User mikhail gaichenkov - MathOverflow

How to draw Archimedean-Galileo spiral?

2013-01-04T18:08:22Z

It is known that some plane curves can be drawn with a tool. For instance, I heard at a web site that Archimedes created his spiral in the third century B.C. by fooling around with a compass and others.

Let’s however look at the spiral defined by the equation: $r'(\theta)^2+r(\theta)^2=\theta^2$, $r(\theta=0)=0$

I am looking for a method ( a tool) which could help to plot the spiral on paper ( I named it as Archimedean-Galileo spiral. For large $\theta$, the curve represents Archimedean spiral: $r=\theta$. When $\theta$ is small it transforms in Galileo spiral $r=\theta^2$) .

The spiral has a property that the junction point of the curve and the ray uniformly rotated in the origin coordinates when the junction point moves with uniform acceleration.

Do you think that there is a way to draw it without computer, but with other special curves (tools)?

I thought about the spiral of Theodorus, but I am not sure how the spiral of Theodurus is connected with the equation.

What is symmetry group of non-linear equation?

2013-02-10T16:13:58Z

I am not very sure if this is a proper question, but I'm trying to investigate what the area of math can offer in researching the differential equation in polar coordinates: $r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const or in other notation: $r'(\theta)^2+r(\theta)^2=\theta^2$, $r(\theta=0)=0$

Ancient method to study Archimedean spiral

2013-02-04T18:10:46Z

It is well-known the properties of Archimedean spiral ($\rho = k\phi$) which is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. For eg any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance. The properties were studied by Archimedes by means of math available at the time.

However, let’s look at the length of Archimedean spiral which requires more advanced tool:
$S(\phi) = \frac{k}{2} \left[ \phi \sqrt{1 + \phi^2} + \ln \left( \phi + \sqrt{1 + \phi^2}\right) \right]$, $lim_{\phi \to \infty} S’’(\phi)=k$ In other words we observe a movement with uniform acceleration along the Archimedean spiral ( the junction point of the line which rotates with constant angular velocity and Archimedean spiral moves along Archimedean spiral with uniform acceleration)

So, my question is to understand if there is a simple way to figure “uniform acceleration” out with a tool available at the times.?

Now, we can calculate the limit, but does that mean that Archimedes failed to discover the property?

I searched in Archimedes Palimpsest, but it is unclear if Archimedes has a tool to realise the uniform acceleration which appears for the curve he studied. http://en.wikipedia.org/wiki/Archimedes_Palimpsest

Radius of convergence to be proved more precisely (differential equation)

2013-01-15T18:34:20Z

There is a differential equation in polar coordinates: $r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const.

It is possible to get a solution which is a power series (see below). However, I am looking for an analytical proof that the radius of convergence of the power series is near $7/2$

Full series can be obtained by substituting the formal power series expansion into the equation and matching the terms at equal powers of t. Coefficients of the resulting power series $r(t)=\sum_{n=1}^\infty R_{2n} t^{2n}$

can be computed using the recurrent formulae $R_2=k/2$, $R_4=-k/32$

$R_{2n}=-\frac {R_{2n-2}} {8n} - \frac {1} {4kn} \sum_{i=1}^{n-2} (R_{2i}+4(i+1)(n-i)R_{2i+2})R_{2n-2i}$, $n=3,4…$

Please note that numerical calculation results in that the radius of convergence is near $7/2$, but I need to be more precise. So, I hope to find an analytical approach and a proof. Any help (ideas) are highly welcomed.

Necklaces and the generating function for inversions

2012-08-01T18:49:59Z

The problem of Necklaces is well-known, i.e "The number of fixed necklaces of length $n$ composed of $a$ types of beads $N(n,a)$" can be calculated: http://mathworld.wolfram.com/Necklace.html

Let us consider the limit $\lim_{n\to \infty}\prod_{p=1}^n N(p,a)$.
It is possible to show that the limit presents the result which looks like the generating function for inversion ( we may exclude one unimportant factor): $\frac {a^n} {n!}$ $\prod_{p=1}^n \frac {1-a^p} {1-a}$

For $n \to \infty$ we have $\prod_{p=1}^n N(p,a) \approx \frac {a^n} {n!}$ $\prod_{p=1}^n \frac {1-a^p} {1-a}$. Then, for eg please see theorem #1 http://www.cs.uwaterloo.ca/journals/JIS/VOL4/MARGOLIUS/inversions.pdf The generating function under theorem 1 looks like $\prod_{p=1}^n \frac {1-a^p} {1-a}$

So, a question appears, how to explain the influence of the symmetric group's properties for the particular case? In other words why and how the connection appears?

