One of the last **Alan Turing**‘s work is about Fibonacci numbers in nature:

One of a number of problems [Alan Turing] was trying to solve was the appearence of Fibonacci numbers in the structure of plants.

^{[1]}Swinton J. (2004). Watching the Daisies Grow: Turing and Fibonacci Phyllotaxis, Alan Turing: Life and Legacy of a Great Thinker, 477-498. DOI: 10.1007/978-3-662-05642-4_20 (pdf)

The problem was knwon as the *Fibonacci phyllotaxis*, and we can state it in this way:

the spiral shapes on the heads of sunflowers seemed to follow the Fibonacci sequence, prompting [Turing’s] proposal that by studying sunflowers we might better understand how plants grow

Turing wrote his interest in a letter to the zoologist **JZ Young**:

About the point (iii) Turing wrote in another letter:

Our new machine is to start arriving on Monday. I am hoping to do something about ‘chemical embyology’. In particular I think I can account for the appearence of Fibonacci numbers in connection with fir-cones.

^{[2]}See note 1.

In 2012, **Jonathan Swinton**, during the *Manchester Science Festival*, announced the results of the great experiment about the Turing’s sunflower:

And finally results are published last year on *Royal Society open science*^{[3]}Swinton, J., Ochu, E., & MSI Turing’s Sunflower Consortium. (2016). Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment. *Royal Society open science*, 3(5), 160091. doi:10.1098/rsos.160091: a great result for Alan Turing and *citizen science*!

References

1. | ↑ | Swinton J. (2004). Watching the Daisies Grow: Turing and Fibonacci Phyllotaxis, Alan Turing: Life and Legacy of a Great Thinker, 477-498. DOI: 10.1007/978-3-662-05642-4_20 (pdf) |

2. | ↑ | See note 1. |

3. | ↑ | Swinton, J., Ochu, E., & MSI Turing’s Sunflower Consortium. (2016). Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment. Royal Society open science, 3(5), 160091. doi:10.1098/rsos.160091 |

Enjoy!

Interestingly, Maxwell’s equations have been drastically reduced into a language of differential geometry. These four sets of equations which perfectly describe the theory of electromagnetism have been reduced to a set of two equations which lay the foundations of most new theories in the physical world today.

The most revolutionary quantum leap in the history of theoretical physics is the birth of general relativity and quantum eld theory (the standard model of elementary particle). These theories describe nature better than any physicist ever had at hand, although they have not been uni ed into a coherent picture of the world. One of the main ingredients of these theories is differential geometry. Euclidean geometry was abandoned in favour of differential geometry and classical field theories had to be quantized.

Maxwell’s equations in the language of differential geometry lead to a generalization to these new theories, and these equations are a special case of Yang-Mills equations (beyond the scope of this essay), which is also gauge invariant and describe not only electromagnetism but also the strong and weak nuclear forces. This essay is nothing but the tip of the iceberg.

(from *Maxwell’s Equations in Terms of Differential Forms* (pdf) by **Solomon Akaraka Owerre**)

**Read also**: The poem of the Maxwell’s equations in pdf written by **Lynda Williams**.

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A distributed system is a set of autonomous computer that comunicate in a network in order to reach a certain goal. So a maximal independent set (MIS) is a distributed system’s subject. But, what we intend for MIS?

In graph theory, a maximal independent set or maximal stable set is an independent set that is not a subset of any other independent set.

Some example of MIS are in the graph of cube:

You can see that every maximal independent set is constituted by point that aren’t adjacent.

The goal of maximum independet set problem is find the maximum size of the maximal independent set in a given graph or network. In other words the problem is the search of the leaders in a local network of connected processors, and forleaderwe intend an active node connected with an inactive node. This problem is a NP-problem.

Following Afek, Alon, Barad, Hornstein, Barkai and Bar-Joseph ^{[1]}Afek, Y., Alon, N., Barad, O., Hornstein, E., Barkai, N., &amp;amp; Bar-Joseph, Z. (2011). A Biological Solution to a Fundamental Distributed Computing Problem Science, 331 (6014), 183-185 DOI: 10.1126/science.1193210,

no methods has been able to efficiently reduce message complexity without assuming knowledge fo the number of neighbours.

