If f is the homomorphism that sends elements of Z_12 into products of powers of roots of unity, there are 12 different possible selections for what f(1) equals. f(1) = e^(pii/6) may be the most obvious. since 1 generates Z_12 f(1) is all you need to define to define f, but when the group is not cyclic you need to use a more general method to find the homomorphism.
first you would have to find the basis of the group by representing it as products of groups of prime power order, in this case represent Z_12 as the direct sum of Z_3 and Z_4.
then you know that Z_12 as ordered pairs of elements Z_3 and Z_4 can be generated as linear combinations of (1, 0) and (0, 1).
So in order to define a function f that sends elements of Z_12 into products of roots of unity, you only need to define f((1, 0)) to be ANY 3rd root of unity and f((0, 1)) to be ANY fourth root of unity.
In my case I arbitrarily defined f((1,0)) = e^(2ipi/3) and f((0, 1)) = e^(2ipi/4), which gives f((1, 1)) (the generator of the cyclic group) to be f((1, 0)) * f((0, 1)) = e^(7ipi/6)
Since there are 3 third roots of unity and 4 fourth roots of unity, there are 3*4 = 12 possible different functions and hence twelve different representations of Z_12 using the exponential function (when the group is not cyclic, the homomorphisms will not be isomorphisms).
“Typhoid & swans - it all comes from the same place.”
The thing about the philosophy of mathematics is that there’s no articulable position so absurd that some influential mathematician hasn’t genuinely held to it. You 100% cannot troll these people; you could walk up to a mathematician and say “I’m a radical finitist, I reject the existence of all numbers larger than 1”, and not only would that turn out to be a real thing, some maniac has worked out how to construct set theory under its constraints.
Of the 230 space groups, the symmetries possible in a crystal lattice, the most complicated is group 214, I4_1 32. In the words of Steve Dutch, “This group looks chaotic, but visualizing it is easy. All you do is sit there until little beads of blood form on your forehead.” In 1980, P. Engel used this group to make a 38-sided space-filling polyhedron. In 2016, Moritz Schmitt did a complete study of lattice-based space-filling polyhedra, and the Engel-38 had more faces than anything else.
So, what does Engel-38 look like? Just use the symmetries of I4_! 32 with the following generator point and find the Voronoi cells
Taking advantage of the recent, simpler classification of three-dimensional crystallographic groups by Conway, Delgado-Friedrichs, Huson and Thurston, in a previous paper we proved that Dirichlet stereohedra for any of the 27 “full” cubic groups cannot have more than 25 facets. Here we study the remaining “quarter” cubic groups. With a computer-assisted method, our main result is that Dirichlet stereohedra for the 8 quarter groups, hence for all three-dimensional crystallographic groups, cannot have more than 92 facets.
do you guys use any kind of quirky/unusual/atypical notation in your notes? I love to use ∃? to signalize that I’m trying to prove existence of something while not knowing if the object I’m searching for exists a priori (saw somebody doing it, so I took it from them)
Your idea of writing a “frantically speculative” article on groupoids seems to me a very good one. It is the kind of thing which has traditionally been lacking in mathematics since the very beginnings, I feel, which is one big drawback in comparison to all other sciences, as far as I know. Of course, no creative mathematician can afford not to “speculate”, namely to do more or less daring guesswork as an indispensable source of inspiration. The trouble is that, in obedience to a stern tradition, almost nothing of this appears in writing, and preciously little even in oral communication. The point is that the disrepute of “speculation” or “dream” is such, that even as a strictly private (not to say secret!) activity, it has a tendency to vegetate - much like the desire and drive of love and sex, in too repressive an environment. Despite the “repression”, in the one or two years before I unexpectedly was led to withdraw from the mathematical milieu and to stop publishing, it was more or less clear to me that, besides going on pushing ahead with foundational work in SGA and EGA, I was going to write a wholly science-fiction kind [of] book on “motives”, which was then the most fascinating and mysterious mathematical being I had come to meet so far. As my interests and my emphasis have somewhat shifted since, I doubt I am ever going to write this book - still less anyone else is going to, presumably. But whatever I am going to write in mathematics, I believe a major part of it will be “speculation” or “fiction”, going hand in hand with painstaking, down-to-earth work to get hold of the right kind of notions and structures, to work out comprehensive pictures of still misty landscapes. The notes I am writing up lately are in this spirit, but in this case the landscape isn’t so remote really, and the feeling is rather that, as for the specific program I have been out for is concerned, getting everything straight and clear shouldn’t mean more than a few years work at most for someone who really feels like doing it, maybe less. But of course surprises are bound to turn up on one’s way, and while starting with a few threads in hand, after a while they may have multiplied and become such a bunch that you cannot possibly grasp them all, let alone follow.