Comparison of two Necklaces at infinity.

2012-05-02T18:09:30Z

It is well know how to calculate the number of fixed necklaces of length $n$ composed of $a$ types of beads $N(n,a)$. http://mathworld.wolfram.com/Necklace.html

You can easily prove Lemma that for very large $n$ ( $n \to \infty$) the number of the necklaces looks like the generating function for the total number of inversion in combinatory ( I get rid of the factor, which includes power and factorial, this is not important in my question, I guess).

The Lemma seems to be true (if possible, please paraphrase it, the idea of Lemma should be clear). So, we have the well-know generating function.

On the other hand, it is known that Kendall-Mann numbers M(n) have the property of $n-1/2$

http://mathoverflow.net/questions/46368/the-property-of-kendall-mann-numbers

So, I just wonder, how we can explain the connection between some symmetry properties for the necklaces problem and the property for Kendal-Mann numbers (looks like a special case $n(n-1)/4$ http://mathworld.wolfram.com/RandomPermutation.html)

In other words I wonder what is behind the facts I tried to explain the problem I am thinking about in simple words. Thank you for any comments and any ideas.

Also, any reference for such cases are highly welcomed

Formulas with "1/2" from different branches of mathematics.

2012-02-29T16:11:59Z

It is well known that Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture about the location of the nontrivial zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2.

What I am looking for ( please do not think that it's very strange question) is to form a set of similar simple formulas from different parts of mathematics. So, any reference for general formulas or hypothesis which presents $1/2$ in a such simple way is highly welcomed.

Actually, I am trying to assosiate the "1/2" with a universal thing. To do this analysis more then few examples are needed from different branches of mathemathics. This is why I would like hear any references for the similar and simple formulas.

( I would like to exclude quantum world for a while, we meet $1/2$ there for sure).

Well, in other words I need final fundamental resutls which have a structure like "something +$1/2$" like in Riemann hypothesis. Thank you for any cooperation

Archimedes’ and Galileo’s spirals in one equation.

2011-12-16T09:34:07Z

The differential equation in polar coordinates $r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const, for large $t$ presents Archimedes’ Spiral and Galileo's spiral for $t \to 0$.

I find it surprisingly, however I failed to find the fact in literature (some more details available at http://mathoverflow.net/questions/54393/analytical-solutions-of-a-differential-equation-from-archimedes-spiral).

So, I wonder if you know where the first publication of the equation appeared and what the context of the research was? I guess it was related to optic research. Any reference and ideas are highly welcomed.

Thank you in advance.

Pólya's Random Walk Constants at infinity

2011-12-13T06:11:48Z

Let be the probability that a random walk on a d-D lattice returns to the origin. In 1921, Pólya proved that $p(1)=p(2)=1$ but $p(d)<1$ for $d>2$. http://mathworld.wolfram.com/PolyasRandomWalkConstants.html

I wonder what we can say about the probability for $d \to \infty$ In other words, if there is a closed formula or approximation which presents the limit of the probability for very big $d$? Does the limit exist?

Thank you in advance for any comments or approach to investigate the question.

Analytical solutions of a differential equation (from Archimedes' Spiral)

2011-02-05T10:22:15Z

There is a differential equation in polar coordinates: $r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const.

I've found that a) if $\phi \in (0,t)$, t is quite small, then $r(\phi) \approx k/2 *\phi^2$ b) if $\phi \in (c, + \infty)$, c is quite large positive number, then $r(\phi)=k\phi+o(\phi)$, as $\phi \to \infty$.

I am looking for approximation between a & b. How the Archimedes' Spiral transforms into other curve. Any ideas are highly welcomed.

A combinatorial proof for the property of KM numbers?

2011-08-29T12:12:54Z

Kendell-Mann numbers $M(n)$ ( see the sequence A000140 http://oeis.org/A000140 ) have the simple property: $M(n+1) \approx (n-1/2)M(n)$.

The property can be proved by different methods. For eg. http://mathoverflow.net/questions/46368/the-property-of-kendall-mann-numbers

What I am looking for is to find out if a combinatorial proof exists?

For eg. Let us start: Suppose we look at all the permutations of $n-1$ in the maximal grouping, then at all the permutation of $n$ in that maximal grouping; is there any simple way in which each permutation in the first set gives rise to $n$ permutations in the second? Better yet, a simple way in which about half the $n-1$-permutations give rise to $n$ $n$-permutations each, and the other half give rise to $n+1$ $n$-permutations each?

