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Now showing items 10-29 of 36

• Evidence for SMAD3 as a modifier of breast cancer risk in BRCA2 mutation carriers

(2010-11-29)
Abstract Introduction Current attempts to identify genetic modifiers of BRCA1 and BRCA2 associated risk have focused on a candidate gene approach, based on knowledge of gene functions, or the development of large genome-wide ...
• Evidence that the 5p12 variant rs10941679 confers susceptibility to estrogen receptor positive breast cancer through FGF10 and MRPS30 regulation

(Elsevier (Cell Press), 2016)
Genome-wide association studies (GWAS) have revealed increased breast cancer risk associated with multiple genetic variants at 5p12. Here, we report the fine-mapping of this locus using data from 104,660 subjects from 50 ...
• Fine scale mapping of the 5q11.2 breast cancer locus reveals at least three independent risk variants regulating MAP3K1

Genome Wide Association Studies (GWAS) revealed SNP rs889312 on 5q11.2 to be associated with breast cancer risk in women of European ancestry. In an attempt to identify the biologically relevant variants, we analysed 909 ...
• Fine scale mapping of the 5q11.2 breast cancer locus reveals at least three independent risk variants regulating MAP3K1.

(2015-12-18)
• Fine-Mapping of the 1p11.2 Breast Cancer Susceptibility Locus

(Public Library of Science, 2016-08-24)
The Cancer Genetic Markers of Susceptibility genome-wide association study (GWAS) originally identified a single nucleotide polymorphism (SNP) rs11249433 at 1p11.2 associated with breast cancer risk. To fine-map this locus, ...
• Fine-mapping of the HNF1B multicancer locus identifies candidate variants that mediate endometrial cancer risk

(Oxford Journals, 2014-11-06)
Common variants in the hepatocyte nuclear factor 1 homeobox B (HNF1B) gene are associated with the risk of type II diabetes and multiple cancers. Evidence to date indicates that cancer risk may be mediated via genetic or ...
• Fine-Scale Mapping at 9p22.2 Identifies Candidate Causal Variants That Modify Ovarian Cancer Risk in $\small \textit{BRCA1}$ and $\small \textit{BRCA2}$ Mutation Carriers

(Public Library of Science, 2016-07-27)
Population-based genome wide association studies have identified a locus at 9p22.2 associated with ovarian cancer risk, which also modifies ovarian cancer risk in $\small \textit{BRCA1}$ and $\small \textit{BRCA2}$ mutation ...
• Fine-scale mapping of 8q24 locus identifies multiple independent risk variants for breast cancer

(Wiley, 2016)
Previous genome-wide association studies among women of European ancestry identified two independent breast cancer susceptibility loci represented by single nucleotide polymorphisms (SNPs) rs13281615 and rs11780156 at 8q24. ...
• Five endometrial cancer risk loci identified through genome-wide association analysis

(Nature Publishing Group, 2016-05-02)
We conducted a meta-analysis of three endometrial cancer genome-wide association studies (GWAS) and two follow-up phases totaling 7,737 endometrial cancer cases and 37,144 controls of European ancestry. Genome-wide imputation ...
• Functional Mechanisms Underlying Pleiotropic Risk Alleles at the 19p13.1 Breast-Ovarian Cancer Susceptibility Locus

(Nature Publishing Group, 2016)
A locus at 19p13 is associated with breast cancer (BC) and ovarian cancer (OC) risk. We analyzed 438 SNPs in this region in 46,451 BC and 15,438 OC cases, 15,252 BRCA1 mutation carriers and 73,444 controls and identified ...
• Gene-Panel Sequencing and the Prediction of Breast-Cancer Risk

(Massachusetts Medical Society, 2015-06-04)
Advances in sequencing technology have made multigene testing, or “panel testing,” a practical option when looking for genetic variants that may be associated with a risk of breast cancer. In June 2013, the U.S. Supreme ...
• Genetic modifiers of CHEK2*1100delC-associated breast cancer risk

(Nature Publishing Group, 2016-10-06)
Purpose: CHEK2*1100delC is a founder variant in European populations that confers a two- to threefold increased risk of breast cancer (BC). Epidemiologic and family studies have suggested that the risk associated with ...
• Genetic overlap between endometriosis and endometrial cancer: evidence from cross-disease genetic correlation and GWAS meta-analyses.

(Wiley, 2018-05)
Epidemiological, biological and molecular data suggest links between endometriosis and endometrial cancer, with recent epidemiological studies providing evidence for an association between a previous diagnosis of endometriosis ...
• Genetic risk score Mendelian randomization shows obesity measured as body mass index, but not waist:hip ratio, is causal for endometrial cancer

(American Association for Cancer Research, 2016-08-22)
$\textbf{Background:}$ The strongest known risk factor for endometrial cancer (EC) is obesity. To determine whether single nucleotide polymorphisms (SNPs) associated with increased body mass index (BMI) or waist-hip ratio ...
• Genetic variation in the immunosuppression pathway genes and breast cancer: a pooled analysis of 42,510 cases and 40,577 controls from the Breast

(Springer, 2015-11-30)
Immunosuppression plays a pivotal role in assisting tumors to evade immune destruction and promoting tumor development. We hypothesized that genetic variation in the immunosuppression pathway genes may be implicated in ...
• Genetically Predicted Body Mass Index and Breast Cancer Risk: Mendelian Randomization Analyses of Data from 145,000 Women of European Descent

(Public Library of Science, 2016-08-23)
$\textbf{Background}$ Observational epidemiological studies have shown that high body mass index (BMI) is associated with a reduced risk of breast cancer in premenopausal women but an increased risk in postmenopausal ...
• Genome-wide association analysis of more than 120,000 individuals identifies 15 new susceptibility loci for breast cancer.

(NPG, 2015-03-09)
Genome-wide association studies (GWAS) and large-scale replication studies have identified common variants in 79 loci associated with breast cancer, explaining ~14% of the familial risk of the disease. To identify new ...
• Identification and characterisation of novel associations in the CASP8/ALS2CR12 region on chromosome 2 with breast cancer risk

(Oxford University Press, 2014-08-28)
Previous studies have suggested that polymorphisms in CASP8 on chromosome 2 are associated with breast cancer risk. To clarify the role of CASP8 in breast cancer susceptibility we carried out dense genotyping of this region ...
• Identification of four novel susceptibility loci for estrogen receptor negative breast cancer

(Nature Publishing Group, 2016)
Common variants in 94 loci have been associated with breast cancer including 15 loci with genome wide significant associations (P<5x10-8) with estrogen receptor (ER)-negative breast cancer and BRCA1-associated breast cancer ...
• Identification of four novel susceptibility loci for oestrogen receptor negative breast cancer

(Nature Publishing Group, 2016-04-27)
Common variants in 94 loci have been associated with breast cancer including 15 loci with genome-wide significant associations (P<5 × 10$^{−8}$) with oestrogen receptor (ER)-negative breast cancer and BRCA1-associated ...


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# Multiply fractions word problems

Solve and interpret fraction multiplication word problems.

### Problem

Eric has started running for exercise. He recorded the number of miles he ran each day this week in the table below.
On Monday, Eric ran start fraction, 5, divided by, 7, end fraction of the distance that he ran on Wednesday.
Fill in the number of kilometers that Eric ran on Monday in the table below.
DayNumber of kilometers run
Monday
Wednesdaystart fraction, 9, divided by, 10, end fraction
Friday1, start fraction, 1, divided by, 2, end fraction

format: plain


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Question

# When a stationary wave is formed, then its frequency is

A
Same as that of the individual waves
B
Twice that of the individual waves
C
Half that of the individual waves
D
That of the individual waves

Solution

## The correct option is A Same as that of the individual wavesLet us consider a progressive wave of amplitude $$a$$ , wavelength $$\lambda$$ and frequency $$f=\dfrac { 1 }{ T }$$ travelling in the direction of $$X$$- axis whose equation is given by-$${ y }_{ 1 }=a\sin { (2\pi [\dfrac { t }{ T } -\dfrac { x }{ \lambda } ] } )$$This wave is reflected from a free end and it travels in the negative direction of $$X$$- axis with frequency $$f=\dfrac { 1 }{ T }$$$${ y }_{ 1 }=a\sin { (2\pi [\dfrac { t }{ T } +\dfrac { x }{ \lambda } ] } )$$According to principle of superposition$$\Rightarrow { y }_{ 1 }+{ y }_{ 2 }=a\sin { (2\pi [\dfrac { t }{ T } -\dfrac { x }{ \lambda } ] } )+a\sin { (2\pi [\dfrac { t }{ T } +\dfrac { x }{ \lambda } ] } )$$$$\Rightarrow { y }_{ 1 }+{ y }_{ 2 }=a\sin { \left( \dfrac { 2\pi t }{ T } \right) } \cos { \left( \dfrac { 2\pi x }{ \lambda } \right) }$$Here we can see the frequency of resultant wave is also $$f=\dfrac { 1 }{ T }$$.Physics

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people:mckenzie:math_220_hw_solutions

Homework solutions are posted a day or so after it is is assigned.