—an extract from a letter dated 14/06/83 from Alexander Grothendieck to Ronnie Brown (Bangor)
Andrey Nikolaevich Kolmogorov was one of the giants of 20th-century mathematics. I’ve always found it amazing that the same man was responsible both for establishing the foundations of classical probability theory in the 1930s, and also for co-inventing the theory of algorithmic randomness (a.k.a. Kolmogorov complexity) in the 1960s, which challenged the classical foundations, by holding that it is possible after all to talk about the entropy of an individual object, without reference to any ensemble from which the object was drawn. Incredibly, going strong into his eighties, Kolmogorov then pioneered the study of “sophistication,” which amends Kolmogorov complexity to assign low values both to “simple” objects and “random” ones, and high values only to a third category of objects, which are “neither simple nor random.” So, Kolmogorov was at the vanguard of the revolution, counter-revolution, and counter-counter-revolution.
Mathematical notation provides perhaps the best-known and best-developed example of language used consciously as a tool of thought. Recognition of the important role of notation in mathematics is clear from the quotations from mathematicians given in Cajori’s A History of Mathematical Notations [2, pp.332,331]. Nevertheless, mathematical notation has serious deficiencies. In particular, it lacks universality, and must be interpreted differently according to the topic, according to the author, and even according to the immediate context. Programming languages, because they were designed for the purpose of directing computers, offer important advantages as tools of thought. Not only are they universal (general-purpose), but they are also executable and unambiguous. Executability makes it possible to use computers to perform extensive experiments on ideas expressed in a programming language, and the lack of ambiguity makes possible precise thought experiments. In other respects, however, most programming languages are decidedly inferior to mathematical notation and are little used as tools of thought in ways that would be considered significant by, say, an applied mathematician.
I’m just archiving this Asian Age summary of a lecture from 9th April 2015, because the newspaper webpage has vanished. [Photos]
Time Out listing:
How would you design an object for a world that does not exist? What does such an object say about the world in which we actually live? This idea tugs at the core of ‘design fiction’ practice. For instance, the iPad first appeared in Kubrick’s 2001: A Space Odyssey. Its writer Arthur C. Clarke was also the first to imagine geostationary satellites. The impact of Minority Report on human-computer interfaces cannot be overstated. Even outside of fully formed fictional worlds, a standalone object can trigger many unexpected narratives, such as the famous 3D-printed gun or the US Army’s “indestructible sandwich”. We will discuss these and many other examples of speculative design in this talk.
Asian Age Article: (16 Apr) For Rohit Gupta, the essential question isn’t “why” but “why not”. He held forth on the concept of “design fiction” at a talk in the city recently. His previous projects include trying to figure out a way to fit astronomical contraptions on top of auto-rickshaws and coming up with a mechanism to type through walking (in which one could type out a whole text message in no less than seven hours!). While many around him may wonder “why”, for Rohit Gupta aka Compasswala aka fadesingh, the only question is “why not”. Giving a talk on design fiction at the Maker’s Asylum, the researcher who studies the history of science and mathematics explained why for him fiction was everywhere, not just in the depiction of future, but even the past. Speaking about what exactly design fiction is, Rohit says, “It’s about the objects. Design fiction deals with how to create objects that describe or imply a story or an aspect about a world that doesn’t exist.” Going on to give us an example in his own style, Rohit says, “Let us consider hypothetically that there was a catastrophic event in Mumbai in 1960 that entirely changed the city. Now let us take a map of Mumbai in 2015 that shows how it looks now in that scenario. We don’t have to describe everything that happened in the time frame between the disaster and now, but just the map, which is an object of design fiction can show or tell us a huge number of details about that world. ‘That’ is design fiction.” Rohit adds, “Design fiction has existed for a long time. Now we may have sci-fi movies and earlier there were books. But those were just the interfaces. It has existed for long before these interfaces came about.” While sci-fi and fiction is usually considered to depict the future or altogether different realities, Rohit contends, it is equally relevant and present in describing the past as well.