Any hints are higly welcomed. I hope that the combinatorial proof will makes the reason for the simple property more transparent.

Random Walk vs Branching process

2011-04-16T17:36:14Z

1) Let us consider the set of all $N!$ permutations of the $N$ elements ${1, 2, . . . ,N}$. In the random state, each permutation of these elements occurs with probability 1/N!. The probability $Pm(N)$ that the inversion number equals $m$ for a random permutation is well known as the Mahonian distribution in probability theory. The statistic is applied to some problems like mixing of diffusing particles. Here is just an example http://arxiv.org/abs/1010.2563

2) On the other hand,it is well known that simple ranadom walk is an example of a Markov chain. What I am looking for is to understand how the MAhonian distribution can be represented in terms of a branching process. In other words I am looking for a branching process which will represents the Mahonian distribution: the basic statistical characteristics of the inversion number including the average, the variance, and more generally, the probability distribution function.

Do you think that the representation is possible? Any ideas on the topic?

Bézier curve -4 model

2011-02-24T18:56:21Z

Let we want to connect A with B: $C(t)=C_0(1-t)^4+4C_1(1-t)^3t+6C_2(1-t)^2t^2+4C_3(1-t)t^3+C_4t^4,\qquad 0\le t\le1,$ where $C_0=(-c;0)$ $C_1=C_0+f(\cos\alpha;\sin\alpha)$ $C_2=(p;q)$ $C_3=C_4-g(\cos\beta;\sin\beta)$ $C_4=(c;0)$ $C_0=A,\; C_4=B,\; |AB|=h=2c$, h-chord Also, there is restriction: $f,g>0$ For known $k(0),k(1),k'_s(0),k'_s(1)$ we'll get four difficult equations ( here k(s) represents natural eq of the curve, $f,g,p,q$ - unknown). So, finnally the question is to find a solutions with restriction f,g>0.

Any hints or references are highly welcomed to simplify the system of eq.

Asymptotic expansion of an integral

2010-12-10T19:25:31Z

I am looking for an asymptotic expansion of J(n)

$J(n)=\frac {2} {\pi} \int_{0}^{\pi/n} \prod_{k=1}^n \frac {\sin kx} {\sin x} dx$, $n=2,3,4,\dots$ The first approximation is managed to get $F_1(n)=\frac {n!} {\sqrt{\pi A}}$, $A=n(n-1)(2n+5)/36$

Is a general expansion known for this?

The meaning of ratio of two sets

2011-02-23T15:34:33Z

I come accros that $(2n-1)!!$ is the number of permutations of 2n whose cycle type consists of n parts equal to 2; these are the involutions without fixed points (A).

Also, for each $n \in N$, let $f(n)$ is the number of subsets of set $[n]=\ {1,2,...,n}$. Then $f(n)=2^n$ (B)

I wonder about the understanding of the "physical" meaning of the ration: A/B? What could be the meaning of $\frac {(2n-1)!!} {2^n}$ in terms of sets?

Also, I noticed that the ratio is very closed to $\Gamma(n+1/2)$. This is why I am trying to understand the ratio.

Finally, the ratio is very closed to recurrence: $a_{n+1}=a_n(n-1/2)$

Subsequence of Kendall-Mann numbers

2011-01-30T11:42:51Z

Let $M(n)=6n!(2\pi)^{-1/2} n^{-3/2} (1-51/(50n)+225937/(98000n^2))+o(1/n^{7/2})$, as $n \to \infty$ http://www.combinatorics.org/Volume_7/PDF/v7i1r50.pdf An Asymptotic Expansion for the Number of Permutations with a Certain Number of Inversions, by Lane Clark, Corollary 2, M(n)=b(n,k), where k=n(n-1)/4) It provides an asymptotic expansion of Kendall-Mann numbers http://oeis.org/A000140

Now, I am looking for K(n) which are selected from A008302 http://oeis.org/A008302

1, 2, 101, 573, 250749, 2409581, 3727542188, 50626553988, 190418421447330, 3344822488498265, 24965661442811799655, 538134522243713149122, 7016726879654720868145951, 179258893496663655603046622, 3741163513205099419577155249749, 110520062557960229937518882573780, 3463861806507747133152591395448445166, 116168034299493079570776886639370348862, 5208610500144919495844659270116658786563011, 195497593731392734506873788376896938168104207

$n\equiv 2,3\pmod{4}$

What is the asymptotic expansion?

