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tarun purohit

Discussion  - 
1
Jerzy Kaltenberg's profile photo
 
...in getting run over.
 The light ahead is likely a train bearing down on you.
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NYT profiles phd mathematician/billionaire/quant Simons, who has also funded significant CS research
The billionaire mathematician James H. Simons has led a life of ferocious curiosity.
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Kenneth Huang's profile photopsher grant's profile photoAndrius Achremcikas's profile photoFreddy Kruger's profile photo
4 comments
 
He is a very likeable billionaire...no offense, but he kind of looks like the evil guys from Sherlock Holmes in Season 3, Episode 3. Anyway,...
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Arjen Meijer

Discussion  - 
 
"when people are free to pursue goals unfettered by presumed limitations on what they can accomplish, they just may manage some extraordinary feats through the combined application of native talent and hard work."

Never listen to people who say something can not be done. 
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Paul Hartzer's profile photoCharlie Richmond's profile photoRob Jongschaap's profile photoMatt Skerritt's profile photo
3 comments
 
Cool. It's good to know that there is a true story behind that urban legend.

I disagree with the OP, that we should never listen to people who say things cannot be done. I would say that sometimes (perhaps even often) we should not listen.

However, sometimes it can be proved that things cannot be done, and we should most certainly pay attention to such things. At the very least we should examine the proof for veracity.

For example:
* It is impossible to compute the halting problem on a Turing machine.
* It is impossible to write the square root of 2 as a ratio of integers.
* It is impossible to find a polynomial with rational coefficients which has Pi as a zero.
* It is impossible to find a method that can trisect all angles using the rules of unmarked ruler and compass constructions.
* It is impossible to find a real number that is both strictly greater than 0 and strictly less than zero.

There is, a +Paul Hartzer pointed out, a difference between "impossible" and "hasn't been done yet". 
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Brasil Germany 1:3 lol
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Anders Karlsson's profile photoDmitry Ulanov's profile photoGideon Moturi's profile photoFrank Stolz's profile photo
5 comments
 
Pawned! Good work Germany!
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It is possible to have 12-sided dice that stack together neatly without gaps. Find out more below.
 
Stackable 12-sided dice

This picture by Robert Fathauer shows the stackable 12-sided dice that he designed with +Henry Segerman. Each of the dice is in the shape of a rhombic dodecahedron, which is a convex polyhedron with 12 faces. Each face has the shape of a rhombus, and is symmetric under rotation by 180 degrees. These dice stack neatly together, without gaps, because the rhombic dodecahedron has the remarkable property that it can tessellate space by translational copies of itself.

The rhombic dodecadron has the same amount of rotational symmetry as a cube or a regular octahedron. Each of the three shapes has 24 rotational symmetries, and in each case, the number of faces multiplied by the number of rotational symmetries of each face is equal to 24. Furthermore, each shape has a group of rotational symmetries that is isomorphic to the symmetric group S4.

The obvious way to design a 12-sided die is to make it in the shape of a dodecahedron, which has 12 pentagonal faces. This does produce a more symmetrical shape, with 60 rotational symmetries, but the disadvantage is that dodecahedra cannot fit together neatly (in Euclidean space) in such a way as not to leave any gaps.

Relevant links
The dice in the picture are for sale at http://goo.gl/n3vpFm (www.mathartfun.com). The site sells many other kinds of unusual dice, including non-transitive dice.

Here's a post by me about non-transitive dice: https://plus.google.com/101584889282878921052/posts/eq9eHus2wQP. My friend +James Grime has also designed sets of non-transitive dice, and as I mentioned in my post https://plus.google.com/101584889282878921052/posts/iQy6GkfD5XP, you can buy the Grime Dice at http://mathsgear.co.uk/.

