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# If negative integer a is multiplied by b and the result is greater tha

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29 Jun 2018, 02:12
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If negative integer a is multiplied by b and the result is greater than 0 but less than |a|, then which of the following must be true of b?

A. $$b > 1$$

B. $$0 < b < 1$$

C. $$-1 < b < 0$$

D. $$b < a$$

E. $$|b| < a$$

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Posts: 6802
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29 Jun 2018, 03:14
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Bunuel wrote:
If negative integer a is multiplied by b and the result is greater than 0 but less than |a|, then which of the following must be true of b?

A. $$b > 1$$

B. $$0 < b < 1$$

C. $$-1 < b < 0$$

D. $$b < a$$

E. $$|b| < a$$

If a*b>0, where a<0, so b is also <0

If ab<|a|, square both sides
$$a^2b^2<a^2.......a^2b^2-a^2<0....a^2(b^2-1)<0$$
Since a^2>0, so $$b^2-1<0........b^2<1$$
Range of b is -1<b<1....
But b is negative, so -1<b<0

Other way
Substitute a as -2
So -2b<|-2|.............-2b<2..........divide both sides by -2...
$$\frac{-2b}{-2}>\frac{2}{-2}...............b>-1$$
Since b<0....-1<b<0

C
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: If negative integer a is multiplied by b and the result is greater tha  [#permalink]

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29 Jun 2018, 08:05
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Bunuel wrote:
If negative integer a is multiplied by b and the result is greater than 0 but less than |a|, then which of the following must be true of b?

A. $$b > 1$$

B. $$0 < b < 1$$

C. $$-1 < b < 0$$

D. $$b < a$$

E. $$|b| < a$$

I think conceptual understanding is a must to work out this question. Few points can be made based on the information given:

1. a is negative . b must also be negative.
2. Absolute value of a is greater than the product of a and b.
3. we know that negative times negative yields positive result.

Let assume the value of a and b.

a = -4 and b = -2

so, ab = 8 but it is stated in the question that |a| > ab . So, b can't be a integer rather it's a fraction.

a = -4
b = -$$\frac{1}{2}$$

a*b = (-4)*(-1/2)
= 2
and |a| > 2.

Thus , b is a negative fraction.

Option C is the best answer.
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Re: If negative integer a is multiplied by b and the result is greater tha  [#permalink]

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29 Jun 2018, 09:05
Keep in mind - a -ve and |a|>ab>0
implies -- b has to be -ve
For a*b to be less than |a| - b has to be -ve fraction < 1
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Posts: 1979
If negative integer a is multiplied by b and the result is greater tha  [#permalink]

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29 Jun 2018, 10:04
1
Bunuel wrote:
If negative integer a is multiplied by b and the result is greater than 0 but less than |a|, then which of the following must be true of b?

A. $$b > 1$$

B. $$0 < b < 1$$

C. $$-1 < b < 0$$

D. $$b < a$$

E. $$|b| < a$$

Use a negative integer for $$a$$
in order to simplify the arithmetic

Let $$a=-3$$
$$|-3|= 3$$

(1) the result of $$a*b$$ is greater than 0
$$a*b>0$$
$$(-3*b)>0$$
= $$0<(-3*b)$$

(2) the result of $$a*b$$ is less than $$|a|$$
$$a*b<|a|$$
$$(-3*b)<3$$

(3) Combine:
$$0<(-3*b)<3$$
Divide all terms by -3, flip the signs
$$0>b>-1$$
= $$-1<b<0$$

_________________

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that within me there lay an invincible summer.

If negative integer a is multiplied by b and the result is greater tha &nbs [#permalink] 29 Jun 2018, 10:04
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# American Institute of Mathematical Sciences

May  2008, 2(2): 223-247. doi: 10.3934/amc.2008.2.223

## A survey of perfect codes

 1 Department of Mathematics, KTH, Stockholm, Sweden S-100 44

Received  December 2007 Revised  April 2008 Published  April 2008

The first examples of perfect $e$-error correcting $q$-ary codes were given in the 1940's by Hamming and Golay. In 1973 Tietäväinen, and independently Zinoviev and Leontiev, proved that if q is a power of a prime number then there are no unknown multiple error correcting perfect $q$-ary codes. The case of single error correcting perfect codes is quite different. The number of different such codes is very large and the classification, enumeration and description of all perfect 1-error correcting codes is still an open problem.
This survey paper is devoted to the rather many recent results, that have appeared during the last ten years, on perfect 1-error correcting binary codes. The following topics are considered: Constructions, connections with tilings of groups and with Steiner Triple Systems, enumeration, classification by rank and kernel dimension and by linear equivalence, reconstructions, isometric properties and the automorphism group of perfect codes.
Citation: Olof Heden. A survey of perfect codes. Advances in Mathematics of Communications, 2008, 2 (2) : 223-247. doi: 10.3934/amc.2008.2.223
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2019 Impact Factor: 0.734

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text:
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text:
# How to Calculate Average Increase

••• wutwhanfoto/iStock/GettyImages
Print

Average increase refers to the average rate of growth that a variable experiences within a given period. You can apply the math and theory behind average increase to many real-life situations, such as speed, finances or population growth. Calculating average growth rate involves basic algebra and is possible as long as there are finite start and end values.

