Knowledge (XXG):Reference desk/Archives/Mathematics/2007 October 15

4798:

small change in weights. And that error flows backard through the network. A nod takes its punishment, looks at the nodes that feed into it, and distributes its punishment to them according to weight. If node C in layer two was punished some amount, and node A and B are connected to it, A with strong influence, B with small influence, the node C punishes A a lot and B a little. Note that if A and B are also connected to D in layer two, then layer D may add it's own punishment to A and B, based on its weight. This way, the nodes that are most responsible for the error, are changed the most. If you think about it for a little bit, you should be able to figure out the backpropagation formula yourself. Or at least, it should make more sense when you read it.

4838:, but unlike the Perceptron can formulate rules within one epoch of the data and distinguish or classify any dependent variable value for subsequent identification. Not only can the Adaptron learn to distinguish one "black box" from another but by formulating rules according to the relationships between the independent and dependent variable values of the "black box", can replace the "black box" in its function. 4770:

input neurons with weights to connect each of the input neurons to each of the hidden layer neurons, and if so, are these weights setup so that one is negative and the other is positive (or is this just for the XOR problem) and are the values of the weights otherwise the same value? I'm not ready to understand backpropagation yet, until I can understand this step.

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true/false inputs. Then comes the hidden layer (let's stick with one hidden layer for now) which are the actual perceptrons. Each perceptron has each of the input nodes as input, and a single output. After that comes the output layer, which is just the outputs from all the perceptrons in the hidden layer.

4797:

A little hint for when you do get into backpropagation. The principle (which tends to get skipped over in explanations) is that you compare the networks output with what the output should have been. You then punish each node according to how wrong it was. Big error, big change in weights, small error

4786:

If you take a simple multi-layer network, you start with the inputs. You can call these input neurons if you want, but they are basically just a bunch of numbers. If your input is a picture, these would be the pixel values. If you're trying to to learn a logical connection (like XOR) they are the two

4911:

Ahh... skip that, I missed the "only four fours" bit. BTW, "one equation for each solution" is redundant. Using those basic operations, I don't think you could find one equation which would account for any two of those solutions. If square root was also an acceptable operation, you could deal with a

4769:

I understand that with a single layer Perceptron, there is an initial weight and a final weight and that the final weight becomes the initial weight for the next iteration. If I want to add a layer to the net then do I use the single layer net input neurons as the hidden layer and add a new layer of

3983:

Right, got it. an equation such as 3sin(x) +sin(2x) has additional zeros in the complex plane - (as complex conjugates - so no imaginary terms are introduced) - so the product series includes the additional terms and so is different. And sin(x) has no zeros outside the reals. Thanks - may your magic

4845:

is critical of artificial neurons “ for not being biologically realistic.” some mathematicians decry the Adaptron as not being an artificial neuron (as they decry the Check sort and the Rapid sorts for not being "true" sort routines) but rather a mere recording device such as a tape player or CD.

2029:

introduce variables to denote the number of blocks in the matrix (row x colums) and the number of elements in each block. Do it for Each of A, B, and AB. Some of these will have to be identical of course. Then redo your proof (which is essentially correct except for some errors in the indices) with

3809:

The issue I have is that I can't convince myself that g(x)=sin(x) - for all I know it might be a triangular wave... My attempts at solving this fail - I get various formula that might give values for pi - but I was looking for something more solid. Can anyone suggest a (simple?) way of proving (if

407:

Thanks for the reply. I tried to prove it myself but am getting nowhere. The problem is that two matrices A and B can be divided into blocks arbitrarily and I can't get a general start to the proof. For example if c_ij is any entry in AB then which blocks of A and B give us c_ij? Can you give me a

2065:

I am still messing up the proof. Here's my version now: Let # of row blocks in A be m and # of column blocks in A be n. Also then # of row blocks in B is n. Let # of column blocks in B be p. (I don't know how to phrase it better but a row/column block means a whole row/column

3971:

ok up to the point of "Weierstrass" I was ok - that could be a little beyond me for a while - however am I along the right lines in my guess that a triangular function (composed of sines) will have zeros at complex values of x as well as at the real multiples of

5046:. There are 59 different results, including infinity (for any nonzero divided by zero) and Not-A-Number (for zero divided by zero). Several results appear in the list only once, and each of the 5 possible parenthesizations does yield at least one unique answer: 3948: 1806: 4939:

Then just cycle through with all possible combinations of oper1,oper2 and oper3. Where oper1 is OPERATOR_1 . Print out the value of oper1,oper2 and oper3 if the result is the desired result(s). There should be no more than 4*4*4=64 possible combinations.

3569: 3003: 4193: 4976:

The original question didn't say whether parentheses are allowed. If they're not, then you only get to choose the operators, not their order. If you get to add parentheses to alter the order, there are 5 possible results from each of the 64 operator

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If you want to add a second hidden layer, each of the perceptron in the second layer has all the outputs of the first layer perceptrons as inputs. The output layer of the network then becomes the outputs of the perceptrons in the second

4846:

What distinguishes the Adaptron from being merely a recording device is that it formulates and records rules and can use the rules to distinguish one "black box" from another; a function that might take the Perceptron many epochs to do.

5015:

Sure, but when the operations are both commutative, several of those expressions are equivalent. I think they fall into two equivalence classes, one containing the middle of the five expressions above, and another for the other four.

3784:

After the above question of getting a sine I was looking at sine estimates such as those involving cubics that work roughly between +pi and -pi.. I decided to try myself to see what I could create - this equation came out:

4094: 4833:

The Adaptron can learn to distinguish one “black box” from another by accepting values (states) of independent variables (sensors, conditions, etc.) and dependent variables (actuators, actions, etc.) as input like the

4949:

Wrong. For example, there is no choice of oper1, oper2, oper3 such that oper3,oper2] = 48. Although 4*(4+(4+4)) = 48. You have to consider each possible parenthesization (is that even a word) of this expression.

701:

is what is given to us already as this is the way we defined block matrix multiplication. The problem is to show it equivalent to regular multiplication. Can you give me a little more hint? Thanks again, Meni and

3305: 2421: 1083: 1380: 5060:(((4+4)/4)-4) is the only way to get -2 ((4-(4*4))/4) is the only way to get -3 ((4*4)-(4/4)) is the only way to get 15 (4/((4/4)-4)) is the only way to get -4/3 (4-(4/(4+4))) is the only way to get 7/2 66: 45: 2604: 51: 4794:

There is no specific way you should set the initial weights. You can just set them randomly if you want. The idea is that the network will gradually learn the correct weights (using backpropagation).

449:

and figuring out to which block it belongs, try starting with a row of blocks in A and a corresponding column of blocks in B, and figure out what you get when you take the "dot product" of those. --

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Note that the partition can't be completely arbitrary, as some of the dimensions have to match. Regardless, I think you should be thinking the other way around - instead of starting with some entry

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so if want the complete list of possible outputs, you definitely need to consider all 5 parenthesizations. Do we really think the original question was homework? I never had homework this fun. --

4896:

I would write a program to produce a table of all possible expressions and simply look up the values. I don't see how you could do it in a more elegant way than trial and error or brute force.

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15:58, 19 October 2007 (UTC) Note: I have checked my steps using a 4 by 4 matrix and it seems okay. Please tell me how to get the last step. Any help will be immensely appreciated. Thanks --

1559: 987: 2485: 1616: 1201: 1471:. Suppose ith block occurs after r rows in A and jth block after s columns in B. Then it will occur in the same position in (AB) too as # of rows(columns) of A(B) match that of AB. Now 3829: 1621: 389:

I couldn't find any online, but it should appear in any comprehensive elementary linear algebra book. But it's actually not that hard to prove it directly using only the definition of

379:

Hi. I'm looking for a general proof that matrix multiplication by dividing the matrices into blocks and then multiplying by using the blocks is justified. Can anyone help me? Thanks.--

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the proper bounds for everything. The very last step is insufficient. You don't prove why the sum gives the same result. But you can only do that with proper indices troughout.

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It has to use four 4s and the four basic operations (+,-,*,/). No concentrations (44, 444 ...), No decimals (.4, .44 ...), No factorals (!), No roots or exponents. Thanks

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It's better now, but why don't you carry out the multiplication (AB) on the right side ? It is a pretty obvious step. Then you just have to check that the indices match.

2287: 1469: 1144: 1990: 1288: 4400: 4318: 518:. You want to prove that taking a dot product of blocks or taking a block out of the product of the original matrices give the same result. That is you want to prove: 493: 447: 2230: 2203: 2176: 2149: 2122: 2095: 119:

Probably a fairly simple and straightforward thing that's been spotted a million times in the past, but is there a logical algebraic proof that this works for all n?

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However, if you reject the Check and Rapid sorts as bonafide sort routines then you will most likely reject the Adaptron as a bonafide artificial neuron as well.

4738:

Now you have the value of a. If you insert that value into one of your equations (for example 7=2a+b) you can isolate b and get b's value. Then you are done.

4118:

This is not unique to Knowledge (XXG) (it has possibly appeared in some new version of LaTeX). However, an alternative code for this (not identical though) is

5138:

You are right. With five 4's they can all be solved. BTW I very much would like to know how you solve for 10 with four 4's since it doesn't seem possible.

