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Thomas Egense
Works at Statsbiblioteket
Attended Århus Universitet
Lives in Aarhus
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Thomas Egense

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New shipment arrived

The one to the right in wrap (Seaweed 120cm*120cm) is sold to +Jacques Riget and is copy #3  and final print of this fractal.

The two to the left (Turtle and Octopus) are replacements of the two sold to +Marselisborg Gymnasium.

The two middle smaller fractals are on printed on glass and due to reflections very hard to catch sharp on camera.
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Thomas Egense

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The counterintuitive Borwein Integral

For an explanation  see:

(Found on Quora)

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Today I learned the universe has limited precision floating point arithmetic
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Article in The New Yorker about Yitang Zhang, this time with more personal information about himself and his life, and not only about his mathematical breakthrough.
Excellent piece in The New Yorker about Yitang Zhang and his groundbreaking work establishing that there is a bound on the gap between two consequent primes no matter how far along we are on the road to infinity.

Zhang's example shows that it's possible to do top level mathematical research outside of the academia. Good new for all of us, and... shame on you, academia!
Unable to get an academic position, Zhang kept the books for a Subway franchise. Credit Photograph by Peter Bohler
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I've always thought along these lines. Science is for everyone
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Combine the golden ratio with a fractal recursion  algorithm and you get the newly invented Hariss Sprial.
Here's me in the Guardian on some fascinating work by +Edmund Harriss, in which I take the liberty of naming this curve after him. He was, after all, the first to draw it....
Inspired by the golden ratio, mathematician Edmund Harriss discovered a delightful fractal curve that no one had ever drawn before. But it’s not just a pretty picture, it contains some lovely theory – and brings the golden ratio into a family of perfect proportions.
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Gaps between prime numbers(large and small)

Last year the big breakthrough in mathematics was the first improvement in centuries on the "Prime number theorem" by Yitang Zhang. His initial gap of 70M is now, by massive collaboration of mathematicans all over the world, pinned down to 246.  Even though the gap is believed to be only 2, this was the first time it was ever proven to be bounded at all!

Instead of proving how small gaps will appear, you can also ask how large a gap you can expect to find. And it seems some new results are about to be revealed about this opposite problem as well.

#math #mathematics  
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Breakthrough Prizes in Math Celebrated

Some of the most talented mathematicians celebrated Hollywood style. The focus and attentention given to mathematics in this glamorized celebration is very welcome and hopefully make pursuing a career as a mathematician more attractive to young people. It is in contrast to the dull and  nerdy way  Hollywood  normally portraits mathematicians in movies.

The whole "Oscar" aka "Nerd"-award thing is a little over the top for my taste, but that is how it is done in Hollywood of course.

TL;DR: Girls please  take a look at the handsome mathematicians!
Internet billionaires flew 14 scientists to Silicon Valley last week to turn them into “superheroes,” into “rock stars.” Tech investor Yuri Milner, who created the first of the Breakthrough Prizes two years ago and co-sponsors all of them, called them the “Oscars of science,” and the black-tie affair honoring the...
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+J.T. Cranston You can blame that at the journalist this time and not Hollywood.
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Thomas Egense

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The counterintuitive Borwein Integral

For an explanation  see:

(Found on Quora)
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Well .499999999999 can be written as .5 which is 1/2
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Yet another utterly futile attempt to find a md5 fixpoint

During lunch discussion at my workplace we started wondering if there existed a fixpoint for the md5 cryptographic hash function.

That is a point such that md5('x')='x' where x is the 32-digit hexadecimal string representation.

To be completely truthful the md5 algorithm works on 512 bit values.  A 32-digit hexidecimal string is 128 bit and is first padded(with zeroes) to 512 bits before the md5 function is taken and
returns a 128 bit value which then is converted to a 32 digit hexidecimal string thus explaining my md5('x')='x' expression.

Some googling showed this is known as the "Kember Identity" and is an unsolved problem. Also a few programmers had tried to brute force it over the years with no luck (not surprisingly as you will see below). Even though I would never succeed I decided to look into the problem anyway.

Interestingly there is a simple mathematical argument that there is ~63.21% probability  for at least one fixpoint. This does not mean there is one of course! But it also reveals there is probably only a few, if any! The argument is based on the same principly that when you shuffle a deck of card, what is the probability that at least one card will end up in the same position in the deck. The probability actually converges to 1-1/e very fast as the number of the elements shufled increases. There is no guarantee that the md5 function is a shuffling though, there could exist different 32-digit hexidecimal strings x,y such that md5(x)=md5(y). But such collisions are extremtly rare and would not change the overall probability by much.

Examples of different x,y with same hashing value does exist , see etc. But this example is 512 bit and not 128 bit (32 hexidecimal). The md5 hash function is not considered safe anymore due to collision vulnerabilities , but this is completly irrelevant to my mission of finding a fix point.
Lets see how futile a brute force attempt will be:
There are 2^128= 32^16 ~ 3.4*10E38 different values.
My brute force program can do 3.5M hash/sec (For each CPU used)
This gives 3*10E24 years to try all combinations (Using 1 CPU).
Not a very uplifting result and now you know why I used the word futile.
So without a cryptographic breakthrough, humanity will never know if such a point indeed does exist.

