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January 22

Cantor's diagonal argument

Could someone explain Cantor's diagonal argument in terms that a relative layman could understand? I tried reading the article, but repeated kept running into terms like "countably infinite" - which links to "countable set," which would make more sense to me if it was "countably finite" instead of "infinite" - and, at any rate, countable set, soon gets into bijections, etc., and before I know it, I'm lost. What does it mean that "real numbers are not countably infinite"? And in what manner is that counterintuitive? Zafiroblue05 03:55, 22 January 2006 (UTC)

I guess people see it as counterintuitive because when you first start thinking about the concept of infinity, it's not at all obvious that there can be more than one size of "infinity". To say that the "real numbers are not countably infinite" basically means that you can't write down the real numbers in a list. No matter how carefully you arrange your list, you'll always miss out some real numbers (in fact you'll miss most of them). Cantor's argument explains why this happens. This situation is in contrast to the natural numbers (0, 1, 2, 3, etc); you can write them down in a list. (I just did.) The term countably infinite pins down more precisely what we mean by "able to write down in a list". Dmharvey 05:15, 22 January 2006 (UTC)

Hmmm... I think I'm starting to get it - what threw me off was the term "countably infinite." But could you explain one more thing - in the actual article - in point number 9 in the proof, it says "x differs in the nth decimal place from rn." How do we know that? Zafiroblue05 01:20, 23 January 2006 (UTC)

Because that's how we defined it. We defined x by saying that it differs on the nth place from rn, by choosing each digit to be different from the corresponding digit on that particular rn. An intuitive way of considering it is that Cantor basically says that we can't make a rule to order the real numbers in such a way that when we are given a real number, we can say that it's the fifth, sixth, or 49th real number, or something like that. (Note this is different from saying that we can make a rule to order the real numbers in a way that when given a number, we can find the 'next' number. That actually is possible)--Fangz 02:58, 25 January 2006 (UTC)

Cantor's diagonal argument compares two sets, and finds that one is strictly larger than the other. Necessarily, it depends on machinery for comparing the sizes of sets. Consider the set S={a,b} and the set T={x,y,z}. It is "obvious" that these two sets are of different sizes, and that the second is larger than the first. But how can we formalize our intuition? A standard approach is to attempt a pairing. Thus we might pair ⟨a,x⟩ and ⟨b,y⟩, so that distinct members of S are paired with distinct members of T. This defines a function, more precisely a monomorphism, from S to T. We can conclude that T is larger than S because our monomorphism is not also an epimorphism; that is, at least one member of T is left unpaired.

Such an argument is straightforward and uneventful for finite sets. For infinite sets, things can get much more interesting. We discover that the set of even postive integers, {2,4,6,8,…}, is exactly the same size as the set of odd postive integers, {1,3,5,7,…}. We also find that the set of even postive integers is exactly the same size as the set of all postive integers! Nor are we restricted to integers. The set of rational numbers, all ratios of integers to positive integers, is again the same size as the set of positive integers alone. In fact, we can define a set as infinite if it contains a subset of the same size.

Having discovered this much, it is tempting to assume that all infinite sets are the same size. Cantor's argument shows that this assumption is false, because the set of real numbers is strictly larger than the set of positive integers. The method of proof is powerful, but peculiar; it is proof by contradiction. For simplicity, restrict attention to positive real numbers less than 1. Cantor says, suppose we're wrong, suppose we have such a real number for every positive integer (a monomorphism) with none left over (an epimorphism). Then we can use the integer paired with each real number to sort them, to list all the real numbers in order. Now write each real number as a (possibly infinite) decimal expansion, such as 0.8537 or 0.333… or 0.7170… or the like. Here the first number has first digit 8; choose a different digit (other than 9, for technical reasons), say 7. The second number has second digit 3; again choose a different second digit, say 2. And so on, through all the real numbers on our (supposedly complete) list. We are constructing a number as we go, something like 0.726…, which is necessarily different from the first number, and from the second, and in fact from any number on the list. And this number is a perfectly valid positive real number less than 1 (the latter because we exclude 9s). Oh, dear. (Actually, oh joy!) We have a contradiction: Our "complete" list necessarily omits our constructed number. Thus our monomorphism (the list) is not an epimorphism (because of the omission), and the set of real numbers is therefore larger than the set of positive integers. --KSmrq^T 06:15, 22 January 2006 (UTC)

Optimization Problem in Economics

A hog weighs 250 pounds. A high-yield diet allows the animal to gain 6 pounds a day at a cost of $0.56 a day. The market price for hogs is currently $0.75 per pounds, but it is falling at a rate of $0.01 per day, and that price decline is expected to remain steady for the foreseeable future. When should the hog be sold in order to provide the farmer with the highest financial gain?

--66.81.193.109 14:19, 22 January 2006 (UTC)

Using the variable t, measured in days, let t=0 be 6 a.m. today. After t days, the hog will weigh 250+6t pounds. The price of hogs after t days will be (0.75-0.01*t) dollars per pound. The cost of the feed for t days of feeding will be 0.56*t. So, the profit after t days will be

$P(t)=(250+6t)\times (0.75-0.01\times t)-0.56\times t$

$P'(t)=(250+6t)\times (-0.01)+(0.75-0.01t)\times 6-0.56=0$

Thus, t=12 days. Here the profit function has the shape of the downward parabola because of the (-t^2) term, and as a result, solving the marginal profit function for t gives the optimal input for the profit function.

[I received help from Dr. Math.] (hmm... answered your own question did you? hydnjo talk 18:28, 22 January 2006 (UTC))

hmm...that was an offensive question. deeptrivia (talk) 15:25, 22 January 2006 (UTC)

modus tollens vs. proof by contradiction

Is modus tollens the same thing as proof by contradiction? They seem to mean the same thing intuitively, but they have separate articles, and they're phrased differently. -- Creidieki 19:02, 22 January 2006 (UTC)

I think the difference is mainly of scope. Modus tollens usually means the specific axiom that if p→q then ~q→~p, and proof by contradiction usually means proving an entire argument by negating its conclusion. The basic idea is the same though. -- Meni Rosenfeld (talk) 19:32, 22 January 2006 (UTC)

Yes. Modus tollens is used in the key step of proof by contradiction. —Keenan Pepper 19:36, 22 January 2006 (UTC)

Just a quibble here: Modus tollens usually denotes, not the axiom (p→q)→(¬q→¬p), but rather the rule of inference

${\frac {A\rightarrow B,\lnot B}{\lnot A}}$

It's a fine distinction, of course, but there are contexts where it matters at least a little. --Trovatore 19:42, 22 January 2006 (UTC)

