DrRocket

Hello fellow hobby-scientists,

the normal differential equation-based definition of the trigonometric functions sin and cos are as solutions to the well-known second order DE y''(x) = -y(x). It occured to me that they can also be defined by a simple first order functional differential equation on the form

 

y'(x) = y(-x),

 

which for y(0) = 1 has the solution y(x) = sin(x) + cos(x). This obviously means it is possible to construct sin(x) as 0.5*(y(x) - y(-x)) and cos(x) as 0.5*(y(x) + y(-x)). As an attempted short section discussing this fact was swiftly and mercilessly rejected from the wikipedia article on trigonometric functions I'd like to ask you guys what you think. Does this definition add to the understanding of the nature of the sine and cosine functions and their apparent relationship to the exponential function (which is defined by y'(x) = y(+x), y(0) = 1..) ?

 

As this is my first entry, try to be nice

The fundamental relationship of the sine and cosine to the exponential function is Euler's formula :

[math] e^{ix} = \displaystyle \sum_0^ \infty \dfrac {(ix)^n}{n!} = cos(x) + i \ sin(x)[/math]

So [math]cos(x) = Re(e^{ix})[/math] and [math]sin(x) = Im(e^{ix})[/math]

This is the way that sine and cosine are defined in a rigorous analytical treatment -- see for instance Rudin's Real and Complex Analysis.

I don't see how defining sine and cosine in terms of your differential equation adds clarity. For instance, without going back to a power series I don't see a clear path via the DE to showing that the solutions are periodic.

Hello everyone,

 

I'm heading to college this upcoming fall and I desire to do research. However, I don't know how to go about getting a position. If any of you have done freshman research, I have a few questions:

 

1. Did you find a professor to research with before or after you started classes? Or were you offered an opportunity to research on your own?

 

2. Did you have prior lab work experience (I've taken AP/IB Biology and AP Chemistry, I'm just assuming that isn't considered experience)? If you did, how did you receive it?

 

3. Are there any books about lab work that I should read before school starts? Any books on scientific thought I should read?

 

I suppose I'm just kind of lost as to where to go, and I figured that since a lot of you have experienced this that it wouldn't be a problem for the community to answer it. Thanks!

The state university here, through the College of Science has a competitive scholarship program for freshmn women that has a significant stipend and arranges for hands-on participation with research faculty that starts in the summer before the fall freshman semester. It has been very successful.

I would start by inquiring if your school has a similar program.

Why does there' have to be infinite decimals? Does cutting something into thirds or 5ths or 7ths not actually exist in nature? There is nothing that is actually exactly 1 third of something else, but it seems like making halves or 4ths is pretty easy in the universe.

Most real numbers have infinite decimal representations. Without the real numbers you could not do the bulk of calculus, or physics.

If you don't want to worry about infinite decinals, become an accoutant.

Wouldn't he be better described as a polymath because he excelled in both, amongst other things?

Nope. We mathematicians claim him. He was, after all, the Lucasian Professor of Mathematics. Mathematicians tend to be pretty versatile.

What's the probability that Feynman was wrong and what's the confidence level of certainty for the answer?

Feynman probably understood quantum yheory as well as anyone who has ever lived.

There are different, valid, ways to view QM. The post above gives you his perspective. To say that Feynman is "wrong" would be pretty sporting.

It can all be traced to Newton anyway though so scientists win! Or maybe mathematicians win...Newton might have struggled without Pythagoras or Descartes.

Yes, mathematicians win.

"To summarize , I would use the words of Jeans, who said that ‘the Great Architect seems to be a mathematician’." . – Richard P. Feynman in The Character of Physical Law

Before the physicists jump in to claim Newton, let me note that 1) I am a mathematician and 2) my academic geneology is directly traceable to Newton. Newton was a mathematician.

I am not a physcist, not even a student. But My average man's mind cannot understand how a particle can act like a wave when we are'nt looking? though I am an atheist, I agree with einsteins words " God does not play dice".

 

 

Here's Feynman on "wave-particle duality.

