- Oct 2020
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qz.com qz.com
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“Many thought, okay to get from A to B there are these three steps, but it turns out there are really five or six,”
Sounds a lot like the mathematicians who came after Perelman to show that his proof of Poincare was correct--they needed help in getting from A to B too.
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danmackinlay.name danmackinlay.name
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A statistician is the exact same thing as a data scientist or machine learning researcher with the differences that there are qualifications needed to be a statistician, and that we are snarkier.
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gowers.wordpress.com gowers.wordpress.com
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there’s nothing like knowing that you’re going to have to justify your decision to a bunch of mathematicians.
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www.aft.org www.aft.org
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If you think mathematics is difficult, tough, or you're scared of it, this article will indicate why and potentially show you a way forward for yourself and your children.
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The low achievers did not know less, they just did not use numbers flexibly—probably because they had been set on the wrong pathway, from an early age, of trying to memorize methods and number facts instead of interacting with numbers flexibly.4
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Unfortunately for low achievers, they are often identified as struggling with math and therefore given more drill and practice—cementing their beliefs that math success means memorizing methods, not understanding and making sense of situations. They are sent down a damaging pathway that makes them cling to formal procedures, and as a result, they often face a lifetime of difficulty with mathematics.
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Unfortunately, many classrooms focus on math facts in isolation, giving students the impression that math facts are the essence of mathematics, and, even worse, that mastering the fast recall of math facts is what it means to be a strong mathematics student. Both of these ideas are wrong, and it is critical that we remove them from classrooms, as they play a key role in creating math-anxious and disaffected students.
This article uses the word "unfortunately quite a lot.
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Notably, the brain can only compress concepts; it cannot compress rules and methods.
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The hippocampus, like other brain regions, is not fixed and can grow at any time,15 but it will always be the case that some students are faster or slower when memorizing, and this has nothing to do with mathematics potential.
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blogs.ams.org blogs.ams.org
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Moreover, the teacher repeatedly asks, “Did anyone get a different answer?” or “Did anyone use a different method?” to elicit multiple solutions strategies. This highlights the connections between different problems, concepts, and areas of mathematics and helps develop students’ mathematical creativity. Creativity is further fostered through acknowledging “good mistakes.” Students who make an error are often commended for the progress they made and how their work contributed to the discussion and to the collective understanding of the class.
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Andrews, P., & Hatch, G. (2001). Hungary and its characteristic pedagogical flow. Proceedings of the British Congress of Mathematics Education, 21(2). 26-40. Stockton, J. C. (2010). Education of Mathematically Talented Students in Hungary. Journal of Mathematics Education at Teachers College, 1(2), 1-6.
references to read
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www.dpmms.cam.ac.uk www.dpmms.cam.ac.uk
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This result of Erd ̋os [E] is famous not because it has large numbers of applications,nor because it is difficult, nor because it solved a long-standing open problem. Its famerests on the fact that it opened the floodgates to probabilistic arguments in combinatorics.If you understand Erd ̋os’s simple argument (or one of many other similar arguments) then,lodged in your mind will be a general principle along the following lines:if one is trying to maximize the size of some structure under certain constraints, andif the constraints seem to force the extremal examples to be spread about in a uniformsort of way, then choosing an example randomly is likely to give a good answer.Once you become aware of this principle, your mathematical power immediately increases.
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Once again, Atiyah writes very clearlyand sensibly on this matter (while acknowledging his debt to earlier great mathematicianssuch as Poincar ́e and Weyl). He makes the point (see for example [A2]) that so muchmathematics is produced that it is not possible for all of it to be remembered. The processesof abstraction and generalization are therefore very important as a means of making senseof the huge mass of raw data (that is, proofs of individual theorems) and enabling at leastsome of it to be passed on. The results that will last are the ones that can be organizedcoherently and explained economically to future generations of mathematicians. Of course,some results will be remembered because they solve very famous problems, but even these,if they do not fit into an organizing framework, are unlikely to be studied in detail by morethan a handful of mathematicians.
bandwidth in mathematics is an important concept
We definitely need ways of simplifying and encoding smaller cases into bigger cases to make the abstractions easier to encapsulate and pass on so that new ground can be broken
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The important ideasof combinatorics do not usually appear in the form of precisely stated theorems, but moreoften as general principles of wide applicability.