But a similar network occurs in the precursors of the fly’s sensory bristles, so researchers idea is to use data from this biological network to solve the starting computational problem!

Such system is called *sensory organ precursors*, SOP.

There are a lot of similarities between MIS and SOP:

*the selection of a particular cell as a SOP is a random event governed by an underlying stochastic process*^{[2]}**P. Simpson**,*Curr. Opin. Genet. Dev.***7**, 537 (1997)^{[3]}**M. E. Fortini**,*Dev. Cell.***16**, 633 (2009);*similar to computational requirements SOP selection is probably constrained in time because the default of all cluster cells is to become SOPs unless they are inhibited*^{[4]}**B. Castro***et al.*,*Development***132**, 3333 (2005);*in computational algorithms*^{[5]}**M. Luby**,*SIAM J. Comput.***15**, 1036 (1986)^{[6]}**N. Alon**,**L. Babai**,**A. Itai**,*J. Algorithms***7**, 567 (1986)*processors send messages only when they propose their candidacy to become leaders, thus reducing communication complexity*.

In particular this properties are developed in biological network studied using Delta and Notch proteins: in this way a cell selected as SOP *inhibits his neighbors* obtaining a situation similar to the first figure:

Researchers so try to describe the biological network:

We assumed a collection of identical processors placed at nodes of an arbitrary synchronous communication network. Nodes can only broadcast one-bit messages. A message broadcasted by a node reaches all of its neighbors that are still active in the algorithm. In each round, a processor can only tell whether or not a message was sent to it. When a processor receives a message, it cannot tell which of its neighboring processors sent it, and it cannot count the number of messages received in a round.

That in terms of algorithms became:

where $n$ is the number of nodes, $D$ an upper bound *on the number of neighbors any node can have*, $M$ a parameter setted to 34, and every node has a probability $p_i$ tosend a message to his neighbors.

In order to select the model Yehuda Afek and collegues confrount experimental data collected from 10 pupas:

between simulated results:

And at the end they can conclude that:

the only way the algorithm may err is by terminating while leaving some nodes that are not in A and are also not connected to nodes in A. Next, we show that when the algorithm terminates all nodes are, with high probability, either in A or connected to a node in A, which solves the MIS problem.

References

1. | ↑ | Afek, Y., Alon, N., Barad, O., Hornstein, E., Barkai, N., &amp;amp; Bar-Joseph, Z. (2011). A Biological Solution to a Fundamental Distributed Computing Problem Science, 331 (6014), 183-185 DOI: 10.1126/science.1193210 |

2. | ↑ | P. Simpson, Curr. Opin. Genet. Dev. 7, 537 (1997) |

3. | ↑ | M. E. Fortini, Dev. Cell. 16, 633 (2009) |

4. | ↑ | B. Castro et al., Development 132, 3333 (2005) |

5. | ↑ | M. Luby, SIAM J. Comput. 15, 1036 (1986) |

6. | ↑ | N. Alon, L. Babai, A. Itai, J. Algorithms 7, 567 (1986) |

The Rieman hypothesis was stated following the 1859 Riemann’s paper *On the Number of Primes Less Than a Given Magnitude*. This is the begin of the paper ^{[1]}Translated by **David R. Wilkins** (pdf):

I believe that I can best convey my thanks for the honour which the Academy has to some degree conferred on me, through my admission as one of its correspondents, if I speedily make use of the permission thereby received to communicate an investigation into the accumulation of the prime numbers; a topic which perhaps seems not wholly unworthy of such a communication, given the interest which

GaussandDirichlethave themselves shown in it over a lengthy period.

For this investigation my point of departure is provided by the observation of Euler that the product

\[\prod \frac{1}{1-\frac{1}{p^s}} = \sum \frac{1}{n^s}\] if one substitutes for $p$ all prime numbers, and for $n$ all whole numbers. The function of the complex variable $s$ which is represented by these two expressions, wherever they converge, I denote by $\zeta (s)$. Both expressions converge only when the real part of $s$ is greater than 1; at the same time an expression for the function can easily be found which always remains valid.