## Step 1: Establish the Initial and Final Values

Locate the starting value and final value for a given time period in your situation. Label the starting value as V1 (first value) and label the final value as V2 (second value).

## Step 2: Determine the Total Change

Subtract V1 from V2. The equation thus far is:

V_2-V_1

## Step 3: Determine the Percentage Change

Divide the value you determined by V1 to get the total percentage change. The equation now looks like this:

\frac{V_2-V_1}{V_1}

## Step 4: Determine Percentage Change as a Function of Time

Divide the value you calculated by the total number of units of time change. This can be in any time unit, such as years, hours or minutes. The equation is now:

\frac{V_2-V_1}{V_1} × \frac{1}{\text{time}}

## Step 5:  Determine the Annual Percent Change

Multipy the final value you calculated to determine the annual increase in percent. The final equation becomes then:

\frac{V_2-V_1}{V_1} × \frac{1}{\text{time}}×100

An example of this calculation would be an investment that increases from $50 to$100 in 10 years. V1 is $50. V2 is$100, and the time is 10 years.

\frac{100-50}{50} × \frac{1}{10 \text{ years}}×100 = 10 \% \text{ average increase per year.}


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# Digging for Buried Insight

The scientific method, as we usually learn it, starts with a hypothesis. The scientist begins with a guess, and asks a question with a clear answer: true, or false? That guess lets them design an experiment, observe the consequences, and improve our knowledge of the world.

But where did the scientist get the hypothesis in the first place? Often, through some form of exploratory research.

Exploratory research is research done, not to answer a precise question, but to find interesting questions to ask. Each field has their own approach to exploration. A psychologist might start with interviews, asking broad questions to find narrower questions for a future survey. An ecologist might film an animal, looking for changes in its behavior. A chemist might measure many properties of a new material, seeing if any stand out. Each approach is like digging for treasure, not sure of exactly what you will find.

Mathematicians and theoretical physicists don’t do experiments, but we still need hypotheses. We need an idea of what we plan to prove, or what kind of theory we want to build: like other scientists, we want to ask a question with a clear, true/false answer. And to find those questions, we still do exploratory research.

What does exploratory research look like, in the theoretical world? Often, it begins with examples and calculations. We can start with a known method, or a guess at a new one, a recipe for doing some specific kind of calculation. Recipe in hand, we proceed to do the same kind of calculation for a few different examples, covering different sorts of situation. Along the way, we notice patterns: maybe the same steps happen over and over, or the result always has some feature.

We can then ask, do those same steps always happen? Does the result really always have that feature? We have our guess, our hypothesis, and our attempt to prove it is much like an experiment. If we find a proof, our hypothesis was true. On the other hand, we might not be able to find a proof. Instead, exploring, we might find a counterexample – one where the steps don’t occur, the feature doesn’t show up. That’s one way to learn that our hypothesis was false.

This kind of exploration is essential to discovery. As scientists, we all have to eventually ask clear yes/no questions, to submit our beliefs to clear tests. But we can’t start with those questions. We have to dig around first, to observe the world without a clear plan, to get to a point where we have a good question to ask.

A couple different things that some of you might like to know about:

Are you an amateur with an idea you think might revolutionize all of physics? If so, absolutely do not contact me about it. Instead, you can talk to these people. Sabine Hossenfelder runs a service that will hook you up with a scientist who will patiently listen to your idea and help you learn what you need to develop it further. They do charge for that service, and they aren’t cheap, so only do this if you can comfortably afford it. If you can’t, then I have some advice in a post here. Try to contact people who are experts in the specific topic you’re working on, ask concrete questions that you expect to give useful answers, and be prepared to do some background reading.

Are you an undergraduate student planning for a career in theoretical physics? If so, consider the Perimeter Scholars International (PSI) master’s program. Located at the Perimeter Institute in Waterloo, Canada, PSI is an intense one-year boot-camp in theoretical physics, teaching the foundational ideas you’ll need for the rest of your career. It’s something I wish I was aware of when I was applying for schools at that age. Theoretical physics is a hard field, and a big part of what makes it hard is all the background knowledge one needs to take part in it. Starting work on a PhD with that background knowledge already in place can be a tremendous advantage. There are other programs with similar concepts, but I’ve gotten a really good impression of PSI specifically so it’s them I would recommend. Note that applications for the new year aren’t open yet: I always plan to advertise them when they open, and I always forget. So consider this an extremely-early warning.