He explains, “Not many might have heard about the Ishango bone. Now the Ishango bone is considered to be the oldest mathematical instrument known to man. But basically it is just a simple bone with hand carved lines drawn on it in varying sequences. Now what these prehistoric humans were trying to do with those lines we don’t know, but researchers have interpreted various reasons ranging from calculating menstrual cycles to lunar calendars. But this is our modern interpretation of what this particular object tells us. It could well have been something else but these are the stories we are interpreting from it. So this is design fiction as well, only in the past.” Design fiction, says Rohit, varies from the miniscule to the astronomical. “You could create a simple toy in a workshop or you could even create an enter solar system like Asimov (Isaac) did in Nightfall.” But while the potential of design fiction could be limitless, it is upto us to ask the questions from whence we can derive the answers says Rohit. “This is increasingly becoming a trend. Researchers in top institutes are taking questions that may sound ridiculous and are coming up with the most scientific explanations for them. For example, 'How does a Muslim astronomer face Mecca while in space’ but believe it or not the Malaysians have actually come up with an entire manual for it.” And progress, says Rohit is all about not shying away from doing what may sound crazy. “One of my friends, a poet named Christian Book is now engaged in a project to create the world’s first indestructible book. How he’s doing it is the most interesting part. He actually took a strain of this microbe called Dienococcus Radiodurans, which is an extremophile (Something which can survive in extreme conditions such nuclear blasts, volcanoes or even in space) and imprinting a poem into its very DNA and is planning to launch it off into space. Now whom he is writing for or what the poem itself is irrelevant. But the only question is 'Why the hell not’,” concludes the Compasswala.
What does “causality” mean, and how can you represent it mathematically? How can you encode causal assumptions, and what bearing do they have on data analysis? These types of questions are at the core of the practice of data science, but deep knowledge about them is surprisingly uncommon.
a paper by Marko Rodriguez called A Methodology For Studying Various Interpretations of the N,N-dimethyltryptamine-Induced Alternate Reality, [suggesting] among other things that you could prove DMT entities were real by taking the drug and then asking the entities you meet to factor large numbers which you were sure you couldn’t factor yourself. So to that end, could you do me a big favor and tell me the factors of 1,522,605,027, 922,533,360, 535,618,378, 132,637,429, 718,068,114, 961,380,688, 657,908,494, 580,122,963, 258,952,897, 654,000,350, 692,006,139?
“Universal love,” said the cactus person.
“Transcendent joy,” said the big green bat.
The sea turned hot and geysers shot up from the floor below. First one of wine, then one of brine, then one more yet of turpentine, and we three stared at the show.
Here, we study the characteristics of functional brain networks at the mesoscopic level from a novel perspective that highlights the role of inhomogeneities in the fabric of functional connections. […] The results show that the homological structure of the brain’s functional patterns undergoes a dramatic change post-psilocybin, characterized by the appearance of many transient structures of low stability and of a small number of persistent ones that are not observed in the case of placebo.
Popping peyote buttons with his assistant in the laboratory, Klüver noticed the repeating geometric shapes in mescaline-induced hallucinations and classified them into four types, which he called form constants: tunnels and funnels, spirals, lattices including honeycombs and triangles, and cobwebs. In the 1970s the mathematicians Jack D. Cowan and G. Bard Ermentrout used Klüver’s classification to build a theory describing what is going on in our brain when it tricks us into believing that we are seeing geometric patterns. Their theory has since been elaborated by other scientists, including Paul Bressloff, Professor of Mathematical and Computational Neuroscience at the newly established Oxford Centre for Collaborative Applied Mathematics.
I very much enjoy reading the “What Is…?” column in the Notices of the AMS. Unfortunately, there seemed to be no index to this column. I have therefore created this one in the hope that it’ll be helpful to others as well.
Excellence comes from qualitative changes in behavior, not just quantitative ones. More time practicing is not good enough. Nor is simply moving your arms faster! A low-level breaststroke swimmer does very different things than a top-ranked one. The low-level swimmer tends to pull her arms far back beneath her, kick the legs out very wide without bringing them together at the finish, lift herself high out of the water on the turn, and fail to go underwater for a long ways after the turn. The top-ranked one sculls her arms out to the side and sweeps back in, kicks narrowly with the feet finishing together, stays low on the turns, and goes underwater for a long distance after the turn. They’re completely different!
This post is a crash course on the notation used in programming language theory (“PL theory” for short). For a much more thorough introduction, I recommend Types and Programming Languages by Benjamin C. Pierce and Semantic Engineering with PLT Redex by Felleisen, Findler, and Flatt. I’ll assume the reader is an experienced programmer but not an experienced mathematician or PL theorist. I’ll start with the most basic definitions and try to build up quickly.
Attention conservation notice: Over 7800 words about optimal planning for a socialist economy and its intersection with computational complexity theory. This is about as relevant to the world around us as debating whether a devotee of the Olympian gods should approve of transgenic organisms. (Or: centaurs, yes or no?) Contains mathematical symbols (uglified and rendered slightly inexact by HTML) but no actual math, and uses Red Plenty mostly as a launching point for a tangent.
I’ve always been intrigued by the sensation of movement in music. And it is fair to say that it was my first calculus class that led me to graduate study in mathematics because, for the first time, I saw movement in mathematics. My fascination with each of these was nudged again by an interview with jazz pianist Vijay Iyer that I heard on NPR’s All Things Considered.