Ch 6 solutions IGNORE #22. I will rewrite solution.

Ch 10 and some Ch 11 solutions These could use a rewrite for clarity. I'll get right on it (ha ha)

Optimization practice problem solutions The first two can be found at the end of this scan, but all of them are there

people/mckenzie/math_220_hw_solutions.txt · Last modified: 2019/04/24 22:08 by mckenzie


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[–][deleted]  (10 children)

[deleted]

[–] 8 points9 points * (2 children)

Physicists do seem to work with math more intuitively than mathematicians. And this is absolutely a good thing: by not sweating the details too much, a physicist can often get to the heart of the matter faster. However, the mathematical intuition that physicists develop is developed in very nice, special, and restricted mathematical contexts. Which means that when a physicist finds himself or herself outside of their accustomed context, their intuition is no longer of much use.

Sometimes, a physicist might not even be aware of the fact that their intuition only applies to very special contexts. In which case he or she isn't even aware of the possibility that his or her intuition might not apply. This is what leads to trouble.

Here's a dangerous example that is sure to upset some physicists: the momentum operator is often not self-adjoint! I'll be very specific here. The momentum operator [; -i \hbar \frac{\partial}{\partial x} ;] acting on (edit: absolutely continuous functions in) L2 [0,1] is not self-adjoint . Anyone familiar with operators on function spaces will know that self-adjoint operators have really nice properties compared to other operators. Thus mistaking a non-self-adjoint operator for a self-adjoint operator is a dangerous mistake.

In then end, I think that physicists' intuitions are wonderful tools. In special contexts, there is simply no need for the full generality and careful attention to detail of mathematics. I just wish that more physicists recognized that their intuitions come from very nice, special, and restricted contexts---and recognized correspondingly that their intuition sometimes fails to apply.

Edit: The momentum operator needs to act on the obvious subspace of L2 as L2 functions are not, in general, even differentiable.

[–] 1 point2 points  (1 child)

What is the adjoint of the momentum operator on L2 [0,1]?

[–] 2 points3 points  (0 children)

The momentum operator acts only on absolutely continuous functions in L2 [0,1].

The adjoint of this operator is the momentum operator, but it is necessarily restricted to act only on absolutely continuous functions in L2 [0,1] which are also zero at 0 and 1.

Which is why they're different operators. You could use the momentum operator to act on the function 1, but you cannot act on this function with the adjoint of the momentum operator!

[–] 4 points5 points  (2 children)

Yeah, I think your example of 'learning' Fubini's theorem is like a Calc 1 student telling you what makes a function continuous or differentiable. They can usually produce a decent result on a given problem but couldn't prove either thing if they wanted to. I don't really like the 'math vs physics' debates that typically come up in topics like this because physics certainly wouldn't be possible without math and math would be meaningless if it didn't describe any real world phenomena.

[–] 11 points12 points  (0 children)

Physics uses math for scientific ends. Physics itself could exist as simply a very large collection of recorded observations of the natural world and recognized patterns. Math simplifies those observations and perceived patterns into very accurate models of the real world... but it's really just the shorthand for a longer explanation of the patterns that the observations seem to imply.

Math definitely wouldn't be meaningless if it didn't describe real world phenomena. There are many areas of math that have very little or no direct applications to behaviors in the natural world. Economics and video games are two examples...

[–] 1 point2 points  (0 children)

Thanks for the advice! Fubini's Theorem is a good example.

[–] 0 points1 point  (1 child)

Mathematics is really the common language of the sciences. Growing up, you probably didn't know the technicalities behind conjugation or what all of the parts of speech were, but you could certainly speak well enough to have a meaningful conversation. When physicists use mathematics, we can get correct results even if our understanding of mathematics is not that great. I like to think that we cheat a little bit, since performing an experiment is much like looking in the back of your calculus book to check your answer.

[–] 0 points1 point  (0 children)

Fubini's theorem is often trivial in the context where physicists "apply" it.

[–] 21 points22 points  (19 children)

I've always joked with other scientists when talking about math, "thank god for mathematicians so we can actually spend our time getting shit done." Math is a tool for scientists and we usually understand how and why and works and all that jazz, but at the end of the day we are trying to prove the science and not the mathematics.

[–] -2 points-1 points  (18 children)

Do you really think math is just a tool? Never is it a direct reflection of the physical reality?

[–] 2 points3 points  (1 child)

Math is built off of axioms. Statements that we can not prove but instead accept as true. If the axioms align with how reality operates then wonderful. Here I will point to euclidean and non euclidean geometry. They have huge differences but both can be used to explain our universe under different conditions.

[–] 2 points3 points  (10 children)

That's a pretty deep question, and there isn't really a concrete argument either way. At levels higher than physics there certainly exist concepts that are not a direct result of math, for example biochemical pathways.

[–] 1 point2 points  (1 child)

Oh great. Now the mathematicians and the physicists are going to maul philosophy.

[–] 1 point2 points  (0 children)

[–] 0 points1 point  (7 children)

I don't necessarily argue either way. But I'll probably argue with anybody that has a non neutral opinion :)

| At levels higher than physics there certainly exist concepts that are not a direct result of math, for example biochemical pathways.
I don't quite get this. If it turned out that we could get complete theories at a the basic level could we not just compose this a zillion times to get a complete(although practically uncomputable) theories of higher scales?

[–] 0 points1 point  (6 children)

(Use the ">" for proper quotes with the blue bar)

If it turned out that we could get complete theories at a the basic level could we not just compose this a zillion times to get a complete(although practically uncomputable) theories of higher scales?

I don't claim to understand the terms or even what I'm about to type out, but this is the "complete lagrangian of the standard model of physics" (which I assume governs kinematics, EM, whatever):

http://nuclear.ucdavis.edu/~tgutierr/files/sml.pdf

Can you use this to formulate consciousness? Physics runs into problems with a system of bodies n > 2.

[–] 1 point2 points  (1 child)

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[–] 0 points1 point  (0 children)

Yeah, I'm getting rather deep into 'topics I don't know much about.' Just trying to illustrate a point. Website I grabbed it from mentioned this too I believe.

[–] 0 points1 point  (3 children)

this is the "complete lagrangian of the standard model of physics"

Sort of. More like the complete lagrangian of the ultimate future evolution of the standard model of physics. You'd be hard pressed to find a physicist who didn't think the standard model is just a stepping stone to something more complete.

As far as consciousness goes, I'm not even going to bother touching that can of worms.

Also, I highly disagree with

Physics runs into problems with a system of bodies n > 2.

In the newtonian limit, physics explains the situation perfectly. Mathematics is where we fail with the n-body problem. We can specify the situation & the rules governing the evolution of the system exactly(at least in the newtonian limit.) Given infinite mathematical sophistication, we can therefore predict the future of such a system with infinite precision. But, as the mathematics are lacking, so is our predictive power.

[–] 0 points1 point  (2 children)

In the newtonian limit, physics explains the situation perfectly. Mathematics is where we fail with the n-body problem.

That's like saying in the organic limit, orgo explains biochemistry perfectly.

[–] 0 points1 point  (1 child)

Yeah, that's sort of what I'm saying :) If the organic chem work is done, now you need to work on a higher scale. I guess I have a precise definition of the line between physics & mathematics. A complete physical theory is a model/system of equations that, given exact initial conditions, defines exactly the equations governing the future evolution of the system. It is a function of the mathematics to actually solve these equations. Any reworking of the physics that yields an easier solution will be equivalent to an (at least partial) mathematical solution of the original problem. Regardless, we may just be arguing semantics.

[–] 0 points1 point  (0 children)

A complete physical theory is a model/system of equations that, given exact initial conditions, defines exactly the equations governing the future evolution of the system.

Even reducing organic chemistry would to math would be really, really tough.

I'm trying to say that at every level of abstraction there exist unique paradigms which are, at the very least, nearly impossible to define with tools from something higher up the ladder.