Are you an amplitudeologist? Registration for Amplitudes 2021 is now live! We’re doing an online conference this year, co-hosted by the Niels Bohr Institute and Penn State. We’ll be doing a virtual poster session, so if you want to contribute to that please include a title and abstract when you register. We also plan to stream on YouTube, and will have a fun online surprise closer to the conference date.

# Who Is, and Isn’t, Counting Angels on a Pinhead

How many angels can dance on the head of a pin?

It’s a question famous for its sheer pointlessness. While probably no-one ever had that exact debate, “how many angels fit on a pin” has become a metaphor, first for a host of old theology debates that went nowhere, and later for any academic study that seems like a waste of time. Occasionally, physicists get accused of doing this: typically string theorists, but also people who debate interpretations of quantum mechanics.

Are those accusations fair? Sometimes yes, sometimes no. In order to tell the difference, we should think about what’s wrong, exactly, with counting angels on the head of a pin.

One obvious answer is that knowing the number of angels that fit on a needle’s point is useless. Wikipedia suggests that was the origin of the metaphor in the first place, a pun on “needle’s point” and “needless point”. But this answer is a little too simple, because this would still be a useful debate if angels were real and we could interact with them. “How many angels fit on the head of a pin” is really a question about whether angels take up space, whether two angels can be at the same place at the same time. Asking that question about particles led physicists to bosons and fermions, which among other things led us to invent the laser. If angelology worked, perhaps we would have angel lasers as well.

“If angelology worked” is key here, though. Angelology didn’t work, it didn’t lead to angel-based technology. And while Medieval people couldn’t have known that for certain, maybe they could have guessed. When people accuse academics of “counting angels on the head of a pin”, they’re saying they should be able to guess that their work is destined for uselessness.

How do you guess something like that?

Well, one problem with counting angels is that nobody doing the counting had ever seen an angel. Counting angels on the head of a pin implies debating something you can’t test or observe. That can steer you off-course pretty easily, into conclusions that are either useless or just plain wrong.

This can’t be the whole of the problem though, because of mathematics. We rarely accuse mathematicians of counting angels on the head of a pin, but the whole point of math is to proceed by pure logic, without an experiment in sight. Mathematical conclusions can sometimes be useless (though we can never be sure, some ideas are just ahead of their time), but we don’t expect them to be wrong.

The key difference is that mathematics has clear rules. When two mathematicians disagree, they can look at the details of their arguments, make sure every definition is as clear as possible, and discover which one made a mistake. Working this way, what they build is reliable. Even if it isn’t useful yet, the result is still true, and so may well be useful later.

In contrast, when you imagine Medieval monks debating angels, you probably don’t imagine them with clear rules. They might quote contradictory bible passages, argue everyday meanings of words, and win based more on who was poetic and authoritative than who really won the argument. Picturing a debate over how many angels can fit on the head of a pin, it seems more like Calvinball than like mathematics.

This then, is the heart of the accusation. Saying someone is just debating how many angels can dance on a pin isn’t merely saying they’re debating the invisible. It’s saying they’re debating in a way that won’t go anywhere, a debate without solid basis or reliable conclusions. It’s saying, not just that the debate is useless now, but that it will likely always be useless.

As an outsider, you can’t just dismiss a field because it can’t do experiments. What you can and should do, is dismiss a field that can’t produce reliable knowledge. This can be hard to judge, but a key sign is to look for these kinds of Calvinball-style debates. Do people in the field seem to argue the same things with each other, over and over? Or do they make progress and open up new questions? Do the people talking seem to be just the famous ones? Or are there cases of young and unknown researchers who happen upon something important enough to make an impact? Do people just list prior work in order to state their counter-arguments? Or do they build on it, finding consequences of others’ trusted conclusions?

A few corners of string theory do have this Calvinball feel, as do a few of the debates about the fundamentals of quantum mechanics. But if you look past the headlines and blogs, most of each of these fields seems more reliable. Rather than interminable back-and-forth about angels and pinheads, these fields are quietly accumulating results that, one way or another, will give people something to build on.

A reader pointed me to Stephen Wolfram’s one-year update of his proposal for a unified theory of physics. I was pretty squeamish about it one year ago, and now I’m even less interested in wading in to the topic. But I thought it would be worth saying something, and rather than say something specific, I realized I could say something general. I thought I’d talk a bit about how we judge good and bad research in theoretical physics.

In science, there are two things we want out of a new result: we want it to be true, and we want it to be surprising. The first condition should be obvious, but the second is also important. There’s no reason to do an experiment or calculation if it will just tell us something we already know. We do science in the hope of learning something new, and that means that the best results are the ones we didn’t expect.