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I mean the distinction between mathematicians who regard their centralaim as being to solve problems, and those who are more concerned with building andunderstanding theories.
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many clever techniquesinvented. Some of these can again be encapsulated in the form of useful principles. Oneof them is the following, known to its friends as Concentration of Measure:if a functionfdepends in a reasonably continuous way on a large number of smallvariables, thenf(x) is almost always close to its expected value.
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their name gives no mnemonic boost whatsoever. Whatever faint associations it might once have held fade away, especially when the discover was neither famous nor narrow, and the reader is several generations removed.
This might be debatable as many of the names in the example are relatively famous names. Any associations they provide might also extend to the dates of the mathematician which also then places the ideas historically as well.
More often I see the problem with some of the bigger greats like Euler and Cauchy who discovered so many things and everything is named after them.
The other problem is mis-attribution of the discovery, which happens all-too-frequently, and the thing is named after the wrong person.
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The average number of coauthors on math papers has gone up since 1900. So has the number of working mathematicians in the world, which raises the odds of independent rediscoveries, separated in time or space. These two trends have opened the door to triple and even quadruple hyphen situations, as in the Albert-Brauer-Hasse-Noether Theorem and the Grothendieck-Hirzebruch-Riemann-Roch Theorem.
But this also gets rid of naming quirks for multiple people like the Cox-Zucker Machine.
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The worst answer I can imagine is the one Pope Gregory VII gave for refusing to let the Holy Scripture be translated out of Latin: “... [I]f it were plainly apparent to all men, perchance it would be little esteemed and be subject to disrespect; or it might be falsely understood by those of mediocre learning, and lead to error.”
I'd push back on this a bit by saying that there are huge swaths of people looking at English translations, of Latin translations, of Greek, Hebrew, and Aramaic translations. Not only is there some detail lost in the multiple levels of translation, but many modern Christians are actively mis-applying the stories in the Bible to apply to their modern lives in radically different ways than was intended.
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www.nytimes.com www.nytimes.com
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It’s easier to “bridge” science and art when you don’t really think there’s a gap between them in the first place, as I don’t. The boundaries between subjects are really artificial constructs by humans, like the boundaries between colors in a rainbow.
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In 1945 Jacques S. Hadamard surveyed mathematicians to determine their mental processes at work by posing a series of questions to them and later published his results in An Essay on the Psychology of Invention in the Mathematical Field.
I suspect this might be an interesting read.
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projecteuclid.org projecteuclid.org
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The parameterization is said to be identifiable if distinct parameter valuesgive rise to distinct distributions; that is,Pθ=Pθ′impliesθ=θ′.
Definition of identifiable parameterization
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Local file Local file
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In fact, if you do the math, if failure equals knowledge and knowledge equals power, then failure equals power by the transitive property.
But first we have to prove that this system has the transitive property to begin with!
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katmat.math.uni-bremen.de katmat.math.uni-bremen.de
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Motivation
I really wish more math textbooks had motivation sections like this one does.
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Kuratowski definition of an ordered pair
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myscript.com myscript.com
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MyScript MathPad
This looks like something I could integrate into my workflow.
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- Sep 2020
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www.scientificamerican.com www.scientificamerican.com
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Hotz, J. (n.d.). Can an Algorithm Help Solve Political Paralysis? Scientific American. Retrieved September 21, 2020, from https://www.scientificamerican.com/article/can-an-algorithm-help-solve-political-paralysis/
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- Jul 2020
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en.wikipedia.org en.wikipedia.org
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twitter.com twitter.com
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You should see me doing a write up for a Ruby feature suggestion only to discover that Array#- is hijacked for non-mathematical reasons
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Oh. Oh no. That's... not a thing that should be. People wonder why we can't get be more popular with science / math academics.
