how you can use a diffusion lens...

Ha. Giving scotch tape the moniker of "lens" -- diffusion or otherwise -- is a great offense to actual lenses everywhere!

 

Ha. Giving scotch tape the moniker of "lens" -- diffusion or otherwise -- is a great offense to actual lenses everywhere!

Hi,

Yeah it is kind of amazing how this works so well. It's like a lot of other stuff we know about though. It's like going backwards through a mixture to 'unmix' it somehow. It depends how much you know about how it was mixed. For these 'lenses', it is assumed that there is some linear transformation at work but we might be able to deviate from that somewhat.

One of the problems though is that digitizing is not a perfect transformation it has serious differences, namely the resolution. If we only had 256 different combinations and the input signal was only between say 200 and 210 for most of the image, we'd end up with a pretty nasty looking output image. Even with today's 16 million color images we see this problem come up when we go to enhance an image. Each color channel has only 8 bits, and that's 256 states. But some images may have all the (say) reds from 150 to 200, which is like 1/5 of the total range. When we go to adjust that, we are already starting with a poor sample. In the analog world, it would not matter as much except for noise. In theory there are an infinite number of levels between 150 and 200, but in the digital world just 50. That's quite a difference. Of course noise messes things up a bit, but that's not always a bad thing.

 

Simply awesome.

 

Simply awesome.

Agreed. One of the best done and delivered videos I have every seen. I hope courses in complex mathematics latch on to it and use it's explanations and make it required viewing.

 

Not so much an algorithm, but a chain of approaches along with some good explanations of how various historical measurements were (possibly/probably) made.

 

Simply awesome.

Hi,

I have his book from back in the early 1970's but not sure when he actually wrote it or did all the many drawings.
The ones I was attracted to most were the 3d to 2d transformations where he takes a 3d object or set of 3d objects and projects them onto a 2d plane, but then that allows more interpretations of the reality so he can show some impossible scenes that look possible in 2d. One example is where people are walking up a staircase and appear to be looping around so they are always walking UP and never down. It's not possible in 3d but in 2d it looks possible.

When I first saw the drawing in the video the first thing I thought of was a transformation because there are a lot of common graphical distortions that are produced by transformations such as punch, skew, cylinder, etc. They all use math transforms to get from the original to the final mostly for special effects. In math though they are used for solving complicated problems that are very hard to solve as first presented. By transforming and solving and then inverse transforming, we get an answer that can be a lot simpler.
The transform in the video is probably z to e^z plus a Mobius transformation. It's amazing that he could come up with this without any complex math (I don't think he used much math but who knows).

One of the things he did not mention for some reason is the preservation of angles in conformal mapping. That's probably one of the most important points. That's why the graphic "little squares" stay little squares, but they actually have to be infinitesimally small in the math.

It is interesting that people are still looking at his work even after all this time. He was pretty amazing and must have had the patience of a god (ha ha). His teacher said he would never become a good architect.

 

I like this kind of stuff:

Imagine a complete FPU built of nested eml(x,y), and NAND gates.

 

I like this kind of stuff:

Imagine a complete FPU built of nested eml(x,y), and NAND gates.

Looks like much ado about nothing. It's not just "one function", it is a function defined as the sum of two other functions -- so right there you have embedded three functions. exponentiation, logarithm, and addition, By taking two arguments, it makes it pretty straightforward to turn on of the embedded functions off. In one of her examples, she showed applying the additive inverse to one of the arguments, which is yet another operation that is needed/embedded in the system.

I'll take her at face value that this might enable different/new/better ways to explore mathematical structures -- I don't have any basis for having an opinion one way or the other. I also don't have any objection to using things like this to underscore how unified mathematics is at its core.

 

Looks like much ado about nothing.

I see it as just another tool for the ol' tool bag.

You can never have enough tools.

 

I like this kind of stuff:

Imagine a complete FPU built of nested eml(x,y), and NAND gates.

Hi,

This 'function' may seem very convoluted at first glance, but because it offers another path to solutions it is very possible and even likely that it could help a lot in theorem proving. I also see a possibility for using it for function searching using an evolutionary type strategy. As quantum computers evolve, it may become even more important.

Part of the key point would be how fast does it work versus using a regular set of functions we normally see, and can we really build ANY function from this or will there be functions that come up that can't be found with this idea. It's an interesting concept though I might have to explore it a little myself.

It does look like some of the higher functions even sin and cos might involve several steps though. That could increase computation time on a regular computer. Note she did not show how to get sin or cos.

This might go pretty high up on the list of "interesting functions" though regardless.

