Riemann Zeta Function

Off Topic
41

[quote]jasmincar wrote:

[quote]thethirdruffian wrote:

[quote]Dr.Matt581 wrote:

[quote]Dr. Pangloss wrote:
I still don’t believe it. It’s make no intuitive sense whatsoever.[/quote]

This is not an intuitive process, and this article is very misleading. This sum is indeed accurate, but it is not the only value that can be assigned to the sum of natural numbers. Under the standard axioms applied to the field of real numbers, this sum is indeed divergent, which is what would be intuitive to most people since the field axioms as applied to real numbers are what are taught in primary, secondary, and post-secondary classes up until complex numbers and other things become a factor. This means that it approaches some kind of infinite value, or the limit in some other way does not exist. In this case it is one form of an infinitely positive value.

The problem is that people are taught these topics without it ever being explained to them that this is not the only way to do things mathematically. Things like addition and multiplication (which when applied to real numbers using the standard field axioms are the exact same thing) are taught as pure, unalterable facts to people and by people who do not understand that the reason that these are true are because we are following arbitrary rules made up by mathematicians for a specific purpose and they can be altered or thrown out altogether under certain circumstances. For example, the field axioms applied to real numbers and the operations that arise from them do not apply at all to imaginary numbers (which, despite the name, are very much real. Now, when we apply the field axioms to other, more complicated groups and fields and such we get new definitions of operations like addition and multiplication and we can define things like the Riemann Zeta function, or Ramanujan summation which we can apply to sum the set of natural numbers and get -1/12 and other values like 1/4 (which is also a valid convergent value for the sum of natural numbers under some conditions) both of these are extremely useful to physicist like myself.

[/quote]

And this is also why we engineers don’t let physicists actually design mechanical objects.[/quote]

Lol you are a douche. Losers who don’t what else to do go in engineering.
[/quote]

Aren’t you an engineer? Or had aspirations to be one recently?

42

[quote]jasmincar wrote:

[quote]thethirdruffian wrote:

And this is also why we engineers don’t let physicists actually design mechanical objects.[/quote]

Lol you are a douche. Losers who don’t what else to do go in engineering.
[/quote]

Wow, little too touchy to have a sense of humor are we? You’re being a prick in an otherwise very civil and enlightening thread. I’m fairly certain Dr. Matt wasn’t offended at what was very clearly a good natured joke.

43

A mathematician and an engineer agreed to take part in a psychological test.
They sat on one side of a room and waited not knowing what to expect. A door
opened on the other side and a naked woman came in the room and stood on the
far side. They were then instructed that every time they heard a beep they
could move half the remaining distance to the woman. They heard a beep and
the engineer jumped up and moved halfway across the room while the
mathematician continued to sit, looking disgusted and bored. When the
mathematician didnt move after the second beep he was asked why. Because I
know I will never reach the woman. The engineer was asked why he chose to
move and replied, Because I know that very soon I will be close enough for
all practical purposes!

Engineering is just applied physics and I have to say some of the most capable techs I have ever met have been physics doctoral students and post docs. Hands on doing whatever is needed around a collidor.

44

[quote]Scotto wrote:
A mathematician and an engineer agreed to take part in a psychological test.
They sat on one side of a room and waited not knowing what to expect. A door
opened on the other side and a naked woman came in the room and stood on the
far side. They were then instructed that every time they heard a beep they
could move half the remaining distance to the woman. They heard a beep and
the engineer jumped up and moved halfway across the room while the
mathematician continued to sit, looking disgusted and bored. When the
mathematician didnt move after the second beep he was asked why. Because I
know I will never reach the woman. The engineer was asked why he chose to
move and replied, Because I know that very soon I will be close enough for
all practical purposes!

Engineering is just applied physics and I have to say some of the most capable techs I have ever met have been physics doctoral students and post docs. Hands on doing whatever is needed around a collidor.[/quote]

Close enough?

From the thread, I’m pretty sure he’d be -1/12 of the original distance inside her.

Certainly the case if TTR was the engineer in question.

45

[quote]jasmincar wrote:

[quote]thethirdruffian wrote:

[quote]Dr.Matt581 wrote:

[quote]Dr. Pangloss wrote:
I still don’t believe it. It’s make no intuitive sense whatsoever.[/quote]

This is not an intuitive process, and this article is very misleading. This sum is indeed accurate, but it is not the only value that can be assigned to the sum of natural numbers. Under the standard axioms applied to the field of real numbers, this sum is indeed divergent, which is what would be intuitive to most people since the field axioms as applied to real numbers are what are taught in primary, secondary, and post-secondary classes up until complex numbers and other things become a factor. This means that it approaches some kind of infinite value, or the limit in some other way does not exist. In this case it is one form of an infinitely positive value.