Wikipedia has some nice articles on (1) the rhombic dodecahedron, (2) the associated tessellation of Eucliean space, and (3) the symmetric group: 
http://en.wikipedia.org/wiki/Rhombic_dodecahedron
http://en.wikipedia.org/wiki/Rhombic_dodecahedral_honeycomb
http://en.wikipedia.org/wiki/Symmetric_group

#mathematics
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nitu joseph's profile photoKonstantin Hartwich's profile photoAdolfo Domínguez's profile photouriel jimenez's profile photo
3 comments
 
Great stuff.
Mathematically it smells like geometric topology.
But This is a topic far beyond my means, but not my intrests.
It is always difficult to find a curriculum path towards a certain mathematics subject. Beginning my first year maths with metric spaces coming from simple calculus calculation was bad pedagogy . We speak 1980. But i would try it again with all the support they get nowadays, and metric and normed linear spaces moved to the 4th semester.
Coursera fives functional analysis but where is real analysis, differential geometry,...Hard to find out.
Thanks 
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Paul Hartzer

Discussion  - 
 
I'm thinking of collecting up examples of fibs and acts of mathematical recklessness committed by elementary and secondary teachers. Here are some examples (to clarify: I feel the statements below are varying degrees of incorrect, but I've run across all of them as absolute truths):

* Lines (and rays) can't be congruent because they don't have a finite length
* An angle consists of two rays or segments that meet at a point
* Asymptotes are values that functions get closer and closer to, but never meet
* 0 cannot be negated (via +Matt McIrvin )
* -1^2 = -1 unambiguously (rather than by definition)
* "And" should not be said when reading integers (201 is "two hundred one"; "two hundred and one" is wrong)
* The obelus (÷) should never be used in high school, only in lower grades

I welcome any other examples. Keep in mind, these are things that teachers say about material that's likely to be met before college. I'm not looking for misinformation about the Reimann-Whorf Complex Metaset Hypothesis here. :)
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Paul Hartzer's profile photoLayra Idarani's profile photo
54 comments
 
Thinking about this some more, I feel like it might be another instance of "restrict too much to prevent something bad from being approached", and that a triangle might be a little too simple to really illustrate this.
Regardless of what order we pick for the three vertices of a triangle, we're going to get the same triangle, so we can be cavalier about which order we list the vertices in. Not so for polygons with more vertices.
If we have a quadrilateral and we label the vertices A, B, C, D by going around the edges, then, assuming nondegeneracy, ACBD is a different polygon than ABCD, as is ABDC. So here the ordering of the points matters a lot, compared to the triangle case where it doesn't matter as much.
Thus, to get away with being able to only specifying the vertices in question, one way is to demand that the vertices be lined up in order. It's too restrictive, sure, but it does prevent someone saying that ABCD and ABDC are congruent without having to explain about dihedral groups. Again, it's a notational matter, but it's more than just pure convenience.
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John Newman

Discussion  - 
 
Sending this to #math
1
Julian Austin's profile photoAlied Perez's profile photo
11 comments
 
Does it relativize? That would be my first test of the claim, to see if it contradicts the baker-gill-solovay theorem. 
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Mathematical Masterpieces: Making Art From Equations at the annual Bridges conference

One of my friends +Marco Garavaglia asked me to share this interesting news.

Artists use math to create works of art to rival gallery masterpieces. 
Mathematicians enthralled with unending fractals and flux patterns have been known to call math beautiful — but, increasingly, they aren't the only ones. For many artists, calculations and numerical analyses provide a rich source of ideas and methods for their creations.

The annual Bridges conference showcases the connections between art and mathematics. The conference features, among other things, a juried art exhibition full of a staggering range of mathematically-inspired artworks, where you can see sunsets, lampshades, and more examples of the golden ratio than you can shake a shell at.

Read the full article >>
http://discovermagazine.com/MathArt#.UysAZD-uTTo

This year, the conference will be held in Seoul, 14-19 August.

Did anyone partecipate in previous Bridges conferences?

Go to>>
http://bridgesmathart.org/bridges-2014/
http://bridgesmathart.org

#math #scientific_art
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edmund ajo's profile photojackie xie's profile photoPaul Zukowski's profile photoFreddy Kruger's profile photo
3 comments
 
Damn, looks like some of my technical drawing in my technical writing papers that I had done. Good work. Good post! I see the platonic solids here too, but i don't see all five of them. I hope non Euclidean geometry, and the mathematical scientific methods of The Theory of Chaos and Order could have a place here too when more of these are ascribed and understood. Bring it up to the 21st century knowledge!
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Paul Hartzer

Discussion  - 
 
Given: A, B, and C are collinear, with B between A and C.

Question: Is it properly rigorous to say that AC is congruent with AB and BC, and if so, what would the appropriate operator be? I'm question the fact that point B is present in both AB and BC, and so would be there twice in their combination but only once in AC. On another level, though, this feels like a version of 0.999... ≟ 1.

