVENTION IN THE PAST TWO THOUSAND YEARS?

I don’t answer such silly questions. Idea? I would have attempted; generalization is possible with ideas. But not with inventions.

1998 : WHAT QUESTIONS ARE YOU ASKING YOURSELF?

Are the reported incidents in which some people could predict the future events true? If yes, what could be the mechanism in the pycho-somatic i.e. yogic terms? What implication does it hold for free will (properly defined, most fundamentally, only as the choice to focus or to evade), and the nature of soul and of consciousness.

2015: WHAT DO YOU THINK ABOUT MACHINES THAT THINK?

It’s a stupid idea. Thinking requires a conceptual (i.e. volitional) consciousness, which in turn requires life.

2014: WHAT SCIENTIFIC IDEA IS READY FOR RETIREMENT?

None. Proper ideas belonging to proper science never retire, they get subsumed away.

There are any number of ideas that should be retired because they were neither proper ideas nor scientific. For instance, the idea of an infinite physical universe. Or, the idea that ascribing anything quantitative to the universe (as a whole) is epistemologically valid. Etc. Also, string theory.

2013 : WHAT *SHOULD* WE BE WORRIED ABOUT?

The String Theory.

More seriously, government control of science and education.

2012: WHAT IS YOUR FAVORITE DEEP, ELEGANT, OR BEAUTIFUL EXPLANATION?

The mathematical aspects of the non-relativistic quantum mechanics. (A proper theory for its physical aspects is not yet available—despite my own attempts to build one.)

2011: WHAT SCIENTIFIC CONCEPT WOULD IMPROVE EVERYBODY’S COGNITIVE TOOLKIT?

The idea that all physical theories should be local.

I wrote that down, and then realized that the question asks for a toolkit.

I think computational modeling as easily realizable using Python and its ecosystem. It should be a part of every scientist’s grooming up. (Python is already being used to introduce programming to the CBSE XI-XII students in India.)

2010: HOW IS THE INTERNET CHANGING THE WAY YOU THINK?

It hasn’t changed *my* way of thinking, much. I always was more conceptual, and always had relatively lower working memory (and therefore also lessser maths skills) as compared to the class-mates who I ended up competing or at least [being] compared with. Internet has made more people forgetful and overall more like me, simply because they can just google up a lot of the concrete information.

2009: WHAT WILL CHANGE EVERYTHING?

I have no idea. Also, I don’t think it really is important or even desirable to change everything.

My view of quantum mechanics. I used to think that physicists were conceptually dumb. I now know that the original discoverers were not dumb—not even conceptually—but that their major flaw was either too much of careless[ness] about basic philosophical ideas, or too much of a diffidence towards bad philosophers.

Also, my view of Objectivists, Americans, and in fact, Western people in general. I no longer think that any of them could be fully relied on, practically speaking. They are not bad. [Here I meant morally bad.] In fact, they are pretty OK. They are just OK in the sense, they can be nice if you already are a practical success yourself. But if you are not a success, [they won’t bother to find that] hidden talent or spark in you. They are too crude and/or dumb and/or isolated, to be able to do that. Especially to an Indian-born. (They are rather like Brahmins in the Indian cities [and in the SF Bay Area]. The Brahmin-borns are not bad [by birth]. It’s just that a Maratha-born growing up rural areas cannot rely on them. He may receive acknowledgement of his talents from them, but not so reliably frequently to be a social norm. [Note today: It’s the elite vs non-elite phenomenon, really speaking, but the point is: I had never imagined as a young man that it would be as bad as it is, esp. in America.]

2007: WHAT ARE YOU OPTIMISTIC ABOUT?

That my new theory of QM will work out.

Actually, on most things, and speaking in terms of how I am habitually like, I am neither optimistic nor pessimistic. Habitually, if there is any one option towards which I do almost habitually get inclined, it is that: I like to know.

2006: WHAT IS YOUR DANGEROUS IDEA?

Dangerous in general? I have none. But if you mean those tiny cultural points of conversation that society people and people of culture (esp. writers of novels like Ayn Rand) use, you know, BS like: “he is too dangerous because he is too independent,” etc., then guess *my* most dangerous idea has been my sharing of the dimmed view of many people (Americans, Objectivists, Brahmins, army people, Indian University people, Westerners). Dangerous to me, that is.

2005: WHAT DO YOU BELIEVE IS TRUE EVEN THOUGH YOU CANNOT PROVE IT?

But I don’t always look for proofs, that’s the point. Validation is one thing, proofs is another. I do make sure to know the meaning and the roots of the concepts that I use. (And, sometimes, I have spent decades understanding their meaning and/or roots.)

But knowing the roots isn’t the same as proving, esp. to someone else, esp. to his satisfaction. You know how increasingly pointless and dangerous this enterprise gets.

Anyway, as the “cultural” kind of a fraud kind of an answer: My new theory of QM.

2016: WHAT DO YOU CONSIDER THE MOST INTERESTING RECENT [SCIENTIFIC] NEWS? WHAT MAKES IT IMPORTANT?

I can’t think of any. It has to be at least interesting, right? And, it has to be recent. Sorry. Can’t think of something that has both. At least not off the hand.

It would also be fun writing my own answers afresh some time later on.

A Song I Like:

(Hindi) “dil lagaa kar hum yeh samajhe”

Singers (separate versions each): Mahendra Kapoor and Asha Bhosale
Music: C. Ramchandra

[Another whimsicality of mine: Unlike most people whose opinions I usually find sensible, here I find that it is tough to decide which singer’s version is better. I mean to say, I don’t automatically find Asha’s version better. In fact, I like Mahendra Kapoor’s version [just] a shade better.]

[E&OE]

# The Mechanical-vs-Metallurgy “Branch-Jumping” Issue—Part II: Not Attending Inter-/Multi-/Trans-Disciplinary Conferences

0. To know the context and the primary intended readership of this post, please see my earlier post in this series, here: [^]. Of course, as mentioned earlier, everyone else is welcome to read this series, too.

1. The 57th (annual) Congress of the Indian Society of Theoretical and Applied Mechanics (An International Meet) was held at Pune this week, from 17th through 20th December, 2012 [^]. The venue was the Defence Institute of Advanced Technology (now a deemed university) [^]. (Caveat: Their Web site is often down, and with the PDF documents almost always missing. For example, try to download their faculty recruitment form.) I attended it, but this time round, without presenting any paper.

2. The conference was inaugurated by Dr. V. K. Saraswat [^], himself a PhD in combustion engineering. [Yes, the stupid primary intended readership [see part I to know exactly who all], this too is a topic common to both metallurgy and mechanical engineering.] The inaugural and valedictory functions were presided over by Dr. Prahlada [^], the vice-chancellor of the host institute (DIAT).

3. Some 180 papers were presented in the parallel sessions, many of them of multi-/trans-/inter-disciplinary nature, and with their authors coming from almost all departments of science and engineering. Even including electronics engineering, and mining engineering, apart from, of course, the usual ones: applied mechanics, mechanical engg., aerospace engg., civil engg., metallurgical/materials engg., mathematics, physics and astrophysics.

4. Even going just by my personal informal observations, people came to this conference from a lot of places: Guwahati, Kharagpur, Coimbatore, Kanpur, Chennai, Bangalore, Visakhapattanam, Hyderabad, Gulbarga, Surat, Mumbai, etc.

The foreign participation was somewhat limited this time round, with just a couple of Americans (both of Indian origin, both well-honored HoDs of mechanical or mechanics departments), and, off-hand, I suppose, one or two leading researchers or professors each from Canada, Germany, Israel, Japan, Taiwan, etc.

But, come, they did.

In contrast, the IIT Bombay QIP PhD D. W. Pande (of mechanical engineering branch from Aurangabad, now lording over at COEP); the meteorology (?) PhD degree holder G. B. Pant (sitting on the board of governors of COEP [a new addition to the stupid intended readership that should have been effected right the last time, and I will explain the reason for his inclusion the next time]); the Dean of the Faculty of Engineering of the University of Pune, PhD degree holder Gajanan Kharate (from Amaravati, now lording over at Pune, and per government, perhaps an OBC); his PhD guide the IIT Bombay QIP PhD Ashok A. Ghatol (formerly, Director, COEP, per government, certainly an OBC) did not come. Neither any of the others of their ilk.

Not even if they all are employed, and even if the places of their employment are all in or around Pune, and the conference was held right in Pune. [And that being academics, they would get discounts for the conference registration fees, and being government/university employed etc., they would get the conference fee refunded back anyway. Unlike me, who borrowed Rs. 3,500/- to attend it. Despite all that discount and its refunds, these characters still did not attend.]

And, of course, they didn’t send a single student of theirs to attend this conference either. Forget for paper presentation, not even for plain attendance.

The acceptance rate this time round was a bit higher, at about 60%. In the earlier ISTAMs which I attended, it has been 50% and lower; in fact, perhaps as low as 33% (if not 25%, but I don’t remember it too well, so let’s say, 33%). Pretty decent. Better than many reputed international journals. Even then, they still didn’t send a single student. [And, I am sure, this evil + stupid primary intended readership, while evaluating my employment application, would immediately pounce on the fact that I have no journal paper to my credit, only conference papers—if they could get past this metallurgy-to-mechanical “branch-jump” issue.]

These stupid idiots (and possibly evil characters—remember, free will as the basis of morality) with government-assured jobs and pensions and prestige, perhaps realized that if they attended the ISTAM conference, they might run into inter-/multi-/trans-disciplinary researches in mechanics and mechanical engineering. They perhaps also further realized that such a fact might then run counter to the one specific belief they fondly cuddle, cherish, openly advocate, defend and profess, and unhesitatingly act on: namely, that metallurgical graduates with PhD in mechanical cannot teach in or be hired by mechanical departments.