4809:

I was going to use a multi-layer perceptron but someone showed me how much less complicated and more accurate an adaptron is, so I've decided to use that.

3797:

Looking at g(x) it's obvious to me that it will have the same zeros as sin(x) ie at 0,pi,2pi etc, expanding the terms gives the x coefficient as 1 as well

85:

The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the

719:

Actually, the right side of the equation uses block multiplication, the left side uses regular multiplication, and equality is what you must prove.

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oper3(4,oper2(4,oper1(4,4))) oper3(4,oper2(oper1(4,4),4)) oper3(oper1(4,4),oper2(4,4)) oper3(oper2(4,oper1(4,4)),4) oper3(oper2(oper1(4,4),4),4)

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I made a mistake. In the first run of my solving program I didn't filter out answers like (44-4)/4 = 10 which are forbidden by the rules. --

3599:

Do you notice the similarity in the indices ? All you have to do is prove that the two sums run over the same indices. That is prove that:

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You can do \usepackage{amsmath} in order to use \begin{cases} in normal LaTeX, although not everyone would consider that normal LaTeX.

3754:

Can you kindly show that the 2 sets are equal. I have tried to do so but to no avail. I am pretty much getting confused now. Cheers--

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Ahhh... don't bother - I've just realised its of the (n+1)(n-1) form, so any (x^n)-1 will be a multiple of x-1. Thanks anyway...

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To prove that two matrices are equal you can simply prove that all their corresponding blocks are equal. Thus to prove that

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Yes, what you have is a system of 2 linear equations in 2 variables, and substitution is one of the ways to solve these.

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If g(x)=sin(x) then a3=1/6 (it is looking like it will do this but I can't prove it. I found a solution of sorts at

4492: 3943:{\displaystyle 1-{\frac {x^{2}}{\pi ^{2}n^{2}}}=\left(1-{\frac {x}{\pi n}}\right)\left(1+{\frac {x}{\pi n}}\right)} 1801:{\displaystyle \sum _{k}\sum _{t=1}^{k}A_{p,t}^{ik}B_{t,q}^{kj}=\sum _{k}\sum _{t=1}^{k}A_{r+p,k+t}B_{k+t,s+q}\;\!} 1474: 923: 2426: 4666: 4630: 4500: 4467: 4223: 4200: 454: 398: 364: 1564: 1149: 4878:

I have to have order of operations equations that equals: 10 11 13 14 18 19 (One equation for each solution)

3564:{\displaystyle \sum _{k=1}^{n}\sum _{t=1}^{y_{k}}A_{x_{i}(i-1)+r,y_{k}(k-1)+t}B_{y_{k}(k-1)+t,z_{j}(j-1)+s}} 2998:{\displaystyle \sum _{k=1}^{n}\sum _{t=1}^{y_{k}}A_{x_{i}(i-1)+r,y_{k}(k-1)+t}B_{y_{k}(k-1)+t,z_{j}(j-1)+s}} 4741: 4680: 4188:{\displaystyle f(x)=\left\{{\begin{array}{ll}1&x\geq 4\\4&{\textrm {otherwise}}\end{array}}\right.} 3310: 339: 331: 155: 4653: 4532: 3973: 3811: 189: 4941: 4210: 3985: 3605: 390: 1904: 4909:

but if you want to gain marks for sheer stubborn obstinacy, there's always (4/4) + (4/4) + (4/4) + ...

4405: 1959:. Hence the result is established. (What does k run on? It should on go from 1 to # of block in which 4323: 4026: 1385: 349:

No, but you can only solve equations with one variable. You can have many variable under a radical.

328:

Is there any limit to the number of variables that you can have in an equation involving radicals?

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The number of ways of parenthesizing an expression involving N+1 terms and N operators is the N'th

4852: 4810: 4771: 4662: 4626: 4496: 4463: 4219: 4196: 450: 394: 360: 159: 4652:(yes - but not there yet) this time subtract - you'll end up with an equation that has no b in it. 2490: 4918: 1107:

Thanks for explaining that to me. How do we go from here? Should we take an arbitrary element of

311: 183: 143: 125: 4586: 4547: 4842: 4688: 2253: 1435: 1110: 2788:{\displaystyle \sum _{t=1}^{y_{k}}A_{x_{i}(i-1)+r,y_{k}(k-1)+t}B_{y_{k}(k-1)+t,z_{j}(j-1)+s}} 108:

Hi all - I was messing around with some numbers earlier today, and I suddenly realised that:

5190: 4240: 4104: 1962: 1260: 350: 172: 4370: 4288: 468: 422: 4749: 3810:

only to me - and preferably with some solid algebra) that g(x) does indeed equal sin(x) .

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I am afraid that I am still dense. The right side, as far as I can think for now is only

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blocks.) Then AB has m row and p column blocks. Also let # of rows in ith block of A be

359:

Perhaps the OP should clarify the question and its context - it is not at all clear. --

5177: 5148: 5121: 5075: 5026: 5017: 4998: 4994: 4960: 4886: 3963: 3955: 900: 498: 168: 5120:, but not any of the others. Maybe it was supposed to be five 4's, and 4 operators? -- 4824:

I've never heard of those. And google doesn't seem to have either. Care to elaborate?

765: 745: 725: 5173: 5039: 5035: 4913: 4825: 4801: 3804: 306: 138: 120: 4885:

We don't do your homework for you. Please look at the top of this page for more. --

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I noticed that Knowledge (XXG)'s implementation of LaTeX has \begin{cases} such as

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Unfortunately not, maths isn't one of my strong points. Do I use substitution? --

619:

Thanks for your help. However I am not still not getting it. It seems to me that

5139: 5105: 4951: 4897: 4234: 4098: 3755: 3743: 3572: 3137: 3108: 3104: 2031: 1993: 1233: 1204: 1086: 703: 607: 409: 380: 295: 2024:"Suppose ith block occurs after r rows in A and jth block after s columns in B" 5186: 4835: 4745: 4705: 4145: 4089:{\displaystyle f(x)={\begin{cases}1&x\geq 4\\4&otherwise\end{cases}}} 4218:

Oh, right, I have forgotten that I have many packages loaded by default. --

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by block multiplication. The definition of block multiplication gives you:

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Silly me, not all of the operations in question are commutative. Oops. -

3103:. I am stuck here. Can you tell me how the proof is finished? Cheers.-- 5104:

Is it just me or are there no solution for any of the numbers given ?

154:

For future reference: you can prove these sorts of statements through

3300:{\displaystyle \sum _{u=1}^{X}A_{x_{i}(i-1)+r,u}B_{u,z_{j}(j-1)+s}} 1290:

represent the (i,j)th entry of the matrix X. We need to show that

79:

Welcome to the Knowledge (XXG) Mathematics Reference Desk Archives

3794:

g(x) = x(x/π -1)(x/π +1)(x/2π -1)(x/2π +1)(x/3π -1)(x/3π +1) etc

4997:. Here N is 3 so it's the 3rd Catalan number, which equals 5. -- 2416:{\displaystyle (AB)_{r,s}^{ij}=(AB)_{x_{i}(i-1)+r,z_{j}(j-1)+s}} 1078:{\displaystyle \forall i,j\,(AB)^{ij}=\sum _{k}A^{ik}B^{kj}\;\!} 305:

Thanks for those ideas. I'd never have thought to use modulo 8!

4715:

O ye a minus and a minus is a plus. Where do i go from here? --

5042:. I've generated the complete list of 320 combinations. It's 2021:

There are some problems with the indices. Instead of saying:

112:

for any positive integer n, (9^n) - 1 = a multiple of eight.

4511:

So what's the simultaneous equation that I need to solve? --

1375:{\displaystyle \forall i,j\,(AB)^{ij}=\sum _{k}A^{ik}B^{kj}} 1257:

Can you tell me whether the following proof is correct: Let

158:, which requires a few initial steps and then some algebra. 4182: 4082: 3984:

wand do what ever wizards magic wands do. (smile) thanks.

3958:

expression for π. I imagine a formal proof would use the

3822:

List_of_trigonometric_identities#Infinite_product_formula

2599:{\displaystyle \sum _{t=1}^{y_{k}}A_{rt}^{ik}B_{ts}^{kj}} 4959:

Wouldn't you miss expressions of the form oper1,4],4]? -

5172:

Those interested in this question might enjoy the game

4544:

I wonder how was I not clear about the equations being

3820:

To see that you are on the right lines, take a look at

4912:

couple of them in one shot, but that's another story.