 

 

Not sure how a 21-year old qualifies as a "teenager".

The video makes it sound like he discovered several "unknown" facts, like the clustering that results from linear probing. But these are well-known issues with a number of well-known ways to mitigate them. Maybe he "discovered" them in the sense that HE was unaware of them and their consequences. The notion of using a non-greedy probing algorithm is not new or particularly novel, in fact most hash systems today use some form of non-greedy algorithm for the same reasons and with the same objectives as Krapivin's approach. What he really brought to the table was a strong mathematical proof of a lower bound for non-greedy algorithms. What he did NOT do was disprove Yao's work, since Yao's analysis was specific to greedy algorithms.

It's unlikely that Krapivin's work will have any significant impact on real world systems any time soon. There are LOTS of practical issues involved, such as memory allocators and cache-efficient indexing as well as stuff like concurrent data structures that, that make the real-world problem much more complex and open up lots of different optimization opportunities. Real-world implementations reflect what the hardware can and can't do well, while new hardware is developed with the aim of facilitating what new algorithms do and don't need to be efficient.

 

It didn't strike me as particularity interesting, an optimization added to an existing concept. Perhaps I missed something but innovative ideas and improvements in software are common, happens every day and few of them get hyped or even noticed much.

 

It didn't strike me as particularity interesting, an optimization added to an existing concept. Perhaps I missed something but innovative ideas and improvements in software are common, happens every day and few of them get hyped or even noticed much.

This could have far reaching consequences so I see it as a milestone in computing THEORY.

At first I was thinking SSDs and Chess computer algorithms, but with the Krapivin algorithm there is too much overwriting so it will be slower for some things especially if the memory is only something like 50 percent full, and I don't think it is (log n)^2, it is probably 2*(log n)^2 because there is the search phase and then the placement phase.

So what does it do for us. For you and I reading here probably nothing, but for someone looking for a new placement algorithm they know that they can't get closer than that log n squared factor, and the closer they get to that the better. So it's good for theory not for practice. That's what it looks like to me anyway.

Also, it was sort of an overly glamorous presentation for someone doing scientific work. I have to wonder who made that video. Maybe someone from his family with the help of 'ai'

 

Simply awesome.

reminded me of Doom

 

Hello,

I remembered another sin(x) algorithm that was used some time in the distant past. I still can't remember or recreate the better one though.

The one I do remember starts with the Taylor series, then uses a correction algorithm to improve on the Taylor solution. The correction algorithm is also a power series, but the advantage is it has small coefficients.
As you probably know, if you want 256 digits for sin(x) and you just use the Taylor series alone, you would need to store numbers with a huge number of digits, at least 256 I think, and you'd need a lot of terms.
With the correction algorithm, you can get away with a smaller number of terms in the Taylor series which means smaller numbers can be stored, and because the correction algorithm has much smaller digits that means the two algorithms become easier to store, so take up less memory space. The factorial function is multiplication with numbers increasing in value that are multiplied, so we have 1*2*3*4*5...etc. and eventually that creates a very large number, and that ends up being in the denominator of the terms that are added in the Taylor series.
It's an interesting way to do it at the very least, and the number of digits can be guaranteed by using a test fraction that tests the seed to make sure it is small enough, and if not it simply divides it another time by the angle divisor (the angle divisor makes the Taylor series more accurate per term). Once the seed is small enough, then the correction algorithm is used and that creates the final outcome for sin(x).
This may have been even more important in the distant past when processing speed was slow and memory bandwidth was low and memory space was severely limited. It's still an interesting way to do it though and you can generate the correction algorithm by solving a set of equations in powers of sin(x).

 

If you want to see some coding art check this out...you don't know what you don't know until you know it!
https://github.com/thealgorithms

 

This could have far reaching consequences so I see it as a milestone in computing THEORY.

At first I was thinking SSDs and Chess computer algorithms, but with the Krapivin algorithm there is too much overwriting so it will be slower for some things especially if the memory is only something like 50 percent full, and I don't think it is (log n)^2, it is probably 2*(log n)^2 because there is the search phase and then the placement phase.

So what does it do for us. For you and I reading here probably nothing, but for someone looking for a new placement algorithm they know that they can't get closer than that log n squared factor, and the closer they get to that the better. So it's good for theory not for practice. That's what it looks like to me anyway.

Also, it was sort of an overly glamorous presentation for someone doing scientific work. I have to wonder who made that video. Maybe someone from his family with the help of 'ai'

I suspect this idea has been coded hundreds of times in the real world by real programmers without them knowing it had some academic history or broader relevance. How many of us must have written neat, imaginative algorithms over the years.