5. As to the research presented in the conference, much of it was not related to my current interests. But still, getting to know about the topics that other people are working on, the ideas they are pursuing, is always intellectually invigorating. I would like to write about the research part separately. Research, in fact any productive work, is such a noble thing. In contrast, for this post, I would not like to dilute the intensity of the focus on my joblessness due to the downright stupidity/evil of these above-mentioned professors/directors/government’s son in laws, etc.

However, I guess I could still mention just a couple of things in the passing.

5.1 One was the mention of the infinite speed of propagation of heat flux in conduction, during the invited lecture by Prof. I. Chung Liu of the National Chi Nan University, Taiwan. (I involuntarily sat up straight from my habitual slump while sitting in that cozy main auditorium at DIAT.) The approach Prof. Liu began with, was already known to me from my arXiv browsing. [No, the stupid intended readership of Mechanical Engineering Professors, Deans and Directors etc., arXiv usually does not have mechanical engineering related articles. So, you need not bother with this research any further, going by your government-funded and -enforced “logic.”] This approach consists of having a hyperbolic equation (the telegrapher’s equation) in place of the usual parabolic one. These days, a fairly neat Wiki page also exists to explain this approach; see here [^]. After his talk, I walked up to him and tried to explain how a particles-based approach makes it possible to remove the instantaneous action at a distance (IAD). However, Prof. Liu was not very well conversant with the Brownian movement/Weiner processes, and so, I could not pursue the conversation further. I just passingly mentioned my own research on diffusion equation to him. [The stupid primary intended readership of government-funded Mechanical Engineering Professors, University Deans and Directors etc., wouldn’t be able to make out why the IAD at all is an issue in the first place. They wouldn’t be able to make out even after being explicitly told twice.] Anyway, even if very brief, this discussion with Prof. Liu did help bring up some of my own thoughts. There is a certain paper on diffusion equation by a Berkeley professor which I had discovered after publishing my paper, and I would like to discuss it. Guess I will write a post at iMechanica (and, naturally, also here) about it, before sending a revised paper on this topic to a journal.

5.2 The second thing was this idea that had struck me while teaching a course on FEM to the COEP undergraduates in Spring 2009. [Yes, stupid/evil intended readership, I did teach the students of the mechanical branch as well, but only as a visiting faculty, and only for one semester. I was not repeated, despite very good student feedback [which Prof. Anil Sahasrabudhe, Director, COEP, didn’t quite share with me, unlike with his practice with other professors, but I do surmise with some pretty good basis—the direct feedback of students to me–that even my official student evaluation/feedback must have been pretty good.]] The idea is concerning finding a physical interpretation for the method of weighted residuals (MWR)—or, at the least, connecting some more mathematical context to MWR, anyway. My idea being too premature, I had not shared it with these undergraduate students back then. However, since the MTech-level students are a bit more mature, I did briefly hint at it while teaching the course on FEM at Symbiosis this year.

SPOILER ALERT: I may write a paper on this idea.

The idea is this: It first struck me that there was some kind of an analog between fitting a straight line to a scatter plot (say, the least-squares fit), and the method of weighted residuals. Sure, the first is an algebraic system and the second one involves differential equations. (Even if the ansatz is algebraic (a polynomial), before getting to the residuals, you still have to differentiate it, thereby changing the nature of the game.) The algebraic vs the infinitesimal is a big difference, and it is there. Yet, the idea of a residual (and setting it to zero) is common.

Then, I recalled that it was basically the same guy who had thought of both of these ideas, at least in their seed form: C. F. Gauss. (Ok, off-hand, I think that the least squares had already been used by someone else, before Gauss, but Gauss reinvented it independently, anyway. (Turns out, that earlier guy was Legendre [^])). The fact that the same mind had invented both the techniques helped gain more confidence in this idea of treating something like the least squares as an analog of the MWR.

In this conference, I got a chance to sound out this idea to two senior professors of mathematics: Prof. Kaloni of University of Windsor, and Prof. Rathish Kumar of IIT Kanpur. Specifically, I asked them if someone had already worked out something following, say, a function spaces-based approach.

Here, I was trying very hard to recall my earlier general reading decades ago concerning topological interpretation of the differentiation operation and all, and its recent mention by Prof. Tim Poston in a brief communication that I had with him. (It was a point which I had not at all understood at all.) Now at this conference, while talking in the hallways and all, I was trying to recollect those words. But somehow, in the hustle and bustle of the conference and the very short time available for those lounge/hall discussions, I could not recall any of such words. So, I tossed the first word I could catch hold of: function spaces.

Prof. Kaloni thought that someone must have worked on it already. In contrast, Prof. Rathish Kumar raised an entirely different point: where is convergence on the algebraic side of it, he asked. According to him, MWR was not limited to just getting to the residual and setting its domain integral to zero. The essence of MWR also had to include the idea of convergence—of a (possibly infinite) sequence of steps, of a systematic process of reducing the discretization error. In contrast, on the algebraic side of it, he observed, it’s just a one-time affair: you just take the fit, and that’s it. There is nothing more to be done; there is no second step; there is no sequence; the idea of convergence doesn’t apply.

In the busy-ness of such sideways discussions, there was no time to explain that I could get (i.e. I already was thinking of) an algebraic system that can still involve the ideas of convergence. In fact, I thought about it and got at an example right on the fly. But I was sure I couldn’t have explained it in the right words—the idea just flashed right during the conversation. So, not to waste his time, I asked him what would he think of it if I could get such a system (a multi-step, converging but algebraic system), and try to establish an analog with the differential equations-involving MWR. He then said that perhaps such a thing has not been done before, and that it would be nice to have a connection like that formally worked out. [I will repeat this part in a separate post, also at iMechanica, but in the meanwhile, if you know that someone has already worked out something along these lines, please drop me a line; thanks in advance.]

So there. The stupid/evil primary intended readership, these discussions, per your government-funded and government-enforced “logic,” had nothing to do with mechanical engineering. After all, both the professors were from the department of mathematics. So, you the stupid/evil primary intended readership (consisting of folks like G. K. Kharate, A. A. Ghatol, D. W. Pande, G. B. Pant, their friends, etc.), you all sit cozy and quiet and keep on drawing your respective 6th-pay commission-enhanced salaries, allowances, refunds, etc. Keep faithfully doing that, you stupids/idiots/evils.

[I remain jobless; the “A Song I Like” section is once again being dropped.]

[E&OE]


text:
# Black Holes and the Information Paradox: a Solution?

Gravity has a good sense of humour. On the one hand, it’s the weakest force we know. The upward push of your chair is more than enough to counteract the pull of the entire planet! Yet gravity has an ace up its sleeve – unlike all other forces, it’s always attractive. For larger objects, the other forces start bickering and cancelling out. But gravity just keeps on getting stronger, until it’s impossible to escape – a black hole!

As a theoretical physicist, I tend to carry a black hole whenever I’m travelling.

As you can see, the top of this bucket is the surface of a black hole, otherwise known as the event horizon. When I release water from a glass above the black hole, it is attracted to the black hole, and falls inexorably towards it, never to be seen again.

Okay, I suppose my bucket isn’t a real black hole. After all, it’s the gravity of the Earth that pulls the water in. And light can definitely escape because I can see inside it! But it does accurately represent the bending of space and time. Albert Einstein taught us that everything in the universe rolls around on the cosmic quilt of spacetime, like balls on an elastic sheet. A heavy ball distorts the sheet, creating my black hole bucket.

You might not feel too threatened by black holes – after all, the nearest one is probably 8 billion billion miles away. But in actual fact you could be falling into a black hole right now without noticing! Turns out that for a large enough black hole, the event horizon is so far away that gravity there is very weak. So there’s no reason why you should experience anything special.

This disturbing fact has an unexpected consequence from the microscopic world of quantum mechanics. Every quantum theory must have a single vacuum, essentially the most boring and lazy state of affairs. If I stand still the vacuum is just empty space. But as soon as I start accelerating something weird happens. Particles suddenly appear from nowhere!

What does that mean for our black hole? Well if you’re not falling in, you must be accelerating away to oppose the huge pull of gravity! This means that you should see the black hole glowing with particles called Hawking radiation. Remember my black hole full of water? Well, you haven’t fallen in. And that means I have to cover you with Hawking radiation!

Luckily for your computer, the Hawking confetti that came out isn’t the same as the water that went in. From your perspective the water has simply disappeared! Exactly the same thing seems to happen for real black holes.

This black hole magic trick has become infamous among scientists, resisting all efforts at explanation. But a solution might be at hand, courtesy of Hawking himself! What if you could slightly change the vacuum every time something dropped into the black hole? Then, if you’re very careful, you might just be able to reconstruct the original water from the confetti of Hawking radiation.

Put another way, the event horizon takes a lock of soft hair from every passing particle as a memento of its existence. This information is eventually carried off by Hawking’s magic particles, reminding us of what we’d lost. It remains for soft experts, like myself, to work out the exact details.

This post is based on a talk given for the Famelab competition. You can read the full paper by Stephen Hawking, Malcolm Perry and Andy Strominger here.

# Romeo and Juliet, through a Wormhole

I spent last week at the Perimeter Institute in Waterloo, Ontario. Undoubtedly one of the highlights was Juan Maldenena’s keynote on resolving black hole paradoxes using wormholes. Matt’s review of the talk below is well worth a read!

Perimeter is hosting this year’s Mathematica Summer School on Theoretical Physics. The school is a mix of lectures on a topic in physics (this year, the phenomenon of quantum entanglement) and tips and tricks for using the symbolic calculation program Mathematica.