4907:

There are simple, elegant solutions to each of these,

4589: 4550: 4408: 4373: 4326: 4291: 4124: 4005: 3832: 3608: 3367: 3313: 3187: 3011: 2801: 2612: 2525: 2493: 2429: 2295: 2256: 2211: 2184: 2157: 2130: 2103: 2076: 1965: 1907: 1894:{\displaystyle \sum _{k,t}A_{r+p,k+t}B_{k+t,s+q}\;\!} 1814: 1624: 1567: 1477: 1438: 1388: 1296: 1263: 1152: 1113: 995: 926: 903: 874: 857:{\displaystyle C^{ij}=\sum _{k}A^{ik}B^{kj}\quad (1)} 790: 768: 748: 728: 625: 526: 501: 471: 425: 236: 192: 1992:

entry occurs where A is m by n. Is that correct?) --

287:{\displaystyle 9^{n}\equiv 1^{n}\equiv 1{\pmod {8}}} 694:{\displaystyle (AB)^{ij}=\sum _{k}A^{ik}B^{kj}\;\!} 595:{\displaystyle (AB)^{ij}=\sum _{k}A^{ik}B^{kj}\;\!} 4617: 4575: 4454: 4394: 4359: 4312: 4187: 4088: 3942: 3800:For x terms I get a3 = 1/π(1+1/4+1/9+1/16.. etc) 3731: 3563: 3353: 3299: 3095: 2997: 2787: 2598: 2511: 2479: 2415: 2281: 2224: 2197: 2170: 2143: 2116: 2089: 1984: 1951: 1893: 1800: 1610: 1553: 1463: 1424: 1374: 1282: 1195: 1138: 1077: 981: 912: 889: 856: 774: 754: 734: 693: 594: 510: 487: 441: 286: 222: 3788:f(x) = x(x-π)(x+π)(x-2π)(x+2π)(x-3π)(x+3π).. etc 4677:3a=2 because the b cancels itself out isnt it? 4531:Best way to go from here is to get rid of the b. 4195:(note also that "otherwise" should be text). -- 3096:{\displaystyle (AB)_{x_{i}(i-1)+r,z_{j}(j-1)+s}} 4613: 4571: 4450: 1947: 1889: 1796: 1606: 1191: 1073: 689: 590: 1554:{\displaystyle (AB)_{p,q}^{ij}=(AB)_{r+p,s+q}} 1146:and prove it equal to an arbitrary element of 169:Mersenne prime#Theorems about Mersenne numbers 4462:. Do you know how to continue from there? -- 4096:. What would be the normal LaTeX for this? 982:{\displaystyle \forall i,j\,(AB)^{ij}=C^{ij}} 8: 920:uses regular multiplication, you prove that 2795:. Thus our problem reduces to showing that 2480:{\displaystyle \sum _{k=1}^{n}A^{ik}B^{kj}} 3571:? Can you explicitly write the step now?-- 2423:. Now let us pick up the (r,s)th entry of 4588: 4549: 4407: 4372: 4325: 4290: 4172: 4171: 4144: 4123: 4021: 4004: 3920: 3889: 3866: 3856: 3845: 3839: 3831: 3717: 3683: 3670: 3659: 3643: 3633: 3622: 3607: 3532: 3498: 3493: 3460: 3426: 3421: 3409: 3404: 3393: 3383: 3372: 3366: 3345: 3335: 3324: 3312: 3268: 3257: 3218: 3213: 3203: 3192: 3186: 3064: 3030: 3025: 3010: 2966: 2932: 2927: 2894: 2860: 2855: 2843: 2838: 2827: 2817: 2806: 2800: 2756: 2722: 2717: 2684: 2650: 2645: 2633: 2628: 2617: 2611: 2587: 2579: 2566: 2558: 2546: 2541: 2530: 2524: 2492: 2468: 2455: 2445: 2434: 2428: 2384: 2350: 2345: 2320: 2309: 2294: 2270: 2255: 2216: 2210: 2189: 2183: 2162: 2156: 2135: 2129: 2108: 2102: 2081: 2075: 1970: 1964: 1921: 1906: 1863: 1835: 1819: 1813: 1770: 1742: 1732: 1721: 1711: 1695: 1684: 1671: 1660: 1650: 1639: 1629: 1623: 1611:{\displaystyle \sum _{k}A^{ik}B^{kj}\;\!} 1595: 1582: 1572: 1566: 1527: 1502: 1491: 1476: 1452: 1437: 1413: 1402: 1387: 1363: 1350: 1340: 1324: 1295: 1268: 1262: 1196:{\displaystyle \sum _{k}A^{ik}B^{kj}\;\!} 1180: 1167: 1157: 1151: 1127: 1112: 1062: 1049: 1039: 1023: 994: 970: 954: 925: 902: 873: 834: 821: 811: 795: 789: 767: 747: 727: 678: 665: 655: 639: 624: 579: 566: 556: 540: 525: 500: 476: 470: 430: 424: 266: 254: 241: 235: 202: 191: 5038:is responsible for more duplicates than 4611: 4569: 4448: 4275:(-1) = -5? Thanks very much everybody -- 2250:Consider the (r,s)th entry of the block 1945: 1887: 1794: 1604: 1189: 1071: 687: 588: 49: 36: 2205:and # of columns in jth block of AB is 2178:. Then # of rows in ith block of AB is 1309: 1008: 939: 606:I'll let you figure it out from there. 65: 3954:Also, take a look at the proof of the 3354:{\displaystyle X=\sum _{k=1}^{n}y_{k}} 722:Now for the long version. Let's write 43: 2151:, # of columns in jth column of B be 2124:, let # of rows in ith block of B be 2097:, # of columns in jth column of A be 7: 4495:is an example of how it is done. -- 393:, and it can be a good exercise. -- 223:{\displaystyle 9\equiv 1{\pmod {8}}} 116:This also works for n=0, of course. 3732:{\displaystyle =\bigcup _{k=1}^{n}} 1432:be the (p,q)th entry in the matrix 4704:No, 7-(-5) is not 2, 7-(-5) = 12. 1952:{\displaystyle (AB)_{r+p,s+q}\;\!} 1561:. The (p,q)th entry in the matrix 1297: 996: 927: 32: 4661:No, 3a + 2b = 2 is not right. -- 4455:{\displaystyle -a+b=f(-1)=-5\;\!} 3960:Weierstrass factorization theorem 3361:. How should I bring it equal to 1085:, which is definitely not given. 275: 211: 3772: 2519:) the (r,s)th entry is given by 1224:Yes. Of course you want to take 989:. Plugging in (1) above you get 4641:3a + 2b = 2? Is that right? -- 4360:{\displaystyle f(2)=a\cdot 2+b} 1425:{\displaystyle (AB)_{p,q}^{ij}} 495:for the block of A at position 5189:article. (Spoiler warning!) - 4436: 4427: 4336: 4330: 4301: 4295: 4267:, a linear function, in which 4134: 4128: 4015: 4009: 3726: 3701: 3689: 3676: 3649: 3609: 3550: 3538: 3516: 3504: 3478: 3466: 3444: 3432: 3286: 3274: 3236: 3224: 3082: 3070: 3048: 3036: 3022: 3012: 2984: 2972: 2950: 2938: 2912: 2900: 2878: 2866: 2774: 2762: 2740: 2728: 2702: 2690: 2668: 2656: 2402: 2390: 2368: 2356: 2342: 2332: 2306: 2296: 2267: 2257: 1918: 1908: 1524: 1514: 1488: 1478: 1449: 1439: 1399: 1389: 1321: 1311: 1124: 1114: 1020: 1010: 951: 941: 851: 845: 636: 626: 537: 527: 280: 269: 216: 205: 18:Knowledge (XXG):Reference desk 1: 2512:{\displaystyle 1\leq k\leq n} 843: 267: 203: 33: 5194:02:51, 16 October 2007 (UTC) 5181:02:07, 16 October 2007 (UTC) 5161:01:59, 16 October 2007 (UTC) 5143:01:51, 16 October 2007 (UTC) 5134:01:34, 16 October 2007 (UTC) 5109:01:29, 16 October 2007 (UTC) 5088:02:20, 16 October 2007 (UTC) 5030:02:05, 16 October 2007 (UTC) 5021:01:51, 16 October 2007 (UTC) 5011:01:43, 16 October 2007 (UTC) 4964:01:38, 16 October 2007 (UTC) 4955:01:44, 16 October 2007 (UTC) 4945:00:52, 16 October 2007 (UTC) 4931:First create the expression 4925:00:26, 16 October 2007 (UTC) 4901:00:16, 16 October 2007 (UTC) 4890:20:25, 15 