John Venn was born #OTD in 1834. He is noted for introducing Venn diagrams, which are used in logic, set theory, probability, statistics, and computer science. Venn then further developed George Boole's theories in the 1881 work Symbolic Logic, where he highlighted what would become known as Venn diagrams. via @wikipedia
"Stock and flow diagrams" are a nice graphical tool for modeling systems. People have had success teaching them to students starting at a young age. It's a way to teach them math, economics, ecology, and other subjects in a unified way.
When you include functions describing the flows - shown as faucets here - you can turn these diagrams into differential equations. But you don't need to do that for young kids: there's a lot you can learn from these models in a purely qualitative way. Basic concepts like feedback, etc.
And once you introduce the flow functions, you can let software solve the resulting differential equations and graph their solutions even before the kids know anything like the definition of derivative! This is a good way to gently get them interested in calculus.
For example, below you can see a model of reindeer population on an island created by middle school students. The population soared and then crashed:
"Students built System Dynamics models to study human population dynamics, non-renewable and renewable resource utilization, economic influences, etc. In these lessons students were asked to build the model, anticipate model behavior, explain discrepancies between anticipated model behavior and actual model output, analyze feedback, then test policies on the model to determine leverage points."
Pushed a huge update of the second chapter of #categorytheoryillustrated - the one where categories are introduced: rewrote a lot of stuff, corrected many mistakes and added an overview of Lawvere's Elementary theory of the category of sets, right before the concept of a category is introduced, section "Defining the rest of set theory using functions".
Nicolaus worked mostly on curves, differential equations, and probability. He was a friend and contemporary of Leonhard Euler, who studied under Nicolaus' father (Johann Bernoulli). He also contributed to fluid dynamics. via @wikipedia
Before I say what it is, I am NOT posting this as clickbait (which is how it's often used)! 😂 I'm posting this as a Maths teacher who knows this topic inside-out and wants to help people to understand it better. There are MANY mistakes that people make and get the wrong answer, and I'm going to cover them in bite-size chunks each week for a few weeks
"Constructive logic is so much more interesting than classical logic and it is so much closer to the true behavior of intelligible things in the world : in classical logic, because of double negation [...]" – Vladimir Voevodsky (1966-2017) #quote#mathematics#math#maths
392 years ago, the English mathematician and clergyman William Oughtred introduced the multiplication sign ✕ for the first time. via @fermatslibrary.
Oughtred was the first to use logarithmic scales and sliding by one another to perform direct multiplication and division. He is credited with inventing the slide rule in about 1622. He also introduced the abbreviations "sin" and "cos" for the sine and cosine functions. via @wikipedia
"Both the honeybees and wasps have solved this problem by mixing in some pairs of five-sided and seven-sided cells, which bridge the gap between different sizes of the six-sided hexagons, researchers report July 27 in PLOS Biology. This fix is close to the optimal solution to this problem, the team says."
"There is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain. [...]" – Alfred North Whitehead (1861-1947) #quote#mathematics#math#maths
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. He made important contributions in the theory of differential equations. He also made contributions to applied mathematics, including the theories of telegraphy and elasticity. via @wikipedia
I remember doing all sorts of crafts with glue & paper learning fractions but never did we try dividing the paper to infinity— I think this should be something everyone tries. I’d do this with calculus students learning series! The idea of an infinite sum having finite value is just more— believable after you really do it. And I’m trying to get these 6th graders to like fractions. So they just can’t be BORING— #mathEducation#mathematics#series#geometricSeries#binaryNumbers#artsAndCrafts
#TIL about the #HyperLogLog algorithm and I think it's a damn brilliant way to estimate the number of unique elements of a potentially gargantuan set of items and only running in O(n) time and O(1) space. The fact that variants of the algorithm can be done in parallel makes it even more awesome!
I want to graphically demonstrate the effect of some algebraic operators in a high-dimensional complex-valued vector space. I will be picking out a small number of example vectors, applying the operators to yield new vectors, and looking at what has happened in terms of the angles between the vectors.
The source vector space may be very high dimensional (say, 1000). Each element of the source vector is constrained to have magnitude 1. That is, each complex value has only one degree of freedom - the phase angle.
I am interested in the angles between vectors in the source space. In these high-dimensional spaces two vectors chosen at random are almost always very close to orthogonal.
I am interested in the angular relations (measured in the high-dimensional source space) between a a small number of vectors and 2 or 3 mutually orthogonal reference vectors.
I want to project the high-dimensional source space onto a real-3-sphere so that the angles of interest are maintained sufficiently well to be visually interpretible. I don't really care what happens to the angle between the other vectors.
I would greatly appreciate any pointers to how I might define and implement such a projection.
(Bonus if you can suggest an R package or code to do this.)
It seems to me that #clarity and #communication is so important in #mathematics, but this need constantly breaks against the language and terminology specific to and embedded in each subfield…
I'm so old I can remember when they wouldn't even let you present slides in a #mathematics talk unless you could find a Mac to borrow, since they were the only things that worked easily. Today I saw my first catastrophic failure in a presentation on a Mac, while my PC set itself up in three seconds and had no issues.