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Rumiana Nikolova
Attended University of Sofia - Medicine
Lives in Veliko Tarnovo, Bulgaria
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Rumiana Nikolova

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~ Red  rose  ~

#photography   #flowerphotography   #roses  
#naturephotography   #flowerslovers  

That's my gift to everyone hwo visit my posts !!
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thanks for your second comment in this post my dear friend +Anna Rimovska Have a lovely evening! Hugs, kiss

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Enjoy the great post from my good friend !!

Thank you +Dominique Keller ! :-)))

Wake up call....good morning from very hot Bulgaria :-D
Réunion de cinq copains d'école de Southend-on-Sea (Essex) désireux, en 2012, de propulser à la face du monde un indie rock qui fait tintinnabuler guitares et batterie sur des mélodies ciselées pour les stades, les Nothing But Thieves se sont d'abord rendus aux States, sur les traces d'un Jeff Buckley dont ils revendiquent l'influence, avant de livrer un premier album (Nothing But Thieves, 16/10), mis sur les rails par une série de concerts épiques. Le futur (grand) groupe que vous adorerez détester.
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Thank you very much, my dear friend! Have a wonderful Friday, +Rumiana Nikolova ;))

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Great post, very interesting !!

H/T +Richard Green 

Fractals, Fibonacci, and factorizations

The rule for generating the famous Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, ... is that each number (after the first two) is the sum of the previous two numbers. The Fibonacci word is an infinite string of zeros and ones with properties reminiscent of the Fibonacci sequence, and the Fibonacci fractal, shown in the picture, is a way to represent the Fibonacci word in the form of a fractal.

One way to generate the Fibonacci word is to define strings of zeros and ones by the rules S(0)=0, S(1)=01 and S(n)=S(n–1)S(n–2) when n is at least 2. This gives rise to the sequence of strings 0, 01, 010, 01001, 01001010, 0100101001001, ..., whose limit, as n tends to infinity, is the Fibonacci word. There are other equivalent, but superficially very different, ways to generate this word, including (a) using an explicit formula for each digit given in terms of the golden ratio; (b) using a substitution rule; and (c) using the Zeckendorf representation of integers in terms of Fibonacci numbers.

By suitably interpreting the digits of the Fibonacci word as turtle graphics instructions in a Logo-like programming language, it is possible to represent the word as a fractal. More precisely, if one reads the digits in order, then the n-th digit corresponds to the following sequence of instructions:
1. draw a segment forwards;
2. if the digit is 0, then turn left 90 degrees is n is even, and turn right 90 degrees if n is odd.

The picture shows the result of this procedure after many iterations. The resulting curve has various interesting mathematical properties, some of which concern the square-shaped gaps. By inspection, we count one large square gap (in the middle, at the bottom); five smaller square gaps, and 21 square gaps of the next size down. The numbers of these gaps, sorted by size, turn out to be given by every third Fibonacci number starting with the second 1 (1, 5, 21, 89...) which means that there are 89 squares of the next size down. Furthermore, each square has a side length that is 1+√2 times the side length of the square of the next size down; the number 1+√2 is known as the silver ratio.

The recent paper Factorizations of the Fibonacci Infinite Word by Gabriele Fici ( surveys some factorizations of the Fibonacci word and shows how to derive these factorizations using elementary properties of the Fibonacci numbers. In some cases, this gives easier derivations of the results than were previously known. An example of such a factorization involves the sequences S(n) from earlier. Proposition 1 of the paper proves that the Fibonacci word can be factorized as the infinite product 0.1.S(0).S(1).S(2)..., where the symbol . is used to separate the factors.

One of the most surprising factorizations in the paper is Proposition 9, which involves the reversals, T(n), of the strings S(n). The strings T(0), T(1) and so on are then given by the sequence 0, 10, 010, 10010, 01010010, ... Remarkably, the concatenation of the strings T(n) also gives the Fibonacci word, even though the ingredients being used to construct it are backwards and generally not palindromic. Another way to say this is that the Fibonacci word can be factorized as the infinite product T(0).T(1).T(2)...

Relevant links

The 2009 paper The Fibonacci Word fractal by Alexis Monnerot-Dumaine is an excellent guide to the mathematical properties of the fractal, and the picture of the fractal here comes from that paper. You can download the paper for free at 
Monnerot-Dumaine's paper explains how to construct the Fibonacci word using a substitution rule, and explores what the fractal looks like if one makes turns at angles other than a right angle. 