If it's not properly rigorous to use "congruent" in this context, is it ever appropriate to say that one object is congruent to some combination of two (or more) others?
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Paul Hartzer's profile photoLayra Idarani's profile photo
18 comments
 
Yeah, that seems to be my impression too. I talked to the head of a math department at a private high school about maybe working there, and he seemed quite surprised that someone with a PhD was even considering teaching a high-school, even one that sometimes taught multivariable calc and linear algebra.
We agreed that I wouldn't be a good fit since I wanted to do at least some research of my own, and apparently teaching high-school doesn't allow enough free time for that. I suspect that most serious mathematicians feel the same.
There are numerous other issues, including the type of personality that usually becomes a serious mathematician, the pay scale for public-school teachers and the people who create materials for public schools, and various cultural biases. Plus teaching at public high-schools and middle-schools usually requires a teaching certification, whereas if you just finished a doctorate then you want a job, not more schooling.
I've had the very good fortune of my formative math years being under the tutelage of Robert Kaplan and James Tanton, both of whom taught at universities but were perfectly happy when they went on to teach high-school and middle-school students. They were definitely exceptions, though.
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Akira Bergman

Discussion  - 
 
FINDING THE PRIMES BY A FACTORIAL TREE

The naturals (n=0,1,2,3,...) can be divided into a binary tree by a self similar process as following;

(n)
(2n,2n+1)
((4n,4n+2),(4n+1,4n+3))
(((8n,8n+4),(8n+2,8n+6)),((8n+1,8n+5),(8n+3,8n+7)))
((((16n,16n+8),(16n+4,16n+12)),((16n+2,16n+10),(16n+6,16n+14))),
(((16n+1,16n+9),(16n+5,16n+13)),((16n+3,16n+11),(16n+7,16n+15))))
...

But this method stops being useful to find the primes at the second step. After that, the primes are divided statistically equally among the odd sets due to the Dirichlet's theorem.

The naturals can be divided into a factorial tree instead of the binary one by a similar self similar process, producing a tree which sorts the primes into tightly concentrated groups. The efficiency of this tree increases by the depth of the tree. This can be proven.

Following demonstrates the sorting of the primes for the factorial tree of depth 6. 
(#)... sets of the form an+b containing the primes, where a,b are coprimes.
(##)... a,b coprime; b prime...only for depth 6

(n)
---
(2n,2n+1#)
---
((6n,6n+2,6n+4),
(6n+1#,6n+3,6n+5#))
---
(((24n,24n+6,24n+12,24n+18),
(24n+2,24n+8,24n+14,24n+20),
(24n+4,24n+10,24n+16,24n+22)),

((24n+1#,24n+7#,24n+13#,24n+19#),
(24n+3,24n+9,24n+15,24n+21),
(24n+5#,24n+11#,24n+17#,24n+23#)))
---
((((120n,120n+24,120n+48,120n+72,120n+96),
(120n+6,120n+30,120n+54,120n+78,120n+102),
(120n+12,120n+36,120n+60,120n+84,120n+108),
(120n+18,120n+42,120n+66,120n+90,120n+114)),

((120n+2,120n+26,120n+50,120n+74,120n+98),
(120n+8,120n+32,120n+56,120n+80,120n+104),
(120n+14,120n+38,120n+62,120n+86,120n+110),
(120n+20,120n+44,120n+68,120n+92,120n+116)),

((120n+4,120n+28,120n+52,120n+76,120n+100),
(120n+10,120n+34,120n+58,120n+82,120n+106),
(120n+16,120n+40,120n+64,120n+88,120n+112),
(120n+22,120n+46,120n+70,120n+94,120n+118))),

(((120n+1#,120n+25,120n+49#,120n+73#,120n+97#),
(120n+7#,120n+31#,120n+55,120n+79#,120n+103#),
(120n+13#,120n+37#,120n+61#,120n+85,120n+109#),
(120n+19#,120n+43#,120n+67#,120n+91#,120n+115)),

((120n+3,120n+27,120n+51,120n+75,120n+99),
(120n+9,120n+33,120n+57,120n+81,120n+105),
(120n+15,120n+39,120n+63,120n+87,120n+111),
(120n+21,120n+45,120n+69,120n+93,120n+117)),

