Tweet from MathType (@MathType)
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Who can tell us more about the Kronekernel Delta process and how, when, where it can be used? MathType (@MathType) Tweeted: The Iverson Bracket is a powerful generalization of the well known Kronecker Delta, very useful for embedding logical conditions and truth values into your formulas #Notation #MathType .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/ |
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Kronecker deltas are effectively selection functions.
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Re: Tweet from MathType (@MathType)
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In reply to this post by Tom Johnson
Suppose one sequences a relevant part of a genome of a bunch of patients -- some who get die from COVID-19 and some that do not -- giving say 10 million columns of data and a few thousand rows per patient. Which minimal set of those columns
are predictive of death? A function that includes these selection functions, each with a cost, and an outcome column would be typical in a variable selection procedure. From: Friam <[hidden email]> on behalf of Tom Johnson <[hidden email]> Who can tell us more about the Kronekernel Delta process and how, when, where it can be used? MathType (@MathType) Tweeted: .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/ |
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In reply to this post by Tom Johnson
Tom - Can you provide a little more context? The Kroneker Delta/Iverson Bracket is a pretty fundamental and generally useful(used) abstraction across lots of domains. If you are trying to use the tweeted advice you quoted, you must be working with "logical conditions and truth values in some formula"? But that only narrows it down a little? - Steve
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Re: Tweet from MathType (@MathType)
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Sorry, I don't have any context. I just saw the reference and reference to
Kroneker Delta. I had not heard of it, so I even before I did research, I thought I would save time by asking this august panel. Tom, Duke of Lazy ============================================ Tom Johnson - [hidden email] Institute for Analytic Journalism -- Santa Fe, NM USA 505.577.6482(c) 505.473.9646(h) NM Foundation for Open Government Check out It's The People's Data ============================================ On Mon, Jun 8, 2020 at 12:54 PM Steve Smith <[hidden email]> wrote:
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Kronecker delta Is trivial. It has two arguments. If they're equal the result is 1. If not the result is 0. --- Frank C. Wimberly 140 Calle Ojo Feliz, Santa Fe, NM 87505 505 670-9918 Santa Fe, NM On Mon, Jun 8, 2020, 2:05 PM Tom Johnson <[hidden email]> wrote:
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For example, the identity matrix (all zeroes except for ones on the diagonal) is given by the ith element of the jth row is the Kronecker delta of i and j. —Barry On 8 Jun 2020, at 16:12, Frank Wimberly wrote:
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Barry wrote:
and Frank wrote:Which makes it all the more enigmatic. Simple things like '0', the identity matrix, 'i', 'e', 'pi', can have profound implications but by themselves seem anywhere from trivial to self-evident. Rather than pontificate (bombasticate, wax reflectively) on my own experiences/random-thoughts/free-associations, I'd be interested in what others here might know of where the Kronecker Delta would be likely applicable to the types of things many of us (might?) assume Tom is interested in (application to big data, natural language processing, etc. in the context of Journalism?) - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/ |
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This post was updated on .