October 2007 (UTC) 4862:20:07, 16 October 2007 (UTC) 4829:00:40, 16 October 2007 (UTC) 4820:00:18, 16 October 2007 (UTC) 4805:23:00, 15 October 2007 (UTC) 4781:20:01, 15 October 2007 (UTC) 4758:20:47, 15 October 2007 (UTC) 4720:19:41, 15 October 2007 (UTC) 4709:19:21, 15 October 2007 (UTC) 4697:19:12, 15 October 2007 (UTC) 4671:18:48, 15 October 2007 (UTC) 4657:18:47, 15 October 2007 (UTC) 4646:18:30, 15 October 2007 (UTC) 4635:18:21, 15 October 2007 (UTC) 4536:18:16, 15 October 2007 (UTC) 4516:18:10, 15 October 2007 (UTC) 4505:17:58, 15 October 2007 (UTC) 4483:17:46, 15 October 2007 (UTC) 4472:17:22, 15 October 2007 (UTC) 4280:17:06, 15 October 2007 (UTC) 4249:12:01, 16 October 2007 (UTC) 4228:17:59, 15 October 2007 (UTC) 4214:17:44, 15 October 2007 (UTC) 4205:15:50, 15 October 2007 (UTC) 4113:15:40, 15 October 2007 (UTC) 3989:13:45, 15 October 2007 (UTC) 3977:11:12, 15 October 2007 (UTC) 3967:11:00, 15 October 2007 (UTC) 3815:10:23, 15 October 2007 (UTC) 3759:14:47, 20 October 2007 (UTC) 3747:13:21, 20 October 2007 (UTC) 3576:11:53, 20 October 2007 (UTC) 3141:11:25, 20 October 2007 (UTC) 3112:04:08, 20 October 2007 (UTC) 2035:13:31, 19 October 2007 (UTC) 1997:05:17, 19 October 2007 (UTC) 1237:07:01, 18 October 2007 (UTC) 1208:06:10, 18 October 2007 (UTC) 1090:05:11, 18 October 2007 (UTC) 707:04:49, 18 October 2007 (UTC) 611:00:13, 17 October 2007 (UTC) 459:16:48, 16 October 2007 (UTC) 413:16:38, 16 October 2007 (UTC) 403:09:18, 15 October 2007 (UTC) 384:07:41, 15 October 2007 (UTC) 369:09:11, 15 October 2007 (UTC) 354:03:36, 15 October 2007 (UTC) 344:02:38, 15 October 2007 (UTC) 318:21:40, 15 October 2007 (UTC) 299:05:59, 15 October 2007 (UTC) 176:03:38, 15 October 2007 (UTC) 163:03:01, 15 October 2007 (UTC) 150:00:20, 15 October 2007 (UTC) 132:00:16, 15 October 2007 (UTC) 4618:{\displaystyle -a+b=-5\;\!} 5209: 4576:{\displaystyle 2a+b=7\;\!} 3770: 324:variable limit in radicals 2282:{\displaystyle (AB)^{ij}} 1464:{\displaystyle (AB)^{ij}} 1139:{\displaystyle (AB)^{ij}} 465:To clarify, let's write 4522:Using y=ax+b you'be got 1985:{\displaystyle A_{m,n}} 1283:{\displaystyle X_{i,j}} 182:Yeah, you can also use 4619: 4577: 4456: 4402:. Similarly, you have 4396: 4395:{\displaystyle 2a+b=7} 4367:on the other hand, so 4361: 4314: 4313:{\displaystyle f(2)=7} 4189: 4090: 3944: 3767:is it the sin (or not) 3733: 3675: 3638: 3565: 3416: 3388: 3355: 3340: 3301: 3208: 3097: 2999: 2850: 2822: 2789: 2640: 2600: 2553: 2513: 2481: 2450: 2417: 2283: 2226: 2199: 2172: 2145: 2118: 2091: 1986: 1953: 1895: 1802: 1737: 1655: 1612: 1555: 1465: 1426: 1376: 1284: 1197: 1140: 1079: 983: 914: 891: 858: 776: 756: 736: 695: 596: 512: 489: 488:{\displaystyle A^{ij}} 443: 442:{\displaystyle c_{ij}} 288: 224: 156:mathematical induction 104:An algebraic proof...? 87:current reference desk 4620: 4578: 4457: 4397: 4362: 4315: 4190: 4091: 3945: 3734: 3655: 3618: 3566: 3389: 3368: 3356: 3320: 3302: 3188: 3098: 3000: 2823: 2802: 2790: 2613: 2601: 2526: 2514: 2482: 2430: 2418: 2284: 2227: 2225:{\displaystyle z_{j}} 2200: 2198:{\displaystyle x_{i}} 2173: 2171:{\displaystyle z_{j}} 2146: 2144:{\displaystyle y_{i}} 2119: 2117:{\displaystyle y_{j}} 2092: 2090:{\displaystyle x_{i}} 1987: 1954: 1896: 1803: 1717: 1635: 1613: 1556: 1466: 1427: 1377: 1285: 1198: 1141: 1080: 984: 915: 892: 859: 777: 757: 737: 696: 597: 513: 490: 444: 391:matrix multiplication 289: 225: 5185:You know, we have a 4587: 4548: 4406: 4371: 4324: 4289: 4285:Well, you know that 4122: 4003: 3830: 3606: 3365: 3311: 3185: 3009: 2799: 2610: 2523: 2491: 2427: 2293: 2254: 2232:. Now for the proof. 2209: 2182: 2155: 2128: 2101: 2074: 1963: 1905: 1812: 1622: 1565: 1475: 1436: 1386: 1294: 1261: 1150: 1111: 993: 924: 901: 890:{\displaystyle AB=C} 872: 788: 766: 746: 726: 623: 524: 499: 469: 423: 234: 190: 4765:multi-layer network 3781:(almost certainly) 2595: 2574: 2328: 1703: 1679: 1510: 1421: 742:for the product of 5118:I can solve for 10 4615: 4614: 4612: 4573: 4572: 4570: 4452: 4451: 4449: 4392: 4357: 4310: 4232:Alright, thanks. 4185: 4180: 4086: 4081: 3940: 3742:Which is trivial. 3729: 3561: 3351: 3297: 3093: 2995: 2785: 2596: 2575: 2554: 2509: 2477: 2413: 2305: 2279: 2222: 2195: 2168: 2141: 2114: 2087: 1982: 1949: 1948: 1946: 1891: 1890: 1888: 1830: 1798: 1797: 1795: 1716: 1680: 1656: 1634: 1618:will similarly be 1608: 1607: 1605: 1577: 1551: 1487: 1461: 1422: 1398: 1372: 1345: 1310: 1280: 1193: 1192: 1190: 1162: 1136: 1075: 1074: 1072: 1044: 1009: 979: 940: 913:{\displaystyle AB} 910: 887: 854: 844: 816: 772: 752: 732: 691: 690: 688: 660: 592: 591: 589: 561: 511:{\displaystyle ij} 508: 485: 439: 284: 276: 268: 220: 212: 204: 184:modular arithmetic 4921: 4859: 4817: 4778: 4760: 4744:comment added by 4699: 4683:comment added by 4320:on one hand, and 4175: 3933: 3902: 3873: 2289:. It is given by 1815: 1707: 1625: 1568: 1336: 1153: 1035: 807: 775:{\displaystyle B} 755:{\displaystyle A} 735:{\displaystyle C} 651: 552: 346: 334:comment added by 314: 146: 128: 93: 92: 73: 72: 5200: 4919: 4857: 4853: 4815: 4811: 4776: 4772: 4739: 4678: 4624: 4622: 4621: 4616: 4582: 4580: 4579: 4574: 4461: 4459: 4458: 4453: 4401: 4399: 4398: 4393: 4366: 4364: 4363: 4358: 4319: 4317: 4316: 4311: 4259:Linear functions 4247: 4246: 4194: 4192: 4191: 4186: 4184: 4181: 4177: 4176: 4173: 4111: 4110: 4095: 4093: 4092: 4087: 4085: 4084: 3949: 3947: 3946: 3941: 3939: 3935: 3934: 3932: 3921: 3908: 3904: 3903: 3901: 3890: 3874: 3872: 3871: 3870: 3861: 3860: 3850: 3849: 3840: 3776: 3775: 3738: 3736: 3735: 3730: 3722: 3721: 3688: 3687: 3674: 3669: 3648: 3647: 3637: 3632: 3570: 3568: 3567: 3562: 3560: 3559: 3537: 3536: 3503: 3502: 3488: 3487: 3465: 3464: 3431: 3430: 3415: 3414: 3413: 3403: 3387: 3382: 3360: 3358: 3357: 3352: 3350: 3349: 3339: 3334: 3306: 3304: 3303: 3298: 3296: 3295: 3273: 3272: 3252: 3251: 3223: 3222: 3207: 3202: 3102: 3100: 3099: 3094: 3092: 3091: 3069: 3068: 3035: 3034: 3004: 3002: 