Paper on global #exponential #stabilization (including the position of the particle) for fluid-particle interaction system: design of a PD control acting on the particle.
by #ModConFlex researcher Zhuo Xu and his supervisor Marius Tucsnak.
Recent searches
Search options
#exponential
0 posts0 participants0 posts today
rk T
#3StrandBraids
#Braids are the last of the #decorative elements on the #IonicScroll, but like #EggsAndDarts, they are not specific to the #IonicOrder.
Braids are a popular design motif that find wide currency in modern #hairstyles, #fashion, and fashion accessories like #belts and #bracelets.
Braids come in infinite varieties with varying number of strands, thickness of strands, roundness or flatness of strands, and how tightly or loosely they are wound together. Here, I focus on the 3-strand variant mentioned in #Vignola's book and previewed in https://pixelfed.social/p/Splines/792015485979791089. The image here is brightly colored to draw attention to the 3 strands.
The geometry of braid strands is not at all obvious despite how familiar they look. Also, a braid strand is the only feature in the entire iconic order whose geometry cannot be captured with straight lines and circular arcs. Instead, a strand geometry must be defined in a series of steps starting with a basic #sinusoidal curve.
A sinusoidal curve or #sinusoid is a wave form whose function belongs to a family of functions known as #transcendentalFunctions that also include #logarithmic and #exponential functions. I mentioned #logarithmicSpirals in https://pixelfed.social/p/Splines/792499765146596723, and in a future post I will show how to construct one and compare it with the spirals used in our implementation of #IonicVolute.
They are called transcendental functions because they transcend the math of finite algebraic polynomials and go beyond geometry into trigonometry. Fortunately, we don't have to go there.
Few #CAD tools have a direct primitive for a sinusoid, but almost all have a primitive for a 3-dimensional round coil shape called a #helix which we can use to create the sinusoids we need for a braid strand. To create s sinusoid, all we need to do is #project a helix on a flat surface to convert it into a 2D waveform.
#Braids are the last of the #decorative elements on the #IonicScroll, but like #EggsAndDarts, they are not specific to the #IonicOrder.
Braids are a popular design motif that find wide currency in modern #hairstyles, #fashion, and fashion accessories like #belts and #bracelets.
Braids come in infinite varieties with varying number of strands, thickness of strands, roundness or flatness of strands, and how tightly or loosely they are wound together. Here, I focus on the 3-strand variant mentioned in #Vignola's book and previewed in https://pixelfed.social/p/Splines/792015485979791089. The image here is brightly colored to draw attention to the 3 strands.
The geometry of braid strands is not at all obvious despite how familiar they look. Also, a braid strand is the only feature in the entire iconic order whose geometry cannot be captured with straight lines and circular arcs. Instead, a strand geometry must be defined in a series of steps starting with a basic #sinusoidal curve.
A sinusoidal curve or #sinusoid is a wave form whose function belongs to a family of functions known as #transcendentalFunctions that also include #logarithmic and #exponential functions. I mentioned #logarithmicSpirals in https://pixelfed.social/p/Splines/792499765146596723, and in a future post I will show how to construct one and compare it with the spirals used in our implementation of #IonicVolute.
They are called transcendental functions because they transcend the math of finite algebraic polynomials and go beyond geometry into trigonometry. Fortunately, we don't have to go there.
Few #CAD tools have a direct primitive for a sinusoid, but almost all have a primitive for a 3-dimensional round coil shape called a #helix which we can use to create the sinusoids we need for a braid strand. To create s sinusoid, all we need to do is #project a helix on a flat surface to convert it into a 2D waveform.
I made a #shader, which shows the #exponential map applied to the #Mandelbrot set
My wife was working through finding the derivative of the #exponential #function #exp(x) from first principles.I was made aware that she hadn’t actually seen why the number e=2.7128… was the #base the of the function and that that’s what you need to start with. In fact, that means one must actually start by finding the first differential of a general #logarithm and find #e from there. Once you’ve find the #Derivative of #lln, the #derivative of the #ExponentialFunction is straightforward. (1/2)
Replied in thread
@hart Men
Ich könnte Kotzen.
Die e(sel)fuels sollten mal in die Kurven gucken:
https://inv.nadeko.net/watch?v=U6V22IWppT8
#Exponential was ist das? Verbrenner schon lange nicht mehr!
#American #Cancer Society found #exponential #increase in cancers in #young #generations of Americans.
... people born in 1975 experience nearly twice the rate of small-intestinal cancers as people born in 1955. For people born in 1990, those cancer rates have grown to nearly four times what they were for people born in the middle of the 20th century. You can see the same general trajectory for kidney, liver, pancreas, and bone marrow cancers.
https://www.vox.com/explain-it-to-me/389508/cancer-early-young-adults-colon-breast-explained
Vox · Why do so many young people suddenly have cancer?
The exponential mapping of the c-plane can be used to speed up the calculation of the Mandelbrot set when rendering a zoom sequence. This is because all the pixels required for a zoom sequence are calculated once, which is much more efficient than creating the same sequence in a Cartesian view. By using the inverse of the exponential map (a logarithmic function), the familiar Cartesian map is recreated by stretching it into a ring shape.
Added perturbation for
Nevertheless, a zoom-in animation into
Replied in thread
Sometimes I pop random 
into the hashtag search bar to see what comes up. It's fun to play around
If there's someone cool you can check their profile to see who else they follow, etc
30 exponential steps
Next I would tell you that you could take 30 #Exponential steps, where the first step was 1 metre, the second 2 metres, third 4 metres, fourth 8 metres, and so on. Then I would ask you how far
30 strides would take you this time. That’s a bit trickier to work out in your head and the answer would probably surprise many of you. In fact your 29th step would take you to the surface of the moon and your 30th step would be long enough to get you back home again.
#maths
For years when #teaching exponential functions I ask “Why is the global human #population over a billion fewer people than what we would predict using a perfectly #exponential model and population figures from the 1950s and 1960s?”
Until about 5 years ago my #students always said “people don’t want big families anymore” an OK answer. But, now? they all say “the model doesn’t account for enough death”
… the model fails in many ways, but I think the #youngPeople are very #depressed 
Replied in thread
@dgoldsmith also don't forget:
exponentials -> !stability
Continued thread
Do you want to know how to fit a probability distribution to data? Watch the latest @LabPlot video tutorial.
https://www.youtube.com/watch?v=g0OQrAVwsTw
#DistributionFitting #ProbabilityDistribution #Distribution #StatisticalDistribution #Gaussian #Log-normal #Probability #Poisson #Binomial #Exponential #MaximumLikelihood #LabPlot
Listened to this #YourUndividedAttention podcast and found it really quite scary:
https://www.humanetech.com/podcast/the-ai-dilemma
One very math-y angle: they mention "double exponentials": it's not just that AI tech is developing at an exponential rate -- it's that the development of AI tech enables *other* things to develop exponentially.
As math folks, we get exponential growth. And when its related to something that plausibly could cause the extinction of humanity (as many AI researchers fear), we should be concerned. Or terrified?
www.humanetech.comThe AI DilemmaAt Center for Humane Technology, we want to close the gap between what the world hears publicly about AI from splashy CEO presentations and what the people who are closest to the risks and harms inside AI labs are telling us. We translated their concerns into a cohesive story and presented the resulting slides to heads of institutions and major media organizations in New York, Washington DC, and San Francisco. The talk you're about to hear is the culmination of that work, which is ongoing.