The Riemann zeta function is connected to the prime numbers distribution, in particular Riemann argued that all of its non trivial zeros ^{[2]}In this case, for trivial zero I intend a negative even integer number. have the form $z = \frac{1}{2} + bi$, where $z$ is complex, $b$real,$i = \sqrt{-1}$. There’s also a general form of the zeros: $z = \sigma + bi$, where $\sigma$ belong to the critical strip (see below and the image at the right).

In the story of the search of the zeta-zeros, **Hugh Montgomery** has an important part ^{[3]}Montgomery, Hugh L. “The pair correlation of zeros of the zeta function.” In *Proc. Symp. Pure Math*, vol. 24, pp. 181-193. 1973. (pdf): in 1972 he investigated the distance between two zeta-zeros, finding a function of this difference. After this paper, in 1979, with **Norman Levinson** ^{[4]}Levinson, Norman, and Hugh L. Montgomery. “Zeros of the derivatives of the Riemann zeta-function.” *Acta mathematica* 133, no. 1 (1974): 49-65. doi:10.1007/BF02392141 he established some others zeta properties, investigating in particular the zeros of zeta derivatives. Obviosly he first of all proofed an equivalence relation between the zeros of Riemann zeta function and the zeros of the derivatives: in particular also these zeros belong to the critical strip, $0 < \sigma < \frac{1}{2}$.

The analitical research around zeta-zeros is not the only way: the first was **Lehmer** (1956 and 1957) who performed the first computational attempt in order to proof the hypothesis. An example of this kind of researches is given by **Richard Brent** ^{[5]}Brent, Richard P. “On the zeros of the Riemann zeta function in the critical strip.” *Mathematics of Computation* 33, no. 148 (1979): 1361-1372. doi:10.1090/S0025-5718-1979-0537983-2: in his work he try to evaluate Riemann zeta using the Gram points, that are the points in which the zeta change its sign ^{[6]}We can also define a Gram point as the value of $b$ such that

\[\zeta \left ( \frac{1}{2} + bi \right ) = Z(b) e^{-i \theta (b)}\]
is a non-zero real.

In the equation

\[\theta(t) \sim \frac{t}{2}\log \frac{t}{2\pi} – \frac{t}{2} – \frac{\pi}{8}+\frac{1}{48t}+ \frac{7}{5760t^3}+\cdots\]
is the Riemann-Siegel theta function and $Z (t)$is the Hardy Z-function.. Brent focused his research on the first 70000000 Gram blocks, veryfing the hypothesis.

But there’s another approach to the problem: physics. In the end of 1990s **Alain Connes** ^{[7]}Connes, Alain. “Trace formula in noncommutative geometry and the zeros of the Riemann zeta function.” *Selecta Mathematica, New Series* 5, no. 1 (1999): 29-106. doi:10.1007/s000290050042 (arXiv) proofed the link of Rieman hypotesis with quantum chaos.

Quantum chaos studies chaotic classical dynamical systems using quantum laws. In particular Connes found a particular chaotic system in which quantum numbers are prime numbers and the energy levels of the system correspond to the zeta-zeros on the critical line $\sigma = \frac{1}{2}$. In physics it could be the better (but not only) suspect to resolve the hypothesis.

So it is interesting to observe the recently paper ^{[8]}Thanks to **Roberto Natalini** for the suggestions about this second part of the article published on *Physical Review Letters* ^{[9]}Bender, Carl M., Dorje C. Brody, and Markus P. Müller. “Hamiltonian for the zeros of the Riemann zeta function.” Physical Review Letters 118, no. 13 (2017): 130201. doi:10.1103/PhysRevLett.118.130201: starting from the Hilbert-Pólya conjecture, **Carl Bender**, **Dorje Brody** and **Markus Müller** proposed a new non-hermitian hamiltonian $\hat H$ which eigenvalues exactly correspond to the non-trivial zeros of the Riemann’s function.