(What about replications? We’ll get there.)

If you’re judging an experiment, you can measure both of these things with statistics. Statistics lets you estimate how likely an experiment’s conclusion is to be true: was there a large enough sample? Strong enough evidence? It also lets you judge how surprising the experiment is, by estimating how likely it would be to happen given what was known beforehand. Did existing theories and earlier experiments make the result seem likely, or unlikely? While you might not have considered replications surprising, from this perspective they can be: if a prior experiment seems unreliable, successfully replicating it can itself be a surprising result.

If instead you’re judging a theoretical result, these measures get more subtle. There aren’t always good statistical tools to test them. Nonetheless, you don’t have to rely on vague intuitions either. You can be fairly precise, both about how true a result is and how surprising it is.

We get our results in theoretical physics through mathematical methods. Sometimes, this is an actual mathematical proof: guaranteed to be true, no statistics needed. Sometimes, it resembles a proof, but falls short: vague definitions and unstated assumptions mar the argument, making it less likely to be true. Sometimes, the result uses an approximation. In those cases we do get to use some statistics, estimating how good the approximation may be. Finally, a result can’t be true if it contradicts something we already know. This could be a logical contradiction in the result itself, but if the result is meant to describe reality (note: not always the case), it might contradict the results of a prior experiment.

What makes a theoretical result surprising? And how precise can we be about that surprise?

Theoretical results can be surprising in the light of earlier theory. Sometimes, this gets made precise by a no-go theorem, a proof that some kind of theoretical result is impossible to obtain. If a result finds a loophole in a no-go theorem, that can be quite surprising. Other times, a result is surprising because it’s something no-one else was able to do. To be precise about that kind of surprise, you need to show that the result is something others wanted to do, but couldn’t. Maybe someone else made a conjecture, and only you were able to prove it. Maybe others did approximate calculations, and now you can do them more precisely. Maybe a question was controversial, with different people arguing for different sides, and you have a more conclusive argument. This is one of the better reasons to include a long list of references in a paper: not to pad your friends’ citation counts, but to show that your accomplishment is surprising: that others might have wanted to achieve it, but had to settle for something lesser.

In general, this means that showing whether a theoretical result is good: not merely true, but surprising and new, links you up to the rest of the theoretical community. You can put in all the work you like on a theory of everything, and make it as rigorous as possible, but if all you did was reproduce a sub-case of someone else’s theory then you haven’t accomplished all that much. If you put your work in context, compare and contrast to what others have done before, then we can start getting precise about how much we should be surprised, and get an idea of what your result is really worth.

# Building One’s Technology

There are theoretical physicists who can do everything they do with a pencil and a piece of paper. I’m not one of them. The calculations I do are long, complicated, or tedious enough that they’re often best done with a computer. For a calculation like that, I can’t just use existing software “out of the box”: I need to program special-purpose tools to do the kind of calculation I need. This means each project has its own kind of learning curve. If I already have the right code, or almost the right code, things go very smoothly: with a few tweaks I can do a lot of interesting calculations. If I don’t have the right code yet, things go much more slowly: I have to build up my technology, figuring out what I need piece by piece until I’m back up to my usual speed.

I don’t always need to use computers to do my calculations. Sometimes my work hinges on something more conceptual: understanding a mathematical proof, or the arguments from another physicist’s paper. While this seems different on the surface, I’ve found that it has the same kinds of learning curves. If I know the right papers and mathematical methods, I can go pretty quickly. If I don’t, I have to “build up my technology”, reading and practicing, a slow build-up to my goal.

The times when I have to “build my technology” are always a bit frustrating. I don’t work as fast as I’d like, and I get tripped up by dumb mistakes. I keep having to go back, almost to the beginning, realizing that some aspect of how I set things up needs to be changed to make the rest work. As I go, though, the work gets more and more satisfying. I find pieces (of the code, of my understanding) that become solid, that I can rely on. I build my technology, and I can do more and more, and feel better about myself in the bargain. Eventually, I get back up to my full abilities, my technology set up, and a wide variety of calculations become possible.

# Doing Difficult Things Is Its Own Reward

Does antimatter fall up, or down?

Technically, we don’t know yet. The ALPHA-g experiment would have been the first to check this, making anti-hydrogen by trapping anti-protons and positrons in a long tube and seeing which way it falls. While they got most of their setup working, the LHC complex shut down before they could finish. It starts up again next month, so we should have our answer soon.

That said, for most theorists’ purposes, we absolutely do know: antimatter falls down. Antimatter is one of the cleanest examples of a prediction from pure theory that was confirmed by experiment. When Paul Dirac first tried to write down an equation that described electrons, he found the math forced him to add another particle with the opposite charge. With no such particle in sight, he speculated it could be the proton (this doesn’t work, they need the same mass), before Carl D. Anderson discovered the positron in 1932.