From the book QED based on the Robb Lectures by Feynman: "Quantum electrodynamics "resolves" this wave-particle duality by saying that light is made up of particles (as Newton originally thought), but the price of this great advancement of science is a retreat by physics to the position of being able to calculate only the probabilities that a photon will hit a detector, without offering a good model of how it actually happens."

I actually thought, at first, that you are all yanking my chain. This entire thing sounds ridiculous to me, honestly, but I must admit, mathematics has this tendency to have paradoxes that I simply don't get.

 

Useful link to anyone who's as confused as I was/am: http://en.wikipedia.org/wiki/0.999... (thanks swanstont)

 

 

So now that I know this is actually REAL, I have a few questions of my own:

 

First this seems to talk about infinite 9s after the decimal. Does this mean that the original post, with

 

as in, there's a 1 at the end and it's not infinite 9s, is *not* the same as infinite 9s? The wiki has examples with limit goes to infinity, and this one doesn't...

 

Second, why is this different than the mathematical paradoxes out there, where I can get a nonsensical result out of mathematical manipulation? Is this not nonsensical? The definition of .999 is that it's not yet 1, isn't it?? so isn't this manipulation resulting in a nonsensical result?

 

I saw a math paradox where 1=0, simply by stating something like

1 = 1 + 0 + 0 + 0 + 0 + ...

and using rules of math, so I can change order of addition:

1 = 0 + 1 + 0 + 0 + 0 + ...

1 = 0 + 0 + 1 + 0 + 0 + ...

1 = 0 + 0 + 0 + 1 + 0 + ...

1 = 0 + 0 + 0 + 0 + 0 + ...

1=0

 

This isn't REAL. It's nonsensical, it's just abusing the laws of math to reach a nonsensical result, hence being called a PARADOX. the 0.999... thing is also a paradox, isn't it? (btw, it's under "math paradoxes" category in wikipedia, if it helps my point).

 

So.. what's the difference between producing nonsensical results that we laugh about and NOT treat seriously like the 1=0 one and this 0.999... one?

 

 

This makes no sense to me.

There is no paradox. .9999999............ = 1

There is a big difference between a finite decimal expansion and an infinite one.

1 = 0 + 1 + 0 + 0 + 0 + ... true

1 = 0 + 0 + 1 + 0 + 0 + ... true

1 = 0 + 0 + 0 + 1 + 0 + ... true

1 = 0 + 0 + 0 + 0 + 0 + ... false , you can't just make the "1" disappear

1=0 ridiculous

 

 

 

That is bad analogy IMHO.

Actually, it is not. GR models spacetime as a Lorentzian manifold without boundary.

Try looking at it this way.

 

Using regular subtraction,

 

0.99999..... ad infinitum = 1 – 0.00000..... ad infitinum .....1

 

The ad infinitum on the right side of the equation means you never get the opportunity to place the one because the zeros go on forever.

 

Therefore,

 

0.99999..... ad infinitum = 1

argh !

I don't understand the problem.

 

Not matter how many 9s you have in your 2.999, it's still, by definition, [math]3-\Delta[/math], where [math]\Delta[/math] is an arbitrarily small unit.

 

Which is also, quite clearly, less than 3.

 

What am I missing?

The only "arbitrarily small" non-zero number is 0.

.999999............ = 1

As can be seen:

x = .9999999.............

10x = 9.9999999..........

9x = 9.99999999..... - .99999999.... = 9

x =1

This is a perfectly legitimate mathematical proof.