[–] 1 point2 points  (0 children)

Depends. What's physical reality?

[–] 0 points1 point  (1 child)

It's an interesting question that, as unanun said, is pretty deep. I mean, calling math just a tool is somewhat dismissive, but it is a tool to help represent physical reality. But what do I know, I'm an experimentalist. I try to discover stuff and let the theoretical people wrap a nice package around it.

[–] 1 point2 points  (0 children)

Remember, the wording of this response is important. People often confuse represent with an actual representation, where physics, as a field of study, helps to build models that produce reproducible and predictive results. The standard model, is just that, a model. This is not some pure language of the universe because things interact at such a level that we can neither influence nor "understand" pure physical reality. Physics is a tool for creating excellent models that help us understand emergent phenomena but I believe it is a mistake to view physics and math as "the language of the universe", where it is more the language that we have created to help us understand the universe and create useful things.

[–] 0 points1 point  (0 children)

Depends on the resolution. I think in general mathematics are not a physical reflection of reality because they are too generalized in order to create realistic functions and models. To some degree this is a product of our tools for measurement and simulation - however it appears that as our tools get better our mathematics have to be refined in order to accurately reflect observed phenomena.

I don't think any theoretical work can match the precise confidence of observed work.

[–] -4 points-3 points  (0 children)

Math reflects reality. Not the other way around. I can get behind that. That's why mathematical rigor is silly (statistical rigor, on the other hand...). It's like arguing English grammar with Hemmingway.

[–] 13 points14 points  (1 child)

As a physicist, I can tell you the answer is no. We abuse mathematics over and over again because the goal of physics is not to describe a concise rigorous system, but rather a messy complex world. What this means is that the math is selected and written in such a way so that it gives the answer which describes the world. It was done thus for calculus (at least by Newton), the delta function, most of the use of tensor calculus.

I don't like saying this (because I would love if more people enjoyed and learned physics at a higher level) but I have by and large not seen mathematicians concerned with rigor do well in graduate physics programs.

[–] 4 points5 points  (0 children)

Curious, because it's the rigorous mathematics that is often complex and messy and it's the physicist's version where the complications are smoothed over or ignored.

[–] 24 points25 points * (40 children)

My physics profs and classmates are quick to dismiss mathematical rigor for the sake of calculation.

Physics would not advanced unless physicists completely neglected rigour. Perfect example is quantum field theory. There exists no mathematically rigorous formulation (and not for lack of trying, no one has successful been able to do it yet, and they have tried!) and it's a bunch of very adhoc mathematical ideas that are completely non-rigorous. And guess what. It works very well with experiment.*

The problem is, rigour is, for the most part in physics, pretty useless. I've never seen it being used nor do I see a need for it in physics. Physicists aren't mathematicians, they are merely using mathematics as a tool along the way to get to the right answer. It doesn't really matter how they get to the answer, for the most part, and you can abuse whatever mathematical techniques you can.

In fact, I would say that most higher level mathematics is kind of useless for doing any practical calculations. For example, consider Maxwell's equations in form notation, dF=0, d*F = *j. This provides a really nice starting point for high level stuff, but for the 99.99% of physicists who don't do really high level theoretical work, it provides no insight into the physics. Heck, to even solve a problem you have to 'undo' most of the nice mathematics you had to get there.

This tends to annoy me because there is often some implicit mathematical theorem working in the background which makes what the prof is talking about work but it's never mentioned.

Can you give me examples?

So I guess the question is there such a thing as mathematically rigorous physics, or maybe is it possible to study theoretical physics by doing grad school in pure mathematics

Nope. You'll learn 0 physics in a pure mathematics graduate school. My friend, who did his undergraduate degree in physics, did his masters in pure mathematics (differential geometry) and he learnt absolutely no new physics.

Anecdotal yes but very common.

There do exist rigorous formulations of many physical theories however. If you want to do rigorous quantum mechanics, be prepared for a full blown course in functional analysis and measure theory. Completely rigorous GR? Get ready to go deep into differential geometry. There exist very rigorous formulations of these things but they simply are useless for the working physicist. In fact, I've read some of them and they really don't provide any deep insight into anything beyond throwing some epsilons and deltas around.

EDIT: Also, how rigorous do you want? I took a course in advanced GR taught by an applied mathematician and we barely mentioned any theorems that would be applicable. We just assumed that all manifolds were nice and well defined and continuous and whatnot.

*for example, consider the Feynman path integral. No one has come up with a rigorous formulation of what it means to take a path integral. Yet, Feynman was able to come up with a very logical and physically motivated interpretation of what it means. He also showed us how to calculate it. Mathematically, it's horrendously ill-defined, yet, it seems to have some good motivation because it gives the right answer. So something must be right mathematically but no one has been clever enough to rigorously define it yet.

[–] 1 point2 points * (4 children)

nor do I see a need for it in physics[rigor]

I get what you're saying, but these are strong words. Physics is clearly the realm of extracting patterns out of physical reality. Mathematics is the language of patterns. It's about building a bridge between reality and mathematics. Do you really think that there's nothing to be gained, nothing to be more deeply understood, by fully building out the foundations of the bridge?
 I guess it really comes down to a philosophical point. Do you believe that the only patterns we'll ever be able to extract out of reality are approximations?

[–] 2 points3 points  (3 children)

I guess it really comes down to a philosophical point. Do you believe that the only patterns we'll ever be able to extract out of reality are approximations?

Indeed!

Physics is clearly the realm of extracting patterns out of physical reality. Mathematics is the language of patterns.

Does nature need to follow those nice patterns though? Of course not. There are people who believe that it should (Dirac) and there are those who do not (Feynman) but it really is a philosophical question.

[–] 1 point2 points * (2 children)

Does nature need to follow those nice patterns though? Of course not.

I'm mostly agnostic on the issue. I wouldn't be necessarily be surprised if nature continued to elude our intellects all the way down the chain of models. But I don't necessarily see any evidence for this. In fact, given theories like (special)relativity, in which we seem to (presumably)have absolute precision based on a few exceedingly elegant philosophical assumptions(homogeneity of spacetime lacking the presence of energy/mass, finite information propagation speed, equivalence of inertial frames), I'm inclined to believe that we may be able to fully map mathematical ideas(at least in some cases) to reality. Who knows though, maybe it's a mixed bag.

[–]Complex Analysis 1 point2 points  (1 child)

If you use > instead of | before a quote you get a nice blue line

like this

[–] 1 point2 points  (0 children)

Thanks! :)

[–] 1 point2 points  (1 child)

There exists no mathematically rigorous formulation (and not for lack of trying, no one has successful been able to do it yet, and they have tried!) and it's a bunch of very adhoc mathematical ideas that are completely non-rigorous. And guess what. It works very well with experiment.*

Don't know if I've misunderstood, doesn't it terrify you that you might just be getting extremely lucky or working with a very special case? Seems kinda scary that an entire field is using mathematics just to hack for an answer :S.

[–] 0 points1 point  (0 children)

Not really. In fact a lot of mathematics done until the 1800s would nowadays be considered a wrong answer. Ideas like continuity, convergence, differentiability were not really well formulated (Euler just assumed things would converge) rigorously defined until the 1800s yet mathematics seems to go alone fine.

In actuality what they are doing is basically using a lot of complex tools (including complex analysis) to approximately solve the problems. A lot of it comes from asymptotic theory in which the resulting expression diverges if you take an infinite number of terms but converges if you take a finite number. It has some rigorous backgrounds but no one has successfully applied it to the Feyman path integral.

[–] 2 points3 points  (4 children)

Path integrals aren't rigorous? Wha?

[–] 8 points9 points  (0 children)

[–] 1 point2 points  (0 children)

Sorry. Clarified that. Most people I know just refer to Feynman path integrals as path integrals.

I normally say line integrals for the vector calculus thing.

[–] 0 points1 point  (0 children)

If you found a rigourous way to construct them then please line up for your Nobel prize.

[–]Geometric Analysis 0 points1 point  (0 children)

The path integral in QM is not the same thing as line/contour integrals in vector calculus/complex analysis - the latter are indeed rigorously defined.

[–] 3 points4 points  (11 children)

Thanks for the reply. QFT is actually an example of the exact situation that tends to rub me the wrong way. My friend who is currently doing the masters in theoretical physics has said his goal for next term is to not taking any more field theory courses because they make his blood pressure rise.

A moderately specific example of "math in the background" is how during my entire fourth year QM class the adjoint of an operator was never actually defined, and a lot of the functional analysis that was relevant to the soundness of the theory was never brought up.