The same math that forced Dirac to add antimatter also tells us which way it falls. There’s a bit more involved, in the form of general relativity, but the recipe is pretty simple: we know how to take an equation like Dirac’s and add gravity to it, and we have enough practice doing it in different situations that we’re pretty sure it’s the right way to go. Pretty sure doesn’t mean 100% sure: talk to the right theorists, and you’ll probably find a proposal or two in which antimatter falls up instead of down. But they tend to be pretty weird proposals, from pretty weird theorists.

Ok, but if those theorists are that “weird”, that outside the mainstream, why does an experiment like ALPHA-g exist? Why does it happen at CERN, one of the flagship facilities for all of mainstream particle physics?

This gets at a misconception I occasionally hear from critics of the physics mainstream. They worry about groupthink among mainstream theorists, the physics community dismissing good ideas just because they’re not trendy (you may think I did that just now, for antigravity antimatter!) They expect this to result in a self-fulfilling prophecy where nobody tests ideas outside the mainstream, so they find no evidence for them, so they keep dismissing them.

The mistake of these critics is in assuming that what gets tested has anything to do with what theorists think is reasonable.

Theorists talk to experimentalists, sure. We motivate them, give them ideas and justification. But ultimately, people do experiments because they can do experiments. I watched a talk about the ALPHA experiment recently, and one thing that struck me was how so many different techniques play into it. They make antiprotons using a proton beam from the accelerator, slow them down with magnetic fields, and cool them with lasers. They trap their antihydrogen in an extremely precise vacuum, and confirm it’s there with particle detectors. The whole setup is a blend of cutting-edge accelerator physics and cutting-edge tricks for manipulating atoms. At its heart, ALPHA-g feels like its primary goal is to stress-test all of those tricks: to push the state of the art in a dozen experimental techniques in order to accomplish something remarkable.

And so even if the mainstream theorists don’t care, ALPHA will keep going. It will keep getting funding, it will keep getting visited by celebrities and inspiring pop fiction. Because enough people recognize that doing something difficult can be its own reward.

In my experience, this motivation applies to theorists too. Plenty of us will dismiss this or that proposal as unlikely or impossible. But give us a concrete calculation, something that lets us use one of our flashy theoretical techniques, and the tune changes. If we’re getting the chance to develop our tools, and get a paper out of it in the process, then sure, we’ll check your wacky claim. Why not?

I suspect critics of the mainstream would have a lot more success with this kind of pitch-based approach. If you can find a theorist who already has the right method, who’s developing and extending it and looking for interesting applications, then make your pitch: tell them how they can answer your question just by doing what they do best. They’ll think of it as a chance to disprove you, and you should let them, that’s the right attitude to take as a scientist anyway. It’ll work a lot better than accusing them of hogging the grant money.

# Redefining Fields for Fun and Profit

When we study subatomic particles, particle physicists use a theory called Quantum Field Theory. But what is a quantum field?

Some people will describe a field in vague terms, and say it’s like a fluid that fills all of space, or a vibrating rubber sheet. These are all metaphors, and while they can be helpful, they can also be confusing. So let me avoid metaphors, and say something that may be just as confusing: a field is the answer to a question.

Suppose you’re interested in a particle, like an electron. There is an electron field that tells you, at each point, your chance of detecting one of those particles spinning in a particular way. Suppose you’re trying to measure a force, say electricity or magnetism. There is an electromagnetic field that tells you, at each point, what force you will measure.

Sometimes the question you’re asking has a very simple answer: just a single number, for each point and each time. An example of a question like that is the temperature: pick a city, pick a date, and the temperature there and then is just a number. In particle physics, the Higgs field answers a question like that: at each point, and each time, how “Higgs-y” is it there and then? You might have heard that the Higgs field gives other particles their mass: what this means is that the more “Higgs-y” it is somewhere, the higher these particles’ mass will be. The Higgs field is almost constant, because it’s very difficult to get it to change. That’s in some sense what the Large Hadron Collider did when they discovered the Higgs boson: pushed hard enough to cause a tiny, short-lived ripple in the Higgs field, a small area that was briefly more “Higgs-y” than average.

We like to think of some fields as fundamental, and others as composite. A proton is composite: it’s made up of quarks and gluons. Quarks and gluons, as far as we know, are fundamental: they’re not made up of anything else. More generally, since we’re thinking about fields as answers to questions, we can just as well ask more complicated, “composite” questions. For example, instead of “what is the temperature?”, we can ask “what is the temperature squared?” or “what is the temperature times the Higgs-y-ness?”.

But this raises a troubling point. When we single out a specific field, like the Higgs field, why are we sure that that field is the fundamental one? Why didn’t we start with “Higgs squared” instead? Or “Higgs plus Higgs squared”? Or something even weirder?