If you desire gory detail here it is ;

[math]\displaystyle \sum_{n=0}^N x^n = 1 + x \displaystyle \sum_{n=o}^N x^n - x^{N+1} [/math]

[math](1-x)\displaystyle \sum_{n=0}^N x^n = 1-x^{N+1}[/math]

[math]\displaystyle \sum_{n=0}^N x^n = \dfrac {1-x^{N+1}}{1-x}[/math]

Similarly

[math]\displaystyle \sum_{n=1}^N x^n = \dfrac {1-x^{N+1}}{1-x} -1 [/math] [math] = \dfrac {x-x^{N+1}}{1-x} [/math]

So, if [math]|x|<1[/math]

[math]\displaystyle \sum_{n=0}^\infty x^n[/math] [math]=\displaystyle \lim_{N \to \infty} \displaystyle \sum_{n=0}^N x^n = \displaystyle \lim_{N \to \infty} \dfrac {1-x^{N+1}}{1-x}[/math] [math] = \dfrac {1}{1-x}[/math]

And

[math]\displaystyle \sum_{n=1}^\infty x^n[/math] [math]= \displaystyle \lim_{N \to \infty} \displaystyle \sum_{n=1}^N x^n = \displaystyle \lim_{N \to \infty} \dfrac {x-x^{N+1}}{1-x}[/math] [math] = \dfrac {x}{1-x}[/math]

[math]0.99999........ = \displaystyle \sum_{n=1}^\infty 9 (\dfrac{1}{10})^n[/math] [math] = 9 \displaystyle \sum_{n=1}^\infty (\dfrac{1}{10})^n[/math] [math] = 9 \dfrac {\frac {1}{10}}{1- \frac{1}{10}}[/math] [math] = 9 \dfrac {1}{9}[/math] [math] = 1[/math]

And Swansont , Is the theory of relativity proven ?

Proof applies to mathematics, not science.

Relativity is supported by a mountain of experimental and observational evidence. But general relativity is also known to be incompatible with quantum mechanics, which is also supported by a large body of evidence. Either or both will probably eventually be supplanted by a theory that will refine and extend both.

I presume this is the one referred to - the pre-print at arxiv , and a few pop-sci articles from New Scientist and Scientific American

maybe this http://arxiv.org/abs/1104.0699

I saw the proof on tv while i flipped channels more specific discovery (science or normal one) and it was michio kaku talking about multiverses.

Short answer to your question no i cant unless you find that clip/episode on the internet somewhere.

 

Hope this clears that up

 

 

 

Ok now i understand thank you

There is no such proof, nor is there likely to ever be. Michio KJaku is a very poor source. He makes flamboyant statements with no basis, apparently in an effort to promote himself and sell books.

If Kaku said that the sky was blue, I would immediately seek independent verification.

I have several hundred science and mathematics books. I have one by Kaku -- a mediocre book on quantum field theory. I am very unlikely to ever own two.

Hm.

Following that logic the Big Bang is the edge of spacetime.

No more so than the North Pole is the edge of the Eaarth.

I think science is more a philosophy or way of thinking. If one applies the scientific method or maybe something close, but more suited to the exact subject in question, then it is a science. For example mathematics can be considered a science by a wide interpretation of the scientific method.

 

So, just about anything can be approached scientifically.

I don't consider mathematics to be science. The major discriminator is that mathematics relies on logical proof, as opposed to evidence from experiment, in establishing "truth'. Mathematics is related to science, but is a discipline unto itself.

I think engineering uses a scientific thought process with a slightly different goal. In my opinion, perhaps only my opinion: the goal of science is to understand and model, while the goal of engineering is to produce and create. Good engineering facilitates good science while good science facilitates good engineering. They are two studies with different goals and similar methods.

Engineering is characterized by an end objective of the production of a useful product, commonly with budget and schedule constraints. The goal of science is the development of understanding. Engineering builds on that understanding to produce products, sometimes in the face of imperfect understanding of the details of the science involved, and often involving very complex systems for which first-principles modeling is impractical or impossible. While the academic subject matter in formal classes is sometimes similar, the ultimate objectives of science and engineering are quite different.

Ha! My functional analysis text is a treasure!

What is the book ? This material is usually treated in a text on complex analysis rather than one on functional analysis.