((120n+5,120n+29#,120n+53#,120n+77#,120n+101#),
(120n+11#,120n+35,120n+59#,120n+83#,120n+107#),
(120n+17#,120n+41#,120n+65,120n+89#,120n+113#),
(120n+23#,120n+47#,120n+71#,120n+95,120n+119#))))
---
(((((720n,720n+120,720n+240,720n+360,720n+480,720n+600),
(720n+24,720n+144,720n+264,720n+384,720n+504,720n+624),
(720n+48,720n+168,720n+288,720n+408,720n+528,720n+648),
(720n+72,720n+192,720n+312,720n+432,720n+552,720n+672),
(720n+96,720n+216,720n+336,720n+456,720n+576,720n+696)),

((720n+6,720n+126,720n+246,720n+366,720n+486,720n+606),
(720n+30,720n+150,720n+270,720n+390,720n+510,720n+630),
(720n+54,720n+174,720n+294,720n+414,720n+534,720n+654),
(720n+78,720n+198,720n+318,720n+438,720n+558,720n+678),
(720n+102,720n+222,720n+342,720n+462,720n+582,720n+702)),

((720n+12,720n+132,720n+252,720n+372,720n+492,720n+612),
(720n+36,720n+156,720n+276,720n+396,720n+516,720n+636),
(720n+60,720n+180,720n+300,720n+420,720n+540,720n+660),
(720n+84,720n+204,720n+324,720n+444,720n+564,720n+684),
(720n+108,720n+228,720n+348,720n+468,720n+588,720n+708)),

((720n+18,720n+138,720n+258,720n+378,720n+498,720n+618),
(720n+42,720n+162,720n+282,720n+402,720n+522,720n+642),
(720n+72,720n+186,720n+306,720n+426,720n+546,720n+666),
(720n+90,720n+210,720n+330,720n+450,720n+570,720n+690),
(720n+114,720n+234,720n+354,720n+474,720n+594,720n+714))),
-
(((720n+2,720n+122,720n+242,720n+362,720n+482,720n+602),
(720n+26,720n+146,720n+266,720n+386,720n+506,720n+626),
(720n+50,720n+170,720n+290,720n+410,720n+530,720n+650),
(720n+74,720n+194,720n+314,720n+434,720n+554,720n+674),
(720n+98,720n+218,720n+338,720n+458,720n+578,720n+698)),

((720n+8,720n+128,720n+248,720n+368,720n+488,720n+608),
(720n+32,720n+152,720n+272,720n+392,720n+512,720n+632),
(720n+56,720n+176,720n+296,720n+416,720n+536,720n+656),
(720n+80,720n+200,720n+320,720n+440,720n+560,720n+680),
(720n+104,720n+224,720n+344,720n+464,720n+584,720n+704)),

((720n+14,720n+134,720n+254,720n+374,720n+494,720n+614),
(720n+38,720n+158,720n+278,720n+398,720n+518,720n+638),
(720n+62,720n+182,720n+302,720n+422,720n+542,720n+662),
(720n+86,720n+206,720n+326,720n+446,720n+566,720n+686),
(720n+110,720n+230,720n+350,720n+470,720n+590,720n+710)),

((720n+20,720n+140,720n+260,720n+380,720n+500,720n+620),
(720n+44,720n+164,720n+284,720n+404,720n+524,720n+644),
(720n+68,720n+188,720n+308,720n+428,720n+548,720n+668),
(720n+92,720n+212,720n+332,720n+452,720n+572,720n+692),
(720n+116,720n+236,720n+356,720n+476,720n+596,720n+716))),
-
(((720n+4,720n+124,720n+244,720n+364,720n+484,720n+604),
(720n+28,720n+148,720n+268,720n+388,720n+508,720n+628),
(720n+52,720n+172,720n+292,720n+412,720n+532,720n+652),
(720n+76,720n+196,720n+316,720n+436,720n+556,720n+676),
(720n+100,720n+220,720n+340,720n+460,720n+580,720n+700)),

((720n+10,720n+130,720n+250,720n+370,720n+490,720n+610),
(720n+34,720n+154,720n+274,720n+394,720n+514,720n+634),
(720n+58,720n+178,720n+298,720n+418,720n+538,720n+658),
(720n+82,720n+202,720n+322,720n+442,720n+562,720n+682),
(720n+106,720n+226,720n+346,720n+466,720n+586,720n+706)),