That kind of swap, from Higgs to Higgs squared, is called a field redefinition. In the math of quantum field theory, it’s something you’re perfectly allowed to do. Sometimes, it’s even a good idea. Other times, it can make your life quite complicated.

The reason why is that some fields are much simpler than others. Some are what we call free fields. Free fields don’t interact with anything else. They just move, rippling along in easy-to-calculate waves.

Redefine a free field, swapping it for some more complicated function, and you can easily screw up, and make it into an interacting field. An interacting field might interact with another field, like how electromagnetic fields move (and are moved by) electrons. It might also just interact with itself, a kind of feedback effect that makes any calculation we’d like to do much more difficult.

If we persevere with this perverse choice, and do the calculation anyway, we find a surprise. The final results we calculate, the real measurements people can do, are the same in both theories. The field redefinition changed how the theory appeared, quite dramatically…but it didn’t change the physics.

You might think the moral of the story is that you must always choose the right fundamental field. You might want to, but you can’t: not every field is secretly free. Some will be interacting fields, whatever you do. In that case, you can make one choice or another to simplify your life…but you can also just refuse to make a choice.

That’s something quite a few physicists do. Instead of looking at a theory and calling some fields fundamental and others composite, they treat every one of these fields, every different question they could ask, on the same footing. They then ask, for these fields, what one can measure about them. They can ask which fields travel at the speed of light, and which ones go slower, or which fields interact with which other fields, and how much. Field redefinitions will shuffle the fields around, but the patterns in the measurements will remain. So those, and not the fields, can be used to specify the theory. Instead of describing the world in terms of a few fundamental fields, they think about the world as a kind of field soup, characterized by how it shifts when you stir it with a spoon.

It’s not a perspective everyone takes. If you overhear physicists, sometimes they will talk about a theory with only a few fields, sometimes they will talk about many, and you might be hard-pressed to tell what they’re talking about. But if you keep in mind these two perspectives: either a few fundamental fields, or a “field soup”, you’ll understand them a little better.

# A Tale of Two Donuts

I’ve got a new paper up this week, with Hjalte Frellesvig, Cristian Vergu, and Matthias Volk, about the elliptic integrals that show up in Feynman diagrams.

You can think of elliptic integrals as integrals over a torus, a curve shaped like the outer crust of a donut.

Integrals like these are showing up more and more in our field, the subject of bigger and bigger conferences. By now, we think we have a pretty good idea of how to handle them, but there are still some outstanding mysteries to solve.

One such mystery came up in a paper in 2017, by Luise Adams and Stefan Weinzierl. They were working with one of the favorite examples of this community, the so-called sunrise diagram (sunrise being a good time to eat donuts). And they noticed something surprising: if they looked at the sunrise diagram in different ways, it was described by different donuts.

What do I mean, different donuts?

The integrals we know best in this field aren’t integrals on a torus, but rather integrals on a sphere. In some sense, all spheres are the same: you can make them bigger or smaller, but they don’t have different shapes, they’re all “sphere-shaped”. In contrast, integrals on a torus are trickier, because toruses can have different shapes. Think about different donuts: some might have a thin ring, others a thicker one, even if the overall donut is the same size. You can’t just scale up one donut and get the other.

My colleague, Cristian Vergu, was annoyed by this. He’s the kind of person who trusts mathematics like an old friend, one who would never lead him astray. He thought that there must be one answer, one correct donut, one natural way to represent the sunrise diagram mathematically. I was skeptical, I don’t trust mathematics nearly as much as Cristian does. To sort it out, we brought in Hjalte Frellesvig and Matthias Volk, and started trying to write the sunrise diagram every way we possibly could. (Along the way, we threw in another “donut diagram”, the double-box, just to see what would happen.)

Rather than getting a zoo of different donuts, we got a surprise: we kept seeing the same two. And in the end, we stumbled upon the answer Cristian was hoping for: one of these two is, in a meaningful sense, the “correct donut”.

What was wrong with the other donut? It turns out when the original two donuts were found, one of them involved a move that is a bit risky mathematically, namely, combining square roots.

For readers who don’t know what I mean, or why this is risky, let me give a simple example. Everyone else can skip to after the torus gif.

Suppose I am solving a problem, and I find a product of two square roots:

$\sqrt{x}\sqrt{x}$

I could try combining them under the same square root sign, like so:

$\sqrt{x^2}$

That works, if $x$ is positive. But now suppose $x=-1$. Plug in negative one to the first expression, and you get,

$\sqrt{-1}\sqrt{-1}=i\times i=-1$

while in the second,

$\sqrt{(-1)^2}=\sqrt{1}=1$

In this case, it wasn’t as obvious that combining roots would change the donut. It might have been perfectly safe. It took some work to show that indeed, this was the root of the problem. If the roots are instead combined more carefully, then one of the donuts goes away, leaving only the one, true donut.