3001: 2996: 2994: 2993: 2971: 2970: 2937: 2936: 2922: 2921: 2899: 2898: 2865: 2864: 2849: 2848: 2847: 2837: 2821: 2816: 2794: 2792: 2791: 2786: 2784: 2783: 2761: 2760: 2727: 2726: 2712: 2711: 2689: 2688: 2655: 2654: 2639: 2638: 2637: 2627: 2605: 2603: 2602: 2597: 2594: 2586: 2573: 2565: 2552: 2551: 2550: 2540: 2518: 2516: 2515: 2510: 2486: 2484: 2483: 2478: 2476: 2475: 2463: 2462: 2449: 2444: 2422: 2420: 2419: 2414: 2412: 2411: 2389: 2388: 2355: 2354: 2327: 2319: 2288: 2286: 2285: 2280: 2278: 2277: 2231: 2229: 2228: 2223: 2221: 2220: 2204: 2202: 2201: 2196: 2194: 2193: 2177: 2175: 2174: 2169: 2167: 2166: 2150: 2148: 2147: 2142: 2140: 2139: 2123: 2121: 2120: 2115: 2113: 2112: 2096: 2094: 2093: 2088: 2086: 2085: 1991: 1989: 1988: 1983: 1981: 1980: 1958: 1956: 1955: 1950: 1944: 1943: 1900: 1898: 1897: 1892: 1886: 1885: 1858: 1857: 1829: 1807: 1805: 1804: 1799: 1793: 1792: 1765: 1764: 1736: 1731: 1715: 1702: 1694: 1678: 1670: 1654: 1649: 1633: 1617: 1615: 1614: 1609: 1603: 1602: 1590: 1589: 1576: 1560: 1558: 1557: 1552: 1550: 1549: 1509: 1501: 1470: 1468: 1467: 1462: 1460: 1459: 1431: 1429: 1428: 1423: 1420: 1412: 1381: 1379: 1378: 1373: 1371: 1370: 1358: 1357: 1344: 1332: 1331: 1289: 1287: 1286: 1281: 1279: 1278: 1202: 1200: 1199: 1194: 1188: 1187: 1175: 1174: 1161: 1145: 1143: 1142: 1137: 1135: 1134: 1084: 1082: 1081: 1076: 1070: 1069: 1057: 1056: 1043: 1031: 1030: 988: 986: 985: 980: 978: 977: 962: 961: 919: 917: 916: 911: 896: 894: 893: 888: 863: 861: 860: 855: 842: 841: 829: 828: 815: 803: 802: 781: 779: 778: 773: 761: 759: 758: 753: 741: 739: 738: 733: 700: 698: 697: 692: 686: 685: 673: 672: 659: 647: 646: 601: 599: 598: 593: 587: 586: 574: 573: 560: 548: 547: 517: 515: 514: 509: 494: 492: 491: 486: 484: 483: 448: 446: 445: 440: 438: 437: 329: 312: 293: 291: 290: 285: 283: 259: 258: 246: 245: 229: 227: 226: 221: 219: 144: 126: 75: 38:Mathematics desk 34: 5208: 5207: 5203: 5202: 5201: 5199: 5198: 5197: 5061: 4986: 4876: 4855: 4813: 4774: 4767: 4585: 4584: 4546: 4545: 4404: 4403: 4369: 4368: 4322: 4321: 4287: 4286: 4261: 4239: 4233: 4179: 4178: 4169: 4163: 4162: 4151: 4140: 4120: 4119: 4103: 4097: 4080: 4079: 4050: 4044: 4043: 4032: 4022: 4001: 4000: 3997: 3925: 3913: 3909: 3894: 3882: 3878: 3862: 3852: 3851: 3841: 3828: 3827: 3795: 3789: 3779: 3778: 3773: 3769: 3713: 3679: 3639: 3604: 3603: 3528: 3494: 3489: 3456: 3422: 3417: 3405: 3363: 3362: 3341: 3309: 3308: 3264: 3253: 3214: 3209: 3183: 3182: 3060: 3026: 3021: 3007: 3006: 3005:is the same as 2962: 2928: 2923: 2890: 2856: 2851: 2839: 2797: 2796: 2752: 2718: 2713: 2680: 2646: 2641: 2629: 2608: 2607: 2542: 2521: 2520: 2489: 2488: 2464: 2451: 2425: 2424: 2380: 2346: 2341: 2291: 2290: 2266: 2252: 2251: 2212: 2207: 2206: 2185: 2180: 2179: 2158: 2153: 2152: 2131: 2126: 2125: 2104: 2099: 2098: 2077: 2072: 2071: 1966: 1961: 1960: 1917: 1903: 1902: 1859: 1831: 1810: 1809: 1766: 1738: 1620: 1619: 1591: 1578: 1563: 1562: 1523: 1473: 1472: 1448: 1434: 1433: 1384: 1383: 1359: 1346: 1320: 1292: 1291: 1264: 1259: 1258: 1176: 1163: 1148: 1147: 1123: 1109: 1108: 1058: 1045: 1019: 991: 990: 966: 950: 922: 921: 899: 898: 870: 869: 830: 817: 791: 786: 785: 764: 763: 744: 743: 724: 723: 674: 661: 635: 621: 620: 575: 562: 536: 522: 521: 497: 496: 472: 467: 466: 426: 421: 420: 377: 326: 250: 237: 232: 231: 188: 187: 106: 101: 30: 29: 28: 12: 11: 5: 5206: 5204: 5170: 5169: 5168: 5167: 5166: 5165: 5164: 5163: 5112: 5111: 5101: 5100: 5099: 5098: 5097: 5096: 5095: 5094: 5093: 5092: 5091: 5090: 5059: 5058: 5057: 5056: 5055: 5054: 5053: 5052: 5051: 5050: 5049: 5048: 5047: 4995:Catalan number 4984: 4983: 4982: 4981: 4980: 4979: 4978: 4969: 4968: 4967: 4966: 4957: 4937: 4936: 4935: 4928: 4927: 4904: 4903: 4893: 4892: 4875: 4872: 4871: 4870: 4869: 4868: 4867: 4866: 4865: 4864: 4850: 4847: 4839: 4799: 4795: 4792: 4788: 4766: 4763: 4762: 4761: 4736: 4735: 4734: 4733: 4732: 4731: 4730: 4729: 4728: 4727: 4726: 4725: 4724: 4723: 4722: 4701: 4700: 4663:Meni Rosenfeld 4654:83.100.252.179 4649: 4648: 4627:Meni Rosenfeld 4610: 4607: 4604: 4601: 4598: 4595: 4592: 4568: 4565: 4562: 4559: 4556: 4553: 4542: 4541: 4540: 4539: 4538: 4533:83.100.252.179 4529: 4526: 4523: 4519: 4518: 4497:Meni Rosenfeld 4489: 4488: 4487: 4486: 4485: 4464:Meni Rosenfeld 4447: 4444: 4441: 4438: 4435: 4432: 4429: 4426: 4423: 4420: 4417: 4414: 4411: 4391: 4388: 4385: 4382: 4379: 4376: 4356: 4353: 4350: 4347: 4344: 4341: 4338: 4335: 4332: 4329: 4309: 4306: 4303: 4300: 4297: 4294: 4263:How do I find 4260: 4257: 4256: 4255: 4254: 4253: 4252: 4251: 4220:Meni Rosenfeld 4207: 4197:Meni Rosenfeld 4183: 4170: 4168: 4165: 4164: 4161: 4158: 4155: 4152: 4150: 4147: 4146: 4143: 4139: 4136: 4133: 4130: 4127: 4083: 4078: 4075: 4072: 4069: 4066: 4063: 4060: 4057: 4054: 4051: 4049: 4046: 4045: 4042: 4039: 4036: 4033: 4031: 4028: 4027: 4025: 4020: 4017: 4014: 4011: 4008: 3996: 3993: 3992: 3991: 3981: 3980: 3979: 3974:83.100.255.190 3956:Wallis product 3952: 3951: 3950: 3938: 3931: 3928: 3924: 3919: 3916: 3912: 3907: 3900: 3897: 3893: 3888: 3885: 3881: 3877: 3869: 3865: 3859: 3855: 3848: 3844: 3838: 3835: 3824:, noting that 3812:83.100.255.190 3793: 3787: 3771: 3768: 3765: 3764: 3763: 3762: 3761: 3740: 3739: 3728: 3725: 3720: 3716: 3712: 3709: 3706: 3703: 3700: 3697: 3694: 3691: 3686: 3682: 3678: 3673: 3668: 3665: 3662: 3658: 3654: 3651: 3646: 3642: 3636: 3631: 3628: 3625: 3621: 3617: 3614: 3611: 3597: 3596: 3595: 3594: 3593: 3592: 3591: 3590: 3589: 3588: 3587: 3586: 3585: 3584: 3583: 3582: 3581: 3580: 3579: 3578: 3558: 3555: 3552: 3549: 3546: 3543: 3540: 3535: 3531: 3527: 3524: 3521: 3518: 3515: 3512: 3509: 3506: 3501: 3497: 3492: 3486: 3483: 3480: 3477: 3474: 3471: 3468: 3463: 3459: 3455: 3452: 3449: 3446: 3443: 3440: 3437: 3434: 3429: 3425: 3420: 3412: 3408: 3402: 3399: 3396: 3392: 3386: 3381: 3378: 3375: 3371: 3348: 3344: 3338: 3333: 3330: 3327: 3323: 3319: 3316: 3294: 3291: 3288: 3285: 3282: 3279: 3276: 3271: 