And now back to my brute force attempt:
I made a small Java program that can be found on Github on the link below.
Thanks to +Toke Eskildsen  for doubling the performance using tricky bit manipulation instead of the standard Java methods as substring, equals etc.
The code is much harder to read now though,  nothing comes for free...
The program starts with a random 32 digit hexadecimal string (for each CPU) and then iterative computes md5(x) and check if it matches x. Using this endless chaining of md5 values there is no performance overhead from generating a a lot of random strings. Of course there is a (tiny!) probability that this would result in an endless loop as it eventually is bound to do! But cycle detection would slow
the program down and detecting a cycle having 1E10 elements would take way too many resources anyway. Also I restarted the program
regulary over the 2 weeks I had it running on 24 CPUs and this would have reset any loops.

Since I would not find a fix-point but still would like to program to output somthing, I logged the maximum prefix and suffix match for md('x')='x'

The maximum match I found was a 12 character suffix and a 12 character prefix match shown below:

prefix 12:
54db1011d76dc70a0a9df3ff3e0b390f -> 54db1011d76d137956603122ad86d762

suffix 12:
df12c1434cec7850a7900ce027af4b78 -> b2f6053087022898fe920ce027af4b78

Remember this used 24*2weeks = 0.9 year CPU time. If you run it and find a 13 character match or more, I would like to hear for you of course.
MD5FixPointSearch - A simple java program that attempts to find a fix point for the MD5 function. It will try find a maximum prefix/suffix match.
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The bitcoin network currently produces 2^58 hashes per second (SHA2 hashes instead of MD5, but the idea is the same). At this rate it would only take 3.8E13 years.

On the other hand it would surely improve on the prefix and suffix matches.
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Article about Ciprian Manolescu who disproved the triangulation conjecture in 2013

Quanta Magazine printed this article a few days ago about Ciprian Manolescu and is worth a read. The mathematical concepts are explained very well for the casual math reader.
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Shinichi Mochizuki speaks to press, express anger over the slow progress in attempting to decipher his proof

It is now over 2 years since he published a  still unconfirmed proof of the ABC-conjecture which was thought  impenetrable hard to crack. The proof was base on years of work in a branch of mathematics  called 'inter-universal Teichmüller theory' that noone know or seems to understand. Following his steps and understand the proof is really difficult and would take years  of dedicated work in a field that may lead nowhere if the proof is wrong.

Up to now he has not been speaking to the press, but he has created a workshop where he will answer questions and help other mathematicians to understand it.

It is not that easy to explain what the ABC conjecture in a few lines, but more information can be found here:
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I don't think he's spoken to the press - my understanding is that this news is based on his December 2014 progress report (in which some frustration is evident):
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Pixel Pioneers: A Brief History of Computer Graphics

Documentary about the history of computer games graphics.
It is divided into five parts where each part is approximate 9 minutes.
If you also are no newbie (no need to mention age) to the computer games scene, you will probably find this documentary as interesting as I did.
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Now You're Playing with Power

This is the beautiful hardware setup (full frontal nudity picture) behind The Danish Internet Archive I wrote about some months ago ( ).

The bottom rack is our index builder having 48 CPU, 256GB ram and 5*1.6 TB SSD discs in RAID 0 for ultra IO performance. The middle rack is the searcher running SolrCloud (25 Tomcat instances) using the 25*1TB distributed indexes in the
top rack. The top rack is non-enterprise storage using 25*1 TB SSD Samsung 840 EVO.

When a new index has been build on the index-builder it will be moved to to the searcher on a new 1 TB disc.
Next week all 25 SSD discs will be full and total number of documents will be 7.3billion in the distributed index, so how do we continue scaling the solution?  We just add another copy of the two top racks, one running all the Solr-servers and one with the storage for the indexes.

In some months I will report if performance is unchanged with this scaling setup :)

For our current performance you can read +Toke Eskildsen post :
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Cool! i got the joke too..and male or female irrelevant no?
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  • Århus Universitet
    Cand. Scient, Mathematics/Physics, 1991 - 2000
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I have a math blog and I am generally interested in all kinds of scientific matters. I use mathematics to create art – you can check out my album. I try to find unique content for my posts,which are mostly of scientific nature.
Keywords that describe me as a person: scientist, mathematician, sceptic, atheist, blogger, and digital artist.

I blog about my mathematical adventures, where I, among other things, solve various puzzles,  which often involves brute force CPU attack on the problems.

In my spare time I use mathematics to create art.

For the last 12 years I have worked as a Java programmer.

I like retro-gaming, and I never become tired of talking about old-school games or computer
music (C64/Amiga).

I love animals - especially cats - and I have two cats of my own.
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