I’m interested in seeing where this goes, how many different donuts we have to understand and how they might be related. But I’ve also been writing about donuts for the last hour or so, so I’m getting hungry. See you next week!

# This Week, at Scattering-Amplitudes.com

I did a guest post this week, on an outreach site for the Max Planck Institute for Physics. The new Director of their Quantum Field Theory Department, Johannes Henn, has been behind a lot of major developments in scattering amplitudes. He was one of the first to notice just how symmetric N=4 super Yang-Mills is, as well as the first to build the “hexagon functions” that would become my stock-in-trade. He’s also done what we all strive to do, and applied what he learned to the real world, coming up with an approach to differential equations that has become the gold standard for many different amplitudes calculations.

Now in his new position, he has a swanky new outreach site, reached at the conveniently memorable scattering-amplitudes.com and managed by outreach-ologist Sorana Scholtes. They started a fun series recently called “Talking Terms” as a kind of glossary, explaining words that physicists use over and over again. My guest post for them is part of that series. It hearkens all the way back to one of my first posts, defining what “theory” means to a theoretical physicist. It covers something new as well, a phrase I don’t think I’ve ever explained on this blog: “working in a theory”. You can check it out on their site!

# Physical Intuition From Physics Experience

One of the most mysterious powers physicists claim is physical intuition. Let the mathematicians have their rigorous proofs and careful calculations. We just need to ask ourselves, “Does this make sense physically?”

It’s tempting to chalk this up to bluster, or physicist arrogance. Sometimes, though, a physicist manages to figure out something that stumps the mathematicians. Edward Witten’s work on knot theory is a classic example, where he used ideas from physics, not rigorous proof, to win one of mathematics’ highest honors.

So what is physical intuition? And what is its relationship to proof?

Let me walk you through an example. I recently saw a talk by someone in my field who might be a master of physical intuition. He was trying to learn about what we call Effective Field Theories, theories that are “effectively” true at some energy but don’t include the details of higher-energy particles. He calculated that there are limits to the effect these higher-energy particles can have, just based on simple cause and effect. To explain the calculation to us, he gave a physical example, of coupled oscillators.

Oscillators are familiar problems for first-year physics students. Objects that go back and forth, like springs and pendulums, tend to obey similar equations. Link two of them together (couple them), and the equations get more complicated, work for a second-year student instead of a first-year one. Such a student will notice that coupled oscillators “repel” each other: their frequencies get father apart than they would be if they weren’t coupled.

Our seminar speaker wanted us to revisit those second-year-student days, in order to understand how different particles behave in Effective Field Theory. Just as the frequencies of the oscillators repel each other, the energies of particles repel each other: the unknown high-energy particles could only push the energies of the lighter particles we can detect lower, not higher.

This is an example of physical intuition. Examine it, and you can learn a few things about how physical intuition works.

First, physical intuition comes from experience. Using physical intuition wasn’t just a matter of imagining the particles and trying to see what “makes sense”. Instead, it required thinking about similar problems from our experience as physicists: problems that don’t just seem similar on the surface, but are mathematically similar.


text:
# lmertest: interaction between a categorical and a continuous variable with random slope

Basically I have the following model with drink type (three drinks) and sugar content of the food as predictors of participants' willingness to pay. I have trouble interpreting the output of the summary (fit2) mainly because I have a continuous variable as a predictor. I pasted my output at the end and these are my interpretations for the fixed effects:

1) drink1 &drink 2: they are the comparisons to baseline drink;

2) sugarp: the regression from sugar to wtp for baseline water; 3) drink1:sugarp: the regression from sugar to wtp for drink1 (beta value) compared with baseline drink beta value;

1. Are those interpretations correct? Can someone explain the fixed effects output for me? Also why do I get different results between anova() and summary()?

2. How can I know which drink is set as a baseline and what can I do if I just want to compare all of the drinks without a baseline?

3. To visualize the interaction, should I draw different regression lines for 3 drinks? I used the emmeans R package to do the post hoc and interaction plots that can take the whole model structure into consideration, but I don't think it can deal with continuous predictors. Do you have any recommended packages for the post hoc and plots making?

4. when reporting the results, what parameters should be included?

5. After some research, I realized I can use contr.sum or contra.treatment in my analysis, I do have a water condition which can be a baseline. But I am interested in comparing all three drinks with each other and I think I only get the same p values as ANOVA when I use contra.sum. Any suggestion for this?