Juan Maldacena is one of the lecturers, which gave me a chance to hear his Romeo and Juliet-based explanation of the properties of wormholes. While I’ve criticized some of Maldacena’s science popularization work in the past, this one is pretty solid, so I thought I’d share it with you guys.

You probably think of wormholes as “shortcuts” to travel between two widely separated places. As it turns out, this isn’t really accurate: while “normal” wormholes do connect distant locations, they don’t do it in a way that allows astronauts to travel between them, Interstellar-style. This can be illustrated with something called a Penrose…

View original post 626 more words

# T-duality and Isometries of Spacetime

I’ve just been to an excellent seminar on Double Field Theory by its co-creator, Chris Hull. You may know that string theory exhibits a meta-symmetry called T-duality. More precisely, it’s equivalent to put closed strings on circles of radius $R$ and $1/R$.

This is the simplest version of T-duality, when spacetime has no background fields. Now suppose we turn on the Kalb-Ramond field $B$. This is just an excitation of the string which generalizes electromagnetic potential.

This has the effect of making T-duality more complicated. In fact it promotes the $R\to 1/R$ symmetry to $O(d,d;\mathbb{Z})$ where $d$ is the dimension of your torus. Importantly for this to work, we must choose a $B$ field which is constant in the compact directions, otherwise we lose the isometries that gave us T-duality in the first place.

Under this T-duality, the $B$ field and $G$ metric get mixed up. This can have dramatic consequences for the underlying geometry! In particular our new metric may not patch together by diffeomorphisms on our spacetime. Similarly our new Kalb-Ramond field $B$ may not patch together via diffeomorphisms and gauge transformations. We call such strange backgrounds non-geometric.

To express this more succintly, let’s package diffeomorphisms and gauge transformations together under the name generalized diffeomorphisms. We can now say that T-duality does not respect the patching conditions of generalized diffeomorphisms. Put another way, the $O(d,d)$ group does not embed within the group of generalized diffeomorphisms of our spacetime!

This lack of geometry is rather irritating. We physicists tend to like to picture things, and T-duality has just ruined our intuition! But here’s where Double Field Theory comes in. The idea is to double the coordinates of your compact space, so that $O(d,d)$ transformations just become rotations! Now T-duality clearly embeds within generalized diffeomorphisms and geometry has returned.

All this complexity got me thinking about an easier problem – what do we mean by an isometry in a theory with background fields? In vacuum isometries are defined as diffeomorphisms which preserve the metric. Infinitesimally these are generated by Killing vector fields, defined to obey the equation

$\displaystyle \mathcal{L}_K g=0$

Now suppose you add in background fields, in the form of an energy-momentum tensor $T$. If we want a Killing vector $K$ to generate an overall symmetry then we’d better have

$\displaystyle \mathcal{L}_K T=0$

In fact this equation follows from the last one through Einstein’s equations. If your metric solves gravity with background fields, then any isometry of the metric automatically preserves the energy momentum tensor. This is known as the matter collineation theorem.

But hang on, the energy momentum tensor doesn’t capture all the dynamics of a background field. Working with a Kalb-Ramond field for instance, it’s the potential $B$ which is the important quantity. So if we want our Killing vector field to be a symmetry of the full system we must also have

$\displaystyle \mathcal{L}_K B=0$

at least up to a gauge transformation of $B$. Visually if we have a magnetic field pointing upwards everywhere then our symmetry diffeomorphism had better not twist it round!

So from a physical perspective, we should really view background fields as an integral part of spacetime geometry. It’s then natural to combine fields with the metric to create a generalized metric. A cute observation perhaps, but it’s not immediately useful!

Here’s where T-duality joins the party. The extended objects of string theory (and their low energy descriptions in supergravity) possess duality symmetries which exchange pieces of the generalized metric. So in a stringy world it’s simplest to work with the generalized metric as a whole.

And that brings us full circle. Double Field Theory exactly manifests the duality symmetries of the generalized metric! Not only is this mathematically helpful, it’s also an important conceptual step on the road to unification via strings. If that road exists.

# Gravity for Beginners

Everyone knows about gravity. Every time you drop a plate or trip on the stairs you’re painfully aware of it. It’s responsible for the thrills and spills at theme parks and the heady hysteria of a plummeting toboggan. But gravity is not merely restricted to us small fry on Earth. It is a truly universal force, clumping mass together to form stars and planets, keeping worlds in orbit around their Suns. Physicists call gravity a fundamental force – it cannot be explained in terms of any other interaction.

The most accurate theory of gravity is Einstein’s General Theory of Relativity. The presence of mass warps space and time, creating the physical effects we observe. The larger the mass, the more curved space and time become, so the greater the gravitational pull. Space and time are a rubber sheet, which a large body like the Sun distorts.

Gravity is very weak compared to other fundamental forces. This might be a bit of a surprise – after all it takes a very powerful rocket to leave Earth’s orbit. But this is just because Earth is so very huge. Things on an everyday scale don’t seem to be pulled together by gravity. But small magnets certainly are attracted to each other by magnetism. So magnetism is stronger than gravity.

The fact that gravity acts by changing the geometry of space and time sets it apart from all other forces. In fact our best theories of particles assume that there is some cosmic blank canvas on which interactions happen. This dichotomy and the weakness of gravity give rise to the conflicts at the heart of physics.

It’s All Relative, After All

“You can’t just let nature run wild.”

Walt Disney

Throughout history, scientists have employed principles of relativity to understand nature. In broad terms these say that different people doing the same scientific experiment should get the same answer. This stands to reason – we travel the world and experience the same laws of nature everywhere. Any differences we might think we observe can be explained by a change in our experimental conditions. Looking for polar bears in the Sahara is obvious madness.

In the early 17th century Galileo formulated a specific version of relativity that applied to physics. He said that the laws of physics are the same for everyone moving at constant speed, regardless of the speed they are going. We notice this in everyday life. If you do an egg and spoon race walking at a steady pace the egg will stay on the spoon just as if you were standing still.

If you try to speed up or slow down, though, the egg will likely go crashing to the ground. This shows that for an accelerating observer the laws of physics are not the same. Newton noticed this and posited that an accelerating object feels a force that grows with its mass and acceleration. Sitting in a plane on takeoff this force pushes you backwards into your seat. Physicists call this Newton’s Second Law.

Newton’s Second Law can be used the opposite way round too. Let’s take an example. Suppose we drop an orange in the kitchen. As it travels through the air its mass certainly stays the same and the only significant force it feels is gravity. Using Newton’s Second Law we can calculate the acceleration of the orange. Now acceleration is just change in speed over time. So given any time in the future we can predict the speed of the orange.

But speed is change in distance over time. So we know the distance the orange has travelled towards the ground after any given time. In particular we can say exactly how much time it’ll be before the orange goes splat on the kitchen floor. One of the remarkable features of the human brain is that it can do approximations to these calculations very quickly, enabling us to catch the orange before disaster strikes.

The power of the principle of relativity is now apparent. Suppose we drop the orange in a lift, while steadily travelling upwards. We can instantly calculate how long it’ll take to hit the lift floor. Indeed by the principle of relativity it must take exactly the same amount of time as it did when we were in the kitchen.

There’s a hidden subtlety here. We’ve secretly assumed that there is some kind of universal clock, ticking away behind the scenes. In other words, everyone measures time the same, no matter how they’re moving. There’s also a mysterious cosmic tape measure somewhere offstage. That is, everyone agrees on distances, regardless of their motion. These hypotheses are seemingly valid for everyday life.

But somehow these notions of absolute space and time are a little unsettling. It would seem that Galileo’s relativity principle applies not only to physics, but also to all of space and time. Newton’s ideas force the universe to exist against the fixed backdrop of graph paper. Quite why the clock ticks and the ruler measures precisely as they do is not up for discussion. And the mysteries only deepen with Newton’s theory of gravity.

A Tale of Two Forces

“That one body may act upon another at a distance through a vacuum without the mediation of anything else […] is to me so great an absurdity that […] no man who has […] a competent faculty of thinking could ever fall into it.”

Sir Isaac Newton

Newton was arguably the first man to formulate a consistent theory of gravitation. He claimed that two masses attract each other with a force related to their masses and separation. The heavier the objects and the closer they are, the bigger the force of gravity.

This description of gravity was astoundingly successful.  It successively accounted for the curvature of the Earth, explained the motion of the planets around the Sun, and predicted the precise time of appearance of comets. Contemporary tests of Newton’s theory returned a single verdict – it was right.

Nevertheless Newton was troubled by his theory. According to his calculations, changes in the gravitational force must be propagated instantaneously throughout the universe. Naturally he sought a mechanism for such a phenomenon. Surely something must carry this force and effect its changes.

Cause and effect is an ubiquitous feature of everyday physics. Indeed Newton’s Second Law says motion and force and inextricably linked. Consider the force which opens a door. It has a cause – our hand pushing against the wood – and an effect – the door swinging open. But Newton couldn’t come up with an analogous explanation for gravity. He had solved the “what”, but the “why” and “how” evaded him entirely.

For more than a century Newton’s theories reigned supreme. It was not until the early 1800s that physicists turned their attention firmly towards another mystery – electricity. Michael Faraday lead the charge with his 1721 invention of an electric motor. By placing a coil of wire in a magnetic field and connecting a battery he could make it rotate. The race was on to explain this curious phenomenon.

Faraday’s work implied that electricity and magnetism were two sides of the same coin. The strange force he had observed became known, appropriately, as electromagnetism. The scientific community quickly settled on an idea. Electricity and magnetism were examples of a force field – at every point in space surrounding a magnet or current, there were invisible lines of force which affected the motion of nearby objects.