Not sure where to start from?

Applications of the property of Kendall-Mann numbers

2011-01-06T15:59:33Z

I am looking for an application of the Kendall-Mann sequence (KM) which uses the property $M(n+1)/M(n) = n - 1/2 + O(1/n)$ ($n \to \infty$) in science ( computer science ( sorting), physics, biology (genetics), etc) to use the property in some particular case with a specific error bound.

The property of Kendall-Mann numbers is proved, but not published in full details ( if you know the property, could you let me know the reference please).

To clarify the question: Kendall-Mann numbers M(n): the maximum number of permutations on n letters having the same number of inversions http://oeis.org/A000140 M(1)=1, M(2)=1, M(3)=2, M(4)=6, M(7)=22, M(8)=101… $M(n+1)/M(n)$ for n=1,… 29

$M(2)/M(1)=1$, $M(3)/M(2)=2$, $M(4)/M(3)=3$,...

1.00000000, 2.00000000, 3.00000000, 3.66666667, 4.59090909, 5.67326733, 6.69458988, 7.61939520, 8.57906801, 9.60953383, 10.6235009, 11.5884536, 12.5657349, 13.5817521, 14.5907723, 15.5704306, 16.5558579, 17.5656455, 18.5718445, 19.5585507, 20.5484134, 21.5549876, 22.5594838, 23.5501133, 24.5426559, 25.5473665, 26.5507683, 27.5438066, 28.5380914

It is proved that $M(n)=\frac {n!} {\sqrt{n(n-1)(2n+5)* \pi}} *(1+Q1/n+….)$, Q1- a constant. It presents the KM numbers. For more details please see http://mathoverflow.net/questions/46368/the-property-of-kendall-mann-numbers

The question is what to do with the fact? Any applications in science just to show how it can be applied with a precise error?

The property of Kendall-Mann numbers

2010-11-17T15:26:05Z

The sequence A000140 is studied http://oeis.org/A000140 (Kendall-Mann numbers: the maximum number of permutations on n letters having the same number of inversions ) and I am looking for a proof that M(n)/M(n-1)=n+1/2 when n= infinity, M(n) - max element in row n. If you have any ideas how to proove or disproove it, even the question is too hard, could you let me know in anyway. Modelling with Pari GP shows for n = 0 to 149 M(n)/M(n-1): 1.00000000, 1.00000000, 2.00000000, 3.00000000, 3.66666667, 4.59090909, 5.67326733, 6.69458988, 7.61939520, 8.57906801, 9.60953383, 10.6235009, 11.5884536, 12.5657349, 13.5817521, 14.5907723, 15.5704306, 16.5558579, 17.5656455, 18.5718445, 19.5585507, 20.5484134, 21.5549876, 22.5594838, 23.5501133, 24.5426559, 25.5473665, 26.5507683, 27.5438066, 28.5380914, 29.5416285, 30.5442887, 31.5389122, 32.5343930, 33.5371446, 34.5392804, 35.5350028, 36.5313400, 37.5335406, 38.5352923, 39.5318079, 40.5287792, 41.5305788, 42.5320411, 43.5291478, 44.5266018, 45.5281005, 46.5293394, 47.5268986, 48.5247283, 49.5259956, 50.5270586, 51.5249718, 52.5230999, 53.5241854, 54.5251073, 55.5233026, 56.5216716, 57.5226117, 58.5234188, 59.5218427, 60.5204088, 61.5212309, 62.5219434, 63.5205550, 64.5192845, 65.5200094, 66.5206430, 67.5194106, 68.5182772, 69.5189212, 70.5194882, 71.5183871, 72.5173696, 73.5179455, 74.5184560, 75.5174660, 76.5165477, 77.5170657, 78.5175276, 79.5166329, 80.5157998, 81.5162682, 82.5166882, 83.5158756, 84.5151165, 85.5155421, 86.5159256, 87.5151843, 88.5144896, 89.5148781, 90.5152297, 91.5145507, 92.5139127, 93.5142686, 94.5145921, 95.5139679, 96.5133798, 97.5137071, 98.5140058, 99.5134299, 100.512886, 101.513188, 102.513465, 103.512932, 104.512428, 105.512707, 106.512964, 107.512470, 108.512001, 109.512260, 110.512499, 111.512039, 112.511602, 113.511843, 114.512067, 115.511637, 116.511229, 117.511454, 118.511663, 119.511261, 120.510879, 121.511090, 122.511285, 123.510909, 124.510550, 125.510748, 126.510932, 127.510578, 128.510241, 129.510426, 130.510599, 131.510267, 132.509949, 133.510123, 134.510286, 135.509973, 136.509673, 137.509838, 138.509992, 139.509696, 140.509412, 141.509568, 142.509713, 143.509434, 144.509165, 145.509312, 146.509450, 147.509185, 148.508930,