Fici's paper explains how to construct the word using the Zeckendorf representation of natural numbers. It is a theorem that any positive integer can be expressed uniquely as the sum of one or more distinct non-consecutive Fibonacci numbers. This is called Zeckendorf's Theorem, even though Zeckendorf was not the first to prove it:'s_theorem

Wikipedia's article on the Fibonacci word gives an explicit formula for the n-th digit of the word and mentions many other interesting properties. For example, the Fibonacci word is often cited as the worst case for algorithms detecting repetitions in a string.

The On-Line Encyclopedia of Integer Sequences on the Fibonacci word:

Wikipedia on turtle graphics:

I have posted about the Fibonacci word twice before, although not recently. 
My post from March 2013 discusses the word in the context of self-shuffling words:
My post from December 2012 discusses Fibonacci snowflakes and some generalizations of the Fibonacci word:

If you're disappointed that I didn't talk about the golden ratio, have a look at the aspect ratio of the accompanying picture.

#mathematics #sciencesunday #spnetwork arXiv:1508.06754
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Rumiana Nikolova

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#Health   #Food   #Recipes  
Healthy lifestyle blog about more than just healthy food recipes
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+Francesco Del Santo I think also, thank you Francesco.

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Wonderful post & wishishing to everyone as well !!!

H/T +Noel Lyons 

.....and live a courageous life....
Wishing Everyone An Unstoppable September!

#unstoppable   #september   #inspire  
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Your good friendship, kind words and support means a lot to me & many thanks my dear friend +Rita Gijbels! Wish you a lovely evening too! Warm greetings and hugs !

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Alzheimer’s disease thought to be accelerated by an abnormal build-up of fat in the brain.

People with Alzheimer’s disease have fat deposits in the brain. For the first time since the disease was described 109 years ago, researchers have discovered accumulations of fat droplets in the brain of patients who died from the disease and have identified the nature of the fat.
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Wishing you to Everyone a Wonderful and Relaxing Weekend :-)))

I want to say ALL a big THANKS for your support with lovely plusses ♦ nice comments ♦ reshares by my posts last days!!!

#hqspflowers   #hqsplandscape   #macroaddict   #flowercolors   #hqspmacro   #flowerphotography  
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Your lovely comment means a lot for me dear Tanja.... Thank you so much! :-)) +Tanja Nopav

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What a beautiful and ellegant....

Feita para brilhar! Vai resistir?
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Waterfalls  Lakes Plitvice, Croatia (National Park)
Is among the 20 most beautiful lakes in the world to 17th place. The park covers an area of 33,000 hectares and includes 16 lakes in succession, connected by waterfalls.Plitvice is the oldest national park in Southeast Europa.All'interno the park there are also many caves of which only a small part is agibile.I lakes are formed by two rivers: the White River and the Black River, which flow in the river Korana. The waters of these rivers are rich in calcareous salts (mostly calcium carbonate and magnesium carbonate), from the dissolution of carbonate rocks forming the geological structure of sito.Questi salts are precipitated by vegetation, forming layers of travertine , a sedimentary rock recently. Over time, these deposits forming real natural dams that act as barriers to water, growing by about a centimeter per year. At one point the water pressure breaks these natural levees, opening new paths in the ground.
This mechanism, in fact common to all the calcareous water, in Plitvice has assumed a particular importance.
The beauty of the National Park Plitvice, Croatia, is increased in the second round of the New Seven Wonders of Nature. ~ DK

#waterfall #waterfallphotography #nature #naturephotography #amazing #awesomeness #colorful #landscape #beautiful #view #photooftheday #summer #whatshot #whatshotnow #whatshotongoogleplus
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Veliko Tarnovo, Bulgaria
Medicine ◊ Science ◊ Macrophotography
            PRACTICE MEDICINE ♥     I am Rumiana / from BULGARIA / for friends - I am  Rumi I graduated from Medical university and then remained as a lecturer at the university..  I worked in a hospital - in the surgical ward.. Now I work in shcool as physican. I'm volunteer in Bulgarian Red Cross                       My interests : - In the medical field - Photography - Fashion and arts - Public relations /PR / - I want to know more all areas of life and science - Politics..Philosophy - Good books and of course good music - Sports ..I love to dance and have fun My Google plus - page My page - Linkedin for Rumiana Nikolova I want to learn more about other countries..and new friends. I can introduce you to  my country and of course...with me...I will sometime talk about myself  and my experiences.  I will share the work of others ...You will get is my sense of laughter.Once  know me  - you will understand that I'm lady with a smile...always want to have fun... My desire is  for good friendship ! Thank you so much for visiting my profile page :))          SOME OF MY FAVORITE QUOTES:. "The only true wisdom is in knowing you know nothing" -  Socrates "Those who trade freedom for the illusion of security deserve neither freedom nor security "  - Benjamin Franklin THE MOST IMPORTANT THING IN LIFE IS TO BE YOURSELF !!!                                                     
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