• Okay, that distinction makes some sense. So then modus tollens wouldn't be included as an inference rule in constructive logics? I'm a bit confused, though; why would you need this as both an axiom and an inference rule? -- Creidieki 19:56, 22 January 2006 (UTC)

• • I don't think intuitionists have any objection to modus tollens. They object when you try to prove positive statements by contradiction; proving negative statements by contradiction is fine with them.
  • As for what you "need", I suppose that depends on what you need it for. Keep in mind that we don't actually do mathematical reasoning in formal deductive systems; rather, we study them as comprehensible analogs of (much messier) actual mathematical reasoning. If you start with a deductive system that includes modus ponens and the axiom schema (A→B)→(¬B→¬A), you won't get any new theorems by adding modus tollens, but some proofs will get slightly shorter. On the other hand, if you add the rule, you can dispense with the axiom, at least as long as you have the rule (I forget what it's called) that allows you to infer (A→B) from a proof of B using A as an assumption. --Trovatore 20:10, 22 January 2006 (UTC)

Modus tollens is an admissible rule of intuitionistic logic: it's the converse, if ~q→~p then p→q which the intuitionists reject (and adding this rule to intuitiontistic logic gives classical logic). Trovatore forgot what the implication introduction rule of natural deduction was called. --- Charles Stewart^(talk) 17:58, 25 January 2006 (UTC)

to decrypt a message

I have a message to decrypt, is there someone can help, thank you. a message: tcbat jcbet cjwjr lybrk pwjad fdzzp orfba by using an affine cipher to encrypt it, and there is no key. the hint is the first and last characters of the plaintext are w and e, respectively. also, a function f(x)=ax+b (mod 26) has been used in it, and I want to find the value a and b.

                                             thanks again.

If you know it's a linear function, and you know two values (f(w) = t and f(e) = a), then you can determine the function. —Keenan Pepper 20:17, 22 January 2006 (UTC)

Actually that's not true for modular arithmetic. Forget I said it. —Keenan Pepper 20:37, 22 January 2006 (UTC)

Well, there may be more than one possible solution, but it sure narrows it down. —Keenan Pepper 20:47, 22 January 2006 (UTC)

is "tcbat" a code if a word? I could not find english words of the form "w..ew". Besides, the equations f("w")="t", or 22a+b=19 mod 26, and f("e")="a" or 4a+b=0 mod 26 can not be solved for a and b, can they? (Igny 21:39, 22 January 2006 (UTC))

Maybe there are extra letters on the end, to make a full group of 5? —Keenan Pepper 22:01, 22 January 2006 (UTC)

I'm with Keenan on the possibility of padding, or alternatively all spaces and punctuation have been removed and the letters grouped into blocks of 5. However (assuming a simple letter<=>mapping a<=>0, b<=>1, ..., z<=>25, or anything else with the letters in alphabetic order), you're right that there cannot be values of a and b which satisfy both equations, since the first implies b is odd while the second requires b to be even. Maybe a typo in the question? -- AJR | Talk 23:15, 22 January 2006 (UTC)

Let's see if we can do this in a mathematical fashion...Letting $a=1$ and so on (ie excahnging each letter with the number of its position in the alphabet), then the functions become $f(5)=5a+b(\mathrm {mod} 26)=1$ and $f(23)=23a+b({\textrm {mod}}26)=20$. Now, we'll let $5a+b=27$ for the moment. Now, using Cramer's Rule, we have...$a=-{\frac {1}{3}},b=27{\frac {2}{3}}$. Okay, so a little bit ofbrute force may be required here, but you have a starting point. --JB Adder | Talk 23:23, 22 January 2006 (UTC)

I think $a$ and $b$ in $f(x)=ax+b{\pmod {26}}$ must be integers, for the following reason: Since we are working mod 26, $x\in \mathbb {Z} _{26}$ i.e. $0\leq x\leq 25,x\in \mathbb {Z}$, so when $x=0$, $f(0)=b{\pmod {26}}$, and so $b$ must be an integer. Now consider $x=1$, giving $f(1)=a+b{\pmod {26}}$ so $a+b$ is an integer, and since $b$ is an integer, so is $a$.

So, $a$ and $b$ must be integers. But, mapping the message letters to numbers in alphabetical order, A->0, B->1, ..., Z->25 (which letter we call 0 doesn't matter, it just changes the value of $b$) we get from the hint in the original question f(W) = T i.e. $f(22)=22a+b=19{\pmod {26}}$(1) and f(E) = A i.e. $f(4)=4a+b=0{\pmod {26}}$(2). Equation (2) tells us that $4a+b=26x$ for some unknown integer $x$, and since $4a$ and $26x$ are both even (because they are integer multiples of even numbers) $b$ must also be even. From equation (1), we get $22a+b=19+26y$ for some unknown integer $y$ i.e. $22a+b$ is odd, which implies $b$ is odd. But $b$ clearly cannot be both even and odd, so there must be an error in one of: the supplied ciphertext, the supplied hint, or my reasoning. And I'm fairly confident I haven't made a mistake - if I am wrong, I'll give a small wikiprize to whoever shows how. -- AJR | Talk 01:51, 23 January 2006 (UTC)

There is a possibility that the position of the letter is also important, $y_{n}=f(x_{n},n)=ax_{n}+bn+c\mod 26$ (Igny 13:09, 23 January 2006 (UTC))

The qeustion specifies that it is an affine cipher, which tells us that the general form of the encryption function is $e(x)=ax+b{\pmod {m}}$, where $m$ is the size of the alphabet. Given that it is an affine cipher, it should be breakable from the information we have - "The cipher's primary weakness comes from the fact that if the cryptanalyst can discover the plaintext of two ciphertext characters then the key can be obtained by solving a simultaneous equation." -- AJR | Talk 17:59, 23 January 2006 (UTC)

Keenan Pepper: you're right, there's indeed an extra letter at the end which decrypts to "e". The last given letter "a" decrypted is "t".