This result isn’t conclusive. Indeed the three authors write in the conclusion:

We hope that further analysis of the properties of $\hat H$, such as identifying its domain and establishing its self-adjointness, will prove the reality of the eigenvalues, and thus the veracity of the Riemann hypothesis

At the other hand, **Jean Bellissard** (Georgia Tech) writes a breaf comment about Bender, Brody and Müller’s paper, finding some problems:

- the space of some mathematical objects is not well defined;
- the operator $\hat p$ proposed in the paper doesn’t admits a selfadjoint extension, so it hasn’t the characteristics required by the authors;
- the eigenvalues don’t occur on the line of the solution of the Riemann hypotesis.

As attractive this idea looks, it does not hold when checking the analysis part of the problem.

Another interesting analysis of the same paper is written by **Alessandro Zaccagni** for the italian site *MaddMaths!* (Google translation).

So we can only wait for further results. In the meanwhile I suggest the following review: *Physics of the Riemann Hypothesis* by **Daniel Schumayer** and **David A. W. Hutchinson**.

Read also: Wikipedia, The Clay Mathematics Institute, MathWorld

References

1. | ↑ | Translated by David R. Wilkins (pdf) |

2. | ↑ | In this case, for trivial zero I intend a negative even integer number. |

3. | ↑ | Montgomery, Hugh L. “The pair correlation of zeros of the zeta function.” In Proc. Symp. Pure Math, vol. 24, pp. 181-193. 1973. (pdf) |

4. | ↑ | Levinson, Norman, and Hugh L. Montgomery. “Zeros of the derivatives of the Riemann zeta-function.” Acta mathematica 133, no. 1 (1974): 49-65. doi:10.1007/BF02392141 |

5. | ↑ | Brent, Richard P. “On the zeros of the Riemann zeta function in the critical strip.” Mathematics of Computation 33, no. 148 (1979): 1361-1372. doi:10.1090/S0025-5718-1979-0537983-2 |

6. | ↑ | We can also define a Gram point as the value of $b$ such that \[\zeta \left ( \frac{1}{2} + bi \right ) = Z(b) e^{-i \theta (b)}\] is a non-zero real. In the equation \[\theta(t) \sim \frac{t}{2}\log \frac{t}{2\pi} – \frac{t}{2} – \frac{\pi}{8}+\frac{1}{48t}+ \frac{7}{5760t^3}+\cdots\] is the Riemann-Siegel theta function and $Z (t)$is the Hardy Z-function. |

7. | ↑ | Connes, Alain. “Trace formula in noncommutative geometry and the zeros of the Riemann zeta function.” Selecta Mathematica, New Series 5, no. 1 (1999): 29-106. doi:10.1007/s000290050042 (arXiv) |

8. | ↑ | Thanks to Roberto Natalini for the suggestions about this second part of the article |

9. | ↑ | Bender, Carl M., Dorje C. Brody, and Markus P. Müller. “Hamiltonian for the zeros of the Riemann zeta function.” Physical Review Letters 118, no. 13 (2017): 130201. doi:10.1103/PhysRevLett.118.130201 |

As you know, the $\pi$ is defined as the ratio of the circumference to its diameter. This number, which is transcendental, was, apparently, known since ancient times. There are, in fact, some Egyptologists who believe that $\pi$, or perhaps $\tau = 2 \pi$, was known to them since the age og the Giza’s pyramid, built between 2589 and 2566 BC, because the relationship between the perimeter and the height is 6.2857.

There are no explicit proof of the fact that, at the time, Egyptian mathematics became aware of a number such as $\pi$, however, between 600 and 1000 years later on a Babylonian tablet it is geometrically established the first value of $\pi$: $25/8 = 3.1250$. From documents written more or less in the same period it can be deduced that also the Egyptians calculated the value of $\pi$, obtaining $(16/9)^2 \simeq 3.1605$.

Indian mathematics, however, seems a little late: in 600 BC on *Shulba Sutras*, it is calculated for the $\pi$ value like $(9785/5568)^2 \simeq 3.088$, which will be updated later in 150 BC as $\sqrt{10} \simeq 3.1622$, which is a value much closer to the value calculated by the Egyptians.

A good approximation of $\pi$ value is in *Mishnat ha-Middot*, a geometric treatise by **Rabbi Nehemiah**: $3 + 1/7 \simeq 3.14286$.