The problem is that people are taught these topics without it ever being explained to them that this is not the only way to do things mathematically. Things like addition and multiplication (which when applied to real numbers using the standard field axioms are the exact same thing) are taught as pure, unalterable facts to people and by people who do not understand that the reason that these are true are because we are following arbitrary rules made up by mathematicians for a specific purpose and they can be altered or thrown out altogether under certain circumstances. For example, the field axioms applied to real numbers and the operations that arise from them do not apply at all to imaginary numbers (which, despite the name, are very much real. Now, when we apply the field axioms to other, more complicated groups and fields and such we get new definitions of operations like addition and multiplication and we can define things like the Riemann Zeta function, or Ramanujan summation which we can apply to sum the set of natural numbers and get -1/12 and other values like 1/4 (which is also a valid convergent value for the sum of natural numbers under some conditions) both of these are extremely useful to physicist like myself.

[/quote]

And this is also why we engineers don’t let physicists actually design mechanical objects.[/quote]

Lol you are a douche. Losers who don’t what else to do go in engineering.
[/quote]

No, they go to law school.

46

[quote]Jewbacca wrote:

[quote]Scotto wrote:
A mathematician and an engineer agreed to take part in a psychological test.
They sat on one side of a room and waited not knowing what to expect. A door
opened on the other side and a naked woman came in the room and stood on the
far side. They were then instructed that every time they heard a beep they
could move half the remaining distance to the woman. They heard a beep and
the engineer jumped up and moved halfway across the room while the
mathematician continued to sit, looking disgusted and bored. When the
mathematician didnt move after the second beep he was asked why. Because I
know I will never reach the woman. The engineer was asked why he chose to
move and replied, Because I know that very soon I will be close enough for
all practical purposes!

Engineering is just applied physics and I have to say some of the most capable techs I have ever met have been physics doctoral students and post docs. Hands on doing whatever is needed around a collidor.[/quote]

Close enough?

From the thread, I’m pretty sure he’d be -1/12 of the original distance inside her.

Certainly the case if TTR was the engineer in question.[/quote]

Actually that one is a convergent series so it does not apply to the original post in regards to the sum of a divergent series.

47

[quote]jasmincar wrote:
Lol you are a douche. Losers who don’t what else to do go in engineering.
[/quote]

48

[quote]Aragorn wrote:

[quote]jasmincar wrote:

[quote]thethirdruffian wrote:

And this is also why we engineers don’t let physicists actually design mechanical objects.[/quote]

Lol you are a douche. Losers who don’t what else to do go in engineering.
[/quote]

Wow, little too touchy to have a sense of humor are we? You’re being a prick in an otherwise very civil and enlightening thread. I’m fairly certain Dr. Matt wasn’t offended at what was very clearly a good natured joke.[/quote]

jasmincar’s always been a little snide. He is from Quebec, if that helps.

49

[quote]Nards wrote:

[quote]Aragorn wrote:

[quote]jasmincar wrote:

[quote]thethirdruffian wrote:

And this is also why we engineers don’t let physicists actually design mechanical objects.[/quote]

Lol you are a douche. Losers who don’t what else to do go in engineering.
[/quote]

Wow, little too touchy to have a sense of humor are we? You’re being a prick in an otherwise very civil and enlightening thread. I’m fairly certain Dr. Matt wasn’t offended at what was very clearly a good natured joke.[/quote]

jasmincar’s always been a little snide. He is from Quebec, if that helps.
[/quote]

It certainly seems to.

50

[quote]DrSkeptix wrote:

[quote]Dr.Matt581 wrote:

[quote]DrSkeptix wrote:
Which leads me to a question in nuclear magnetic resonance…[/quote]

By all means, ask away, but it is way past my bedtime so my answer will have to wait until tomorrow.[/quote]

Oh! I would so like to learn how to even ask the question. And please, forgive in advance, my naive ignorance.
I will leave it to you: "Polar atoms are in a perfectly alternating high Gauss field. They either are spinning “up” or “down” in it. + or -. 1 or -1. In what state of the magnetic field are they “1/2”?[/quote]

This is a rather odd question. The spin of an atom is determined by the spin of the particles (the ones that have spin anyway,) that make up that atom. You really just add up the spin of each particle (i.e. an H-1 nucleus has just one proton and thus a spin of 1/2). In terms of nuclear magnetic resonance the nuclear spin is what we are primarily concerned with. Now, in order to change the spin of an atom, one must change the makeup of the particles that make it up, which can arise from a lot of different effects.