In regards to how rigorous I would like my studies to be, I am currently reading Isham's Lectures on Quantum Theory. I find this type of exposition to be acceptable since he makes note of the relevant mathematics, even if it's just in passing reference. I've been told that he is trying to develop of version of QM using category theory, which is an example of the type of thing I'd like to learn in graduate school.

[–] 4 points5 points  (8 children)

QFT is actually an example of the exact situation that tends to rub me the wrong way.

Rather unfortunately you aren't really going to get much better then that. Like I said, people have tried to make a rigorous formulation but no one has succeed. There do exist "mathematical" treatments of the subject but most of the ones I've found are very technical and assume a knowledge of QFT all ready. Search around mathoverflow a bit for some references.

A moderately specific example of "math in the background" is how during my entire fourth year QM class the adjoint of an operator was never actually defined, and a lot of the functional analysis that was relevant to the soundness of the theory was never brought up.

Huh? How did they not define the adjoint of an operator?

Also, the functional analysis isn't brought up for, I think, two good reasons. The first is that isn't actually that useful in the context (and honestly I've talked to guys who work on this stuff and they agree it isn't that insightful) and secondly because they simply don't know it.

I've been told that he is trying to develop of version of QM using category theory, which is an example of the type of thing I'd like to learn in graduate school.

The types of folks who focus on a lot of the mathematical aspects of quantum mechanics are quantum foundational guys. They're basically trying to re-write quantum mechanics in a different way and attempting to discern what is necessary and unnecessary about the theory. Also interesting is the formulation of open quantum systems which uses a lot of the more applied functional analysis (projectors, POVMs, PVMs, and so on) however this is taught as an advanced graduate course at my university and is pretty difficult. It's effectively a research level field and it takes a fair amount of work to get there.

As for category theory, I've heard it come up a bit in terms of quantum computing and computer science but and made some stuff from IAS but it's fairly high level stuff (even in mathematics where it too is regarded as too abstract!). Frankly it's at the cutting edge of research and I'm more of the type of person who likes to apply the mathematics (I can be rigorous however!) and use it to calculate things as opposed to just coming up theories.

Also, how is Isham's book? I've heard of it but never read it.

[–] 0 points1 point  (7 children)

Huh? How did they not define the adjoint of an operator?

Well I guess the prof sort of did, she basically said it's the hermition conjugate of the matrix representation of the operator. I guess that's an operational (I think that's a pun) definition but I was expecting at least the definition in terms of the relation <Ax,y> = <x,A*y>. However she never even introduced what an inner product was in the course, just sorta used it.

The types of folks who focus on a lot of the mathematical aspects of quantum mechanics are quantum foundational guys.

Perhaps I should look into quantum foundations then, it seems like my type of thing.

Also, how is Isham's book? I've heard of it but never read it.

I had it recommended to me by the aforementioned physics grad student as well as another grad student in pure math. I haven't gotten into the meat of the book yet, but so far it has been quite good. He assumes a working knowledge of linear algebra although he does review the necessary stuff briefly. It's been right up my alley since for example he explains the difference between the measure theoretic probabilities used in classical mechanics (using ratios of generalized volumes) vs the probabilities used in Quantum Mechanics, which arise essentially from the Pythagorean Theorem (the probabilities are the cosines of the angles between the appropriate unit vectors and the vector in question, of course these vectors are states in your Hilbert space). He also takes care to note when an argument isn't mathematically valid but is good enough as far as physics is concerned, and usually briefly explains how it could be made valid, (thus allowing you to see why physicists need not be concerned with it)

[–] 0 points1 point  (6 children)

Well I guess the prof sort of did, she basically said it's the hermition conjugate of the matrix representation of the operator. I guess that's an operational (I think that's a pun) definition but I was expecting at least the definition in terms of the relation <Ax,y> = <x,A*y>. However she never even introduced what an inner product was in the course, just sorta used it.

Ah. My professors all defined it that way in terms of an inner product.

(thus allowing you to see why physicists need not be concerned with it)

the existence of a dual for example; Reisz representation theorem.

Perhaps I should look into quantum foundations then, it seems like my type of thing.

Based off what I've seen, definitely. As jjCyberia said, many of the physicists don't really care about the full measure theoretical definitions of things (since that takes time to develop which physicists never really get to) and they don't really care. But take a look at foundational quantum mechanics stuff as they are interested in really finding out what quantum mechanics tells us about nature. For example, some people believe that Born's rule isn't logically independent of the other axioms of quantum mechanics and search for alternative formulations.

See, for example, this paper for the type of work they do.

[–] 5 points6 points  (5 children)

Trouble is, that isn't what the adjoint of an operator is! Here, even the "good" definition is still bad!

This is exactly the sort of thing that gets physicists into trouble! This definition is actually for whether an operator is symmetric, not self-adjoint. In finite dimensions, these two concepts are equivalent. In infinite dimensions---for example, the Hilbert spaces in which quantum mechanics take place---these are different concepts.

For a physicist to not worry too much about the technical definition of self-adjoint is pretty bad; but for a physicist to not even mention that a technical definition exists is shameful.

[–] 1 point2 points  (1 child)

True. And I did have the professors mention problems with take the infinite limit of finite dimensional quantum mechanics however they said such issues were to be brought up in a mathematical class as technicalities.

Like Hilbert spaces. We want our functions to be L2 but they aren't. Do physicists care? Not really. They just shove them into the set of functions and continue along their merry way.

For a physicist to not worry too much about the technical definition of self-adjoint is pretty bad

Eh, I disagree. I don't think it's really that important until you're at a pretty high level. In terms of adjoint operators, the only time it really became an issue was when were in a general relativity course where, since we're using a psuedo-Riemannian manifold, things don't work as nicely.

[–] 0 points1 point  (0 children)

Okay, okay, I get it: physicists do the math with a wink, wink, nudge, nudge, say no more. At least you get there in the end, right? So if you end up in the right place with or without the technical details, then why would you ever worry about them?

Why? Because the technical details can tell you about the physics!

Here's an example that I posted about before: consider the momentum operator acting on L2 [0,1]. Basically, we're talking about a particle in a box. Now, you want the momentum operator to be self-adjoint. As it's a derivative operator, you might think that you have to restrict to acting only on differentiable functions in L2 [0,1]. But when you compute the adjoint of the momentum operator restricted to differentiable functions, you find that it does not equal its adjoint! When you work through the math, you find that you have restricted the momentum operator to too small a class of functions.

In fact, you need only restrict to so-called absolutely continuous functions. These functions are only almost-everywhere differentiable, so their derivatives are allowed to be discontinuous. Among other things, this means that wave functions are allowed to have cusps. Wavefunctions with cusps should sound familiar---you encounter them when you solve the attractive delta function potential. You also find discontinuities in the derivatives when you solve the particle in the box.

Even more, the fact that we need consider only absolutely continuous functions hints at some really wild physics: for example, if f(s) is the function equal to 1 for every rational x between 0 and 1, and zero everywhere else, then [; \int_0^x f(s) ds ]; is absolutely continuous. Which means that wavefunctions can be much more wild than you would otherwise have thought.

[–] 0 points1 point * (2 children)

Careful. Those Hilbert spaces aren't necessarily two*infinite dimensional.

[–] 0 points1 point  (1 child)

I suspect you meant that they're not necessarily infinite-dimensional. Which is true. My bad---I so rarely work with finite dimensional spaces that I often forget that they even exist.

[–] 0 points1 point  (0 children)

Yes, I did. The first counterexample I thought of was two dimensional. :P

Though you can pretty trivially come up with one dimensional ones, too.

[–] 0 points1 point  (1 child)

I can say from personal experience that if you're looking for guys in the physics department who apply spectral theory and non-commutative operator algebras to quantum mechanics, you're going to spend a long time looking. (there are math guys who do this, and do kind of contribute to the field. if you want I can pass on names. )

almost all of the physicists working on quantum foundations work in finite dimensional systems, mainly because they're easier to think about as well as the simple fact that's the direction quantum computing is taking it.

In my office the term POVM (positive operator valued measure) get's thrown around a lot but almost all of them don't know, and don't really care to know, its full measure theoretic origin.

There are some people who are looking at trying to merge some of the ideas from topology with the field of quantum error correction.

But really as its been said earlier in this thread, physicists use math to describe how nature works. And the mathematical model you develop usually fails in two ways. the first is that the math breaks down in some limit. The second, and MUCH more likely is that your model is too simple and doesn't really capture the way the world actually works. So you can either test for mathematical robustness, OR you can test to see if the thing just bloody well works in the typical but realistic case.