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ncatlab.org ncatlab.orgnLab1
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www.youtube.com www.youtube.com
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Net-COVID Session4A: Math Models of Epidemic Spreading in the Time of COVID-19 by Ginestra Bianconi. (2020, May 1). https://www.youtube.com/watch?v=vZ2Ezsffkqs&list=PLVWaQYnj_BZVQal-KQ0rf8CcZJPIhpuO3&index=4
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- Jun 2020
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thinkmath.edc.org thinkmath.edc.org
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In many ways, though, mathematicians treat the problems they are attempting to solve—problems that require highly specialized background and sophisticated thinking and technique—in much the way that non-mathematicians treat puzzles.
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thinkmath.edc.org thinkmath.edc.org
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Research tells us that for skills like the ones students need for mathematics short practices that recur frequently are far more effective than the same amount of time packed into one session.
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www.nature.com www.nature.com
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Viglione, G. (2020). Has Twitter just had its saddest fortnight ever? Nature. https://doi.org/10.1038/d41586-020-01818-3
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www.humblebundle.com www.humblebundle.com
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PAY $1 OR MORE TO ALSO UNLOCK!
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Psych Experiments https://www.amazon.com/Psych-Experiments-Rorschachs-psychologys-fascinating-ebook/dp/B01M3R7RVN/ 4.6/5 $12/$11
DNA is You! https://www.amazon.com/DNA-You-Marvelous-One-Kind-ness/dp/1721400176/ 3.8/5 $11/$14 Want: 8/10
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- May 2020
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psyarxiv.com psyarxiv.com
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Williams, A. E. (2020, April 20). The Global Response to COVID-19 as an Example of a One-Sided Problem Definition in the Absence of General Collective Intelligence. https://doi.org/10.31234/osf.io/emgxc
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wellcomeopenresearch.org wellcomeopenresearch.org
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Endo, A., Centre for the Mathematical Modelling of Infectious Diseases COVID-19 Working Group, Abbott, S., Kucharski, A. J., & Funk, S. (2020). Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China. Wellcome Open Research, 5, 67. https://doi.org/10.12688/wellcomeopenres.15842.1
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www.nature.com www.nature.com
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Vespignani, A., Tian, H., Dye, C. et al. Modelling COVID-19. Nat Rev Phys (2020). https://doi.org/10.1038/s42254-020-0178-4
Tags
- isolation
- COVID-19
- quarentine
- epidemiology
- modeling
- lang:en
- open data
- is:article
- mathematics
- prediction
- superspreading
- pharmaceutical
- China
- emergency
- computational modeling
- contact tracing
- intervention
- antibody testing
- policy
- transmission dynamics
- forecast
- war time
- infection
- challenge
- public health
- complex network
- containment measures
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ai.googleblog.com ai.googleblog.com
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Tsitsulin, A. & Perozzi B. Understanding the Shape of Large-Scale Data. (2020 May 05). Google AI Blog. http://ai.googleblog.com/2020/05/understanding-shape-of-large-scale-data.html
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en.wikipedia.org en.wikipedia.org
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www.bmj.com www.bmj.com
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Response to “Modelling the pandemic”: Reconsidering the quality of evidence from epidemiological models. (2020). https://www.bmj.com/content/369/bmj.m1567/rr-0
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- Apr 2020
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en.wikipedia.org en.wikipedia.org
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The programming language is augmented with natural language description details, where convenient, or with compact mathematical notation.
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en.wikipedia.org en.wikipedia.org
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en.wikipedia.org en.wikipedia.org
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In mathematical jargon, one says that two objects are the same up to an isomorphism.
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arxiv.org arxiv.org
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Velásquez, N., et al. (2020, April 1). Hate multiverse spreads malicious COVID-19 content online beyond individual platform control. Cornell University. arXiv:2004.00673.