  • just fyi,“up” and “down” refer specifically to spin 1/2 particles, which are the ones that we are concerned with as far as nuclear magnetic resonance since spin 1/2 particles have a permanent magnetic moment.
51

[quote]DrSkeptix wrote:

[quote]Dr.Matt581 wrote:

[quote]DrSkeptix wrote:

S = 1 - 1 + 1 - 1 �??�??�??�?�¢?�??�??�??�?�¦
S = 1 - 1 + 1�??�??�??�?�¢?�??�??�??�?�¦.

2S = 1, and
S = 1/2

which of course works for only sequences S of infinite length.[/quote]

I do not know if you are trying to use some kind of operator on S to transform it or not, if you are pleas tell me what it is and I can tell you whether it is valid or not, but the way you have it written you have two different sets.

S = 1 - 1 + 1 - 1 and
S’ = 1 - 1 + 1

They are not the same, so the sum of the two would be (S + S’) = 2, not 2S = 2.

Edit: now I really am going to bed. My wife hates it when I stay up this late.[/quote]

The forum hieroglyphics could not record a space or an ellipsis.
I edited the original.
The “frame shift” is the maneuver of adding the infinite sequence, but displaced by one digit for visual clarity.

Good night, Professor.
[/quote]

Ah, okay. The way you are writing it, then no that is not a valid way of finding a convergent value for that series. What you are doing are adding up partial sums of the series that are chosen arbitrarily, which are not equal to the original sum.

I ran through a few methods that I am familiar and did find a way to assign a value of 1/2 to the sequence that you introduced. It turns out that it is weakly, (C, 1), Cesaro summable. The way that (C, 1) Cesaro summation works is you sum all of the partial sums of the original sequence and take the average of that sum as the limit goes to infinity. This does indeed give your value of 1/2.

Edit: It should be noted that the manipulations you were doing are valid, but only for convergent series, which this is not.

52

[quote]Dr.Matt581 wrote:

[quote]DrSkeptix wrote:

[quote]Dr.Matt581 wrote:

[quote]DrSkeptix wrote:

S = 1 - 1 + 1 - 1 �??�??�??�??�??�??�?�¢?�??�??�??�??�??�??�?�¦
S = 1 - 1 + 1�??�??�??�??�??�??�?�¢?�??�??�??�??�??�??�?�¦.

2S = 1, and
S = 1/2

which of course works for only sequences S of infinite length.[/quote]

I do not know if you are trying to use some kind of operator on S to transform it or not, if you are pleas tell me what it is and I can tell you whether it is valid or not, but the way you have it written you have two different sets.

S = 1 - 1 + 1 - 1 and
S’ = 1 - 1 + 1

They are not the same, so the sum of the two would be (S + S’) = 2, not 2S = 2.

Edit: now I really am going to bed. My wife hates it when I stay up this late.[/quote]

The forum hieroglyphics could not record a space or an ellipsis.
I edited the original.
The “frame shift” is the maneuver of adding the infinite sequence, but displaced by one digit for visual clarity.

Good night, Professor.
[/quote]

Ah, okay. The way you are writing it, then no that is not a valid way of finding a convergent value for that series. What you are doing are adding up partial sums of the series that are chosen arbitrarily, which are not equal to the original sum.

I ran through a few methods that I am familiar and did find a way to assign a value of 1/2 to the sequence that you introduced. It turns out that it is weakly, (C, 1), Cesaro summable. The way that (C, 1) Cesaro summation works is you sum all of the partial sums of the original sequence and take the average of that sum as the limit goes to infinity. This does indeed give your value of 1/2.

Edit: It should be noted that the manipulations you were doing are valid, but only for convergent series, which this is not.[/quote]

Thanks! I understand. Among other errors, I had conflated the notions finite/infinite and convergent/divergent.
Also, I did know about Cesaro summation. (I was intuitively taking a stab at that with the construction of finite sets within the sequence. Points for naivete?) And as much as you hate wiki, the example it offers for Cesaro summation is precisely the sequence under discussion, = 1/2.

53

I am over halfway through a book by Jim Baggott called Farewell to Reality: Why Fairy Tale Physics Has Betrayed the Search For Scientific Truth

Man that’s hard for me to understand…do you know of it DrSkeptix?

54

[quote]Nards wrote:
I am over halfway through a book by Jim Baggott called Farewell to Reality: Why Fairy Tale Physics Has Betrayed the Search For Scientific Truth

Man that’s hard for me to understand…do you know of it DrSkeptix?[/quote]

Nope. But maybe I should Kindle it.

I know an old physicist–more of a hydraulic engineer, I guess–who has written a book revising the laws of fluid dynamics. But his proofs requiring re-thinking physics in toto. I can understand a only a fraction of it; I don’t know if he may have kernel of understanding, of if he is totally bonkers.

There are two sorts that make a skeptic uncomfortable: a believer and another skeptic.