All they needed now were some equations. This would allow them to predict the behaviour of currents near magnets, and verify that this revolutionary force field idea was correct. Without a firm mathematical footing the theory was worthless. Several preeminent figures tried their hand at deriving a complete description, but in 1861 the problem remained open.

In that year a young Scotsman named James Clerk Maxwell finally cracked the issue. By modifying the findings of those before him he arrived at a set of equations which completely described electromagnetism. He even went one better than Newton had with gravity. He found a mechanism for the transmission of electromagnetic energy.

Surprisingly, Maxwell’s equations suggested that light was an electromagnetic wave. To wit, solutions showed that electricity and magnetism could spread out in a wavelike manner. Moreover the speed of these waves was determined by a constant in his equations. This constant turned out to be very close to the expected speed of light in a vacuum. It wasn’t a giant leap to suppose this wave was light itself.

At first glance it seems like these theories solve all of physics. To learn about gravity and the motion of uncharged objects we use Newton’s theory. To predict electromagnetic phenomena we use Maxwell’s. Presumably to understand the motion of magnets under gravity we need a bit of both. But trying to get Maxwell’s equations to play nice with Galileo’s relativity throws a big spanner in the works.

A Breath of Fresh Air

“So the darkness shall be the light, and the stillness the dancing.”

T.S. Eliot

Galileo’s relativity provides a solid bedrock for Newtonian physics. It renders relative speeds completely irrelevant, allowing us to concentrate on the effects of acceleration and force. Newton’s mechanics and Galileo’s philosophy reinforce each other. In fact Newton’s equations look the same no matter what speed you are travelling at. Physicists say they are invariant for Galileo’s relativity.

We might naively hope that Maxwell’s equations are invariant. Indeed a magnet behaves the same whether you are sitting still at home or running around looking for buried treasure. Currents seem unaffected by how fast you are travelling – an iPod still works on a train. It would be convenient if electromagnetism had the same equations everywhere.

Physicists initially shared this hope. But it became immediately apparent that Maxwell’s equations were not invariant. A change in the speed you were travelling caused a change in the equations. Plus there was only one speed at which solutions to the equations gave the right answer! Things weren’t looking good for Galileo’s ideology.

To solve this paradox, physicists made clever use of a simple concept. Since antiquity we’ve had a sense of perspective – the world looks different from another point of view. Here’s a simple example in physics. Imagine you’re running away from a statue. From your viewpoint the statue is moving backwards. From the statue’s viewpoint you are moving forwards. Both descriptions are right. They merely describe the same motion in contrary ways.

In physics we give perspective a special name. Any observer has a frame of reference from which they see the world. In your frame of reference the statue moves backwards. In the statue’s frame of reference you move forwards. We’ve seen that Newton’s physics is the same in every frame of reference which moves with a constant speed.

We can now rephrase our discovery about Maxwell’s equations. Physicists found that there was one frame of reference in which they were correct. Maxwell’s equations somehow prefer this frame of reference over all other ones! Any calculations in electromagnetism must be done relative to this fixed frame. But why is this frame singled out?

Faced with this question, physicists spotted the chance to kill two birds with one stone. The discovery that light is a wave of electromagnetism raises an immediate question. What medium carries the light waves? We’re used to waves travelling through some definite substance. Water waves need a sea or ocean, sound waves require air, and earthquakes move through rock. Light waves must travel through a fixed mysterious fog pervading all of space. It was called aether.

The aether naturally has it’s own frame of reference, just as you and I do. When we measure the speed of light in vacuum, we’re really measuring it relative to the aether. So the aether is a special reference frame for light. But light is an electromagnetic wave. It’s quite sensible to suggest that Maxwell’s preferred reference frame is precisely the aether!

Phew, we’ve sorted it. Newtonian mechanics works with Galilean relativity because there’s nothing to specify a particular reference frame. Maxwell’s equations don’t follow relativity because light waves exist in the aether, which is a special frame of reference. Once we’ve found good evidence for the aether we’ll be home and dry.

So thought physicists in the late 19th century. The gauntlet was down. Provide reliable experimental proof and win instant fame. Two ambitious men took up the challenge – Albert Michelson and Edward Morley.

They reasoned that the Earth must be moving relative to the aether. Therefore from the perspective of Earth the aether must move. Just as sound moves faster when carried by a gust of wind, light must move faster when carried by a gust of aether. This prediction became known as the aether wind.

By measuring the speed of light in different directions, Michelson and Morley could determine the direction and strength of the aether wind. The speed of light would be greatest when they aligned their measuring apparatus with the wind. Indeed the gusts of aether would carry light more swiftly. The speed of light would be slowest when they aligned their apparatus at right angles to the wind.

The expected changes in speed were so minute that it required great ingenuity to measure them. Nevertheless by 1887 they had perfected a cunning technique. With physicists waiting eagerly for confirmation, Michelson and Morley’s experiment failed spectacularly. They measured no aether wind at all.

The experiment was repeated hundreds of times in the subsequent years. The results were conclusive. It didn’t matter which orientation you chose, the speed of light was the same. This devastating realisation blew aether theory to smithereens. Try as they might, no man nor woman could paper over the cracks. Physics needed a revolution.

How Time Flies

“Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute. That’s relativity.”

Albert Einstein

It is the most romanticised of fables. The task of setting physics right again fell to a little known patent clerk in Bern. The life and work of Albert Einstein has become a paradigm for genius. But yet the idea that sparked his reworking of reality was beautifully simple.

For a generation, physicists had been struggling to reconcile Galileo and Maxwell. Einstein claimed that they had missed the point. Galileo and Maxwell are both right. We’ve just misunderstood the very nature of space and time.

This seems a ridiculously bold assumption. It’s best appreciated by way of an analogy. Suppose you live in a rickety old house and buy a nice new chair. Placing it in your lounge you notice that it wobbles slightly. You wedge a newspaper under one leg. At first glance you’ve solved the problem. But when you sit on the chair it starts wobbling again.

After a few more abortive attempts with newspapers, rags and other household items you decide to put the chair in another room. But the wobble won’t go away. Even when you sand down the legs you can’t make it stand firm. The chair and the house seem completely incompatible.

One day a rogue builder turns up. He promises to fix your problem. You don’t believe he can. Neither changing the house nor the chair has made the slightest difference to you. When you return from work you are aghast to see he’s knocked the whole house down. He shouts up to you from a hole in the ground, “just fixing your foundations”!

More concretely Einstein said that there’s no problem with Galileo’s relativity. The laws of physics really are the same in every frame moving at constant speed. Physicists often rename this idea Einstein’s special principle of relativity even though it is Galileo’s invention! Einstein also had no beef with Maxwell’s equations. In particular they are the same in every frame.

To get around the fact that Maxwell’s equations are not invariant for Galileo’s relativity, Einstein claimed that we don’t understand space and time. He claimed that there is no universal tape measure, or cosmic timepiece. Everybody is responsible for their own measurements of space and time. This clears up some of the issues that annoyed Newton.

In order to get Maxwell’s equations to be invariant when we change perspective, Einstein had to alter the foundations of physics. He postulated that each person’s measurements of space and time were different, depending on how fast they were going. This correction magically made Maxwell’s equations work with his principle of special relativity.

There’s an easy way to understand how Einstein modified space and time. We’re used to thinking that we move at a constant speed through time. Clocks, timers and watches all attest to our obsession with measuring time consistently. The feeling of time dragging on or whizzing by is merely a psychological curiosity. Before Einstein space didn’t have this privilege. We only move through space as we choose. And moving through space has no effect on moving through time.

Einstein made everything much more symmetrical. He said that we are always moving through both space and time. We can’t do anything about it. Everyone always moves through space and time at the same speed – the speed of light. To move faster through space you must move slower through time to compensate. The slower you trudge through space, the faster you whizz through time. Simple as.

Einstein simply put space and time on a more even footing. We call the whole construct spacetime. Intuitively spacetime is four dimensional. That is, you can move in four independent directions. Three of these are in space, up-down, left-right, forward-backward. One of these is in time.

You probably haven’t noticed it yet, but special relativity has some weird effects. First and foremost, the speed of light is the same however fast you are going. This is because it is a constant in Maxwell’s equations, which are the same in all frames. You can never catch up with a beam of light! This is not something we are used to from everyday life.

Nevertheless it can be explained using Einstein’s special relativity. Suppose you measure the speed of light when you are stationery. You do this by measuring the amount of time it takes for light to go a certain distance. For the sake of argument assume you get the answer 10 mph.

Now imagine speeding up to 5 mph. You measure the speed of light again. Without Einstein you’d expect the result to be 5 mph. But because you’re moving faster through space you must be moving slower through time. That means it’ll take the light less time to go the same distance. In fact the warping of spacetime precisely accounts for the speed you’ve reached. The answer is again 10 mph.

Einstein’s spacetime also forces us to forget our usual notions of simultaneity. In everyday physics we can say with precision whether two events happen at the same time. But this concept relies precisely on Newton’s absolute time. Without his convenient divine timepiece we can’t talk about exact time. We can only work with relative perspectives.

Let’s take an example. Imagine watching a Harrison Ford thriller. He’s standing on the top of a train as it rushes through the station. He positions himself precisely in the center of the train. At either end is a rapier wielding bad guy eager to kill him. Ford is equipped with two guns which he fires simultaneously at his two nemeses. These guns fire beams of light that kill the men instantly.

In Ford’s frame both men die at the same time. The speed of light is the same in both directions and he’s equidistant from the men when he shoots. Therefore the beams hit at the same moment according to Ford. But for Ford’s sweetheart on the platform the story is different.