It's not difficult to find the solution knowing only that the first letter of the message is "w", as that leaves only $\varphi (26)$ possibilities. As an addittional clue, the most frequent letters after a "w" beginning a word are (in order starting with most frequent) "hiaeorw". Most of those are impossible for the same parity reason as mentioned above. That leaves two: "wrgvwqrgnwrqdqamzgabedqvcyckketaygv", and the real solution which I won't spoil here. – b_jonas 21:16, 23 January 2006 (UTC)

Oh, and there seems to be one more typo in the middle of the cyphertext. The correct one should be "tcbatjcbejcjwjrlybrkpwjadfdzzporfbaj" I guess. – b_jonas 21:24, 23 January 2006 (UTC)

January 23

Name of a kind of 3D space

I was wondering if there's a name for this 3D space I thought of when playing with some functions. In a way, the space would be $\mathbb {R} \times \mathbb {R} \times (\mathbb {C} -\mathbb {R} )$ (probably wrong notation), that is, a complex plane with an extra real axis perpendicular to it. This space would allow functions returning imaginary numbers such as $f(x)={\sqrt {r^{2}-x^{2}}}$ to be seen in three dimensions. ☢ Ҡieff⌇↯ 01:24, 23 January 2006 (UTC)

"A complex plane with an extra real axis perpendicular to it" should be just $\mathbb {C} \times \mathbb {R}$, no? —Keenan Pepper 01:34, 23 January 2006 (UTC)

I wasn't so sure about this point (hence the "wrong notation" note), because one of the axis would be exclusively for the imaginary part, where the other two would be real. If $\mathbb {C}$ automatically defines a 2D space (which seems to be about right from the complex number article), then I guess this space would indeed be $\mathbb {C} \times \mathbb {R}$. I just thought that the complex plane would be $\mathbb {R} \times \mathbb {C} -\mathbb {R}$ (reals in one axis, imaginary on the other). Made sense to me. ☢ Ҡieff⌇↯ 01:49, 23 January 2006 (UTC)

Are you sure "extra real" means imaginary. To me extra-real means exactly $(\mathbb {C} -\mathbb {R} )$ , i.e., numbers Z such that Imag(Z) <> 0 (all numbers that are not real, like 5i and 2+3i ). deeptrivia (talk) 05:03, 23 January 2006 (UTC)

Take the example (fixed now, actually): $f(x)={\sqrt {r^{2}-x^{2}}}$. If x > r, the function turns into an imaginary number, so you can imagine a third imaginary axis perpendicular to the x and y on the cartesian plane. That's what I meant... ☢ Ҡieff⌇↯ 05:11, 23 January 2006 (UTC)

is that it? Two branchs of a hyperbola escaping in the imaginary direction, and a semicircle in the real direction. -lethe ^talk 05:26, 23 January 2006 (UTC)

Yep, that was the space I meant! Where did you plot this? ☢ Ҡieff⌇↯ 06:46, 23 January 2006 (UTC)

I'd thought of this before, a long time ago. I believe you can take your space to be R³, with your regular real-valued function remaining on the xy plane, and then having the complex values on the xz plane, since C is isomorphic to R². Dysprosia 05:38, 23 January 2006 (UTC)

... though you wouldn't be restricted to xz. x is in R, so is the real part of f(x). The imaginary part of f(x) would just be mapped on the z axis. Example: $f(x)=x+{\sqrt {r^{2}-x^{2}}}$ wouldn't be restricted to two planes (xy & xz). ☢ Ҡieff⌇↯ 06:46, 23 January 2006 (UTC)

You can choose different axes if you wish, but three should be enough. Dysprosia 22:55, 25 January 2006 (UTC)

How do you memorize your multiplication table?

Greetings:

I was wondering how all of you Wikipedians rote your multiplication table when you were small? Is there any tricks or English nursery rhymes used to facilitate this learning proces?

Regards,

129.97.252.63 03:06, 23 January 2006 (UTC)

Well, whenever I forgot one I would figure it out by successive addition, which was a pain, so that taught me to remember them instead. =P —Keenan Pepper 03:11, 23 January 2006 (UTC)

I don't recall how I memorized mine. There are a number of famous tricks, though. You might also consider using software - like this freeware - to learn them. Alas, I came around before the days of ubiquitous home PCs. --George 03:58, 23 January 2006 (UTC)

There is no magic substitute for repetition. There are tricks for confirmation or reconstruction. For example, any product with an even number must be even, and the product of two odd numbers must be odd. Any product with 9 must give digits that sum to 9. (Thus 9×8 = 72, with 7+2 = 9.) Any product with 5 must end in 5 or 0; and the product with an even number halves the number and appends 0, or with an odd number appends 5. (Thus 6×5 = 30, 7×5 = 35.) To reconstruct, say, 3 times 7, count by threes, as in 3, 6, 9, 12, 15, 18, 21. Commutativity applies; thus 3×7 must equal 7×3, and we can count 7, 14, 21. For multiplying large numbers, it can be helpful to use an entirely different procedure such as that of Trachtenberg. --KSmrq^T 05:32, 23 January 2006 (UTC)

Since I learned them before PCs were common, I used flash cards. They are very helpful as you can remove cards you have already learned and concentrate on the rest. One pitfall to avoid is always memorizing them in the same order. When you do this, you may only be able to regurgitate them in the same order, which is frequently too slow to pass a test. StuRat 15:54, 23 January 2006 (UTC)

We used songs like "she'll be coming around the mountain" for multiples of 8's. And "you are my sunshine" for 6's. I still remember them well :) Keepitrude 02:24, 24 January 2006 (UTC)

Zero

To whom it may concern,

I’m sorry if I was rude “Ksmrq” and thank you “The Infidel”, for allowing me chance to post the question again.

According to history, the Mayans did not create the number Zero (0), but the concept of Zero (0). This discovery, is very important because, by definition, the number Zero (0), not created, should not have a point of origin where infinity exists, but it does.

Infinity, having no beginning and no end but at the same time, the number Zero (0), gives infinity a point of origin, a beginning.

Examples: The number Zero (0), equal to infinity, is the point of origin, in which a number can go positive or negative, to infinity and back to Zero (0).

I had time to sleep on it as to give an example as to how Zero (0) can be the Biggest number & the Smallest number. Here goes nothing.

Example: 1. Zero (0) is the Biggest number when going from Zero (0) to -1, -2, -3, -4, > to Infinity.

2. Zero (0) is the Smallest number when going from Zero (0) to +1, +2, +3, +4, > to Infinity.

I believe this sounds right.

Based on this findings, I'm I correct to believe that: Zero (0 is equal to Infinity?

I love waking up to a great tasting smell of coffee.

Sincerely,

Guadalupe Guerra, Jr.

I'm really sorry, but you still haven't quite understood how this works. Please use the [edit] link on your previous post to continue discussion. enochlau (talk) 09:54, 23 January 2006 (UTC)

I'm sorry, but this is still just rambling. 37 is the smallest number when going from 37 to +Infinity and the biggest when going from 37 to -Infinity. Zero possesses special significance in cases such as the additive identity but is wholly unremarkable on the number line. — Lomn Talk 20:00, 23 January 2006 (UTC)

You mix up definitions (ideas). Firstly, lets fix this: the numbers in your example are integers.