3267: 3263: 3260: 3256: 3250: 3247: 3244: 3241: 3238: 3235: 3232: 3229: 3226: 3221: 3217: 3212: 3206: 3201: 3198: 3195: 3191: 3160: 3159: 3158: 3157: 3156: 3155: 3154: 3153: 3152: 3151: 3150: 3149: 3148: 3147: 3146: 3145: 3144: 3143: 3116: 3090: 3087: 3084: 3081: 3078: 3075: 3072: 3067: 3063: 3059: 3056: 3053: 3050: 3047: 3044: 3041: 3038: 3033: 3029: 3024: 3020: 3017: 3014: 2992: 2989: 2986: 2983: 2980: 2977: 2974: 2969: 2965: 2961: 2958: 2955: 2952: 2949: 2946: 2943: 2940: 2935: 2931: 2926: 2920: 2917: 2914: 2911: 2908: 2905: 2902: 2897: 2893: 2889: 2886: 2883: 2880: 2877: 2874: 2871: 2868: 2863: 2859: 2854: 2846: 2842: 2836: 2833: 2830: 2826: 2820: 2815: 2812: 2809: 2805: 2782: 2779: 2776: 2773: 2770: 2767: 2764: 2759: 2755: 2751: 2748: 2745: 2742: 2739: 2736: 2733: 2730: 2725: 2721: 2716: 2710: 2707: 2704: 2701: 2698: 2695: 2692: 2687: 2683: 2679: 2676: 2673: 2670: 2667: 2664: 2661: 2658: 2653: 2649: 2644: 2636: 2632: 2626: 2623: 2620: 2616: 2593: 2590: 2585: 2582: 2578: 2572: 2569: 2564: 2561: 2557: 2549: 2545: 2539: 2536: 2533: 2529: 2508: 2505: 2502: 2499: 2496: 2487:. For each k ( 2474: 2471: 2467: 2461: 2458: 2454: 2448: 2443: 2440: 2437: 2433: 2410: 2407: 2404: 2401: 2398: 2395: 2392: 2387: 2383: 2379: 2376: 2373: 2370: 2367: 2364: 2361: 2358: 2353: 2349: 2344: 2340: 2337: 2334: 2331: 2326: 2323: 2318: 2315: 2312: 2308: 2304: 2301: 2298: 2276: 2273: 2269: 2265: 2262: 2259: 2248: 2247: 2246: 2245: 2244: 2243: 2242: 2241: 2240: 2239: 2238: 2237: 2236: 2235: 2234: 2233: 2219: 2215: 2192: 2188: 2165: 2161: 2138: 2134: 2111: 2107: 2084: 2080: 2048: 2047: 2046: 2045: 2044: 2043: 2042: 2041: 2040: 2039: 2038: 2037: 2027: 2026: 2025: 2008: 2007: 2006: 2005: 2004: 2003: 2002: 2001: 2000: 1999: 1979: 1976: 1973: 1969: 1942: 1939: 1936: 1933: 1930: 1927: 1924: 1920: 1916: 1913: 1910: 1884: 1881: 1878: 1875: 1872: 1869: 1866: 1862: 1856: 1853: 1850: 1847: 1844: 1841: 1838: 1834: 1828: 1825: 1822: 1818: 1791: 1788: 1785: 1782: 1779: 1776: 1773: 1769: 1763: 1760: 1757: 1754: 1751: 1748: 1745: 1741: 1735: 1730: 1727: 1724: 1720: 1714: 1710: 1706: 1701: 1698: 1693: 1690: 1687: 1683: 1677: 1674: 1669: 1666: 1663: 1659: 1653: 1648: 1645: 1642: 1638: 1632: 1628: 1601: 1598: 1594: 1588: 1585: 1581: 1575: 1571: 1548: 1545: 1542: 1539: 1536: 1533: 1530: 1526: 1522: 1519: 1516: 1513: 1508: 1505: 1500: 1497: 1494: 1490: 1486: 1483: 1480: 1458: 1455: 1451: 1447: 1444: 1441: 1419: 1416: 1411: 1408: 1405: 1401: 1397: 1394: 1391: 1369: 1366: 1362: 1356: 1353: 1349: 1343: 1339: 1335: 1330: 1327: 1323: 1319: 1316: 1313: 1308: 1305: 1302: 1299: 1277: 1274: 1271: 1267: 1246: 1245: 1244: 1243: 1242: 1241: 1240: 1239: 1228:elements, not 1215: 1214: 1213: 1212: 1211: 1210: 1186: 1183: 1179: 1173: 1170: 1166: 1160: 1156: 1133: 1130: 1126: 1122: 1119: 1116: 1099: 1097: 1096: 1095: 1094: 1093: 1092: 1068: 1065: 1061: 1055: 1052: 1048: 1042: 1038: 1034: 1029: 1026: 1022: 1018: 1015: 1012: 1007: 1004: 1001: 998: 976: 973: 969: 965: 960: 957: 953: 949: 946: 943: 938: 935: 932: 929: 909: 906: 886: 883: 880: 877: 866: 865: 864: 853: 850: 847: 840: 837: 833: 827: 824: 820: 814: 810: 806: 801: 798: 794: 771: 751: 731: 720: 712: 711: 710: 709: 684: 681: 677: 671: 668: 664: 658: 654: 650: 645: 642: 638: 634: 631: 628: 614: 613: 604: 603: 602: 585: 582: 578: 572: 569: 565: 559: 555: 551: 546: 543: 539: 535: 532: 529: 507: 504: 482: 479: 475: 462: 461: 451:Meni Rosenfeld 436: 433: 429: 417: 416: 415: 395:Meni Rosenfeld 376: 375:Block Matrices 373: 372: 371: 361:Meni Rosenfeld 325: 322: 321: 320: 303: 302: 301: 282: 279: 274: 271: 265: 262: 257: 253: 249: 244: 240: 218: 215: 210: 207: 201: 198: 195: 180: 179: 178: 114: 113: 105: 102: 100: 97: 95: 91: 90: 82: 81: 71: 70: 64: 48: 41: 40: 31: 15: 14: 13: 10: 9: 6: 4: 3: 2: 5205: 5196: 5195: 5192: 5188: 5183: 5182: 5179: 5175: 5162: 5158: 5154: 5150: 5146: 5145: 5144: 5141: 5137: 5136: 5135: 5131: 5127: 5123: 5119: 5116: 5115: 5114: 5113: 5110: 5107: 5103: 5102: 5089: 5085: 5081: 5077: 5073: 5072: 5071: 5070: 5069: 5068: 5067: 5066: 5065: 5064: 5063: 5062: 5045: 5041: 5040:commutativity 5037: 5036:associativity 5033: 5032: 5031: 5028: 5024: 5023: 5022: 5019: 5014: 5013: 5012: 5008: 5004: 5000: 4996: 4992: 4991: 4990: 4989: 4988: 4987: 4977:combinations: 4975: 4974: 4973: 4972: 4971: 4970: 4965: 4962: 4958: 4956: 4953: 4948: 4947: 4946: 4943: 4942:202.168.50.40 4938: 4933: 4932: 4930: 4929: 4926: 4923: 4922: 4915: 4910: 4906: 4905: 4902: 4899: 4895: 4894: 4891: 4888: 4884: 4883: 4882: 4879: 4873: 4863: 4860: 4858: 4851: 4848: 4844: 4840: 4837: 4832: 4831: 4830: 4827: 4823: 4822: 4821: 4818: 4816: 4808: 4807: 4806: 4803: 4800: 4796: 4793: 4789: 4785: 4784: 4783: 4782: 4779: 4777: 4764: 4759: 4755: 4751: 4747: 4743: 4737: 4721: 4718: 4714: 4713: 4712: 4711: 4710: 4707: 4703: 4702: 4698: 4694: 4690: 4686: 4682: 4676: 4675: 4674: 4673: 4672: 4668: 4664: 4660: 4659: 4658: 4655: 4651: 4650: 4647: 4644: 4640: 4639: 4638: 4637: 4636: 4632: 4628: 4608: 4605: 4602: 4599: 4596: 4593: 4590: 4566: 4563: 4560: 4557: 4554: 4551: 4543: 4537: 4534: 4530: 4527: 4524: 4521: 4520: 4517: 4514: 4510: 4509: 4508: 4507: 4506: 4502: 4498: 4494: 4490: 4484: 4481: 4477: 4476: 4475: 4474: 4473: 4469: 4465: 4445: 4442: 4439: 4433: 4430: 4424: 4421: 4418: 4415: 4412: 4409: 4389: 4386: 4383: 4380: 4377: 4374: 4354: 4351: 4348: 4345: 4342: 4339: 4333: 4327: 4307: 4304: 4298: 4292: 4284: 4283: 4282: 4281: 4278: 4274: 4270: 4266: 4265:f(x) = ax + b 4258: 4250: 4245: 4242: 4238: 4237: 4231: 4230: 4229: 4225: 4221: 4217: 4216: 4215: 4212: 4211:84.239.133.38 4208: 4206: 4202: 4198: 4166: 4159: 4156: 4153: 4148: 4141: 4137: 4131: 4125: 4117: 4116: 4115: 4114: 4109: 4106: 4102: 4101: 4076: 4073: 4070: 4067: 4064: 4061: 4058: 4055: 4052: 4047: 4040: 4037: 4034: 4029: 4023: 4018: 4012: 4006: 3994: 3990: 3987: 3986:87.102.47.243 3982: 3978: 3975: 3970: 3969: 3968: 3965: 3961: 3957: 3953: 3936: 3929: 3926: 3922: 3917: 3914: 