I think most physicists stop worrying about mathematical rigor when they realize that the approximations they made to get to something tractable will break down long before any function ceases to be analytic.

[–] 0 points1 point  (0 children)

Those names would be great. I spent a portion of my summer reading about spectral theory and C* algebra's specifically because I was aware of the connections to QM (although it was neat enough on it's own anyways)

[–] -1 points0 points * (14 children)

If one ignores the mathematics in attempting to come up with the answers to problems, how can you even claim to have "explained" anything? Why even bother with physics at all if "it works" is the ultimate justification for anything? Why not make a totally phenomenological theory that doesn't explain anything at all?

There is really no justification for ignoring the math if you're a physicist, except for the sake of time. To ignore the math is to ignore the physics, because the physics includes the math.

EDIT: I'm talking here about fundamental physical theories. I might accept that it's not really that useful to be super rigorous about a random condensed matter theory whose existence is a total practical matter, but when it comes to something as fundamental as quantum field theory, ignoring rigour (in the long run: I understand that one must be tentatively unrigorous to get anywhere) is just undermining the entire subject.

[–] 5 points6 points  (11 children)

but when it comes to something as fundamental as quantum field theory, ignoring rigour (in the long run: I understand that one must be tentatively unrigorous to get anywhere) is just undermining the entire subject.

What do you mean by rigour? As I said, the path integral is not rigorously defined and no one has done it. I mean in QFT people completely neglect all matters of convergence and interchange integration and summation and limits (suck on that dominated convergence!) because it gives the right answer and agrees with experiment. People are working on a rigorous quantum field theory but no one has found it and it's a difficult problem. Since most people who use QFT are simply using it as a tool to calculate things, it being rigorous or non-rigorous is irrelevant. It gives the right answer

If one ignores the mathematics in attempting to come up with the answers to problems, how can you even claim to have "explained" anything? Why even bother with physics at all if "it works" is the ultimate justification for anything? Why not make a totally phenomenological theory that doesn't explain anything at all?

For what it's worth people do work on phenomenological theories, see hep-ph, as you're probably aware.

I think there is a difference between ignoring rigorous mathematics and mathematics in general when solving a problem. Perhaps I wasn't clear but I was referring to mathematical rigour, not mathematics in general. Did Einstein care about the partition of unity on a manifold? Not really. Would a mathematician? Definitely. Does it help you calculate the precession of the perihelion of Mercury? No.

EDIT: I'm talking here about fundamental physical theories.

For this, I would say some attention to rigour should be paid but, then again, not as much. Insight into nature has been gotten through looking at very "mathematically rigorous" things in terms of gauge theories for example, but I wouldn't say that qualifies at rigour. I think what is at issue here is that to me, as a physicist, I don't really care about the assumptions that my manifold is Hausdorff or second countable. Einstein did his theory perfectly well without it and to me, it's completely irrelevant to the discussion at hand. It doesn't really offer any physical insight into the problem other than it allows it to behave in ways I would like it to behave which I assumed it would. If I put on my geometers hat, I would definitely care about these things since I could construct many strange situations without these assumptions that I, as a physicist, could disregard for being unphysical.

[–] 1 point2 points  (10 children)

The problem is that you're basically defending the idea of coming to a conclusion on wrong justification and thinking it's ok just because it's right. If I guessed the perihelion precession of mercury by a wrong calculation but it turned out by luck to be right, is that ok?

[–] 4 points5 points  (9 children)

I don't really understand what you're disagreeing about. I am arguing that mathematical rigour is not needed in physics. I'm not saying to ignore the mathematics, I'm saying that ignoring rigour is perfectly and regularly justified.

[–] 1 point2 points  (8 children)

How? Rigour IS by definition "the justification of your steps" in arriving at a (mathematical) conclusion. If you just skip that justification, how have you explained anything when you conclude that g = whatever?

[–] 0 points1 point  (7 children)

Oh, I see. Rigour, in my mind, and I think many other people, is being very careful to define and prove everything along the way. For example, when a physicist does general relativity they implicitly assume an underlying manifold that is C\infty but a mathematician would go crazy. They would need to talk about the topological properties of a manifold, prove existence and uniqueness, and so on.

Another example. When a physicist defines a tensor they simply define it as something that transforms under a certain law. When a mathematician defines it, it's a multi-linear mapping from vector spaces. There are tons of properties you can prove about these things that a physicist just takes for granted.

Or when a physicist interchanges integration and summation. We do so without a care for convergence and just assume it does. A mathematician would go crazy because you can't do that!

I think what you're talking about is the actual steps when solving a mathematical problem which is different from rigour.

For your example, a physicist can normally guess an answer based off physical reasoning (for example Feynman did this with liquid helium) and then later go through and show their reasoning to be correct by deriving it from the equations.

[–] 0 points1 point  (6 children)

So what's wrong with proving the necessary facts along the way? That's precisely what "justification" is. I'm not suggesting that we be GENERAL, just that we prove the necessary things. Also:

1. Why would a mathematician go crazy assuming nice properties of the manifold? As long as you STATE that you're assuming that, there's no problem.

2. The physicist's definition is un-intuitive IMO, but that's just opinion. The real problem is that physicist's don't define what they mean by "transforming under a certain law". They don't even MENTION manifolds once when they talk about tensor(-fields) (to undergraduates, at least), which is completely messed up because it's completely confusing to be talking about changes of coordinates then. They refuse to give their definition in a clear manner. This irks me, but it's not really an issue of rigour.

3. Exchanging integration and summation temporarily to see if the calculation works out is fine. But how could you say your answer is justified if you don't justify it? Just because the SYMBOLS work out aesthetically is by no means justification for the conclusion, unless you prove that the aesthetics follow the logic.

So what I'm saying is that it's ok to be specific, as a physicist, but it's really messed up in my opinion that they act as if it's ok to charge ahead with calculations without figuring out why it should even be true. What's impressive about that? You haven't shown anything!

[–] 1 point2 points  (1 child)

Exchanging integration and summation temporarily to see if the calculation works out is fine. But how could you say your answer is justified if you don't justify it? Just because the SYMBOLS work out aesthetically is by no means justification for the conclusion, unless you prove that the aesthetics follow the logic.

Because it gives the right answer that agrees with experiment.

Recall the problem of infinities in quantum field theory. Our calculations diverged. So what did physicists do? Simply subtract out the infinities. A horrible thing that is completely undefined and doesn't make sense. But it gives the right answer that agrees with experiment to 12 decimal places.

The thing is physicists aren't in the nature of ensuring the mathematics is rigourously defined. All the time in my physics courses the professor would be "we're going to do this even though we can't really". Just like how physicists divide by dx and whatnot.

The physicist's definition is un-intuitive IMO, but that's just opinion. The real problem is that physicist's don't define what they mean by "transforming under a certain law". They don't even MENTION manifolds once when they talk about tensor(-fields) (to undergraduates, at least), which is completely messed up because it's completely confusing to be talking about changes of coordinates then. They refuse to give their definition in a clear manner. This irks me, but it's not really an issue of rigour.

I am agnostic on this issue. I learnt GR from an applied mathematician so he did give the definition of a tensor as a multilinear mapping from tangent and cotangent spaces. And technically we're dealing with tensor fields but whatever. As for the transformation law, I have no qualms with that. I don't really understand what you mean by how it isn't defined. A tensor is something that under a change of co-ordinates transforms according to a certain law.

Why would a mathematician go crazy assuming nice properties of the manifold? As long as you STATE that you're assuming that, there's no problem.

Most physicists do. It's typically stated in the introduction or equivalent that everything is as many times continuous or differentiable as necessary.

[–] 0 points1 point  (0 children)

See my response to nobaru below this in regards to why "giving the right answer" isn't really acceptable imo.

As for tensors, they simply weren't defined clearly. Compare the definitions to a math book (one that mentions the classical definition) and a physics book and you should see a pretty big different. I really can't understand the definitions given in physics books without already knowing what it is. But I can't really articulate it very clearly here since I don't have examples at my immediate disposal.

And physicists definitely don't state their assumptions of niceness very often, they pretty much just start doing things.

Basically, my opinion is that physicists need to learn to respect math more and appreciate its clarity. They don't have to be mathematicians, but the fields will always be divided in an unhealthy way (in places where there should be interaction) if physicists refuse to acknowledge math as more than technical pedantry. Math SEEMS like technical pedantry if you don't work at it, but the reality is that it actually makes things easier, IF you put the (difficult) effort in to understanding it.