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arxiv.org arxiv.org
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Taleb, N. N. (2019). On the Statistical Differences between Binary Forecasts and Real World Payoffs. ArXiv:1907.11162 [Physics, q-Fin]. http://arxiv.org/abs/1907.11162
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Resnick, B. (2020 April 10). Why it's so hard to see into the future of Covid-19. Vox. https://www.vox.com/science-and-health/2020/4/10/21209961/coronavirus-models-covid-19-limitations-imhe
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www.nature.com www.nature.com
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Gog, J. R. (2020). How you can help with COVID-19 modelling. Nature Reviews Physics, 1–2. https://doi.org/10.1038/s42254-020-0175-7
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jamanetwork.com jamanetwork.com
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Jewell, N. P., Lewnard, J. A., & Jewell, B. L. (2020). Predictive Mathematical Models of the COVID-19 Pandemic: Underlying Principles and Value of Projections. JAMA. https://doi.org/10.1001/jama.2020.6585
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en.wikipedia.org en.wikipedia.org
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This type relation is sometimes written S <: T
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- Jan 2020
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en.wikipedia.org en.wikipedia.org
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en.wikipedia.org en.wikipedia.org
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en.wikipedia.org en.wikipedia.org
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math.stackexchange.com math.stackexchange.com
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en.wikipedia.org en.wikipedia.org
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www.amazon.com www.amazon.com
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Norbert Wiener was a mathematician with extraordinarily broad interests. The son of a Harvard professor of Slavic languages, Wiener was reading Dante and Darwin at seven, graduated from Tufts at fourteen, and received a PhD from Harvard at eighteen. He joined MIT's Department of Mathematics in 1919, where he remained until his death in 1964 at sixty-nine. In Ex-Prodigy, Wiener offers an emotionally raw account of being raised as a child prodigy by an overbearing father. In I Am a Mathematician, Wiener describes his research at MIT and how he established the foundations for the multidisciplinary field of cybernetics and the theory of feedback systems. This volume makes available the essence of Wiener's life and thought to a new generation of readers.
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en.wikipedia.org en.wikipedia.org
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en.wikipedia.org en.wikipedia.org
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en.wikipedia.org en.wikipedia.org
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mathoverflow.net mathoverflow.net
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en.wiktionary.org en.wiktionary.org
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en.wikipedia.org en.wikipedia.org
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www.red-gate.com www.red-gate.com
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(quotient CROSS JOIN divisor = dividend) is like (a/b = c) implies (b * c = a) in integer maths
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- Dec 2019
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hybridpedagogy.org hybridpedagogy.org
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“The pupil is thereby ‘schooled’ to confuse teaching with learning, grade advancement with education, a diploma with competence, and fluency with the ability to say something new. His imagination is ‘schooled’ to accept service in place of value.” (1)
I think this issue is particularly important in mathematics. One of the seminal researchers in my field, Les Steffe, distinguishes "school mathematics" from the mathematics of students as a modeling construct; others have conceptualized situated cognition; informal mathematics,
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- Oct 2019
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en.wikipedia.org en.wikipedia.org
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Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "determinacy" is the property of a game whereby such a strategy exists.
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www.youtube.com www.youtube.com
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Harvard Science of Psychedelics Club: The Hyperbolic Geometry of DMT Experiences
Craft 5 mathematics & psychedelics tokens for this video.
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- Feb 2019
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neuralnetworksanddeeplearning.com neuralnetworksanddeeplearning.com
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And so it makes most sense to regard epoch 280 as the point beyond which overfitting is dominating learning in our neural network.
I do not get this. Epoch 15 indicates that we are already over-fitting to the training data set, on? Assuming both training and test set come from the same population that we are trying to learn from.
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If we see that the accuracy on the test data is no longer improving, then we should stop training
This contradicts the earlier statement about epoch 280 being the point where there is over-training.
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It might be that accuracy on the test data and the training data both stop improving at the same time
Can this happen? Can the accuracy on the training data set ever increase with the training epoch?
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What is the limiting value for the output activations aLj
When c is large, small differences in z_j^L are magnified and the function jumps between 0 and 1, depending on the sign of the differences. On the other hand, when c is very small, all activation values will be close to 1/N; where N is the number of neurons in layer L.