((720n+16,720n+136,720n+256,720n+376,720n+496,720n+616),
(720n+40,720n+160,720n+280,720n+400,720n+520,720n+640),
(720n+64,720n+184,720n+304,720n+424,720n+544,720n+664),
(720n+88,720n+208,720n+328,720n+448,720n+568,720n+688),
(720n+112,720n+232,720n+352,720n+472,720n+592,720n+712)),

((720n+22,720n+142,720n+262,720n+382,720n+502,720n+622),
(720n+46,720n+166,720n+286,720n+406,720n+526,720n+646),
(720n+70,720n+190,720n+310,720n+430,720n+550,720n+670),
(720n+94,720n+214,720n+334,720n+454,720n+574,720n+694),
(720n+118,720n+238,720n+358,720n+478,720n+598,720n+718)))),
--
((((720n+1#,720n+121#,720n+241##,720n+361#,720n+481#,720n+601##),
(720n+25,720n+145,720n+265,720n+385,720n+505,720n+625),
(720n+49#,720n+169#,720n+289#,720n+409##,720n+529#,720n+649#),
(720n+73##,720n+193##,720n+313##,720n+433##,720n+553#,720n+673##),
(720n+97##,720n+217#,720n+337##,720n+457##,720n+577##,720n+697#)),

((720n+7##,720n+127##,720n+247#,720n+367##,720n+487##,720n+607##),
(720n+31##,720n+151##,720n+271##,720n+391#,720n+511#,720n+631##),
(720n+55,720n+175,720n+295,720n+415,720n+535,720n+655),
(720n+79##,720n+199##,720n+319#,720n+439##,720n+559#,720n+679#),
(720n+103##,720n+223##,720n+343#,720n+463##,720n+583#,720n+703#)),

((720n+13##,720n+133#,720n+253#,720n+373##,720n+493#,720n+613##),
(720n+37##,720n+157##,720n+277##,720n+397##,720n+517#,720n+637#),
(720n+61##,720n+181##,720n+301#,720n+421##,720n+541##,720n+661##),
(720n+85,720n+205,720n+325,720n+445,720n+565,720n+685),
(720n+109##,720n+229##,720n+349##,720n+469#,720n+589#,720n+709##)),

((720n+19##,720n+139##,720n+259#,720n+379##,720n+499##,720n+619##),
(720n+43##,720n+163##,720n+283##,720n+403#,720n+523#,720n+643##),
(720n+73##,720n+187#,720n+307##,720n+427#,720n+547##,720n+667#),
(720n+91#,720n+211##,720n+331##,720n+451#,720n+571##,720n+691##),
(720n+115,720n+235,720n+355,720n+475,720n+595,720n+715))),
-
(((720n+3,720n+123,720n+243,720n+363,720n+483,720n+603),
(720n+27,720n+147,720n+267,720n+387,720n+507,720n+627),
(720n+51,720n+171,720n+291,720n+411,720n+531,720n+651),
(720n+75,720n+195,720n+315,720n+435,720n+555,720n+675),
(720n+99,720n+219,720n+339,720n+459,720n+579,720n+699)),

((720n+9,720n+129,720n+249,720n+369,720n+489,720n+609),
(720n+33,720n+153,720n+273,720n+393,720n+513,720n+633),
(720n+57,720n+177,720n+297,720n+417,720n+537,720n+657),
(720n+81,720n+201,720n+321,720n+441,720n+561,720n+681),
(720n+105,720n+225,720n+345,720n+465,720n+585,720n+705)),

((720n+15,720n+135,720n+255,720n+375,720n+495,720n+615),
(720n+39,720n+159,720n+279,720n+399,720n+519,720n+639),
(720n+63,720n+183,720n+303,720n+423,720n+543,720n+663),
(720n+87,720n+207,720n+327,720n+447,720n+567,720n+687),
(720n+111,720n+231,720n+351,720n+471,720n+591,720n+711)),

((720n+21,720n+141,720n+261,720n+381,720n+501,720n+621),
(720n+45,720n+165,720n+285,720n+405,720n+525,720n+645),
(720n+69,720n+189,720n+309,720n+429,720n+549,720n+669),
(720n+93,720n+213,720n+333,720n+453,720n+573,720n+693),
(720n+117,720n+237,720n+357,720n+477,720n+597,720n+717))),
-
(((720n+5,720n+125,720n+245,720n+365,720n+485,720n+605),
(720n+29##,720n+149##,720n+269##,720n+389##,720n+509##,720n+629#),
(720n+53##,720n+173##,720n+293##,720n+413#,720n+533#,720n+653##),
(720n+77#,720n+197##,720n+317##,720n+437#,720n+557##,720n+677##),
(720n+101##,720n+221#,720n+341#,720n+461##,720n+581#,720n+701##)),