[–] 0 points1 point * (3 children)

So what I'm saying is that it's ok to be specific, as a physicist, but it's really messed up in my opinion that they act as if it's ok to charge ahead with calculations without figuring out why it should even be true. What's impressive about that? You haven't shown anything!

Physics is an experimental science in the end : talking of oughtness/trueness of a theory, or simply showing (ie, proving) something is meaningless, without experimental verification : asking for such a thing is to ask for the "why?" (Why is it, that CP-Symetry is broken ? Why my carefully crafted, all-symetrical, model is broken ?), where physics can only provide the "How?" (CKM-matrix). Mathematics, on the other hand, is deduction from axioms and logical principles that cannot be changed (by experiment), and thus, mathematicians kindof know the end of the story : rigour can be defined as the characteristic of a fully consistent logical deduction, and the "grand scheme of thing" has in fact an answer to the "Why".

Also, this comment.

[–] 0 points1 point  (2 children)

I'm not saying to talk about oughtness WITHOUT experimental verification, but rather to talk about it PROPERLY in addition to experimentally verifying. Otherwise what's the point in theoretical physics at all? This is a philosophical question with many answers, but in my opinion one should try to deduce the results of experiments within a small number of physical postulates and a mathematical framework, without saying "well the method of looking at my magic 8-ball tells me that atoms DO exist, and it gives the right experimental answer, so I'm satisfied with using this method in my theory".

[–] 1 point2 points * (1 child)

It's a philosphical viewpoint. Some may say that physics is only models. That we can never encode absolutely true statements about the structure of the universe in rigorous mathematics, that all we have are algorithms for attempting to predict the future. In this viewpoint, why bother with the mathematical rigor? You've got something that works, who cares what else is there. We're only working with approximations anyways. Although its hard to argue that a bit more rigor would have no bearing on the accuracy of the models

Others may try to absolutely map mathematical space to physical reality. In this case it seems like mathematical rigor has a distinct necessity.

[–] 0 points1 point  (0 children)

Just because you are working with approximations, doesn't mean you should undermine your model even if it won't be 100% true to reality.

[–] 0 points1 point  (0 children)

This tends to annoy me because there is often some implicit mathematical theorem working in the background which makes what the prof is talking about work but it's never mentioned.

Can you give me examples?

As a simple example, Newton's second law is a second order differential equation. How do you know there exists a general solution to it, say specifically for a constant gravitational field (d2 y/dt2 = g)? That is, how do you know y(t) = y0+v0t-(1/2)g*t2 is the most general solution? Maybe some funny sine/exponential function combination satisfies that differential equation. No it can't? Why not? Existence and uniqueness is something you'll see treated suitably in a differential equations class, but not a physics class.

[–] 12 points13 points  (0 children)

Physics isn't math, if something is known to be proven then there isn't a reason to dive into it during class. There are other topics to discuss and the professor can't spend precious lecture time discussing the mathematics in rigor. This is not an issue, you need to relax. Do the math on your own time if its important to you.

[–] 30 points31 points  (1 child)

Bit of a rhetorical question. Us physicists will stop abusing math when you stop hitting your wife.

[–] 7 points8 points  (0 children)

But she likes it.

[–] 4 points5 points  (0 children)

Can you give a more specific example? I'm trying to understand the problem.

[–] 4 points5 points  (0 children)

Physics deals with theories that are approximations of reality based on experiment. Those theories are rarely complete or totally precise, so there's usually some amount of uncertainty or disagreement between theory and observation. When we try to apply mathematical constructs to those theories, the precise rigor of a mathematically complete solution is pretty irrelevant, since what you gain in rigor is swamped in magnitude by the errors in the theory. Rigor is often not worth the added complexity in notation and symbolic manipulation.

Other times it's simply not worth the time to explain the finer points of the mathematical underpinnings of a theory given the limited time allotment of an undergraduate class. (Graduate classes might even go into less detail but over a broader range of topics, rather than greater detail on a specific topic.)

[–][deleted]  (2 children)

[deleted]

[–]Algebraic Geometry 6 points7 points  (0 children)

Easier =/= less rigorous. Think of a race car driver as a physicist and the guy who designed and built the race car as a mathematician. A great driver doesn't need to know how the car works, and building the car doesn't mean you can drive it faster. It's just a matter of preference.

[–] 0 points1 point  (0 children)

If the physicists are doing things correctly in an easier way than you would like to do them, then the way you'd like to do them is worse.

As long as it works out that's fine, and since most of what you do has been done many times before and verified experimentally, of course it works out fine despite the fact that you don't understand why.

The problems come when you're trying to do something fundamentally new.

When physicists' "intuition" fails they don't notice. That's how you get General Relativity "experts" that don't understand how there can be flat space forms that are compact.

[–] 3 points4 points  (0 children)

Man, if you think physicists abuse math, you should see what astronomers do to it.

"Eh, that's good to an order of magnitude..."

[–] 2 points3 points  (1 child)

Physics is constrained by reality, so certain mathematical rigours can be discarded. It drives the mathematicians crazy, but it would be a huge waste of time otherwise. Also, physicists in the past "invented" mathematical tools to suit their needs and later discovered that those tools already existed in mathematics. In situations like that, the physicists didn't use the same rigorous techniques mathematicians did.

[–]Physics 2 points3 points  (0 children)

There is, but they don't teach it in undergrad. In undergrad physics they teach you the math you need to solve undergrad physics problems. As evidenced by Ed Witten winning the Fields medal, there are physicists who do good math in the course of their physics.

[–] 2 points3 points  (1 child)

Great rant!

I ran into this when I was an Astrophysics Major 20 years ago (I'm now a software engineer).

I've always been told that the science ladder progresses like this (from purest to most practical):

mathematics->physics->chemistry->biology->engineering (all)

Each progression moves further away from the 'purity' of mathematics. This used to bother me, but i got over it fast because it turns out, physicists rely on mathematicians to develop the tools needed to do physics, chemists rely on the physics tools, and biologist rely on the chemist tools, etc.

Another way to look at it is that you move from the purely theoretical (mathematics) to the purely real (engineering).

If engineers had to know all of the pure mathematical proofs and theorems, all the physics, all the chemistry, and all the biology to build a fMRI machine, we would never get anything done.

If you look at the work of Einstein and Plank, Plank had a much better understanding of discrete mathematics and formed the foundation of Quantum Mechanics, an area of physics that is much more reliant on mathematical theory than Relativity. (This is my opinion and will be flamed)

In fact, if it really bothers you that much, stay in the physics areas closer to advanced theory such as QM, Manifold Mechanics, N-dimensional space mechanics. Hell, even gravity is proving to be a much more mathematically involved area than Einstein ever predicted.

[–] 1 point2 points  (0 children)

I don't know if I agree with your ordering there, I'd put engineering before chemistry or biology (I'd put it miles before biology, hell there are social sciences courses that use more math than most bio students learn). By far, the most talented pure mathematician I've studied with was an engineering student who treated it like a hobby. You at least get a fair number of engineering students with an understanding of pure math, especially with computer and electrical engineering, the same which can not be said for bio or chem.

[–] 2 points3 points  (0 children)

Relevant, as always.

[–] 1 point2 points  (1 child)

Well, Im a CompSci minor and Ive had my CS profs dismiss my arguments that something wasnt true, or mathematically can be defined much more simply.

Not saying this as an authority but, I guess you gotta just grin and bare with the butchering of math by other majors, and use your mathematical rigor to be a better physicist than your less rigorous counter parts.

Also, another problem Ive run into in CS is profs explaining to me (after class) that theyre trying to teach CS not CS/Math. I dont know if this is relevant to you, but thats my experience.

[–] 0 points1 point  (0 children)

There are some pretty nonsensical papers by physicists, particularly when they're trying to branch out into other fields, armed with only their "physics intuition" on how stuff should work mathematically.

[–]Physics 1 point2 points  (0 children)

Only when it stops working.

[–] 1 point2 points  (0 children)

I hope not. Physics is a science, and our knowledge of it is refined by experiment. It may be guided by math, but you could fill entire libraries with well-reasoned mathematical hypotheses that have been proven wrong by experiment. When physicists truncate a Taylor series at 3 terms, which they do all the time, it's often because that's all they can be completely certain of, not just because they're lazy. To say that the phenomenon is defined by the function those terms approximate is just a guess, and insisting on applying that guess rigorously is false precision, no matter how elegant the theory behind it.