It first states the Thm of Cauchy: If [math]f(z)[/math] is analytic within and on the closed contour [math]C[/math] and the derivtive [math]f'(z)[/math] is continuous throughout this region, then

 

[math]\oint_C f(z) \,dz = 0[/math]. (Parenthetically, given the amazing power of this thm., the proof is surprisingly straightforward. In fact I have seen two, one using Green's Thm and the other using the closely related Thm of Stokes. Um well - neither of these is so easily proved I guess)

Those hypotheses are a bit strange. Once you know that f is analytic, you know that it has continuous derivatives of all orders. See below

But if the authopr is starting from scratch an intends to use Stokes Theorem then he needs the hypothesis of continuous derivatives. There are better ways to do this.

It then points out that Goursat showed that the statement about continuity is superfluous, and re-states the Thm as Cauchy-Goursat.

Yes, see above. Most treatments simply call the theorem Cauchy's Theorem. The usual proof is done first for triangles in a convex region and then generalized to arbitrary curves in a convex region.

It then comes out with this priceless gem:

 

"Some authors (never mathematicians!) define an analytic function as a differentiable function with continuous derivatives.......But this is a mathematical fraud of cosmic proportions"

 

No words minced there, then.

An analytic function is one that is locally representable by a power series. This is considerably more restrictive that even having continous derivatives of all orders ( [math]C^\infty[/math])

What is striking about complex-valued functions of a complex variable is that any differentiable function is automatically analytc. This makes complex analysis very different from real analysis. Analytic functions are, in a sense, "nice". But you give up existence of partitions of unity, so complex manifolds are so difficult that very little theory exists, and Kahler manifolds (about which I know very little) are what are studied.

So do me another kindness, and see if the following floats your goat:

 

We know from elementary analysis that [math]\overline{\sin(z)} = \sin (\overline{z})[/math] and is analytic (indeed entire) and also that where [math]f(z) = z^n[/math] for any nonzero real [math]n[/math] that [math]f(z)[/math] is analytic also. Using fingers and toes only, it appears that for small [math]n[/math] then [math]\overline{f(z)} \equiv \overline{f}(z) = f(\overline{z})[/math]. Likewise any other polynomial function.

 

I somewhat rashly propose the following generalization: if a function is analytic then [math]\overline{f}(z) = f(\overline{z})[/math] always. True or false?

I don't follow your statement regarding sine as following from "elementary analysis" since extending the domain to complex numbers requires complex analysis, but the statement is true.

You have outlined the proof that [math]\overline{f}(z) = f(\overline{z})[/math]. To wit:

[math] \overline{z_1z_2} = \overline {z_1} \overline{z_2} [/math] clearly extends to show that [math]\overline{f}(z) = f(\overline{z})[/math] for any polynomial function [math]f[/math]. Since conjugation is continuous and since any analytic function is the uniform limit on compacta of polynomials the theorem now follows for entire functions. Since sine is entire, it follows for sine.

I confess I am having trouble with the converse, namely that analycity is required for this equality to hold. I use as an example [math]f(z) = |z|^2 \equiv z\overline{z}[/math] which clearly not analytic, but where the equality seems to hold (unless I made a mistake).

Analyticity is not required for equality. Your counterexample is valid. The only real-valued analytic functions are constant.

Is this gibberish?

no

This should be laughably easy, but I am a little confused here.

 

Suppose that the function [math]f: \mathbb{C} \to \mathbb{C}[/math]. Suppose further that [math]z \in \mathbb{C} = x +iy,\,\, x,\, y \in \mathbb{R}[/math].

 

What is meant by the complex conjugate of this function?

 

My thoughts (such as they are!). Set [math]z = x +iy[/math], and set [math]f(z) = ax+iby[/math] and [math]\overline{f(z)}= ax-iby[/math]

 

Apparently this can be written as the identity [math]\overline{f}(z) = \overline{f(z)}= ax-iby[/math], which I don't quite get. Moreover,......

 

...... how does this differ from, say, [math] f(\overline{z}) =\overline{ax+iby}[/math]?

Just think of conjugation as a function then

[math] \overline {f}(z) = \overline {f(z)} = (conjugation \circ f)(z)[/math]

[math] f(\overline {z}) = (f \circ conjugation) (z) [/math]

Can anyone help me I am having a hard time finding the solution?

 

 

How many directed graphs with 5 nodes, among which one node is isolated, can exist?