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- Nov 2018
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iphysresearch.github.io iphysresearch.github.io
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Stochastic Gradient Descent Optimizes Over-parameterized Deep ReLU Networks
这个哥们的文章和这个月内好几篇的立意基本一致(1811.03804/1811.03962/1811.04918) [抓狂] ，估计作者正写的时候，内心是崩溃的~[笑cry] 赶快强调自己有着不同的 assumption~
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Accelerating Natural Gradient with Higher-Order Invariance
每次看到研究梯度优化理论的 paper，都感觉到无比的神奇！厉害到爆表。。。。
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Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers
全篇的数学理论推导，意在回答2/3层过参的网络可以足够充分地学习和有良好的泛化表现，即使在简单的优化策略（类SGD）等假定下。（FYI： 文章可谓行云流水，直截了当，标准规范，阅读有种赏心悦目的感觉~）
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Gradient Descent Finds Global Minima of Deep Neural Networks
全篇的数学理论证明：深度过参网络可以训练到0。（仅 train loss，非 test loss）+（GD，非 SGD）
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A Convergence Theory for Deep Learning via Over-Parameterization
又一个全篇的数学理论证明，但是没找到 conclusion 到底是啥，唯一接近的是 remark 的信息，但内容也都并不惊奇。不过倒是一个不错的材料，若作为熟悉DNN背后的数学描述的话。
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- Apr 2018
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wiki.c2.com wiki.c2.com
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ConvexHull
In mathematics, the convex hull or convex envelope of a set X of points in the Euclidean plane or in a Euclidean space (or, more generally, in an affine space over the reals) is the smallest convex set that contains X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around X. -Wikipedia
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- Nov 2017
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blog.mrmeyer.com blog.mrmeyer.com
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As Uri Treisman said, “The most common use of algebra in the adult world is helping their kids with algebra.”
CUNY found that more students went on to graduate when they were allowed to take statistics instead of remedial algebra. And the results were the same regardless of race or ethnicity.
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- Jul 2017
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www.pobonline.com www.pobonline.com
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there are two different measurements for the length of a foot in the United States: the International Foot (also commonly called the foot) and the U.S. Survey Foot. The International Foot (which we were all taught in school) is defined as 0.3048 meters, whereas the U.S. Survey Foot is defined as 0.3048006096 meters. The difference of the two equates to 2 parts per million.
For example, in a measurement of 10,000 feet, the difference would be 0.02 feet (just less than one-quarter of an inch). In a measurement of 1 million feet, the difference is 2 feet.
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- Apr 2017
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waitbutwhy.com waitbutwhy.com
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Two people can have one conversation. Three people have four unique conversation groups (three different two-person conversations and a fourth conversation between all three as a group). Five people have 26. Twenty people have 1,048,554.
what's the equation for that?
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- Jan 2017
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betterexplained.com betterexplained.com
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Whether you're a student, parent, or teacher, this book is your key to unlocking the aha! moments that make math click -- and learning enjoyable.
You had me already at the Coffee Cup picture over the equations! :)
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- Nov 2016
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www.scottaaronson.com www.scottaaronson.com
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If you accept this, thenit seems fair to say that untilPversusNPis solved, the story of Hilbert's Entscheidungsproblem|itsrise, its fall, and the consequences for philosophy|is not yet over.
If you accept this bizarre interpretation, then you can suspend the belief in the fact, known to everybody, that \(P \ne NP\), because it hasn't been mathematically proved, and say the question isn't solved yet. Wow, how interesting!
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- Oct 2016
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bullshit.ist bullshit.ist
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Sunil Singh asks us to stop promoting mathematics based on its current applications in business and science. Math is an art that should be enjoyed for its own sake.
This reminded me of A Mathematician's Lament by Paul Lockhart. This is a 25-page essay which was later worked into a 140-page book. (And Sunil Singh has read at least one of them. He credits Lockhart in one of the replies.)
It also reminds me of this article on the history of Gaussian elimination and the birth of matrix algebra. Newton's algebra text included instructions for solving systems of equations -- but it didn't have much practical use until later. (Silly word problems are as old as mathematics.)