My model and the output:

   > anova(fit1)
Type III Analysis of Variance Table with Satterthwaite's method
Sum Sq Mean Sq NumDF  DenDF F value  Pr(>F)
drink        1.7208 0.86038     2  40.69  1.6381 0.20690
sugarp       2.2398 2.23978     1 825.71  4.2645 0.03923 *
drink:sugarp 3.5459 1.77296     2 829.86  3.3757 0.03466 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> summary(fit1)
Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: wtp ~ drink * sugarp + (drink | ID/food)
Data: data

REML criterion at convergence: 16300.4

Scaled residuals:
Min      1Q  Median      3Q     Max
-4.4593 -0.2890 -0.0406  0.2584  4.4765

Random effects:
Groups   Name           Variance Std.Dev. Corr
food:ID  (Intercept)    1.8874   1.3738
drinkGlucose   0.7650   0.8747   -0.41
drinkSucralose 0.8203   0.9057   -0.44  0.52
ID       (Intercept)    0.5507   0.7421
drinkGlucose   0.2593   0.5092   -0.36
drinkSucralose 0.4603   0.6785   -0.52  0.68
Residual                0.5252   0.7247
Number of obs: 5494, groups:  food:ID, 840; ID, 28

Fixed effects:
Estimate Std. Error         df t value Pr(>|t|)
(Intercept)             1.753803   0.163613  38.867724  10.719  3.6e-13 ***
drinkGlucose           -0.105242   0.116089  41.809531  -0.907  0.36983
drinkSucralose         -0.258070   0.144398  36.061829  -1.787  0.08231 .
sugarp                  0.004529   0.001670 824.021966   2.711  0.00685 **
drinkGlucose:sugarp    -0.003332   0.001288 836.809293  -2.586  0.00987 **
drinkSucralose:sugarp  -0.001466   0.001316 812.753050  -1.114  0.26573
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
(Intr) drnkGl drnkSc sugarp drnkG:
drinkGlucos -0.390
drinkSucrls -0.508  0.632
sugarp      -0.412  0.203  0.175
drnkGlcs:sg  0.186 -0.447 -0.189 -0.453
drnkScrls:s  0.196 -0.230 -0.368 -0.476  0.513

• R uses the first factor level as baseline. If you don't set them manually, factor levels are in lexical order. Nov 12 '19 at 8:42
• You have to define more clearly what "compare all drinks" means to you. It's best if you write down the exact null hypotheses you want to test. Nov 12 '19 at 8:43
• I have a sugar preload manipulation (water vs sucralose vs glucose), and I guess my hypothesis will be there is willingness to pay difference (my DV) between three drink conditions. Nov 12 '19 at 9:15
• I was asking for formal hypotheses, i.e., expressed with mathematical notation. Nov 12 '19 at 9:36
• the willingness to pay after drinking water is more than the willingness to pay after taking glucose; also wtp (sucralose)> wtp(glucose); and after taking glucose, the willlingness to pay is less sensitive to the sugar content of the food than after taking water and sucralose. Nov 12 '19 at 17:32

Your questions in point (3) can be answered using the emmeans package. To visualize the fitted lines, use

library("emmeans")
emmip(fit1, drink ~ sugarp, cov.reduce = range)

To obtain the slopes of the fitted lines, use

emt <- emtrends(fit1, "drink", var = "sugarp")
emt

To compare those slopes, use

pairs(emt)


text:
• CN：11-2187/TH
• ISSN：0577-6686

›› 2008, Vol. 44 ›› Issue (4): 113-117.

• 论文 •

### 惩罚函数的构造及多模态平稳过渡策略

1. 浙江大学现代制造工程研究所;杭州机床集团有限公司;浙江纺织服装学院
• 发布日期:2008-04-15

### Construction of Penalty Function and Multi-modal Smooth Transition Strategy

HU Xuxiao;PAN Xiaohong;HE Wei;CHEN Gang

1. Institute of Modern Manufacturing Engineering, Zhejiang University Hangzhou Machine Tool Group Co., Ltd. Zhejiang Texile & Fashion College
• Published:2008-04-15

Abstract: In order to realize steady transition between the modes in intelligent control, a new penalty function is found combined with a kind of multi-mode divided method, and a multi-mode steady transition strategy is proposed based on the penalty function. Different from the usual way that each control strategy in view of each mode, different control strategies are promoted to all modes in the multi-mode steady transition strategy, the control quantity is the sum of all control strategies by weight, the function of the penalty function is made the weight of corresponding control strategy bigger, and that of other control strategies is punished in this mode. Finally, the steady transition strategy is confirmed by the experiment on the nano precision drive parts which have large feeding force.


text:
## Algebra and Trigonometry 10th Edition

y = $-\frac{1}{3}x$ + $\frac{4}{3}$
Point: [4,0]; m = $-\frac{1}{3}$ y - 0 = $-\frac{1}{3}x$(x - 4) y = $-\frac{1}{3}x$ + $\frac{4}{3}$