## Profile

eden andes

136 followers|37,237 views

AboutPostsPhotosVideos

## Stream

### eden andes

Shared publicly -still waiting in oncology for an assessment and treatment plan. more slides. 99¢store toys from my hunney :*)

5 photos

1

Add a comment...

### eden andes

Shared publicly -science is magic

Science Creates A Quantum Link Between Photons that Don't Exist at the Same Time

Physicists have created a quantum link between photons that don’t exist at the same time. You read that right, scientists have entangled two photons together that didn’t exist at the same time. Yep, it’s officially official – quantum mechanics is a freaky weird place.

To learn more about this crazy experiment, see http://goo.gl/M8KV78

Image credit: NASA/Sonoma State University/Auore Simonnet

Physicists have created a quantum link between photons that don’t exist at the same time. You read that right, scientists have entangled two photons together that didn’t exist at the same time. Yep, it’s officially official – quantum mechanics is a freaky weird place.

To learn more about this crazy experiment, see http://goo.gl/M8KV78

Image credit: NASA/Sonoma State University/Auore Simonnet

1

Add a comment...

### eden andes

Shared publicly -The Right to Know is the Right to Choose!

Shop Smart!

For a full list of companies that are against GMO Labeling visit: http://goo.gl/R8Cw4v

Shop Smart!

For a full list of companies that are against GMO Labeling visit: http://goo.gl/R8Cw4v

1

Add a comment...

### eden andes

Shared publicly -were too busy defending corporations, to defend something as low on the agenda as survival

2

Add a comment...

In her circles

209 people

### eden andes

Shared publicly -Eden Andes Plus 1this post

1

Hi I saw your comments from a few months ago asking where I was and if I was ok yep I'm fine thanks just got out of using google plus not so easy on I pad I still use Flickr if you are on there let me know thank you so much al Shep

Add a comment...

### eden andes

Shared publicly -Eden +1 this post

Pinto de mi memoria las impresiones de mi infancia...

Edvard Munch

+10,000,000 Artists & Artlovers

#buenosaires #laboca

Edvard Munch

+10,000,000 Artists & Artlovers

#buenosaires #laboca

1

Add a comment...

### eden andes

Shared publicly -great post via +James Hastings

totally cool ! it's like a music of Fractals

totally cool ! it's like a music of Fractals

**The Fractal Music of Per Nørgård**

Many of the musical pieces written by the Danish composer

**Per Nørgård**(1932-) make prominent use of a musical sequence that Nørgård calls the

*Uendelighedsrækken,*or

**infinity series**.

The infinity series is based on an infinite sequence of integers {s(n)}, defined recursively by the conditions s(0) = 0, s(n) = - s(n/2) if n is even, and s(n) = s((n-1)/2) + 1 if n is odd. The sequence can be rendered as musical notes in various ways; for example, an integer

*r*occurring in the sequence may be interpreted to refer to the note

*r*semitones (half steps) above the C above middle C. The illustration shows the first 20 notes of the sequence interpreted in this way.

Nørgård has stated that he discovered this series rather than inventing it. The infinity series has some remarkable mathematical properties; for example, if one takes every fourth note in the sequence (the 4th, 8th, 12th notes in the sequence, and so on) the net effect is to transpose the music up by one tone. This property of self-similarity is a well-known feature of

**fractals**. Remarkably, Nørgård invented the infinity series in 1959, sixteen years before the introduction of the word

*fractal*by Benoît Mandelbrot in 1975.

The recent paper

*Notes and Note-Pairs in Noergaard's Infinity Series*by Christopher Drexler-Lemire and Jeffrey Shallit (http://www.arxiv.org/abs/1402.3091) proves several new results about the infinity series. For example, they determine exactly which pairs of notes can appear consecutively in the sequence, and they prove that if a sequence of notes repeats, then the repetitions are widely separated. For example, the second bar (measure) shown in the picture is repeated later, but not until the fifth bar.

Another result in the paper discusses the problem of finding all the positions at which a given note appears. For example, the C above middle C appears at positions 0, 5, 10, 17, and in infinitely many other places. Given a binary representation of one of these positions (for example, 10001 for 17), how difficult would it be computationally to determine whether the note occurring at this position in the sequence is this particular C?

The answer to this question turns out to be related to the

**Chomsky hierarchy**of languages, named after the linguist

**Noam Chomsky**(1928-). Given a note, such as the C above middle C, we regard the collection of all binary positions of that C in the sequence as words in a “language”, where each word is a string of zeros and ones. For example, the language corresponding to the aforementioned C is an infinite set of words, starting {0, 101, 1010, 10001, ...}. The authors prove that for any fixed note, this language is a

**context-free language**; such languages are the second simplest languages in the Chomsky hierarchy, and are the basis for the phrase structure in many programming languages.

The simplest languages in the Chomsky hierarchy are the

**regular languages**, which are the ones that can be recognised by a finite-state automaton. Drexler-Lemire and Shallit prove that the languages arising from the infinity sequence are not examples of regular languages. The most complicated languages in the hierarchy are the

**recursively enumerable languages**, which can be recognised by a Turing machine, and the second most complicated are the

**context-sensitive languages**, which can be recognised by a certain type of Turing machine with a bounded tape length.

**Related links:**

Details of the Chomsky hierarchy: http://en.wikipedia.org/wiki/Chomsky_hierarchy

Per Nørgård's website: http://pernoergaard.dk/

The infinity series in the

*On-line Encyclopedia of Integer Sequences:*http://oeis.org/A004718

Read modulo 2, the infinity series gives the

*Thue-Morse word*. Here's a post by me about that: https://plus.google.com/101584889282878921052/posts/CTw4UKqDnY1

Here's a recent post by me about a finite-state automaton: https://plus.google.com/101584889282878921052/posts/RQrjPGQ6nwm

Here's a (rather loud) YouTube rendition of the Infinity Series: Per Norgard -- Infinity Sequence -- A004718

And last, but not least, here's Noam Chomsky saying

*“Oppan Chomsky style”:*Oppan Chomsky Style

#mathematics #musictheory #linguistics #sciencesunday #spnetwork arXiv:1402.3091

1

1

Add a comment...

### eden andes

Shared publicly -"After Antea"... by the painter Parmigianino (1503)Girolamo Francesco Maria Mazzola (also known as Francesco Mazzola or, more commonly, as Parmigianino ('the little one from Parma') or Parmigiano; 11 January 1503 – 24 - 1540 was an Italian Mannerist painter and poet.. ph.detail cr. en wikipedia

2

Add a comment...

People

In her circles

209 people

Links

YouTube

Basic Information

Gender

Female