3910: 3905: 3898: 3895: 3891: 3886: 3883: 3879: 3875: 3867: 3863: 3857: 3853: 3846: 3842: 3836: 3833: 3826: 3825: 3823: 3819: 3818: 3817: 3816: 3813: 3807: 3806: 3805:Basel problem 3801: 3798: 3792: 3786: 3782: 3766: 3760: 3757: 3753: 3752: 3751: 3750: 3749: 3748: 3745: 3723: 3718: 3714: 3710: 3707: 3704: 3698: 3695: 3692: 3684: 3680: 3671: 3666: 3663: 3660: 3656: 3652: 3644: 3640: 3634: 3629: 3626: 3623: 3619: 3615: 3612: 3602: 3601: 3600: 3577: 3574: 3556: 3553: 3547: 3544: 3541: 3533: 3529: 3525: 3522: 3519: 3513: 3510: 3507: 3499: 3495: 3490: 3484: 3481: 3475: 3472: 3469: 3461: 3457: 3453: 3450: 3447: 3441: 3438: 3435: 3427: 3423: 3418: 3410: 3406: 3400: 3397: 3394: 3390: 3384: 3379: 3376: 3373: 3369: 3346: 3342: 3336: 3331: 3328: 3325: 3321: 3317: 3314: 3292: 3289: 3283: 3280: 3277: 3269: 3265: 3261: 3258: 3254: 3248: 3245: 3242: 3239: 3233: 3230: 3227: 3219: 3215: 3210: 3204: 3199: 3196: 3193: 3189: 3180: 3179: 3178: 3177: 3176: 3175: 3174: 3173: 3172: 3171: 3170: 3169: 3168: 3167: 3166: 3165: 3164: 3163: 3162: 3161: 3142: 3139: 3135: 3134: 3133: 3132: 3131: 3130: 3129: 3128: 3127: 3126: 3125: 3124: 3123: 3122: 3121: 3120: 3119: 3118: 3117: 3114: 3113: 3110: 3106: 3088: 3085: 3079: 3076: 3073: 3065: 3061: 3057: 3054: 3051: 3045: 3042: 3039: 3031: 3027: 3018: 3015: 2990: 2987: 2981: 2978: 2975: 2967: 2963: 2959: 2956: 2953: 2947: 2944: 2941: 2933: 2929: 2924: 2918: 2915: 2909: 2906: 2903: 2895: 2891: 2887: 2884: 2881: 2875: 2872: 2869: 2861: 2857: 2852: 2844: 2840: 2834: 2831: 2828: 2824: 2818: 2813: 2810: 2807: 2803: 2780: 2777: 2771: 2768: 2765: 2757: 2753: 2749: 2746: 2743: 2737: 2734: 2731: 2723: 2719: 2714: 2708: 2705: 2699: 2696: 2693: 2685: 2681: 2677: 2674: 2671: 2665: 2662: 2659: 2651: 2647: 2642: 2634: 2630: 2624: 2621: 2618: 2614: 2591: 2588: 2583: 2580: 2576: 2570: 2567: 2562: 2559: 2555: 2547: 2543: 2537: 2534: 2531: 2527: 2506: 2503: 2500: 2497: 2494: 2472: 2469: 2465: 2459: 2456: 2452: 2446: 2441: 2438: 2435: 2431: 2408: 2405: 2399: 2396: 2393: 2385: 2381: 2377: 2374: 2371: 2365: 2362: 2359: 2351: 2347: 2338: 2335: 2329: 2324: 2321: 2316: 2313: 2310: 2302: 2299: 2274: 2271: 2263: 2260: 2217: 2213: 2190: 2186: 2163: 2159: 2136: 2132: 2109: 2105: 2082: 2078: 2069: 2064: 2063: 2062: 2061: 2060: 2059: 2058: 2057: 2056: 2055: 2054: 2053: 2052: 2051: 2050: 2049: 2036: 2033: 2028: 2023: 2022: 2020: 2019: 2018: 2017: 2016: 2015: 2014: 2013: 2012: 2011: 2010: 2009: 1998: 1995: 1977: 1974: 1971: 1967: 1940: 1937: 1934: 1931: 1928: 1925: 1922: 1914: 1911: 1882: 1879: 1876: 1873: 1870: 1867: 1864: 1860: 1854: 1851: 1848: 1845: 1842: 1839: 1836: 1832: 1826: 1823: 1820: 1816: 1789: 1786: 1783: 1780: 1777: 1774: 1771: 1767: 1761: 1758: 1755: 1752: 1749: 1746: 1743: 1739: 1733: 1728: 1725: 1722: 1718: 1712: 1708: 1704: 1699: 1696: 1691: 1688: 1685: 1681: 1675: 1672: 1667: 1664: 1661: 1657: 1651: 1646: 1643: 1640: 1636: 1630: 1626: 1599: 1596: 1592: 1586: 1583: 1579: 1573: 1569: 1546: 1543: 1540: 1537: 1534: 1531: 1528: 1520: 1517: 1511: 1506: 1503: 1498: 1495: 1492: 1484: 1481: 1456: 1453: 1445: 1442: 1417: 1414: 1409: 1406: 1403: 1395: 1392: 1367: 1364: 1360: 1354: 1351: 1347: 1341: 1337: 1333: 1328: 1325: 1317: 1314: 1306: 1303: 1300: 1275: 1272: 1269: 1265: 1256: 1255: 1254: 1253: 1252: 1251: 1250: 1249: 1248: 1247: 1238: 1235: 1231: 1227: 1226:corresponding 1223: 1222: 1221: 1220: 1219: 1218: 1217: 1216: 1209: 1206: 1184: 1181: 1177: 1171: 1168: 1164: 1158: 1154: 1131: 1128: 1120: 1117: 1106: 1105: 1104: 1103: 1102: 1101: 1100: 1091: 1088: 1066: 1063: 1059: 1053: 1050: 1046: 1040: 1036: 1032: 1027: 1024: 1016: 1013: 1005: 1002: 999: 974: 971: 967: 963: 958: 955: 947: 944: 936: 933: 930: 907: 904: 884: 881: 878: 875: 867: 848: 838: 835: 831: 825: 822: 818: 812: 808: 804: 799: 796: 792: 784: 783: 769: 749: 729: 721: 718: 717: 716: 715: 714: 713: 708: 705: 682: 679: 675: 669: 666: 662: 656: 652: 648: 643: 640: 632: 629: 618: 617: 616: 615: 612: 609: 605: 583: 580: 576: 570: 567: 563: 557: 553: 549: 544: 541: 533: 530: 520: 519: 505: 502: 480: 477: 473: 464: 463: 460: 456: 452: 434: 431: 427: 418: 414: 411: 406: 405: 404: 400: 396: 392: 388: 387: 386: 385: 382: 374: 370: 366: 362: 358: 357: 356: 355: 352: 347: 345: 341: 337: 333: 323: 319: 316: 315: 308: 304: 300: 297: 277: 272: 263: 260: 255: 251: 247: 242: 238: 213: 208: 199: 196: 193: 185: 181: 177: 174: 170: 166: 165: 164: 161: 157: 153: 152: 151: 148: 147: 140: 136: 135: 134: 133: 130: 129: 122: 117: 111: 110: 109: 103: 98: 96: 88: 84: 83: 80: 77: 76: 68: 61: 57: 53: 47: 42: 39: 35: 27: 23: 19: 5184: 5171: 5117: 4934:oper3,oper2] 4917: 4908: 4880: 4877: 4854: 4812: 4773: 4768: 4272: 4271:(2) = 7 and 4268: 4264: 4262: 4235: 4099: 3998: 3808: 3802: 3799: 3796: 3790: 3783: 3780: 3741: 3598: 3115: 2249: 2067: 1229: 1225: 1098: 378: 348: 327: 310: 142: 124: 118: 115: 107: 94: 78: 5191:Rainwarrior 4740:—Preceding 4679:—Preceding 351:A math-wiki 336:75.30.68.22 330:—Preceding 173:PrimeHunter 26:Mathematics 5187:four fours 4874:Four fours 4843:Izhikevich 4836:Perceptron 4525:7=2a+b and 3995:LaTeX help 1382:. Let the 1232:elements. 99:October 15 67:October 16 46:October 14 5178:GTBacchus 5149:tcsetattr 5122:tcsetattr 5076:tcsetattr 5027:GTBacchus 5018:GTBacchus 4999:tcsetattr 4961:GTBacchus 4887:Trovatore 4174:otherwise 3964:Gandalf61 1230:arbitrary 702:Morana.-- 167:See also 50:<< 5157:contribs 5130:contribs 5084:contribs 5034:I think 5007:contribs 4914:Grutness 4841:Just as 4754:contribs 4742:unsigned 4693:contribs 4681:unsigned 4528:-5=-1a+b 3777:Resolved 897:, where 332:unsigned 307:Grutness 139:Grutness 121:Grutness 24:‎ | 22:Archives 20:‎ | 4717:Hadseys 4685:Hadseys 4643:Hadseys 4513:Hadseys 4480:Hadseys 4277:Hadseys 408:hint?-- 89:pages. 56:October 5174:Krypto 5140:Morana 5106:Morana 4952:Morana 4898:Morana 4791:layer. 