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We should let people learn at their own pace. We should neither rush them, nor hold them back. If they show a talent, then encouraging them to push themselves is fine.
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medium.com medium.com
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Math isn't for everyone, and that's fine. The same is true of any other subject. We should help people learn what they are interested in learning.
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- Jul 2016
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www.quantamagazine.org www.quantamagazine.org
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hyperuniform distribution - Appears random at smaller scales, but more predictable at larger scales.
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backchannel.com backchannel.com
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I always found it incredible. He would start with some problem, and fill up pages with calculations. And at the end of it, he would actually get the right answer! But he usually wasn’t satisfied with that. Once he’d gotten the answer, he’d go back and try to figure out why it was obvious. And often he’d come up with one of those classic Feynman straightforward-sounding explanations. And he’d never tell people about all the calculations behind it. Sometimes it was kind of a game for him: having people be flabbergasted by his seemingly instant physical intuition, not knowing that really it was based on some long, hard calculation he’d done.
Straightforward intuition isn't just intuition.
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- Jun 2016
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www.phdontrack.net www.phdontrack.net
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The Hardy-Littlewood Rule
"The rule states that anyone that joins collaboration in good faith will be listed equally as an author, regardless of the relative contributions they end up making.
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- Apr 2016
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bartoszmilewski.com bartoszmilewski.com
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Category Theory for Programmers<br> Haskell and C++, Bartosz Milewski
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- Mar 2016
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www.quantamagazine.org www.quantamagazine.org
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New property of prime numbers discovered. Primes greater than 5 can end with 1, 3, 7, or 9. The next prime is less likely to end with the same digit, and biased toward one of the remaining three. For instance, a prime ending in 3 is most likely to be followed by a prime ending in 9. The bias evens out as the primes get larger, but only very slowly.
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- Feb 2016
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www.math-cs.gordon.edu www.math-cs.gordon.edu
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Number Theory: In Context and Interactive, Karl-Dieter Crisman (online textbook)
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linear.pugetsound.edu linear.pugetsound.edu
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A First Course in Linear Algebra, Robert A. Beezer<br> Free as PDF or online.
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www.brainpickings.org www.brainpickings.org
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Borges “Like works of literature, mathematical ideas help expand our circle of empathy, liberating us from the tyranny of a single, parochial point of view.”
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www.rdmiddlebrook.com www.rdmiddlebrook.com
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Design-Oriented Analysis Rules and Tools, Dr. R. David Middlebrook
Real problems usually have more variables than equations. So approximations and inequalities are essential.
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www.complexityexplorer.org www.complexityexplorer.org
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Free courses and tutorials from the Santa Fe Institute on subjects related to complex systems science.
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- Jan 2016
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www.muslimheritage.com www.muslimheritage.com
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Site on pre-Renaissance Muslim contributions to knowledge. https://twitter.com/HeritageMuslim
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category-theory.mitpress.mit.edu category-theory.mitpress.mit.edu
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Category Theory for the Sciences by David I. Spivak<br> Creative Commons Attribution-NonCommercial-ShareAlike 4.0<br> MIT Press.
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Open source textbooks approved by the American Institute of Mathematics.
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www.golf-simulators.com www.golf-simulators.comPhysics1
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The Physics of Golf, by Martin Paul Gardiner
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- Dec 2015
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math.mit.edu math.mit.eduCT4S.pdf1
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All this time, however, category theory was consistently seen by much of the mathe-matical community as ridiculously abstract. But in the 21st century it has finally cometo find healthy respect within the larger community of pure mathematics. It is the lan-guage of choice for graduate-level algebra and topology courses, and in my opinion willcontinue to establish itself as the basic framework in which mathematics is done
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- May 2015
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oxfordstudent.com oxfordstudent.com
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an infamous number conundrum
One of the central problems in these articles is the glaring lack of math. These personal choices and the environment they have set up for themselves tells us rather only the minuscule part of the story!
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eccentricities of Grigori Perelman
What eccentricities? He just lives with his parents and did not accept either the Fields' Medal or the Claymath's Millennium Prize.
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