There are two different concepts of "bigger than" for integers, which for natural numbers cannot be destinguished. One is the concept of "order", in the sense that one comes before two and six comes before seven. :The other is the concept of "magnitude" (see absolute value). This is "five sheep are more than four sheep".

But with negative values possible, minus four comes before minus three. On the other hand, if have one billion dollars, this is a whole lot of money. If you have one billion dollars in debts, that's also an awful lot of money and in no respect small, althoug in mathematical notation it is "<" (spoken "less than") +1.000.000.000. The Infidel 20:25, 23 January 2006 (UTC)

Area vs. Perimeter and Surface Area vs. Volume

I was thinking about these questions, and have come up with answers for some, others in bold I have no answer to.

In two dimensional space, which shape gives:

• The largest perimeter to a constant area?
  • The perimeter could be infinite, as you could take a rectangle with the constant area, and repeatedly half the width and double the length.
• The smallest perimeter to a constant area?
  • The isoperimetric inequality tells you that this must also be a circle -lethe ^talk 09:59, 23 January 2006 (UTC)
• The largest area to a constant perimeter?
  • A circle
• The smallest area to a constant perimeter?
  • A straight line with the length of the perimeter
    • Well, any open curve with the length of the perimeter. Or half the perimeter, if you look at it the right way. Black Carrot 01:27, 29 January 2006 (UTC)

In three dimensional space, which shape gives:

• The largest surface area (SA) to a constant volume?
  • Just like with the rectangle with constant area, this is unbounded -lethe ^talk 09:59, 23 January 2006 (UTC)
• The smallest SA to a constant volume?
  • A sphere.
• The largest volume to a constant SA?
  • sphere, by 3 dimensional isoperimetry -lethe ^talk 09:59, 23 January 2006 (UTC)
• The smallest volume to a constant SA?
  • Once again, would be a straight line?
    • No, a straight line will have 0 SA. But a planar square will work. -- Meni Rosenfeld (talk) 10:09, 23 January 2006 (UTC)

Thanks.

dual spaces of (non) locally convex spaces

So according to a couple places here, the L^p space for 0<p<1 has no nontrivial continuous functionals. Another example of a space that's not locally convex is stated to have no nontrivial continuous functionals as well. I got to thinking that it's probably true in general that if a space isn't locally convex, then it has to have a trivial continuous dual space. So I tried to prove it.

Proof: Suppose f is a linear functional on TVS V. Let U be an open set in space V. Assuming the underlying field is locally convex (since R and C are the only topological fields worth spit), I may choose a convex open set O in f(U) (an interval if the field is R or a disc if the field is C). Define p(x) to be |f(x)|. Then p^–1(O) is a convex, balanced, and absorbing subset of U, provides a base, and hence V is a locally convex space.

OK, so as a sanity check, for example, the Dirac delta functional should not be continuous on L^1/2([–1,1]). I should be able to find a convergent sequence of functions f_n → f so that f(0) ≠ lim f_n(0). In other words, it converges in the ||*||_1/2 norm, but not pointwise.

So I want someone to look at my proof and tell me it looks OK, and to help me find a sequence to show the failure of continuity. I know some examples of sequence of functions which converge to something in the uniform norm but not in the pointwise topology, but those examples, at least the ones I know, the two limits disagree on sets of measure zero, so those examples don't help with L^p spaces. -lethe ^talk 09:52, 23 January 2006 (UTC)

I haven't dealt with TVS's for a long time. It looks okay except that I am too stupid to see why this is a basis of the topology right now. About your example: How do you even define the Dirac delta on a space that is not a subspace of C⁰? You need a different type of example to start with if you don't want the set of measure zero problem to come in, maybe try something in the dual space of $L^{\infty }$? Kusma (討論) 02:26, 25 January 2006 (UTC)

I don't think you're too stupid, I think I am. What I mean is, since writing that, I've come to learn that in fact, my sets are not a base. I didn't update it, because I thought no one was looking at it, but I should have. OK, so I write that p^–1(O) ⊆ U. If this were true, then for any open set U, I would have a convex subset, which is the definition of a base. Except of course it's not true. Consider basically f(x₁,x₂) = x₁, and U the unit disc in R². p^–1(O) is not in U. I think I blundered.

I've come to know that my conjecture is wrong, so we know my proof must be as well (it's like you say, those sets don't comprise a base). There are spaces which are not locally convex but still have nontrivial dual spaces. I'm still struggling with it, but when I've got it all internalized, I'm going to add a section to locally convex spaces about it.

As far as the Dirac delta, I guess you're right. Members of L^p spaces are only defined up to almost everywhere, so the delta functional isn't well-defined. Is that your point? That was sloppy of me as well. (Won't L^∞ have the same problem though?)

Anyway, thanks for your input. I'm struggling my way through all this. -lethe ^talk 02:42, 25 January 2006 (UTC)

The (continuous) dual spaces of C⁰ and L^∞ are two different spaces of measures, with the dual space of L^∞ being measures that are slightly more regular (the Dirac delta is not regular enough), hence can be integrated against functions that are only defined almost everywhere. I can't remember the details, though. Kusma (討論) 04:39, 25 January 2006 (UTC)

I guess there should be some list with three columns:

functions	dual space	measure
continuous functions of compact support	distributions of order 0	Borel measures
smooth functions of rapid decrease	tempered distributions	?
smooth functions of compact support	distributions	?

where each space of functions has a dual space of certain distributions, and each distribution gives a measure by the Riesz-Markov theorem. Where can we find a list like this? And then your point will be that for the right choice of distribution, it will be defined on equivalence classes of functions that are equal almost everywhere. Somehow, I think the Dirac delta distribution can still be defined over functions defined almost everywhere by using the Dirac measure, but then of course that's a different notion of almost everywhere. Hmm -lethe ^talk 07:07, 25 January 2006 (UTC)

mathematics

Hello there,my name is Fatima and i am an undergrad student i have two questions 1. whats the difference between Δx ,δx , dx (as written in differential equations) and dx ( as written partial differential equations) since they all signify small change in the variable under consideration ( which is x in this case) 2.whats iota. i know thats a strange question, but i know its the under root of -1, but whats its physical significance, and why do we treat imaginary parts of numbers and equations if its imaginary?? i hope i have been able to explain myself.