n+0.5 for n = infinity gaichenkov@yandex.ru

Comment by Mikhail Gaichenkov

2013-03-28T17:50:10Z

@Kapiton Could you remove unrelated message please. You may see my profile in case contacts are required.

Comment by Mikhail Gaichenkov

2012-08-04T05:05:32Z

@Gerry Myerson Sorry, I deleted it and the account as well ( I failed to recover my account).

Comment by Mikhail Gaichenkov

2012-08-03T17:17:47Z

@ Gjergji Zaimi For $n \to \infty$ we have $\prod_{p=1}^n N(p,a) \approx \frac {a^n} {n!}$ $\prod_{p=1}^n \frac {1-a^p} {1-a}$. Then, for eg please see theorem #1:cs.uwaterloo.ca/journals/JIS/VOL4/MARGOLIUS/…

Comment by Mikhail Gaichenkov

2012-08-01T19:06:48Z

@Qiaochu Yuan Thank you, I've added the approximation of the limit

Comment by Mikhail Gaichenkov

2011-12-14T13:38:52Z

Thank you very much!Thank you for the reference. @ Noam D. Elkies Thank you very much, amazing example and good to see how OEIS helps. I am not sure why the result is related to the cubic lattice but not other type of lattice (hexagonal etc)? Probably I've asked a "stupid" question, but would it be possible to explain it for me please?

Comment by Mikhail Gaichenkov

2011-08-30T08:15:26Z

Well, let $I_n(k)$ - the number of permutations of $n$ objects with precisely $k$ inversions. We should study the property of the numbers $M(n)=I_n([n(n-1)/4])$.

Comment by Mikhail Gaichenkov

2011-02-23T17:11:49Z

Thank you, yes, I know that and got it with no difficulties. but the question is about the meaning in terms of sets.

Comment by Mikhail Gaichenkov

2011-02-06T07:48:19Z

Thank you, very intresting!

Comment by Mikhail Gaichenkov

2011-02-06T07:41:21Z

Thank you, the spiral curve which is the solution of the equation is of great intrest for me. The length is $d^2S/dt^2=C^2$, C is a const. The similar movement but along a line will be: $x(t)=ch(t)-1$, $y(t)=sh(2t)/4-t/2$, $t \in (0, +\infnty)$ ( the natural equation: $k=1/(2(s+b))*\sqrt{b/s}$, b - const). Do you find it intresting?

Comment by Mikhail Gaichenkov

2011-02-02T19:30:16Z

$2!,3!,6!,7!...$ will be $(2n+(-1-(-1)^n)/2)!$

Comment by Mikhail Gaichenkov

2011-01-30T15:44:44Z

Sorry, I missed n!

Comment by Mikhail Gaichenkov

2011-01-08T09:46:28Z

Well, could you advise me what the question to ask about the property please? I would not like to ask odd questions. If the property is uknown, then I would publish it in a paper adding a particular case with a specific error bound. I know the property, but I know nothing where to apply it

Comment by Mikhail Gaichenkov

2010-12-03T19:29:47Z

error from n=2 to 18: 0,595769122 0,250044867 0,084111637 0,066041844 0,068568453 0,054024687 0,037563243 0,032785381 0,032782518 0,028992661 0,023807067 0,021830376 0,021573067 0,019891303 0,017496158 0,016414384 0,016145111

Comment by Mikhail Gaichenkov

2010-12-03T19:26:10Z

$M(n)=n!(1+O (n^{-1+\epsilon}))/ \sqrt{A\pi}$. $ A = n(n-1)(2n+5)/36$

Comment by Mikhail Gaichenkov

2010-11-27T15:43:27Z

Good point to think too. Stein’s method is applied to show that W (inversions of a permutation) satisfies a central limit theorem with error rate $n^{-1/2}$. On the other hand, the basic complex analysis and classical analysis with Euler Product representation of Sin function should present an error rate for the particular case too, and the M(n) with O(1). The error is being investigated. Any comments at this side are highly welcomed.