[–] 1 point2 points  (0 children)

There are really only ever two things going on in physics Re: rigor. Either -

1. It is possible to justify things rigorously, but doing so requires introducing a bunch of extra heavy machinery to cover special cases. For example, for finite dimensional hilbert spaces, adjoints are easy to define, eigenvalues and eigenvectors exist for self adjoint operators, etc. Operationally, all of this logic carries over to the infinite dimensional case. To prove this rigorously, though, requires a deep excursion into functional analysis. It gets even worse when you try to talk about eigenvectors of the position and momentum operator (distributions, rigged hilbert spaces, etc). Given any such issue, there are always some physicists who have to deal with the math for their particular research problem, at which point they learn it (usually because something that seemed like it should be true spectacularly fails).

2. The math just doesnt exist yet. Throughout history, physics has always played a role in pushing the frontiers of mathematics. Calculus is a classic example, but even today we have the Yang-Mills mass gap as a millennium prize problem, path integrals need a rigorous definition ,etc. In these cases, the nonrigorous math of physicists can be viewed, if you like, as a systematic way of formulating conjectures.

[–] 1 point2 points  (0 children)

Look at the history of the two subjects: concepts bounce back and forth between them: sometimes a physicists will hand-wave their way through some new maths needed to solve a problem (Newton's calculus would be the big e.g. here) and then mathematicians back fill the rigour. And sometimes mathematicians come up with some new abstract, useless but interesting thing (quaternions, say) and it sits around for a while until physicists find a physical problem that it helps solve.

This is actually very interesting, because the physicists' hand-wavy problem solving approach to maths, intimately tied in to experiment, so often reveals a new route for mathematicians to rigorously explore and the maths does all hang together in the end.

[–] 1 point2 points  (0 children)

In my experience (I don't know if yours is similar, not having attended your university or lectures), physicists often skip over details if it's either a straightforward extension (e.g: this and that behaves similar to normal 3D vectors) or intuitive (usually followed by some physical example or an appeal to intuition, i.e. the eigenvalues of this system should be negative, since we don't expect concentrations of ions to increase in a diffusion process).

This is done so they can focus on the physics. They want to transfer an intuition of the physics to their students, so they get a feel for the problem, it's solutions and the methods with which to solve it. The details (and the ability to solve related problems) comes from studying and doing exercises. If you prove everything mathematically rigorously, then you're switching back and forth between physics and mathematics thereby obscuring the point you're trying to make.

So from the physics students (and prof) point of view, they're doing a straightforward calculation that they intuitively understand in order to explain some sort of physical behavior. The prof want to transfer an intuition about this physical system to the students, and then you but in and bury it in (in their eyes) redundant mathematical rigor.

Another way to see it is that physicists typically like to study a problem from large scale to small. They first want to get an idea of the behavior of a system, then an intuition on why, then some insight in underlying laws of nature, then an idea how these interact etc. etc. Typically lectures are also given at this large scale, and so focusing on mathematical rigor (which is small scale to them) obscures the point being made about the large scale being discussed.

That's the reason they roll eyes etc. when you make the remarks described in your post.

[–] 13 points14 points  (4 children)

Sounds like you need to chill on the self praise.

[–] 6 points7 points  (1 child)

I'm sorry if I came across as self aggrandizing, that was not my intention.

[–] 3 points4 points  (0 children)

You didn't come across that way at all. You asked a legitimate and interesting question with great details that helped the discussion.

[–]Algebraic Geometry 4 points5 points  (0 children)

Sounds like you have an inferiority complex.

[–] -2 points-1 points  (0 children)

Ah, the physicist who refuses to accept his butchery of mathematics. Classic.

[–] 1 point2 points  (0 children)

Physics and math are different subjects. Physics uses math, but they are not and shouldn't be the same thing. Physics is about describing reality by building models that make predictions. Math is the language we use to describe the models, but mathematical rigor isn't the point. Complaining about a lack of rigor when mathematics is used in physics is like taking an accounting class and complaining that the professor didn't show the underpinnings of arithmetic with sufficient rigor.

That said, there are some physicists who do focus on mathematical rigor. String theorists in particular are often at the cutting edge of math (Ed Witten even has a Fields Medal). So it is possible to focus on theoretical physics in a mathematically rigorous fashion.

[–]Representation Theory 0 points1 point  (0 children)

Here's my perspective on the issue. On one hand, QFT works. If you study it, you will learn to do the calculations correctly, understand what the calculations mean, and be able to accurately model fundamental processes and occurrences in the natural world. This makes it a successful physical theory, and, even if you are a mathematician, you should learn this well before attacking rigorous mathematical foundations.

Additionally, trying to ground physical theories with mathematical axioms is ABSOLUTELY USELESS to physicists if it does not aid physical intuition.

HOWEVER, as a math/physics major applying to grad school in pure math, I believe that pure math has an indispensable role in physics. I believe that many mathematical formalisms have a significant role in understanding and creating physical theories. I think that the search for mathematical foundations in physics can be incredibly productive, if the search is done with an eye toward exposing the symmetries and basic properties occurring in physics.

An example: Hilbert Space formalism has done wonders for our understanding of QM. On the other hand, 'Rigged Hilbert Spaces' (an attempt to make rigorous some common QM calculations) appear somewhat ad-hoc and are completely unhelpful for the physics.

Good luck to you with whatever you do!

[–] 0 points1 point  (0 children)

If someone omits unnecessary rigor, that's one thing, but I hate when classes are taught in such a way that I can't see how I would come up with a similar result myself. If you show people how to do a particular calculation, but you've obscured enough of the motivation that they won't be able to do a closely related calculation, you haven't really taught anything.

[–] 0 points1 point  (0 children)

Math is to physics what physics is to engineering: a tool.

[–] 0 points1 point  (0 children)

I've actually gotten the impression that there's not such a clear divide between physics and mathematics as you get higher up. For example, Witten did some amazing work with topological quantum field theory that put the Jones polynomial in the context of physics (mathematically rigorously, of course). Then Reshetikhin and Turaev took the cue from Witten's work and came up with a way to view knot invariants as maps between representations of quantum groups, and in so doing found (as per Witten's conjecture) that knot invariants yielded 3-manifold invariants at a root of unity - pretty amazing stuff.

I would think that most serious theoretical physicists pay attention to mathematical rigor. But when they're teaching a course they usually don't have the time to develop the rigor behind the results they're using, since then they would effectively have to teach multiple courses at once. For instance, it would be impossible to teach quantum mechanics in a reasonable amount of time if one were to delve into all the subtleties of functional analysis i.e the world of L2 (Hilbert) spaces. Or for a more extreme example: the mathematics of classical mechanics is captured by symplectic geometry, but it's highly unreasonable to expect someone teaching Hamiltonian mechanics to first cover all of smooth manifolds and symplectic forms beforehand.

[–]Discrete Math 0 points1 point  (2 children)

getting a BSc Hons in Pure Mathematics and Physics, as well as a BA in Computer Science

Just out of curiosity, what does this mean? Are you triple-majoring, or is one degree in "Pure Math and Physics?. Because I've never seen a major that combined "Pure Math" and "Physics" together. Pure math usually kind of requires it not be paired with anything. :-P

[–] 1 point2 points  (1 child)

Okay well technically the BSc degree is in Applied Mathematics and Physics and the BA is Computer Science. However the guy who designed the Applied Mathematics and Physics program at my university made it extremely flexible (via lots of electives) so that it can effectively be turned into a Pure Math and Physics degree. The only truly applied courses I have to do are one in Numerical Analysis, and a couple differential equations courses. He said he did this so people could take courses like Group Theory, Topology, etc which are often useful to theoretical physicists and still meet graduation requirements.

[–]Discrete Math 0 points1 point  (0 children)

Ah, gotcha. That sounds nifty.

Just make sure that universities know your classes were purely-oriented. BS vs BA can have carry connotations with them. If your CS is a BA, that kind of sets the bar for be BS in math is going to be pretty applied. Surely they'll be smart enough to actually look at your classes, though, so never-mind, on second thought.

[–] 1 point2 points  (3 children)

What does it mean to abuse math?

[–] 0 points1 point  (2 children)

My physics profs and classmates are quick to dismiss mathematical rigor for the sake of calculation. This tends to annoy me because there is often some implicit mathematical theorem working in the background which makes what the prof is talking about work but it's never mentioned.