Well, 1) Δx isn't a derivative, it usually represents a change in x. δx is usually the same, only for a smaller change. Sometimes it's used for a partial derivative. The difference between the 'd' in 'dx' in a diff equation and the 'd' in a partial diff equation is to signfify that it's a partial derivative (the derivative of only one variable of several). That's my experience, although someone might know a more formal definition. 2) Ok, the root of -1. Well.. it doesn't in itself have any physical significance. It's just an abstract mathematical object with a certain property, namely that it's the square root of -1. Now, this gains physical significance when you're modelling different problems. For instance, if you've got a second-order diff equation which you solve the characteristic equation for, the real roots will represent exponential solutions and the imaginary ones will represent periodic ones. And these periodic solutions to diff equations show up everywhere in nature, from the vibration of strings to the Schrödinger equation. --BluePlatypus 17:39, 23 January 2006 (UTC)

And for your last question, I once wondered that myself a lot. The issue is really that "imaginary" is an unfortunate name. Imaginary numbers aren't imaginary, they're just as real (or unreal) as other numbers. The natural numbers are an abstract thing (five cows are real, five apples are real, 'five' isn't), negative numbers are more abstract (how do you have minus five cows?), and imaginary numbers are even more abstract. Natural numbers work well for representing everyday countable objects. Decimal numbers or fractions are needed once you're not dealing with whole objects. Negative numbers are useful for balance sheets of money. And imaginary numbers are really good for things that are representable by a differential equation. None of them are more or less "real", they just follow different rules. Remember grade school when you had to re-learn addition and multiplication with negative numbers? Complex numbers are just another extension, one which has turned out to be very useful (Which is why you learn it.) There are other extensions too, like Quaternions, which didn't turn out to be terribly useful, and aren't used as much. --BluePlatypus 18:53, 23 January 2006 (UTC)

A protest is warranted by the characterization of quaternions as not terribly useful. They are widely used in many applications that need to manipulate 3D rotations and orientations. Examples include robotics, satellite control, guidance systems, six-degree-of-freedom input devices, and 3D computer graphics (including most games). Also, quaternions paved the way for today's wider concept of abstract algebra, where we can define an algebra of matrices, or choose formalisms like groups or rings or p-adic numbers or whatever we choose. They also live on in physics, disguised as quantum spin, and generalize within Clifford algebra as spin groups. Maxwell's equations for electromagnetism, one of the cornerstones of modern physics, are originally quaternion-based, though that dependency is often suppressed. Also, we have the notable theorem of Frobenius that there are exactly three real associative division algebras: real numbers, complex numbers, and quaternions. A typical reason for stopping with complex numbers is because they are enough to solve any polynomial equation; that hardly justifies denigrating quaternions. --KSmrq^T 01:23, 25 January 2006 (UTC)

I believe Δx is used for finite changes in x, as in slope = Δy/Δx, while dx is typically used for the infinitely small divisions of x used in calculus. The square root of -1, i, is also used in some common problems in electronics dealing with reactance. However, it is called j there, since i is used in electronics to mean current. StuRat 18:55, 23 January 2006 (UTC)

See this site for an excellent visualization of complex numbers:

http://www.st-andrews.ac.uk/~jcgl/Scots_Guide/info/signals/complex/cmplx.html

StuRat 19:01, 23 January 2006 (UTC)

File:Dxddx.jpg

Δx vs. δx

Hopefully this illustration helps with Δx and δx. Both tend to dx as the independent variable becomes infinitesimal. deeptrivia (talk) 19:24, 23 January 2006 (UTC)

The names "real" and "imaginary" are not the best. Perhaps "visible" and "hidden" numbers would be a better way to describe the differences. Of course, in French hidden is "occult", and having "occult numbers" might attract a lot of weirdos. LOL. StuRat 19:39, 23 January 2006 (UTC)

;-) O Fatima, if your question is really about mathematics and not about lowely computations, then do not let yourself get befooled by the crude understanding and missleading talk of the engineers ;-)

To be serious now: if you want not only the ability to correctly solve equations but to have a deeper understanding, it is wrong to think of the imaginary unit as "the root of -1", $i={\sqrt {-1}}$, as this leads to $-1=({\sqrt {-1}})^{2}={\sqrt {(-1)^{2}}}={\sqrt {1}}=+1$.

Instead, think of i as a (one of two) solutions (roots) of the equation $x^{2}+1=0$, which just happen not to be included in the set of real numbers but require an extension thereof, the complex numbers.

In case you want a hint of the real beauty of complex numbers, look at Euler's formula and as it expands using Taylor series. The Infidel 20:59, 23 January 2006 (UTC)

I'm wondering if there is a font/character problem with your first question. Usually the character used for partial differentials is a curly d, "∂". I'll assume that's what you meant.

1. • The notation "Δx" is used consistently to mean a finite change in x, however small. So we could say that if y is a function of x, then Δy/Δx gives a number, a slope, which approximates the rate at which y changes proportional to a given change in x.
   • The notation dx is often used to mean an infinitesimal change in x, a subtle concept which takes great care to formalize. Thus dy/dx is not an approximation to the proportion of change, but exact. In more sophisticated mathematical contexts the notation dx is used for a wider range of purposes, which we needn't explore today.
   • The notation δx is typically only seen in variational calculus. This is difficult to explain succinctly. For ordinary differential calculus, dy/dx indicates that a numeric quantity, y, varies with a change in another numeric quantity, x, by some definite proportion. But in variational calculus we essentially change all the values of a function proportional to an input change. The function itself varies. An example is where we begin with a collection of functions that give position in a plane in response to distance along a path. If we want to find which of those functions gives the shortest path from one fixed position to another fixed position, we use variational calculus, and the "δ" symbol denotes variation in this sense.
   • The notation ∂x is used in multivariable calculus, where a function like f(x,y) = x²+3y is a function of more than one variable. Thus ∂f/∂x describes the proportion by which f changes when x alone changes, holding y fixed.
2. Imaginary numbers are written using the letter lower-case " i ", not Greek iota (" ι "); the latter has no dot. By definition, this letter denotes a quantity that squares to −1. Where ordinary numbers allow us to step forwards and backwards along the real number line, imaginary numbers allow us to step to the side. More generally, complex numbers allow us to step at any angle and by any amount we wish. The need for such numbers arises naturally when we look for roots of polynomials. For example, the roots of x²−1, the values of x that cause the polynomial to evaluate to zero, are twofold in number: +1 and −1. The roots of x⁵−1 are fivefold in number, and step off a regular pentagon around zero. Such numbers are just as normal and physical as, say, the square root of 2; the name "imaginary" is a holdover from earlier times when they were poorly understood and treated with suspicion. --KSmrq^T 01:47, 24 January 2006 (UTC)