Let’s assume she is aligned with Ford at the moment he shoots. She sees the back of the train catching up with the point where Ford took the shot. Moreover she sees the front of the train moving away from the point of firing. Now the speed of light is constant in all directions for her. Therefore she’ll see the man at the back of the train get hit before the man at the front!

Remarkably all of these strange effects have been experimentally confirmed. Special relativity and spacetime really do describe our universe. But with our present understanding it seems that making genuine measurements would be nigh on impossible. We’ve only seen examples of things we can’t measure with certainty!

Thankfully there is a spacetime measurement all observers can agree on. This quantity is known as proper time. It’s very easy to calculate the proper time between events A and B. Take a clock and put it on a spaceship. Set your spaceship moving at a constant speed so that it goes through the spacetime points corresponding to A and B. The proper time between A and B is the time that elapses on the spaceship clock between A and B.

Everyone is forced to agree on the proper time between two events. After all it only depends on the motion of the spaceship. Observers moving at different speeds will all see the same time elapsed according to the spaceship clock. The speed that they are moving has absolutely no effect on the speed of your spaceship!

You might have spotted a potential flaw. What if somebody else sets off a spaceship which takes a different route from A to B? Wouldn’t it measure a different proper time? Indeed it would. But it turns out this situation is impossible. Remember that we had to set our spaceship off at a constant speed. This means it is going in a straight line through spacetime. It’s easy to see that there’s only one possible straight line route joining any two points. (Draw a picture)!

That’s it. You now understand special relativity. In just a few paragraphs we’ve made a huge conceptual leap. Forget about absolute space and time – it’s plain wrong. Instead use Einstein’s new magic measurement of proper time. Proper time doesn’t determine conventional time or length. Rather it tells us about distances in spacetime.

Einstein had cracked the biggest problem in physics. But he wasn’t done yet. Armed with his ideas about relativity and proper time, he turned to the Holy Grail. Could he go one better than Newton? It was time to explain gravity.

The Shape Of Space

“[…] the great questions about gravitation. Does it require time? […] Has it any reference to electricity? Or does it stand on the very foundation of matter – mass or inertia?”

James Clerk Maxwell

Physicists are always happy when they are going at constant speed. Thanks to Galileo and Einstein they don’t need to worry exactly how fast they are going. They can just observe the world and be sure that their observations are true interpretations of physics.

We can all empathise with this. It’s much more pleasant going on a calm cruise across the Aegean than a rocky boat crossing the English Channel. This is because the choppy waters cause the boat to accelerate from side to side. We’re no longer travelling at a constant speed, so the laws of physics appear unusual. This can have unpleasant consequences if we don’t find our sea legs.

Given this overwhelming evidence it seems madness to alter Einstein and Galileo’s relativity. But this is exactly what Einstein did. He postulated a general principle of relativity – the laws of physics are the same in any frame of reference. That is, no matter what your speed or acceleration.

To explain this brazen statement, we’ll take an example. Imagine you go to Disneyland and queue for the Freefall Ride. In this terrifying experience you are hoisted 60 metres in the air and then dropped. As you plunge towards the ground you notice that you can’t feel your weight. For a few moments you are completely weightless! In other words there is no gravity in your accelerating frame.

Let’s try a similar thought experiment. Suppose you wake up and find yourself in a bare room with no windows or doors. You stick to the floor as you usually would on Earth. You might be forgiven for thinking that you were under the influence of terrestrial gravity. In fact you could equally be trapped in a spaceship, accelerating at precisely the correct rate to emulate the force of gravity. Just think back to Newton’s Second Law and this is obviously true.

Hopefully you’ve spotted a pattern. Acceleration and gravity are bound together. In fact there seems to be no way of unpicking the knots. Immediately Einstein’s general relativity becomes more credible. An accelerating frame is precisely a constant speed frame with gravity around it.

So far so good. But we haven’t said anything about gravity that couldn’t be said about other forces. If we carry on thinking of gravity as a conventional interaction the argument becomes quite circular. We need a description of gravity that is independent of frames.

Here special relativity comes to our rescue. Remember that proper time gives us a universally agreed distance between two points. It does so by finding the length of a straight line in spacetime from one event to the other. This property characterises spacetime as flat.

To see this imagine you are a millipede, living on a piece of string. You can only move forwards or backwards. In other words you have one dimension of space. You also move through time. Therefore you exist in a two dimensional spacetime.

One day you decide to make a map of all spacetime. You travel to every point in spacetime and write down the proper time it took to get there. Naturally you are very efficient, and always travel by the shortest possible route. When you return you try to make a scale drawing of all your findings on a flat sheet of paper.

Suppose your spacetime is exactly as we’d expect from special relativity. Then you have no problem making your map. Indeed you always travelled by the shortest route to each point, which is a straight line in the spacetime of special relativity. On your flat sheet of paper, the shortest distance between two points is also a straight line. So your map will definitely work out.

You can now see why special relativistic spacetime is flat. Just for fun, let’s assume that you had trouble making your map. Just as you’ve drawn a few points on the map you realise there’s a point which doesn’t fit. Thinking you must be mistaken you try again. The same thing happens. You go out for another trek round spacetime. Exhausted you return with the same results. How immensely puzzling!

Eventually you start to realise that something fundamental is wrong. What if spacetime isn’t flat? You grab a nearby bowling ball and start drawing your map on its surface. Suddenly everything adds up. The distances you measured between each point work out perfectly. Your spacetime isn’t flat, it’s spherical.

Let’s look a bit closer at why this works. When you went out measuring your spacetime you took the shortest route to every point. On the surface of a sphere this is not a straight line. Rather it is part of a circle – a line of longitude or latitude. If you aren’t convinced, find a globe and measure it yourself on the Earth’s surface.

When you try to make a flat map, it doesn’t work. This is because the shortest distance between two points on the flat paper is a straight line. The two systems of measurement are just incompatible. We’re used to seeing this every time we look at a flat map of the world. The distances on it are all wrong because it has to be stretched and squished to fit the flat page.

Gallivanting millipedes aside, what has this got to do with reality? If we truly live in the universe of special relativity then we don’t need to worry about such complications. Everything is always flat! But Einstein realised that these curiosities were exactly the key to the treasure chest of gravity.

In his landmark paper of 1915, Einstein told us to forget gravity as a force. Instead, he claimed, gravity modifies the proper time between events. In doing so, it changes the geometry of spacetime. The flat, featureless landscape of special relativity was instantly replaced by cliffs and ravines.

More precisely Einstein came up with a series of equations which described how the presence of mass changes the calculation of proper time. Large masses can warp spacetime, causing planets to orbit their suns and stars to form galaxies. Nothing is immune to the change in geometry. Even light bends around massive stars. This has been verified countless times during solar eclipses.

Einstein had devised a frame independent theory of gravity which elegantly explained all gravitational phenomena. His general principle of relativity was more than vindicated; it became the title of a momentous new perspective on the cosmos. The quest to decipher Newton’s gravity was complete.

In the past century, despite huge quantities of research, we have made little progress beyond Einstein’s insights. The experimental evidence for general relativity is tumultuous. But yet it resists all efforts to express it as a quantum theory. We stand at an impasse, looking desperately for a way across.

# Affine Parameters and Euler-Lagrange Equations

Earlier today I was struggling to see why I couldn’t derive the general geodesic equation (*)

$\frac{\textrm{d}}{\textrm{d}\lambda}(g_{ab}\dot{x}^b)-\frac{1}{2}g_{bc,a}\dot{x}^b\dot{x}^c=f(\lambda)g_{ab}\dot{x}^b$

where $f$ a general nonzero function. I had been trying to do this by varying the action

$S=\int L \textrm{d}\lambda$ with Lagrangian $L=\frac{\textrm{d}s}{\textrm{d}\lambda}=\sqrt{g_{ab}\dot{x}^a\dot{x}^b}$.

The standard technique is use as a Lagrangian $L'=L^2$ instead, and claim that this produces the same results as we would have got with $L$. I’d always accepted this as gospel, but a simple calculation shows that we need an additional assumption for this to be true.

Indeed the Euler-Lagrange equations for $L'$ give

$\frac{\textrm{d}}{\textrm{d}\lambda}(2L\frac{\partial L}{\partial\dot{x}^a})=2L\frac{\partial L}{\partial x^a}$

To regain the Euler-Lagrange equations for $L$ we want to cancel our $2L$ factors. Clearly a sufficient condition for this is that $L$ is constant. But by the definition of $L$ this means that $\frac{\textrm{d}s}{\textrm{d}\lambda}$ is constant, which precisely says that $\lambda$ is an affine parameter.

Hence, by the use of this method we lose the generality needed to obtain equation (*).

Nevertheless it is easy to derive (*) from the affine geodesic equation (**)

$\frac{\textrm{d}}{\textrm{d}\lambda}(g_{ab}\dot{x}^b)-\frac{1}{2}g_{bc,a}\dot{x}^b\dot{x}^c=0$

Indeed let $\mu$ be a general parameter and write $\lambda = \lambda(\mu)$. Then rewriting the geodesic equation (**) in terms of $\mu$ using the chain rule yields (*) with $f(\mu)=\frac{\lambda ''}{\lambda '}$ (try it).

Alternatively you can reach a manifestly reparameterisation invariant version of (*) by directly varying our original action $S$. It’s not pretty though!

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8
Q:

Adam and Smith are working on a project. Adam takes 6 hrs to type 36 pages on a computer, while Smith takes 5 hrs to type 40 pages. How much time will they take, working together on two different computers to type a project of 120 pages?