[–] 3 points4 points  (1 child)

Oh, right. Have you tried asking them to elaborate on that background theorem they ignore?

edit nevermind...

[–] 0 points1 point  (1 child)

There was a guy in one of my physics classes who got upset if we used the wrong symbol in a secular determinant. In math classes, you use lambda. Every time we didn't he got upset, to the point of him actually having to say something about it.

It was actually kind of annoying.

It seems to me that people in physics classes who are more concerned with mathematical formalism than actually learning the physics involved in a situation are completely missing the point.

[–] 0 points1 point  (0 children)

In math classes, you use lambda. Every time we didn't he got upset, to the point of him actually having to say something about it.

Sounds like an idiot. I'm sure you also met some idiot physics students in your time.


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d If the volume and surface area of a sphere JEMrare.com

# Volumes MCQ 1: If the volume and surface area of a sphere are numerically the same, then its radius is…….?

a) 1 unit                 b) 2 units                        c) 3 units                   d) 4 units

Solution:

In this problem the 3D shape of which area and volume is discussed – is a round  sphere !

We should know the formulas for surface area and volume of a sphere if radius is given.

If radius of a sphere be “r” then;

$Volume = \frac{4 }{3}\pi r^3$

and

$Surface Area=4\pi r^2$

Since it is given that volume and surface area are equal to each other – (as per question statement )i-e

$4\pi r^2 =\frac{4}{3}\pi r^3$

from which after calculation we have,

$3 =r$

i-e its radius is 3 units !

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# A doubt regarding the proof of kirchoff's law of thermal radiation

1. Apr 6, 2012

### ashutoshd

In the proof that I have studied, there is a Isothermal cavity that behaves as a black body. there is another small opaque body inside the cavity at the same temperature with emissivity e, absorptivity a and area s. now the irradiation on the small body is Eb = σ times T raised to 4. now the small body absorbs "a" times Eb. And emissive power is e times Eb. Then by conservation of energy we have a*Eb= e* Eb. therefore a = e. What i havent understood is that since the cavity emits Eb watts per unit area, shouldnt we multiply it by the internal area of the cavity to get energy flow in watts and then equate it with the emissive power of the small body also multiplied by its own area. This way we will be equating the rates of energy streaming rather than the radiation fluxes as has been done in the proof. But since the areas of the small body and the internal area of the cavity are obviously not equal, will a not be equal to e then?

2. Apr 7, 2012

### conquest

are emissivity and absorptivity extensive or intensive properties?

3. Apr 7, 2012

### Dickfore

I thought that Kirchoff's Law claims that:

$$e/a = f(\nu, T)$$

where f is a function function which coincides with the emissivity of a black body.

4. Apr 7, 2012

### Staff: Mentor

Some parts of this energy flow will hit the cavity again, as long as your body in the cavity is not really close to the cavity walls (and in that case, the areas are equal).
In the general case, it can be tricky to calculate how much of the emitted radiation of the cavity hits the cavity again. But you can use the surface of your (convex?) body to determine this. Or just ignore the geometric details and assume a constant energy density in the volume.


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# Tile Based - "Correct way"

This topic is 4970 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Im making a top down view 2d tile game, i draw the entire map image to a buffered image, now when the user uses the arrow keys i simply redraw the map croping it differently. Is this the correct way of doing this is a game? I dont have any experience in game programming and i cant see into the future and see problems i might have atm. Is it ok doing wat i am currently doing, or to make things easier in the long run should i change my method to another method lyk: Giving the player a position in the map array, then redrawing the entire map and player to the buffered image and showing that. Cheers, Nick

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I don't see any problems with your way, as long as you divide a large map into chunks. Then, just don't let the, "camera" go past them and when the player goes off screen, load the new map.

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The way im currently doing it the player never actually leaves the middle of the screen, is that ok?

And also the reason i was asking the first question is because ive started thinking about collision detection, and im just wondering if the way i do it will work still? its much easier putting the user in the 2d array for the map, then u can see if they are going to walk into lyk a wall of sumtin.

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Ok, couple of points. Firstly, I'm not sure you want to be drawing the entire map as that almost defeats the purpose of using a tile-based system. The idea being that the map is broken down into managable chunks so you only need to be checking and drawing those around immediately around you (within the camera view). Bare in mind, if you know your camera location and you know how many tiles fit on the screen you can determine which tiles you should be drawing at any frame.

Secondly, you'd never need the player inside your array of tiles. When it comes to collision detection you will want to query individual tiles but that can be done without physically placing a player in with them. So let's say you wanted a collision check for the left side of your player (assuming this is a top-down view - the theory still applies to side-on though)

1. Determine the world coordinates of the players left side. This would probably be
// determine world position of players left sideplayer_location_x - player_width/2

Obviously your variable names will be different.

2. Determine which tile that point is inside.
// determine which tile players left edge is inside/overplayer_leftside/tile_width// take note: that would only determine which collumn the left side is in.// you'd need another check with the player center to detemrine the y value

3. Act appropriately. If the tile is solid, push them back to a point where they are no longer colliding.

As for the player never leaving the center of the screen that isn't a problem. Given time you could add some smoothing to the camera but for now just keep it functional. Hopefully your camera is contained within a class so you can easily update it's behaviour without too much hassle.

Anyway, that's the basic gist of the process so I hope it helps.

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The reason i and drawing the map the way i am is because i only need to draw the 2d array to the buffered image once (at program start), from then on i just draw a crop of that image to the bufferedStratergy. Is this still incorrect or not as efficient?

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If this is DirectDraw7 or GDI then prepare for counter-intuition.

Blitting one screensized subrectangle from a very large surface (many times the screen size) is usualy a lot slower than blitting many smaller rectangles stored in seperate small surfaces.

The idea here is that the very large surface cannot be (or is not in the case of GDI) stored in video memory yet the driver is designed to let the hardware perform all actions involved in the blit. Thus the entire large surface is shuttled through the agp/pci bus every time you blit even though you only wish to blit a small portion of it.

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Ok, well the main reason im drawing the complete map to buffered image then drawing a crop of that is because i think it is the easiest way to draw my bullets, it draws them to the map so that if i move it still goes to that position on the map, not that position on the screen? I draw a bullet to the map at the user x + cropX and y + crop Y, and it destination is x + cropX and y + cropY.

You get wat im saying?

EG:

FULL MAP

|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||....................C......................................................||||....................C......................................................||||............1....0..C....................0.................................||||....................C......................................................||||....................C......................................................||||C.C.C.C.C.C.C.C.C.C.C..........................0...........................||||...........................................................................||||...........................................................................||||...........................................................................||||...........................................................................||||...........................................................................|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

. Is grass
1 is the player
0 is a bullet
C is the cropped image

All bulets have been shot from the current pos of the player.
The are drawn to the entire map, only the crop is shown, this is the simplest way i could think of.

Any better way?

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Yeah, I get ya and to be quite honest it is probably rather inneficient to crop such a large image each frame. Although you have only 'drawn' it the once, as the AP points out, the clipping process is not going to be too efficient (irrespective of the API). It will be trimming down a large amount of image data which is not adviseable. Furthermore, if you wanted a larger map, that would mean a larger initial image which would further slow the cropping process.

What I would suggest is a system that only draws the tiles currently in the view of the camera. This will mean a loop of drawing a couple of dozen tiles each frame but that will actually be faster, and more flexible too.

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Any objects (i.e. bullets are renderd after the tile's have been drawn, you draw them at theur current position _minus_ the camera coordinates. This allows the objects to scroll around the screen as well as the map.

I dont like the idea of cropping a huge image. I've seen a guy do both techniques (clipping and tile-based rendering) and the clipping one runs slower generally and takes longer to load (loading/rendering a huge image takes longer than loading a couple of tiles), so all round is just slow. Stay clear of it and do a nice tile rendering loop.

Keeping the player in the middle of the screen... uhmm... the player should be able to move anywhere, i think it works out better if the player has coordinates relative to the map, then the camera is positioned in such a way that it always appears that the player is in the middle of the screen... Maybe its just the way of how i think objects exist, but i think object collsion etc will be easy as you wont have to keep adding camera coords to the player's position every time you wanna compare it agains another object in the level...

I'd check out tutorials on 2d-tile-based-rendering techniques, google it.
You should pick it up in no time!

Chris

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I agree that this is a very poor way to do this. The only time this would be a good idea was if the map was very small (only a couple screens) and you're doing multiplayer-split-screen or something. Start each frame from a blank slate and draw everything on the screen, blit, and repeat.

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