To illustrate the above x⁵−1 example, we get 5 roots:

• +1

• +cos(72) + sin(72)i = +.309 + .951i

• +cos(72) - sin(72)i = +.309 - .951i

• -cos(36) + sin(36)i = -.809 + .588i

• -cos(36) - sin(36)i = -.809 - .588i

Now let's graph them on the complex plane:

     ........................+i.........................
     .........................^.........................
     .......................+1|.........................
     .........................|.----O...................
     .....................----|.......\.................
     ................----.....|.........................
     .........O----...........|..........\..............
     .........|...............|.........................
     .........|...............|.............\...........
     .........|...............|.........................
     .........|...............|................\........
     .........|...............|.........................
     -R <-----+---------------+-------------------O-> +R
     ....-1...|...............|..................+1.....
     .........|...............|................/........
     .........|...............|.........................
     .........|...............|............./...........
     .........|...............|.........................
     .........O----...........|........../..............
     ................----.....|.........................
     .....................----|......./.................
     .........................|.----O...................
     .......................-1|.........................
     .........................v.........................
     ........................-i.........................

StuRat 10:15, 24 January 2006 (UTC)

decrypt a message

given f:Z26->Z26, f(x)=23x+10 (mod 26) is a bijection(one to one and onto) that it can be used as a subtitution cipher, then decrypt the message ZYCU was enctypted by using the function."

Sounds like homework to me. --BluePlatypus 17:49, 23 January 2006 (UTC)

Homework! Homework! Please try the problem out and you can always come again and write up where your actual difficulty is, or rather where your gettin held up. Then maybe we wikipedians can help you out. But never post the whole homework problem, that spoils the whole point of it! Right? -- Rohit 18:18, 23 January 2006 (UTC)

I guessed the answer was CODE but it turns out that was a wrong guess. As a hint, the inverse of 23 with the modulus 26 is 17. – b_jonas 20:30, 23 January 2006 (UTC)

January 24

mathematical symbols

What is the notation used to signify Infinity, please? A.Hortin

It is somewhat like an "8" turned 90° on its side, like this: $\infty$ ☢ Ҡieff⌇↯ 04:11, 24 January 2006 (UTC)

It is a lemniscate. -lethe ^talk 04:12, 24 January 2006 (UTC)

In case lethe's answer wasn't clear: Yes, $\infty$ is used to denote infinity, but only in those cases when we refer to "just infinity". When we want to distinguish different kinds of infinity, there is a different symbol for each kind. -- Meni Rosenfeld (talk) 07:41, 24 January 2006 (UTC)

Meni says true. He's been discussing elsewhere the need to distinguish between +∞, –∞ and unsigned ∞. Among the ordinals and cardinals, there's also alef ℵ, beth ℶ, omega ω, epsilon ε, and more. Many different kinds of infinities, many different symbols! -lethe ^talk 07:46, 24 January 2006 (UTC)

Mathematical news

Anyone knows a good source of news on the field of mathematics? Even better if they have a newsfeed ☢ Ҡieff⌇↯ 09:19, 24 January 2006 (UTC)

Googling 'math news' or 'math news RSS feed' brings up quite a few promising looking results, amongst them, http://mathworld.wolfram.com and http://www.mathforge.net --Noodhoog 13:15, 24 January 2006 (UTC)

I sometimes look at the news section of the AMS website (of course, they concentrate on AMS and US news). -- Jitse Niesen (talk) 12:00, 25 January 2006 (UTC)

Perhaps we could have a news section at the mathematics portal here on Wikipedia? Fredrik Johansson - talk - contribs 14:58, 25 January 2006 (UTC)

January 25

(no questions today)

this sucks ☢ Ҡieff⌇↯

January 26

elementry math

please define simple balance.

daw

Do you mean balance as in a checkbook or as in a see-saw or something else ? StuRat 02:33, 26 January 2006 (UTC)

Perhaps equals on both sides of an equation. User:AlMac|^(talk) 10:04, 27 January 2006 (UTC)

Canonical Painlevé ODEs

From Painlevé transcendents: In a landmark achievement, they found that up to certain transformations, every such equation [2nd order ODEs with Painlevé property] can be put into one of fifty canonical forms.

The page then covers the 6 equations that do not have elementary solutions. What I would like to know is where I can see a list of all 50 forms, preferably with at least some information on the solutions to each. I can find nothing on Wiki and far too many pages on Google covering Painlevé property-related study that is far beyond what I want to know. TIA, Confusing Manifestation 16:41, 26 January 2006 (UTC)

According to my notes, it is covered in Ordinary Differential Equations by Edward Lindsay Ince, originally published in 1926, reprinted by Dover in 1956. -- Jitse Niesen (talk) 17:18, 26 January 2006 (UTC)

Sequence

i am trying to find out the sequence to the following mathematic sequence: 11235813.....what is the proper name to this sequence? thank you

Assuming you mean 1, 1, 2, 3, 5, 8, 13, ..., the sequence you're looking for is the Fibonacci sequence. Fredrik Johansson - talk - contribs 20:58, 26 January 2006 (UTC)

You should search for it on the OEIS. --cesarb 16:19, 28 January 2006 (UTC)

Support (geometric)

I asked a question a while back on Talk:Support (mathematics) and haven't gotten a response. In short: is the usage of the term "support function" meaning "a function taking an N-directional vector and returning the (any) point P on the N-dimensional convex hull of an object for which the function N dot P is maxed" a valid definition to add to the Support (mathematics) page, or is this usage specific to geometric computer algorithms?