 A) 8 hrs 45 min B) 8 hrs 42 min C) 8 hrs D) 8 hrs 34 min

Answer:   D) 8 hrs 34 min

Explanation:

Number of pages typed by Adam in 1 hour = $\inline \fn_jvn \small \frac{36}{6}$ = 6
Number of pages typed by Smith in 1 hour = $\inline \fn_jvn \small \frac{40}{5}$ = 8
Number of pages typed by both in 1 hour = (6 + 8) = 14
Time taken by both to type 110 pages = (120 * 1/14) = 8 $\inline \fn_jvn \small \frac{4}{7}$ = 8 hrs 34 min.

Q:

Four pipes A,B, C and D can fill a cistern  in 20,25, 40 and 50 hours respectively.The first pipe A was opened at 6:00 am, B at 8:00 am, C at 9:00 am and D at 10:00 am. when will the Cistern be full?

 A) 4:18 pm B) 3:09 pm C) 12:15 pm D) 11:09 am

Explanation:

Efficiency of P= 5%

Efficiency of Q= 4%

Efficiency of R= 2.5%

Efficiency of S= 2%

$\inline&space;\left.\begin{matrix}&space;Till&space;\;&space;10\:&space;am&space;\;&space;pipe\;&space;P&space;\;&space;filled\;&space;20\;&space;percent\\&space;Till&space;\;&space;10\:&space;am&space;\;&space;pipe\;&space;Q&space;\;&space;filled\;&space;8\;&space;percent\\&space;Till&space;\;&space;10\:&space;am&space;\;&space;pipe\;&space;R&space;\;&space;filled\;&space;2.5\;&space;percent&space;\end{matrix}\right\}30.5$ %

Thus, at 10 am pipe P,Q and R filled 30.5% of the cistern.

Now, the time taken by P,Q,R and S together to fill the remaining capacity of the cistern

=$\inline&space;\frac{69.5}{13.5}$ = $\inline&space;\frac{139}{27}$ =5 Hours and 9 minutes(approx)

Therefore, total time =4 hrs + 5hrs 9 mins

= 9 hrs and 9 mins

It means cistern will be filled up at 3:09 pm

2 1382
Q:

A group of workers was put on a job. From the second day onwards, one worker was withdrawn each day. The job was finished when the last worker was withdrawn. Had no worker been withdrawn at any stage, the group would have finished the job in 55% of the time. How many workers were there in the group?

 A) 50 B) 40 C) 45 D) 10

Explanation:

It can be solved easily through option.

$\inline&space;\left&space;(&space;10+9+8+....+1&space;\right&space;)=10\left&space;(&space;10\times&space;\frac{55}{100}&space;\right&space;)$

55 = 55     Hence correct

Alternatively:

$\inline&space;\frac{n(n+1)}{2}=n\times&space;\frac{55n}{100}$

=> n= 10

In Both cases total work is 55man-days.

10 2632
Q:

Mr. stenley employed a certain number of typist for his project. 8 days later 20% of the typist left the job and it was found that it took as much time to complete the rest work from then as the entire work needed with all the employed typists. The average speed of a typist is 20 pages/hour. Minimum how many typist could be employed?

 A) 10 B) 5 C) 15 D) 4

Explanation:

Since 20% $\left&space;(&space;i.e,\frac{1}{5}&space;\right&space;)$ typists left the job. So, there can be any value which is multiple of 5 i.e, whose 20% is always an integer. Hence, 5 is the least possible value.

2 679
Q:

A Contractor employed a certain number of workers  to finish constructing a road in a certain scheduled time. Sometime later, when a part of work had been completed, he realised that the work would get delayed by three-fourth of the  scheduled time, so he at once doubled the no of workers and thus he managed to finish the road on the scheduled time. How much work he had been completed, before increasing the number of workers?

 A) 10 % B) 14 2/7 % C) 20 % D) Can't be determined

Explanation:

Let he initially employed x workers which works for D days and he estimated 100 days for the whole work and then he doubled the worker for (100-D) days.

D * x +(100- D) * 2x= 175x

=>  D= 25 days

Now , the work done in 25 days = 25x

Total work = 175x

therefore, workdone before increasing the no of workers = $\frac{25x}{175x}\times&space;100=14\frac{2}{7}$ %

17 2683
Q:

A single reservoir supplies the petrol to the whole city, while the reservoir is fed by a single pipeline filling the reservoir with the stream of uniform volume. When the reservoir is full and if 40,000 liters of petrol is used daily, the suply fails in 90 days.If 32,000 liters of petrol is used daily, it fails in 60 days. How much petrol can be used daily with out the supply ever failing?

 A) 64000 liters B) 56000 liters C) 78000 liters D) 60000 liters

Explanation:

Let x liter be the per day filling and v litr be the capacity of the reservoir, then

90x + v = 40000 * 90     -----(1)

60x + v= 32000 * 60     ------(2)

solving eq.(1) and (2) , we get

x = 56000

Hence , 56000 liters per day can be used without the failure of supply.


text:
# When an open-drain output asserts low, where does the line capacitance discharge?

I was looking at the auto-retry functionality of the TI part TPS2553 (datasheet http://www.ti.com/lit/ds/symlink/tps2553.pdf) and came across something I didn't understand and thought it a good opportunity to ask a general question.

I am going to refer to the picture below to ask this question using it as an example but really I am unsure about this in any situation where you have a line pulled up with a resistor and connected to an open drain output with some similar capacitance.

Referring to the picture below, when the part senses an overcurrent condition, the FAULT pin pulls low. My question is, when this happens where does Cretry discharge to? The only place I could see it discharging is directly into the FAULT pin. However there is no resistance between Cretry and that pin and the FAULT pin specifes a 10mA max current from the datasheet. So wont there be a big current spike when it discharges since there is nothing to limit the current, and this would damage the pin as it's outside the recommended sink current for the pin (datasheet actually specifies absolute max of 25mA sink current for FAULT)? Am I wrong in my understanding about how/where the cap discharges, the fact that the current spike exists assuming I am right about the previous point, or am I misunderstanding the significance of said discharging current spike if I am right about previous 2 points?

Again I am not just asking about this specific case but using this as an example to ask this question. I wonder too about I2C lines. When the I2C driver pulls the line low, where does the charge that was on that line discharge to?

• currents travel in loops from an internal switch (Fault_) to the return path of ground, dissipating in the switch. – Tony Stewart EE75 Sep 17 '17 at 1:51
• The switch ( drain when closed to source =gnd) has low enough resistance to discharge the cap quickly. ( usually around 50 ohms for 74HCxx) – Tony Stewart EE75 Sep 17 '17 at 2:07
• But spec actual says VOL Output low voltage, FAULT I/ FAULT = 1 mA 180 mV thus 180 ohms (Max) – Tony Stewart EE75 Sep 17 '17 at 2:10
• You are right. The cap discharges directly into the FAULT pin. In this case, since the vendor is telling you to do it, it is probably OK. In other cases, you normally would not put 0.1uF on an open collector output. If you did, you could also insert a small resistor (10 to 100 Ohms) between cap and output to limit the current. To be honest, there is not much energy in a 0.1uF cap charged to 3 or 5V or whatever. All the energy will be dissipated as heat by the open collector output. But it may not be enough energy to destroy the internal structure. – mkeith Sep 17 '17 at 2:12
• Yes but not ONLY limiting the current. The other point is controlling where the dissipation occurs. The cap internal series R, for a ceramic cap, is negligible. So the energy stored in the cap will be dissipated between the external series R and the internal structure of the open collector output. The bigger the external resistor, the larger the percentage of energy will be dissipated in the external resistor. Hope that makes sense. – mkeith Sep 17 '17 at 2:22

The charge on $C_{RETRY}$ goes to ground through the $\overline{FAULT}$ pin.
The absolute max current specification on that line specifies a continuous current. This puts a lower limit on the size of $R_{FAULT}$.
The energy on $C_{RETRY}$ ends up being dissipated as heat in the pin driver transistor. This is a small and limited amount of energy which has to be distributed in the thermal mass of the transistor, unlike the energy due to a continuous current which is unlimited, so has to be dissipated.


text:
# Exercises - Hypothesis Tests for Proportions (One Sample)

1. In a presidential election, 308 out of 611 voters surveyed said that they voted for the candidate who won. Use a 0.01 significance level to test the claim that among all voters, the percentage who believe that they voted for the winning candidate is equal to 43%, which is the actual percentage of votes for the winning candidate. What does the result suggest about voter perceptions?

1. Identify the claim
2. State the null and alternative hypotheses
3. Check any relevant assumptions
4. Find the value of the relevant test statistic
5. Determine if this is a one-tailed or two-tailed test
6. Identify the significance level
7. Find any relevant critical value(s)
8. Make a sketch of the relevant distribution, indicate the location of the rejection region, the test statistic, and any critical values for the test
9. Find the p-value for the test
10. Conclude whether or not to reject the null hypothesis
11. Make an appropriate inference

(a) The claim is that among all voters, the percentage who believe that they voted for the winning candidate is 43%.

(b) The null hypothesis: $H_0 : p = 0.43$
The alternative hypothesis: $H_1 : p \neq 0.43$

(c) Assumptions: $np = (611)(0.43) = 262.73 \ge 5$, $nq = (611)(0.57) = 348.27 \ge 5$, so the requirements/assumptions of the test are met.

(d) Test Statistic: $$z = \frac{\frac{308}{611} - 0.43}{\sqrt{\frac{(.43)(.57)}{611}}} \doteq 3.699$$

This is a two-tailed test (as the alternative hypothesis involves "$\neq$")

Significance level: $\alpha = 0.01$

Critical values: $\pm2.5758$ (on a TI-calculator, find $\pm$invNorm(0.005))

$p$-value $\doteq 0.000216$ (on a TI-calculator, find 2*normalcdf(3.699,999999999))

Reject the null as $p$-value $\lt \alpha$ (or as the test statistic falls in the rejection region)

There is significant evidence that the percentage of voters that believed they voted for the winning candidate was not 43%.