((720n+11##,720n+131##,720n+251##,720n+371#,720n+491##,720n+611#),
(720n+35,720n+155,720n+275,720n+395,720n+515,720n+635),
(720n+59##,720n+179##,720n+299#,720n+419##,720n+539#,720n+659##),
(720n+83##,720n+203#,720n+323#,720n+443##,720n+563##,720n+683##),
(720n+107##,720n+227##,720n+347##,720n+467##,720n+587##,720n+707#)),

((720n+17##,720n+137##,720n+257##,720n+377#,720n+497#,720n+617##),
(720n+41##,720n+161#,720n+281##,720n+401##,720n+521##,720n+641##),
(720n+65,720n+185,720n+305,720n+425,720n+545,720n+665),
(720n+89##,720n+209#,720n+329#,720n+449##,720n+569##,720n+689#),
(720n+113##,720n+233##,720n+353##,720n+473#,720n+593##,720n+713#)),

((720n+23##,720n+143#,720n+263##,720n+383##,720n+503##,720n+623#),
(720n+47##,720n+167##,720n+287#,720n+407#,720n+527#,720n+647##),
(720n+71##,720n+191##,720n+311##,720n+431##,720n+551#,720n+671#),
(720n+95,720n+215,720n+335,720n+455,720n+575,720n+695),
(720n+119#,720n+239##,720n+359##,720n+479##,720n+599##,720n+719##)))))
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Kyl Justin Perez's profile photo
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Niyati bhuta

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This community is helping me a lot by solving my math problems thank you to give this new person to engineering a hope to solve maths
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Hugo A. G. V. Rosa's profile photoNiyati bhuta's profile photo
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We don't have ans to copy we have to solve questions and find out the ans on our own 
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John Cook

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Using generating functions, you can show that the number of ways to make change for an amount of n cents using 1,  5, 10, 25, and 50 cent coins is the coefficient of z^n in the series for the rational function below.

This post also gives a Scheme program for computing the same result.

http://www.johndcook.com/blog/2014/07/09/making-change/
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Bhairu Jirage's profile photowei zhang's profile photoarian vc's profile photoChristel Davies's profile photo
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"using generating functions, you can show..." Well, this is essentially what generating functions are in the first place; there isn't anything to show :P
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Linked is a great discussion about math after this question was posed:

"Q: 'I’ve always wanted to be an astronautical engineer, but I am horrible at math, but I’ve got lots of passion. Can this dream ever be a reality and where do I start?'"

I especially enjoyed it since I'm an older student returning to school, the discussion evolves into a debate about the current state of math and if it's even the right "language".  The transcript is just a portion, there are much longer videos in the page:

http://chycho.blogspot.com/2013/11/bill-nye-brian-greene-neil-degrasse.html
Math lovers and aficionados will find the following discourse both entertaining and informative. Below you will find the video and partial transcript of Arizona State University’s Origins Project’s Q&A segment from their ‘The...
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Jerzy Kaltenberg's profile photoBrian Schrom's profile photoEdward Corrigan's profile photoLayra Idarani's profile photo
 
I have to say that, as always, I find Greene's stance awfully positivist. Asking why something we created should be intrinsic to the universe begs the question of it being intrinsic to the universe and completely glosses over the process of making models based on observation..
If you want to describe something you don't string a bunch of random words together and hope that it makes a description, you look at the thing you want to describe and you choose words that describe it. It's basic empiricism. The thing we created describes the universe because we created it to describe the universe, not because the universe is necessarily mathematical at its core.
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The level of understanding of stats by powerful professionals is truly shocking.
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Chris Dye's profile photoEliza James's profile photo
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Things that look like derivatives:

If I had to pick a favorite thing from calculus, it would have to be the product rule. Mostly because all I do these days is look at things that resemble the product rule.
Recall that if we have functions f and g then the product rule tells us
(f * g) ' = f ' * g + f * g '
And from this rule spring things like the derivatives for polynomials or quotients, or the setup for partial integration.
But there are things that look a lot like the product rule that don't have anything to do with functions or derivatives.
For instance, suppose we have three n by n  matrices, A, B and C. We define the bracket [ , ] by saying that [A, B] = A * B - B * A.
Now we have the identity
[A, B * C] = [A, B] * C + B * [A, C]
Check this yourself.
This looks like the product rule, in that if we consider [A, D] to be the "derivative of D", then it has the exact same form as the product rule from calculus. The derivative of a product is the derivative of the first factor times the second factor plus the first factor times the derivative of the second factor. But this is all matrix multiplication, pure algebra with no calculus, no functions, no variables and no limits.
What other rules does this obey? Well, it obeys the addition rule:
[A, B + C] = [A, B] + [A, C]
as we expect, and the multiplication-by-constants rule, in that if we have a scalar c, then
[A, c * B] = c * [A, B]
We have a version of the constants rule, in that if c is a scalar and I is the identity matrix (1s on the upper-left-to-lower-right diagonal and 0s everywhere else) then
[A, c * I] = 0.
Note: We can't say that [A, D] is the derivative of D with respect to A, because [A, A] = 0, whereas the derivative of a thing with respect to itself ought to be 1.So this type of derivative means something quite different from the usual derivative, even if it obeys a bunch of the same rules.
Alas, because the matrices are not functions and the derivatives are not with respect to variables, there's no chain rule, which is the other interesting rule for derivatives.

The operation that takes a matrix D to [A, D] acts like a derivative with respect to the usual matrix product. But it also acts like a derivative with respect to the bracket! Explicitly, if we say that the bracket-product of B and C is [B, C], and the bracket-derivative of D is [A, D]. then the bracket-product rule for bracket-derivatives is
[A, [B, C] ] = [ [A, B], C] + [B, [A, C] ]
Check this yourself!
This last equation is sometimes called the Jacobi identity and has all sorts of fun consequences for algebra and geometry. Certainly we don't have anything like this for calculus derivatives.
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Leo Stein's profile photoLayra Idarani's profile photo
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There's no manifold at all. It's all done in terms of the algebra. We start with a set of "coordinate functions" but without any domain for them to be functions on, and go from there.
http://arxiv.org/abs/1109.1085 It's vaguely like some setups for more mainstream noncommutative geometry, but with a different interpretation of what the noncommutativity means geometrically, and more focus on differential rather than more classical-ish algebro-geometric aspects.
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John Cook

Discussion  - 
 
Michael Keith rewrote Edgar Allen Poe's poem The Raven to turn it into a mnemonic for the first 740 digits of pi.

More here: http://www.johndcook.com/blog/2014/07/03/pi-and-the-raven/
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patrick tinkham's profile photoThomas Smith's profile photo
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Carl Knox

Discussion  - 
 
Has anyone wondered the time taken
for the Minute & Hour hands to cross.
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Hugo A. G. V. Rosa's profile photoMike Aben's profile photo
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+Hugo A. G. V. Rosa I like yours too.  I love problems that have multiple ways to approach them.
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Circle DNA under US patent !
 
The Most Absurd Things People Have Claimed Ownership of in the History of Science

Intellectual property laws were originally formulated to protect the rights of writers, graphic designers, coders, inventors and other types of content creators. The laws that are meant to protect intellectual property, not unlike pretty much every other law known to man, are oftentimes taken advantage of. There are a number of reasons for this. Sometimes, it's about money. Other times, it's about politics and investors. Unfortunately, science isn't immune to any of these scenarios.

The following intellectual property issues show that: http://bit.ly/1qDz5Ga
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سمير شيخ's profile photosreeraj rajan's profile photoRichard Cronin's profile photoAustin Burch's profile photo
 
Jean DAVID -  I'm intrigued by the suggestions offered by Nobel prize-winning economist Joseph Stiglitz on the topic of intellectual property and current patent laws that "lock up" new innovations. Currently, innovations are purchased and smothered by Global Corporate Monoliths. 
http://opinionator.blogs.nytimes.com/2013/07/14/how-intellectual-property-reinforces-inequality/?_php=true&_type=blogs&_r=0
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The birthday paradox

With 23 people there is a 50% chance that 2 or more people will have the same birthday. If in a football match there are 23 people, that means that one in every 2 matches there are going to be 2 or more people that have the same birthday.
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Joel Tan's profile photo
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But it's quite amazing how math is sometimes counterintuitive
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