As my math-ese is rough, some examples of (local-space) support functions for simple shapes:

• sphere: support(V) = normal(V) * radius
• convex hull: support(V) = the vertex of the hull which is farthest along vector V

--Kyle Davis 23:06, 26 January 2006 (UTC)

According to my EDM2, this thing is called a supporting functional. The space has to be locally convex and Hausdorff, and the set itself must be convex. Then a linear functional on C is called a supporting functional if sup f(C) = f(x) for some x in the boundary of C, and x itself is called a supporting point of C. So that's about all EDM2 has about it. I'm not sure if this agrees with your expecation. Sounds like it might, but this definition doesn't assume an inner product, so I don't know what normal would mean in that context. -lethe ^talk 23:53, 26 January 2006 (UTC)

PS Of course, in the case that your space is an inner product space, then every linear functional is of the form f(x)=(x,n) for some n. -lethe ^talk 23:57, 26 January 2006 (UTC)

Assuming I'm understanding what you wrote correctly, it's related to what I'm describing, but not quite a match. Specifically, the topology requirements agree, but the function itself isn't a mapping from vector space into linear space but is instead a mapping from an input vector (representing a direction in localspace for the shape) to an output vector (representing a location on the hull). In looking online, I'm finding many examples in source code, but nothing formal. --Kyle Davis 00:14, 27 January 2006 (UTC)

It sounds like your map is related to my map in the following way: you assign to every vector the supporting point of the functional defined by the vector. -lethe ^talk 00:31, 27 January 2006 (UTC)

That would make sense. So would it be preferable for me to put this at Support (mathematics), or create a new Supporting functional and put a note at the former regarding the terminology "support(ing) function"? --Kyle Davis 00:36, 27 January 2006 (UTC)

I don't have a strong opinion about what name it should go under. If you know this concept as a "support" rather than a "supporting functional", then use that name. If you don't use the name "supporting functional", you should of course make a redirect. -lethe ^talk 10:23, 27 January 2006 (UTC)

January 27

Descriptive term

Howdy! I'm in the middle of creating an article on "Pin art/Pinpressions, but I'm struggling to come up with a term for the toy's inability to create shapes with a concave surface in the plane perpendicular to the original "blank" plane. I figure there must be a term for this type of function, and that I'd be better off asking here than on the Language Desk. Any ideas? GeeJo _{(t) (c)} • 10:31, 27 January 2006 (UTC)

The only thing I can think of to say is that the height must be a function of the position (because each pin can have only one height). —Keenan Pepper 23:38, 27 January 2006 (UTC)

Yeah, a function or, more specifically, multidimensional function (because it's a contour plane rather than a curve) is the only thing that seems to describe it. What exactly are you trying to do with this word, phrase or description? Are you hoping for a noun? Verb? Something for laymen or jargonauts? Black Carrot 03:16, 28 January 2006 (UTC)

Under some assumptions about continuous differentiability, if you want an intrinsic definition you could say that the normal to the surface must (be able to be defined continuously so as to) point upwards everywhere, never downwards, or equivalently that the signed Jacobian thingy dR / dx dy must be positive everywhere (where x, y are coördinates of the "original plane"). —Blotwell 03:20, 28 January 2006 (UTC)

vector sums?

This question no verb. —Keenan Pepper 23:39, 27 January 2006 (UTC)

I would say see vector addition, but that redirect doesn't seem to have one good example in it, so here are four decent examples:

Let's say vector A is 4 units long and vector B is 3 units long...

• If they are pointed in exactly the same direction, then just do straight addition to get a vector in that direction with a unit length of 7.

• If they are pointed in exactly the opposite direction, then subtract the smaller from the other. In this case, the unit length is 1 and it points in the same direction as the longer vector.

• If the vectors are at a right angle, use the Pythagorean Theorem to find the length of the diagonal vector. In this case, we would get $3^{2}+4^{2}=D^{2}$, so D = 5. This means the unit length is 5. To find the angle of the resultant vector, use trig: arctan(3/4) = 36.87 deg from the unit length 4 vector toward the unit length 3 vector.

• If the vectors are at some other angle, it is necessary to find the X and Y components of each vector (each of which may be positive or negative), then add those together. At this point it has been reduced to the previous case of two vectors at a right angle and you can use the method above to combine them into a single resultant vector.

StuRat 02:02, 28 January 2006 (UTC)

Our article for this seems to be Vector (spatial). It's linked at the bottom of vector space where noöne will find it and noöne who does will realize it's what they want. Should we have a dab link at the top of vector space? Should we change the redirect on vector addition? —Blotwell 03:15, 28 January 2006 (UTC)

Sounds good to me ! StuRat 06:42, 28 January 2006 (UTC)

Principal component analysis

I've tried to read principal component analysis and I can't understand it at all. Can someone give me a one or two line summary in lay man's terms? (This is not homework -- it is a statistical method which comes up in some historical work I am doing but I'm having trouble figuring out exactly what it is.) The best I can come up with is that it looks like a very fancy way to average a lot of data points. Right? Wrong? I'm completely clueless here. --Fastfission 04:09, 28 January 2006 (UTC)

Let me try...Take a dataset generated from a bunch of variables. Can you reasonably represent this dataset with fewer variables than the original? That is to say, can you choose a fewer number of new variables which would describe most of the details of the original data set? PCA answers this. It also tells you which new variables to use, and it also tells you how much these new variables contributes to describing the original data set. --HappyCamper 05:42, 28 January 2006 (UTC)

Hmm. Maybe if I told you the context it would help more: a number of geneticists use PCA to compile gene frequency data (plotted on a map) of 82 genes into one massive "synthetic" map, which is a combinationof all of them (the point is to look for human ancestry trends and soforth). So... what they are doing then with the data is "simplifying it" into one coherent dataset that reflects the combined influence of all of the data? (Basically, I need to describe this in about one sentence in a paper I am writing which mentions this gene mapping project.) --Fastfission 13:30, 28 January 2006 (UTC)

What if I said... "They compiled the map using principal component analysis, a common statistical technique which simplified the frequency data from the other 82 genes into one composite map." Would that be true enough? --Fastfission 15:23, 28 January 2006 (UTC)

It seems quite okay to me, but it is a bit hard to say without knowing what data the map represents. Instead of "simplified", I'd use "combined" or "aggregated", though neither of these words indicates that you lose some data in the process. And I don't understand why you included the word "other" in your sentence. -- Jitse Niesen (talk) 16:00, 28 January 2006 (UTC)

Thanks for the suggestions! Yeah, the "other" just crept in there somehow. --Fastfission 19:57, 28 January 2006 (UTC)

January 28

names for symbols used to record time

From the mathematical perspective, does the colon (as it is used to write time) have a specific technical name - or is it merely called a colon as it would be in composition? Our school has a desire to be precise when we teach mathematical concepts, and time is currently in lesson plans for first and second grade. Thanks for your help! --216.63.217.100 20:34, 28 January 2006 (UTC)

I think colon is fine. That's the only word I've ever heard used. —Keenan Pepper 21:24, 28 January 2006 (UTC)

In English class, the lesson on colons always included timekeeping, so yeah, I figure that's the right word. Black Carrot 01:22, 29 January 2006 (UTC)

January 29

Lyapunov stability for state space models

The article says:

A state space model

${\dot {\textbf {x}}}=A{\textbf {x}}$ is asymptotically stable if

$A^{T}M+MA+N=0$

has a solution where $N=N^{T}>0$ and $M=M^{T}>0$ (positive definite matrices).