4236:x42bn6 4100:x42bn6 3756:Shahab 3744:Morana 3573:Shahab 3307:where 3138:Morana 3109:Shahab 3105:Shahab 2606:or by 2032:Morana 1994:Shahab 1234:Morana 1205:Shahab 1087:Morana 704:Shahab 608:Morana 410:Shahab 381:Shahab 296:Spoon! 4746:Aenar 4706:Aenar 4625:. -- 230:, so 160:Strad 69:: --> 63:: --> 62:: --> 44:< 16:< 5153:talk 5126:talk 5080:talk 5044:here 5003:talk 4920:wha? 4856:Clem 4826:risk 4814:Clem 4802:risk 4775:Clem 4750:talk 4689:talk 4667:talk 4631:talk 4583:and 4501:talk 4493:here 4468:talk 4244:Mess 4241:Talk 4224:talk 4201:talk 4108:Mess 4105:Talk 1203:. -- 762:and 455:talk 399:talk 365:talk 340:talk 313:wha? 145:wha? 127:wha? 5176:. - 4916:... 3972:pi? 3791:or 1901:or 1808:or 309:... 273:mod 209:mod 141:... 123:... 60:Nov 52:Sep 5159:) 5155:/ 5132:) 5128:/ 5086:) 5082:/ 5009:) 5005:/ 4756:) 4752:• 4695:) 4691:• 4669:) 4633:) 4606:− 4591:− 4503:) 4470:) 4443:− 4431:− 4410:− 4346:⋅ 4226:) 4203:) 4157:≥ 4038:≥ 3927:π 3896:π 3887:− 3854:π 3837:− 3696:− 3657:⋃ 3620:∑ 3545:− 3511:− 3473:− 3439:− 3391:∑ 3370:∑ 3322:∑ 3281:− 3231:− 3190:∑ 3077:− 3043:− 2979:− 2945:− 2907:− 2873:− 2825:∑ 2804:∑ 2769:− 2735:− 2697:− 2663:− 2615:∑ 2528:∑ 2504:≤ 2498:≤ 2432:∑ 2397:− 2363:− 2068:of 1817:∑ 1719:∑ 1709:∑ 1637:∑ 1627:∑ 1570:∑ 1338:∑ 1298:∀ 1155:∑ 1037:∑ 997:∀ 928:∀ 809:∑ 653:∑ 554:∑ 457:) 401:) 367:) 342:) 294:-- 261:≡ 248:≡ 197:≡ 186:. 171:. 58:| 54:| 5151:( 5124:( 5078:( 5016:- 5001:( 4748:( 4687:( 4665:( 4629:( 4609:5 4603:= 4600:b 4597:+ 4594:a 4567:7 4564:= 4561:b 4558:+ 4555:a 4552:2 4499:( 4466:( 4446:5 4440:= 4437:) 4434:1 4428:( 4425:f 4422:= 4419:b 4416:+ 4413:a 4390:7 4387:= 4384:b 4381:+ 4378:a 4375:2 4355:b 4352:+ 4349:2 4343:a 4340:= 4337:) 4334:2 4331:( 4328:f 4308:7 4305:= 4302:) 4299:2 4296:( 4293:f 4273:f 4269:f 4222:( 4199:( 4167:4 4160:4 4154:x 4149:1 4142:{ 4138:= 4135:) 4132:x 4129:( 4126:f 4077:e 4074:s 4071:i 4068:w 4065:r 4062:e 4059:h 4056:t 4053:o 4048:4 4041:4 4035:x 4030:1 4024:{ 4019:= 4016:) 4013:x 4010:( 4007:f 3962:. 3937:) 3930:n 3923:x 3918:+ 3915:1 3911:( 3906:) 3899:n 3892:x 3884:1 3880:( 3876:= 3868:2 3864:n 3858:2 3847:2 3843:x 3834:1 3727:] 3724:k 3719:k 3715:y 3711:; 3708:1 3705:+ 3702:) 3699:1 3693:k 3690:( 3685:k 3681:y 3677:[ 3672:n 3667:1 3664:= 3661:k 3653:= 3650:] 3645:k 3641:y 3635:n 3630:1 3627:= 3624:k 3616:; 3613:1 3610:[ 3557:s 3554:+ 3551:) 3548:1 3542:j 3539:( 3534:j 3530:z 3526:, 3523:t 3520:+ 3517:) 3514:1 3508:k 3505:( 3500:k 3496:y 3491:B 3485:t 3482:+ 3479:) 3476:1 3470:k 3467:( 3462:k 3458:y 3454:, 3451:r 3448:+ 3445:) 3442:1 3436:i 3433:( 3428:i 3424:x 3419:A 3411:k 3407:y 3401:1 3398:= 3395:t 3385:n 3380:1 3377:= 3374:k 3347:k 3343:y 3337:n 3332:1 3329:= 3326:k 3318:= 3315:X 3293:s 3290:+ 3287:) 3284:1 3278:j 3275:( 3270:j 3266:z 3262:, 3259:u 3255:B 3249:u 3246:, 3243:r 3240:+ 3237:) 3234:1 3228:i 3225:( 3220:i 3216:x 3211:A 3205:X 3200:1 3197:= 3194:u 3089:s 3086:+ 3083:) 3080:1 3074:j 3071:( 3066:j 3062:z 3058:, 3055:r 3052:+ 3049:) 3046:1 3040:i 3037:( 3032:i 3028:x 3023:) 3019:B 3016:A 3013:( 2991:s 2988:+ 2985:) 2982:1 2976:j 2973:( 2968:j 2964:z 2960:, 2957:t 2954:+ 2951:) 2948:1 2942:k 2939:( 2934:k 2930:y 2925:B 2919:t 2916:+ 2913:) 2910:1 2904:k 2901:( 2896:k 2892:y 2888:, 2885:r 2882:+ 2879:) 2876:1 2870:i 2867:( 2862:i 2858:x 2853:A 2845:k 2841:y 2835:1 2832:= 2829:t 2819:n 2814:1 2811:= 2808:k 2781:s 2778:+ 2775:) 2772:1 2766:j 2763:( 2758:j 2754:z 2750:, 2747:t 2744:+ 2741:) 2738:1 2732:k 2729:( 2724:k 2720:y 2715:B 2709:t 2706:+ 2703:) 2700:1 2694:k 2691:( 2686:k 2682:y 2678:, 2675:r 2672:+ 2669:) 2666:1 2660:i 2657:( 2652:i 2648:x 2643:A 2635:k 2631:y 2625:1 2622:= 2619:t 2592:j 2589:k 2584:s 2581:t 2577:B 2571:k 2568:i 2563:t 2560:r 2556:A 2548:k 2544:y 2538:1 2535:= 2532:t 2507:n 2501:k 2495:1 2473:j 2470:k 2466:B 2460:k 2457:i 2453:A 2447:n 2442:1 2439:= 2436:k 2409:s 2406:+ 2403:) 2400:1 2394:j 2391:( 2386:j 2382:z 2378:, 2375:r 2372:+ 2369:) 2366:1 2360:i 2357:( 2352:i 2348:x 2343:) 2339:B 2336:A 2333:( 2330:= 2325:j 2322:i 2317:s 2314:, 2311:r 2307:) 2303:B 2300:A 2297:( 2275:j 2272:i 2268:) 2264:B 2261:A 2258:( 2218:j 2214:z 2191:i 2187:x 2164:j 2160:z 2137:i 2133:y 2110:j 2106:y 2083:i 2079:x 1978:n 1975:, 1972:m 1968:A 1941:q 1938:+ 1935:s 1932:, 1929:p 1926:+ 1923:r 1919:) 1915:B 1912:A 1909:( 1883:q 1880:+ 1877:s 1874:, 1871:t 1868:+ 1865:k 1861:B 1855:t 1852:+ 1849:k 1846:, 1843:p 1840:+ 1837:r 1833:A 1827:t 1824:, 1821:k 1790:q 1787:+ 1784:s 1781:, 1778:t 1775:+ 1772:k 1768:B 1762:t 1759:+ 1756:k 1753:, 1750:p 1747:+ 1744:r 1740:A 1734:k 1729:1 1726:= 1723:t 1713:k 1705:= 1700:j 1697:k 1692:q 1689:, 1686:t 1682:B 1676:k 1673:i 1668:t 1665:, 1662:p 1658:A 1652:k 1647:1 1644:= 1641:t 1631:k 1600:j 1597:k 1593:B 1587:k 1584:i 1580:A 1574:k 1547:q 1544:+ 1541:s 1538:, 1535:p 1532:+ 1529:r 1525:) 1521:B 1518:A 1515:( 1512:= 1507:j 1504:i 1499:q 1496:, 1493:p 1489:) 1485:B 1482:A 1479:( 1457:j 1454:i 1450:) 1446:B 1443:A 1440:( 1418:j 1415:i 1410:q 1407:, 1404:p 1400:) 1396:B 1393:A 1390:( 1368:j 1365:k 1361:B 1355:k 1352:i 1348:A 1342:k 1334:= 1329:j 1326:i 1322:) 1318:B 1315:A 1312:( 1307:j 1304:, 1301:i 1276:j 1273:, 1270:i 1266:X 1185:j 1182:k 1178:B 1172:k 1169:i 1165:A 1159:k 1132:j 1129:i 1125:) 1121:B 1118:A 1115:( 1067:j 1064:k 1060:B 1054:k 1051:i 1047:A 1041:k 1033:= 1028:j 1025:i 1021:) 1017:B 1014:A 1011:( 1006:j 1003:, 1000:i 975:j 972:i 968:C 964:= 959:j 956:i 952:) 948:B 945:A 942:( 937:j 934:, 931:i 908:B 905:A 885:C 882:= 879:B 876:A 852:) 849:1 846:( 839:j 836:k 832:B 826:k 823:i 819:A 813:k 805:= 800:j 797:i 793:C 770:B 750:A 730:C 683:j 680:k 676:B 670:k 667:i 663:A 657:k 649:= 644:j 641:i 637:) 633:B 630:A 627:( 584:j 581:k 577:B 571:k 568:i 564:A 558:k 550:= 545:j 542:i 538:) 534:B 531:A 528:( 506:j 503:i 481:j 478:i 474:A 453:( 435:j 432:i 428:c 397:( 363:( 338:( 281:) 278:8 270:( 264:1 256:n 252:1 243:n 239:9 217:) 214:8 206:( 200:1 194:9

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Knowledge (XXG)

:Reference desk/Archives/Mathematics/2007 October 15 - Knowledge (XXG)

Index