The approximation, however, the most amazing not only for accuracy but also for the method is that proposed by **Archimedes**, the mathematician who invented the method of polygons in order to calculate $\pi$, a constant that for a millennium became known simply as the *Archimedes’ constant*.

He simply calculated the perimeter of polygons inscribed and circumscribed in a circle, thus obtaining a lower and an upper estimate of the value of the constant:

\[223/71 < \pi < 22/7\]
or

\[3.1408 < \pi < 3.1429\]
It’s clear that his method of calculation is very modern and above suggests that Archimedes was well aware of the transcendental nature of the constant, which could be known only through approximations.

Today $\pi$ is known to 5 trillion digits and if you try to type the symbol $\pi$ on modern scientific calculators, the value they give you is, to the first decimal place, 3.14159265…

Calculating $\pi$ digits is like a mathematical art, that combines the technique of iterative algorithms with the convergent series. For example the achievment of 5 trilions of digits it could be possible thanks to * Chudnovsky formula*:

\[\frac{1}{\pi} = 12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}}\]

\[\pi = \sum_{k = 0}^{\infty} \frac{1}{16^k}

\left( \frac{4}{8k + 1} – \frac{2}{8k + 4} – \frac{1}{8k + 5} – \frac{1}{8k + 6}\right)\] and the

\[\pi=\frac{1}{2^6}\sum_{n=0}^\infty \frac{(-1)^n}{2^{10n}} \left (-\frac{2^5}{4n+1}-\frac{1}{4n+3}+\frac{2^8}{10n+1}\right.\] \[\left.-\frac{2^6}{10n+3}-\frac{2^2}{10n+5}-\frac{2^2}{10n+7}+\frac{1}{10n+9}\right )\] It’s interesting to observe that the previous series are based, in some way, to the series developed since 1914 by

\[\frac{1}{\pi} = \frac{2 \sqrt{2}}{9801} \sum_{k=0}^{\infty} \frac{(4k!) (1103 + 26390k)}{(k!)^4 396^{4k}}\] Other mathematicians developed other series, but one of the most curious is the

\[\pi = 4 \sum_{k=1}^{500000} \frac{(-1)^{k-1}}{2k – 1}\] The last series that I would propose you is dued by

\[\frac{\pi}{2} = \sum_{i=0}^\infty \frac{i!}{(2i+1)!!}\] where $k!!$ is the product of all the odd numbers up to $k$.

Chudnovsky D.V. (1989). The Computation of Classical Constants, Proceedings of the National Academy of Sciences, 86 (21) 8178-8182. DOI: 10.1073/pnas.86.21.8178

Bailey D., Borwein P. & Plouffe S. (1997). On the rapid computation of various polylogarithmic constants, Mathematics of Computation, 66 (218) 903-914. DOI: 10.1090/S0025-5718-97-00856-9

Fabrice Bellard. Computation of the $n-th$ digit of pi in any base in $O(n^2)$

Borwein J.M. & Borwein P.B. (1988). Ramanujan and Pi, Scientific American, 258 (2) 112-117. DOI: 10.1038/scientificamerican0288-112 (pdf)

Borwein J.M., Borwein P.B. & Dilcher K. (1989). Pi, Euler Numbers, and Asymptotic Expansions, The American Mathematical Monthly, 96 (8) 681. DOI: 10.2307/2324715 (pdf)

Rabinowitz S. & Wagon S. (1995). A Spigot Algorithm for the Digits of π, The American Mathematical Monthly, 102 (3) 195. DOI: 10.2307/2975006 (pdf)

** Pi-links**: Pi su en.wiki | Digits of Pi | Pi to 1,000,000 places

Neural network is one of the most powered method to analize data. It can be use in most research subject, for example in astronomy: in this case, we can use NNs to examine astronomical images or also the red shift effect. For example, in 2003 **Jorge Núñez** (Universitad de Barcelona) and **Jorge Llacer** (EC Engineering Consultants LLC) published a paper ^{[1]}Núñez J. & Llacer J. (2003). Astronomical image segmentation by self-organizing neural networks and wavelets, Neural Networks, 16 (3-4) 411-417. DOI: 10.1016/S0893-6080(03)00011-X in which they describe the develop of an algorithm to study *astronomical image segmentation that uses a self-organizing neural network as basis*. In their work, the scientists examine the separation between some stars and also a Saturn’s image: the alghoritm seems quite robust against noise and fragmentation.