2. The Gallup Poll reported that 45% of individuals felt they were worse off than 1 year ago. A politician feels that this is too high for her district, so she commissions her own survey and finds that, out of 150 randomly selected citizens, 58 feel they are worse off today than 1 year ago. At significance level 0.05, is the politician correct about her district?

$n = 150$

$H_0 : p = 0.45$
$H_1 : p \lt 0.45$

Check assumptions:
$np = (150)(0.45) = 67.5 \ge 5$ and
$nq = (150)(0.55) = 82.5 \ge 5$,
so requirements are met.

$\widehat{p} = 58\,/\,150$

Test statistic: $$z = \frac{\frac{58}{150} - 0.45}{\sqrt{\frac{\left(0.45\right)\left(0.55\right)}{150}}} \doteq -1.5591$$

$p$-value = $0.0595 \gt \alpha = 0.05$

Conclusion: fail to reject the null hypothesis

Inference: there is no statistically significant evidence the politician is correct

3. Adults were randomly selected for a Newsweek poll. They were asked if they "favor or oppose using federal tax dollars to fund medical research using stem cells obtained from human embryos." Of those polled, 481 were in favor, 401 were opposed, and 120 were unsure. A politician claims that people don't really understand the stem cell issue and their responses to such questions are random responses equivalent to a coin toss. Exclude the 120 subjects who said that they were unsure, and use a 0.01 significance level to test the claim that the proportion of subjects who respond in favor is equal to 0.5. What does the result suggest about the politician's claim?

481 in favor, 401 opposed $\rightarrow n = 882$

$\widehat{p} = 481\,/\,882$

$H_0 : p = 0.5$
$H_1 : p \neq 0.55$

Checking assumptions: $np = nq = (882)(0.5) = 441 \ge 5$, so the requirements are met.

Test statistic:

$$z = \frac{\frac{481}{882} - 0.5}{\sqrt{\frac{(0.5)(0.5)}{882}}} \doteq 2.6937$$

$p$-value $= 0.00706 \lt \alpha = 0.01$, so reject the null hypothesis.

There is statistically significant evidence the politician is wrong.


text:
# source:docs/HPCA2011/00-abstract.tex@1320

Last change on this file since 1320 was 1302, checked in by lindanl, 8 years ago

Create a directory for HPCA

File size: 2.0 KB
Line
1XML is a set of rules for the encoding of documents in machine-readable form.
2The simplicity and generality of the rules make it widely used in web services and database
3systems.  Traditional XML parsers are built around a
4byte-at-a-time processing model where each character token
5of an XML document is examined in sequence.  Unfortunately, the byte-at-a-time
6sequential model is a performance barrier in more demanding applications,
7is energy-inefficient, and makes poor use of the wide SIMD registers
8and other parallelism features of modern processors.
9
10This paper assesses the energy and performance of a new approach
11to XML parsing, based on parallel bit stream technology, and as implemented on successive
12software generations of the Parabix XML parser.
13In Parabix, we first convert character streams into sets of parallel
14bit streams. We then exploit the SIMD operations prevalent on commodity-level hardware for performance.
15The first generation Parabix1 parser exploits the processor built-in $bitscan$ instructions
16over these streams to make multibyte moves but follows an otherwise sequential
17approach.  The second generation Parabix2 technology adds further
18parallelism by replacing much of the sequential
19bit scanning with a parallel scanning approach based on bit stream
20addition.  We evaluate Parabix1 and Parabix2
21against two widely used XML parsers, James Clark's Expat and Apache's Xerces, and
22across three generations of x86 machines, including the new Intel
23\SB{}.  We show that Parabix2's speedup is 2$\times$--7$\times$
24over Expat and Xerces.  In stark contrast to the energy expenditures necessary
25to realize performance gains through multicore parallelism, we also show
26that our Parabix parsers deliver energy savings in direct proportion
27to the gains in performance.  In addition, we assess the scalability advantages
28of SIMD processor improvements across Intel processor generations,
29culminating with an evaluation of the 256-bit AVX technology in
30\SB{} versus the now legacy 128-bit SSE technology.
31
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text:
# A Testable Conspiracy: Simulating Baryonic Effects on Self-Interacting Dark Matter Halos [GA]

We investigate the response of self-interacting dark matter (SIDM) halos to the growth of galaxy potentials using idealized simulations, each run in tandem with standard collisionless Cold Dark Matter (CDM). We find a greater diversity in the SIDM halo profiles compared to the CDM halo profiles. If the stellar gravitational potential strongly dominates in the central parts of a galaxy, then SIDM halos can be as dense as CDM halos on observable scales. For extreme cases with highly compact disks core collapse can occur, leading to SIDM halos that are denser and cuspier than their CDM counterparts. If the stellar potential is not dominant, then SIDM halos retain constant density cores with densities far below CDM predictions. When a disk potential is present, the inner SIDM halo becomes \em{more flattened} in the disk plane than the CDM halo. These results are in excellent quantitative agreement with the predictions of Kaplinghat et al. (2014). We also simulated a galaxy cluster halo with a central stellar distribution similar to the brightest central galaxy of the cluster A2667. A SIDM halo simulated with cross section over mass $\sigma/m = 0.1\ \mathrm{cm^2 g^{-1}}$ provides a good match to the measured dark matter density profile of A2667, while an adiabatically-contracted CDM halo is denser and cuspier. The cored profile of the same halo simulated with $\sigma/m = 0.5\ \mathrm{cm^2 g^{-1}}$ is not dense enough to match A2667. Our findings are in agreement with previous results that $\sigma/m \gtrsim 0.1\ \mathrm{cm^2 g^{-1}}$ is disfavored for dark matter collision velocities in excess of about 1500 km/s. More generally, the predictive cross-talk between baryonic potentials and SIDM density distributions offers new directions for constraining SIDM cross sections in massive galaxies where baryons are dynamically important.

O. Elbert, J. Bullock, M. Kaplinghat, et. al.
Thu, 29 Sep 16
3/76


text:
Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > DETAIL:

### Paper:

TR21-007 | 14th January 2021 20:36

#### Almost Optimal Inapproximability of Multidimensional Packing Problems

TR21-007
Authors: Sai Sandeep
Publication: 18th January 2021 18:02
Keywords:

Abstract:

Multidimensional packing problems generalize the classical packing problems such as Bin Packing, Multiprocessor Scheduling by allowing the jobs to be $d$-dimensional vectors. While the approximability of the scalar problems is well understood, there has been a significant gap between the approximation algorithms and the hardness results for the multidimensional variants. In this paper, we close this gap by giving almost tight hardness results for these problems.
1. We show that Vector Bin Packing has no $\Omega( \log d)$ factor asymptotic approximation algorithm when $d$ is a large constant, assuming $P \neq NP$. This matches the $\ln d + O(1)$ factor approximation algorithms (Chekuri, Khanna SICOMP 2004, Bansal, Caprara, Sviridenko SICOMP 2009, Bansal, Eliáš, Khan SODA 2016) upto constants.
2. We show that Vector Scheduling has no polynomial time algorithm with an approximation ratio of $\Omega\left( (\log d)^{1-\epsilon}\right)$ when $d$ is part of the input, assuming NP has no quasipolynomial time algorithms. This almost matches the $O\left( \frac{\log d}{\log \log d}\right)$ factor algorithms(Harris, Srinivasan JACM 2019, Im, Kell, Kulkarni, Panigrahi SICOMP 2019). We also show that the problem is NP-hard to approximate within $(\log \log d)^{\omega(1)}$.
3. We show that Vector Bin Covering is NP-hard to approximate within $\Omega\left( \frac{\log d}{\log \log d}\right)$ when $d$ is part of the input, almost matching the $O(\log d)$ factor algorithm (Alon et al., Algorithmica 1998).

Previously, no hardness results that grow with $d$ were known for Vector Scheduling and Vector Bin Covering when $d$ is part of the input and for Vector Bin Packing when $d$ is a fixed constant.

ISSN 1433-8092 | Imprint


text:
• Browse all
Comprehensive measurements of $t$-channel single top-quark production cross sections at $\sqrt{s} = 7$ TeV with the ATLAS detector

The collaboration
Phys.Rev.D 90 (2014) 112006, 2014.

Abstract (data abstract)
- CERN-LHC. This article presents measurements of the $t$-channel single top-quark $(t)$ and top-antiquark $(\bar t)$ total production cross sections $\sigma(tq)$ and $\sigma(\bar tq)$, their ratio $R_{t}=\sigma(tq)/\sigma(\bar tq)$, and a measurement of the inclusive production cross section $\sigma(tq+\bar tq)$ in proton-proton collisions at $\sqrt{s}~=~7~\mathrm{TeV}$ at the LHC. Differential cross sections for the $tq$ and $\bar tq$ processes are measured as a function of the transverse momentum and the absolute value of the rapidity of $t$ and $\bar t$, respectively. The analyzed data set was recorded with the ATLAS detector and corresponds to an integrated luminosity of $4.59~\mathrm{fb}^{-1}$. Selected events contain one charged lepton, large missing transverse momentum, and two or three jets. The cross sections are measured by performing a binned maximum-likelihood fit to the output distributions of neural networks. The resulting measurements are $\sigma(tq)=46 \pm 1(\mathrm{stat})\pm 6(\mathrm{syst})~\mathrm{pb}$, $\sigma(\bar tq)=23 \pm 1(\mathrm{stat})\pm 3(\mathrm{syst})~\mathrm{pb}$, $R_{t}=2.04 \pm 0.13(\mathrm{stat})\pm 0.12(\mathrm{syst})$, and $\sigma(tq+\bar tq)=68 \pm 2(\mathrm{stat})\pm 8(\mathrm{syst})~\mathrm{pb}$, consistent with the Standard Model expectation. The uncertainty on the measured cross sections is dominated by systematic uncertainties, while the uncertainty on $R_{t}$ is mainly statistical. Using the ratio of $\sigma(tq+\bar tq)$ to its theoretical prediction, and assuming that the top-quark-related CKM matrix elements obey the relation $|V_{tb}|\gg |V_{ts}|$, $|V_{td}|$, we determine $|V_{tb}|=1.02 \pm 0.07$.