What have you done to try to solve this problem ?

1) How to show that if W is a subspace of a finite-dimensional vector space V, then W is finite-dimensional and dim W<= dimV.

 

2) How to show that if a subspace of a finite-dimensional vector space V and dim W = dimV, then W = V.

 

3) How to prove that the subspace of R^3 are{0}, R^3 itself, and any line or plane passing through the origin.

 

How to approach these three Questions?

 

Thanks

What have you tried ?

These should be pretty simple if you were paying attention in class.

D0 you know the definitions of: 1) linearly independent set, 2) spanning set, 3) basis, and 4) dimension ?

Care to elaborate? I've not been following the story.

He was opining on difficultiy with relativity and stated that [math] F = m \frac {dv}{dt} [/math] which is not correct. In special relativity the correct relation is [math] F = \frac {d(\gamma m_0v)}{dt}[/math]. This makes me think he was simply reciting to the camera words that he did not understand. Yet he is being portrayed in some stories as the 12 year old genuis who is challenging Einstein's theory of relativity.

 

Dr.Rocket , As an example , the purpose of language is to ask for a bag of chips for one's snack when needs require so , not inconsistent with your impression , shouldn't I utter ?

If you were to never utter again, that would be all right with me.

There are four masses hanging by a rope from the

ceiling in the simplest arrangement possible, mass 4 is attached by the rope to mass three right above it,

mass three is attached by a rope to mass 2 right above it, mass 2 is attached by the rope to mass 1 right above it

and mass one is attached by the rope to the ceiling. So the masses are hanging vertically from the ceiling attached

by the rope. Two of the tensions and three of the masses have been measured.

We know: T1 T2 m1 m2 m3 Show that the fourth mass can be expressed as

 

m4 = (m1T2/T1 - T2) - m2 - m3

 

Solution:

 

We know that m4g + m3g + m2g = T2

so m4 = (T2/g) - m2 - m3 since multiplying the first term by m1/m1 is the same as multiplying the term by one,

we get m4 = (m1T2/m1g) - m2 - m3 using the fact that T1 - T2 = m1g and substituting this equation in the denominator

we get m4 = (m1T2/T1 - T2) - m2 - m3 QED

 

Is this right? Did I answer the question properly? Just seems like I cheated. If you can point me in the direction of a better answer I'd greatly appreciate it.

That is just fine. I have no idea why you might feel that you cheated.

Imatfaal's stylistic comments are helpful and might make your explanation easier to follow, but that is just icing on the cake.

You might want to take a look at the idea of "free body diagrams" as a device to make the relations among forces clear. For some reason they are not always taught in physics courses (I recall showing the technique to a nuclear physicist who was teaching a basic non-majors class and getting a "gee that's neat" response), but they are routine in engineering mechanics. See an engineering text on mechanics or statics, or just talk to an engineering student. Engineering calculations quite often have to be shown to a large, critical audience for approval and therefore clarity of exposition is as important as is getting the right answer. Free body diagrams are a big help.

I was wondering what peoples opinions were on this kid:

 

http://www.indystar....CAL01/103200369

 

i know next to nothing in mathematics and physics so i have no gauge on how far along he actually is compared to his peers (university level) or how much potential he has, but he sounds impressive enough to a layperson. are people like him rare, or is this something (slightly) more common that the media is playing up? does being a prodigy really give the person a benefit over an entire career, or does it just put them at a headstart, before his catch up?

 

also the reporter is a little grating, just bear with it lol.

As a result of conversations elsewhere what I have seen and learned is:

1. The kid was diagnosed with Asperger's syndrome which tends to result in some odd behavior. I have a nephew with Asperger's and he is also a bit odd, but not a prodigy.

2. He has made some statements about relativity that, though wrong, are getting some press.

3. He talked about integration by parts, coming from the product rule for differentiation, and got that right. That puts him ahead of most college freshmen, but does not make him a genius.

4. He has a pushy mother who seems to be behind the hoopla.

5. Time will tell if he ever produces anything original and startling. I am not holding my breath.