In the same year, a group of italian astronomers published a review ^{[2]}Tagliaferri R., Longo G., Milano L., Acernese F., Barone F., Ciaramella A., Rosa R.D., Donalek C., Eleuteri A., Raiconi G. & Sessa S. (2003). Neural neZtworks in astronomy, Neural Networks, 16 (3-4) 297-319. DOI: 10.1016/S0893-6080(03)00028-5 (pdf) of the models used in astronomy and examined some data used by **AstroNeural** collaboration. Finally in 2004 a collaboration between researchers in Italy, Germany and France perform an application of NNs to redshift calculations ^{[3]}Vanzella E., Cristiani S., Fontana A., Nonino M., Arnouts S., Giallongo E., Grazian A., Fasano G., Popesso P. & Saracco P. & (2004). Photometric redshifts with the Multilayer Perceptron Neural Network: Application to the HDF-S and SDSS, Astronomy and Astrophysics, 423 (2) 761-776. DOI: 10.1051/0004-6361:20040176 (arXiv).

Now you can quest: *What is a neural network?*

It is a system of *node* and *link*. In the structure we can eventually distinguished between different layers, and every layers are fully connected. Such networks are usually called <e,>Multilayer Perceptron NN, and the interaction between every node is modelled by the following funcition

\[f(x) = k \sum_i w_i g_i(x)\]
where $k$ is some predefined function, $g_i$ is a vector of functions.

A more simple model is

\[n_j = h \sum_i w_{ij} z_i\]
where $n_j$ is the $j$-th node, $h$ is a constant, $w_{ij}$ the weights of the link, $z_i$ the previous node.

There is also the self-organizing maps, in which every node, or neurons, are in competition with each other:

It’s possible implement NNs with fuzzy logic and perform image segmentation, object detection, noise identification (for example in detection of gravitational waves), estimation of redshifts. The great advantage using NNs in astronomy is, however, to perform complex calculations with usually alghoritms. For example, we can see the plots in Vanzella’s et al. paper^{(3)}. In the work, astronomers used a *Multilayer Perceptron* NN, using real data from Hubble Deep Field North dataset.

This is the first comparison between

spectroscopic redshift in the HDF-S and the neural redshift using the colors as an input pattern. The training has been done on the HDF-N spectroscopic sample, the estimation of the redshift for each object is the median of 100 predictions and the error bars represent 1- interval. Open circles represent objects with unreliable photometry and triangles are objects with uncertain spectroscopic redshift.

And here there is a comparison after adding informations.

One of the most important tool in NNs is the ability of these type of networks to learn from data.

The last plot that I propose you is the *redshift distribution of the spectroscopic sample*:

And now some tutorials: **How to build a brain with neural networks** and, for all geek, **Evolving Neural Networks with SharpNEAT**: part 1 and part 2. And, in conclusion, read the following post about NNs and tea leaves.

Enjoy!

Gianluigi Filippelli

References

1. | ↑ | Núñez J. & Llacer J. (2003). Astronomical image segmentation by self-organizing neural networks and wavelets, Neural Networks, 16 (3-4) 411-417. DOI: 10.1016/S0893-6080(03)00011-X |

2. | ↑ | Tagliaferri R., Longo G., Milano L., Acernese F., Barone F., Ciaramella A., Rosa R.D., Donalek C., Eleuteri A., Raiconi G. & Sessa S. (2003). Neural neZtworks in astronomy, Neural Networks, 16 (3-4) 297-319. DOI: 10.1016/S0893-6080(03)00028-5 (pdf) |

3. | ↑ | Vanzella E., Cristiani S., Fontana A., Nonino M., Arnouts S., Giallongo E., Grazian A., Fasano G., Popesso P. & Saracco P. & (2004). Photometric redshifts with the Multilayer Perceptron Neural Network: Application to the HDF-S and SDSS, Astronomy and Astrophysics, 423 (2) 761-776. DOI: 10.1051/0004-6361:20040176 (arXiv) |