Does anyone have a proof for this? deeptrivia (talk) 23:37, 28 January 2006 (UTC)

You might get better responses if you try Wikipedia:Reference desk/Mathematics. Grutness...wha? 00:42, 29 January 2006 (UTC)

I think that $V(x)=x^{T}Mx$ is a Lyapunov function under the condition. By the way, I'd write the condition as $A^{T}M+MA$ is negative definite and M is positive definite (and symmetric). -- Jitse Niesen (talk) 02:01, 29 January 2006 (UTC)

So the task is to prove that ${\dot {V}}(x)$ is negative definite, i.e.,

${\dot {V}}(x)={\dot {x}}^{T}Mx+x^{T}M{\dot {x}}$
$=x^{T}A^{T}Mx+x^{T}MA^{T}x$
$=x^{T}(A^{T}M+MA^{T})x$ is negative definite.

And that implies $A^{T}M+MA^{T}$ is negative definite. Is that what you mean? deeptrivia (talk) 02:47, 29 January 2006 (UTC)

Yes: then ${\dot {V}}(x)={\dot {x}}^{T}Mx+x^{T}M{\dot {x}}=x^{T}A^{T}Mx+x^{T}MAx=-x^{T}Nx$, so $V(X)>0$ and ${\dot {V}}(x)<0$ away from 0.

Okay...got it! Thanks! deeptrivia (talk) 02:49, 29 January 2006 (UTC)

Can you demonstrate how to use this? Let's say we have the case of a damped harmonic oscillator (which we know is asymptotically stable.) We have:

$A={\begin{bmatrix}0&1\\{-M^{-1}K}&{-M^{-1}C}\\\end{bmatrix}}$

M, K and C are matrices. How can we use this property to prove stability? Thanks! deeptrivia (talk) 03:00, 29 January 2006 (UTC)

Beautiful

this : eiπ + 1 = 0 is supposed to be the most beautiful (or one of the most beautiful)formula, no, not formula, theorem, but ...since I don't know any math, I don't find it beautiful because I do not understand it. lol. --Cosmic girl 05:16, 29 January 2006 (UTC)

Its beauty stems from the fact that it relates seemingly unrelated constants like the base of natural logarithm, square root of negative one, ratio of circumference to diameter of a circle, multiplicative identity and additive identity. deeptrivia (talk) 05:20, 29 January 2006 (UTC)

BTW, it's ${e^{i\pi }}+1=0$ deeptrivia (talk) 05:22, 29 January 2006 (UTC)

It's a special case of Euler's formula, which is an extension of the exponential function to complex numbers. I highly recommend the book Visual Complex Analysis (ISBN 0-19-853446-9) if you can afford it. It has a geometric explanation of why the exponential function behaves so. —Keenan Pepper 07:07, 29 January 2006 (UTC)

Beauty is in the mind, as well as the eye, of the beholder. We can look at a flower and see its beauty immediately. But it also has a beauty as a part of an ecosystem, using shape and color and nectar to attract and feed selected insects (bees) or mammals (bats), simultaneously spreading its pollen. This more sophisticated beauty only an educated mind can see.

The beauty of e^iπ+1 = 0 is not like the obvious color and shape of a flower, but more sophisticated, more ecological. It ties together diverse mathematical objects in a surprising and remarkable way. It links shapes (a circle, a line), constants, functions, and number systems. And it's useful.

For those whose minds are trained to see, this deeper beauty of both nature and mathematics as irresistible as the song of the sirens (but with a kinder reward). --KSmrq^T 07:15, 29 January 2006 (UTC)

$\sin \left(x+y\right)=\sin x\cos y+\cos x\sin y$

$\sin \left(x-y\right)=\sin x\cos y-\cos x\sin y$

$\cos \left(x+y\right)=\cos x\cos y-\sin x\sin y$

$\cos \left(x-y\right)=\cos x\cos y+\sin x\sin y$

Did you have to learn that by hart? That's about the same as having to divide 3639 by 3 when forced to use Roman numbers. That's a kind of cruel and unusual punishment.

Now look at this:

${\begin{matrix}\cos \left(x+y\right)+i\sin \left(x+y\right)&=&e^{i\left(x+y\right)}\\&=&e^{ix}e^{iy}\\&=&\left(\cos x+i\sin x\right)\left(\cos y+i\sin y\right)\\&=&\cos x\cos y-\sin x\sin y+i\left(\cos x\sin y+\cos y\sin x\right)\end{matrix}}$

The Infidel 09:14, 29 January 2006 (UTC)

I don't understand general relativity theory

... and I don't expect this can be cured in just one go, but to start with a simple question:

In a spacetime with Schwarzschild metric and Schwarzschild coordinates outside the Schwarzschild radius, how does a (small) spacetime vector (t, r, φ, θ) transform when the origin of the locale coordinate system is shifted from (0, R, 0, 0) to (0, R+ΔR, 0, 0) ?

Is there an answer that doesn't need tensors, Christoffel symbols and the like? (Maybe the proof of the answer needs to involve these, but the transformation as such should be possible with vectors, matrices and calculus alone, I hope.

The Infidel 11:11, 29 January 2006 (UTC)

I'm not sure if this is really what you want to ask. First of all, the origin is usually at (0,0,0,0), not (0,R,0,0). Certainly that's the origin in Schwarzschild coordinates. So maybe you want to ask what happens when you shift the origin to (0,ΔR,0,0)? It's not a hard question to answer in principle, but it's a bit messy, since we're in polar coordinates and you lose the spherical symmetry if you change the origin. And of course, while the various components of the invariant distance may change, the total invariant distance itself must of course remain invariant.

But let's see what we can say. I'm fooling around with it on paper, and doing it in spherical coordinates in 3d is too much paperwork for me, so I'm going to outline in 2d, and you can fill in the details if you want. I've drawn the triangle with the new origin, and from the law of sines, I can see that

${\frac {r'}{\sin \phi }}={\frac {r}{\sin \phi '}}$

and from the law of cosines I have

$r^{2}=r'^{2}+\Delta r^{2}-2r'\Delta r\cos \phi '.$

Now solve those two equations for r and φ, take the derivatives, and stick them in the Schwarzschild metric and you're done. But you know, that actually still looks like a lot of work. Ugh. Maybe it can be done more easily, but who would want to? The whole point of coordinates is that they're arbitrary, and you should use them to take advantage of the symmetry. I don't think you'd even learn anything from this. -lethe ^talk 11:55, 29 January 2006 (UTC)