• Table 1

Table I on page 112006-7 of the paper

10.17182/hepdata.64385.v2/t1

Predicted and observed events yields for the 2-jet and 3-jet channels considered in this measurement. The multijet background is estimated...

• Table 2

Table III on page 112006-18 of the paper

10.17182/hepdata.64385.v2/t2

Detailed list of the contribution of each source of uncertainty to the total uncertainty on the measured values of $\sigma(tq)$,...

• Table 3

Page 112006-19 of the paper

10.17182/hepdata.64385.v2/t3

The cross sections for top-quark and top-antiquark production in the t-channel, together with the cross-section ratio.

• Table 4

Table IV on page 112006-20 of the paper

10.17182/hepdata.64385.v2/t4

Parametrization factors for the $m_{t}$ dependence [see Eq. (4) in the paper] of $\sigma(tq)$, $\sigma(\bar tq)$, and $\sigma(t q+\bar t... • Table 5 Page 112006-20 of the paper 10.17182/hepdata.64385.v2/t5 The value of$|V_{tb}|^{2}$is extracted by dividing the measured value of$\sigma(t q+\bar t q)$by the prediction of... • Table 6 Table V on page 112006-21 of the paper 10.17182/hepdata.64385.v2/t6 Event yields for the 2-jet-$\ell^{+}$and 2-jet-$\ell^{-}$HPR channels. The expectation for the signal and background yields correspond to the... • Table 7 FIG. 14(a) on page 112006-22 of the paper 10.17182/hepdata.64385.v2/t7 Migration matrix relating the parton level$p_{\mathrm{T}}(t)$shown on the$y$axis and the reconstruction level$p_{\mathrm{T}}(\ell \nu b)$shown... • Table 8 FIG. 14(b) on page 112006-22 of the paper 10.17182/hepdata.64385.v2/t8 Migration matrix relating the parton level$p_{\mathrm{T}}(\bar t)$shown on the$y$axis and the reconstruction level$p_{\mathrm{T}}(\ell \nu b)$... • Table 9 FIG. 14(c) on page 112006-22 of the paper 10.17182/hepdata.64385.v2/t9 Migration matrix relating the parton level$|y(t)|$shown on the$y$axis and the reconstruction level$|y(\ell \nu b)|$shown... • Table 10 FIG. 14(d) on page 112006-22 of the paper 10.17182/hepdata.64385.v2/t10 Migration matrix relating the parton level$|y(\bar t)|$shown on the$y$axis and the reconstruction level$|y(\ell \nu b)|$... • Table 11 Table VI on page 112006-23 of the paper 10.17182/hepdata.64385.v2/t11 Differential t-channel top-quark production cross section as a function of$p_{\mathrm{T}}(t)$with the uncertainties for each bin given in percent. • Table 12 Table VI on page 112006-23 of the paper 10.17182/hepdata.64385.v2/t12 Differential t-channel top-quark production cross section as a function of$p_{\mathrm{T}}(\bar t)$with the uncertainties for each bin given in... • Table 13 Table VI on page 112006-23 of the paper 10.17182/hepdata.64385.v2/t13 Differential t-channel top-quark production cross section as a function of$|y(t)|$with the uncertainties for each bin given in percent. • Table 14 Table VI on page 112006-23 of the paper 10.17182/hepdata.64385.v2/t14 Differential t-channel top-quark production cross section as a function of$|y(\bar t)|$with the uncertainties for each bin given in... • Table 15 Table VII on page 112006-23 of the paper 10.17182/hepdata.64385.v2/t15 Normalized differential t-channel top-quark production cross section as a function of$p_{\mathrm{T}}(t)$with the uncertainties for each bin given in... • Table 16 Table VII on page 112006-23 of the paper 10.17182/hepdata.64385.v2/t16 Normalized differential t-channel top-quark production cross section as a function of$p_{\mathrm{T}}(\bar t)$with the uncertainties for each bin given... • Table 17 Table VII on page 112006-23 of the paper 10.17182/hepdata.64385.v2/t17 Normalized differential t-channel top-quark production cross section as a function of$|y(t)|$with the uncertainties for each bin given in... • Table 18 Table VII on page 112006-23 of the paper 10.17182/hepdata.64385.v2/t18 Normalized differential t-channel top-quark production cross section as a function of$|y(\bar t)|$with the uncertainties for each bin given... • Table 19 Table VIII on page 112006-25 of the paper 10.17182/hepdata.64385.v2/t19 Comparison between the measured differential cross sections and the predictions from the NLO calculation using the MSTW2008 PDF set. For... • Table 20 Table IX on page 112006-26 of the paper 10.17182/hepdata.64385.v2/t20 Detailed list of the contribution of each source of uncertainty to the total relative uncertainty on the measured$\dfrac{\mathrm{d}\sigma}{\mathrm{d}p_{\mathrm{T}}(t)}$distribution... • Table 21 Table X on page 112006-26 of the paper 10.17182/hepdata.64385.v2/t21 Detailed list of the contribution of each source of uncertainty to the total relative uncertainty on the measured$\dfrac{\mathrm{d}\sigma}{\mathrm{d}p_{\mathrm{T}}(\bar t)}$... • Table 22 Table XI on page 112006-27 of the paper 10.17182/hepdata.64385.v2/t22 Detailed list of the contribution of each source of uncertainty to the total relative uncertainty on the measured$\dfrac{\mathrm{d}\sigma}{\mathrm{d}|y(t)|}$distribution... • Table 23 Table XII on page 112006-27 of the paper 10.17182/hepdata.64385.v2/t23 Detailed list of the contribution of each source of uncertainty to the total relative uncertainty on the measured$\dfrac{\mathrm{d}\sigma}{\mathrm{d}|y(\bar t)|}$... • Table 24 Table XIII on page 112006-28 of the paper 10.17182/hepdata.64385.v2/t24 Detailed list of the contribution of each source of uncertainty to the total relative uncertainty on the measured$\dfrac{1}{\sigma}\dfrac{\mathrm{d}\sigma}{\mathrm{d}p_{\mathrm{T}}(t)}$distribution... • Table 25 Table XIV on page 112006-28 of the paper 10.17182/hepdata.64385.v2/t25 Detailed list of the contribution of each source of uncertainty to the total relative uncertainty on the measured$\dfrac{1}{\sigma}\dfrac{\mathrm{d}\sigma}{\mathrm{d}p_{\mathrm{T}}(\bar t)}$... • Table 26 Table XV on page 112006-29 of the paper 10.17182/hepdata.64385.v2/t26 Detailed list of the contribution of each source of uncertainty to the total relative uncertainty on the measured$\dfrac{1}{\sigma}\dfrac{\mathrm{d}\sigma}{\mathrm{d}|y(t)|}$distribution... • Table 27 Table XVI on page 112006-29 of the paper 10.17182/hepdata.64385.v2/t27 Detailed list of the contribution of each source of uncertainty to the total relative uncertainty on the measured$\dfrac{1}{\sigma}\dfrac{\mathrm{d}\sigma}{\mathrm{d}|y(\bar t)|}$... • Table 28 FIG. 17(a) on page 112006-30 of the paper 10.17182/hepdata.64385.v2/t28 Statistical correlation matrices for the differential cross section as a function of$p_{\mathrm{T}}(t)$. • Table 29 FIG. 17(b) on page 112006-30 of the paper 10.17182/hepdata.64385.v2/t29 Statistical correlation matrices for the differential cross section as a function of$p_{\mathrm{T}}(\bar t)$. • Table 30 FIG. 17(c) on page 112006-30 of the paper 10.17182/hepdata.64385.v2/t30 Statistical correlation matrices for the differential cross section as a function of$|y(t)|$. • Table 31 FIG. 17(d) on page 112006-30 of the paper 10.17182/hepdata.64385.v2/t31 Statistical correlation matrices for the differential cross section as a function of$|y(\bar t)|$. • Table 32 FIG. 18(a) on page 112006-31 of the paper 10.17182/hepdata.64385.v2/t32 Statistical correlation matrices for the normalized differential cross section as a function of$p_{\mathrm{T}}(t)$. • Table 33 FIG. 18(b) on page 112006-31 of the paper 10.17182/hepdata.64385.v2/t33 Statistical correlation matrices for the normalized differential cross section as a function of$p_{\mathrm{T}}(\bar t)$. • Table 34 FIG. 18(c) on page 112006-31 of the paper 10.17182/hepdata.64385.v2/t34 Statistical correlation matrices for the normalized differential cross section as a function of$|y(t)|$. • Table 35 FIG. 18(d) on page 112006-31 of the paper 10.17182/hepdata.64385.v2/t35 Statistical correlation matrices for the normalized differential cross section as a function of$|y(\bar t)|\$.

Version 2 modifications: Detailed tables of uncertainties and statistical correlation matrices have been added.