Steve, Tom,
The Kronecker delta (or Dirac delta or indicator function depending on context) appears in the technical machinery of mathematics and so does not usually show up meaningfully in the target science of the mathematical theory. The delta is a lot like a projection map (likely dual for those playing at home) in that it is useful for selecting data out of larger data, but not in any magical way. It is exactly like when we select a column in a Google doc, maybe I move the mouse over to the column and then click the mouse button. This process is internal to how I work with the data mechanistically and does not really tell me anything about the content. Seeming exceptions do arise, like when one is working with expectations in probability theory, but even these cases just make the process of 'counting' easier. The reason we perhaps wish to use something like the Iverson bracket is so that we can keep track of types. By mapping a truth value to a number, like claiming True to be 1, we can count how many people have their hands raised, say. Many people don't really concern themselves with these differences and are somehow ok with it when we write stuff like 3 * True = 3, but they are usually javascript programmers. Knuth advocates for the use of the Iverson bracket (see Concrete Mathematics) because concerning oneself with types often leads to more clear and powerful expressions of thought. Jon -- Sent from: http://friam.471366.n2.nabble.com/ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/ |
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OK. The Kronecker delta on a set A is a function or set of ordered pairs. The arguments of the function are ordered pairs of the elements of A. The elements of the function are defined by <<x,y>, z> where x and y are elements of A and z is in {0, 1}. In other words the domain of the Kronecker delta is the set of ordered pairs of elements of A and it's range is the set {0, 1} and the function is evaluated as delta(x, x) = 1 for all x and delta(x, y) = 0 if x != y. Is that better? I stand by my original post --- Frank C. Wimberly 140 Calle Ojo Feliz, Santa Fe, NM 87505 505 670-9918 Santa Fe, NM On Mon, Jun 8, 2020, 3:33 PM Jon Zingale <[hidden email]> wrote: Steve, Tom, - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/ |
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In reply to this post by Tom Johnson
Tom, Reflecting a bit more, there are other places in mathematics where similar ideas arise. Consider a series like: 1 + 1/2 + 1/4 + 1/8 + ... mathematicians will often wish to treat these infinite sums as if they were lists. One thing to do with a list is to pop the head from the list and return just the tail. Multiplying the list by 1/2 does exactly this: 1/2 (1 + 1/2 + 1/4 + 1/8 + ...) 1/2 + 1/4 + 1/8 + ... Now I have just the same list but without the head. Manipulations like this one are more-or-less part of the machinery for working with a series. Another classic example are the pair of functions div and mod. The first of these acts like division but only gives back the integer part (quotient), thus div 4 25 is 6. mod on the other hand is a function which returns the remainder, 1 here. Now given a number like 273427893045 in base 10, we can use mod 10 to pop the 5 off the end of the number as-if-it-were a list and div 10 to return the rest of the number as-if-it-were a list. Examples like these are ubiquitous in mathematics and are in part what makes the whole project seem like black magic or index twiddling. Really, they are perhaps just conveniences that arise via analogies between one domain of mathematics and another. The Iverson bracket is similarly one such device, connecting logic (Bool) to number (Int). Jon - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/ |
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In reply to this post by Frank Wimberly-2
I have only seen them inside of summations, Σs, but I'm sure they are used elsewhere. They are used like a filter. For instance, if you want to sum weights of butterflies in an insect database you would say "for every insect in X, if it is a butterfly then add it's weight to the sum". When it gets translated into an equation, the part "if its a butterfly" gets turned into a Kronecker delta function where it outputs 1 when it is a butterfly and 0 otherwise. So in some sort of pseudo equation, it might look like y=Σ_of_i_in_X( Kron_delta( Label(i), "butterfly") * Weight(i) ) I hope this doesn't muddy the water too much, Cody Smith On Mon, Jun 8, 2020 at 4:00 PM Frank Wimberly <[hidden email]> wrote:
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In reply to this post by Tom Johnson
ps. Further down that twitter stream, there is a math problem presented by the UK mathematics trust. The problem is to find the smallest prime which divides (300^300)-1. Using the ideas in my post above we can see that (300^300)-1 is a very large number: 136891479058588375991326027382088315966463695625337436471480190078368997177499076593800206155688941388250484440597994042813512732765695774566000999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 For those that have learned the 3's trick, we can quickly verify that 3 does NOT divide this number because the sum of the digits here gives 6092 which is NOT divisible by 3. This is an example of considering the large number above to be a list of numbers. Next, applying the Pollard Rho method, I quickly found that 7 DOES divide this number. To verify, I use a method similar to the 3's trick (well really the same) and pop the last 9 off the list, multiply it by 2 and then subtract it from the remaining list. This gives another really large number, but iterating through the list eventually gives a much smaller number that can easily be verified to be divisible by 7. Therefore the whole number is divisible by 7. Now part of the beauty of the div/mod characterization I mentioned earlier is that we can then arithmetically define Kronecker deltas for numbers by defining functions that act on numbers as lists, which I do here. Techniques like this appear almost everywhere and deltas are a similar such thing. Consider the Dirac Delta in particular. There we have a generalized function that is tremendously useful for selecting values out of a time-series and yet really isn't a function at all. Jon - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/ |
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In reply to this post by cody dooderson
Not at all. --- Frank C. Wimberly 140 Calle Ojo Feliz, Santa Fe, NM 87505 505 670-9918 Santa Fe, NM On Mon, Jun 8, 2020, 5:30 PM cody dooderson <[hidden email]> wrote:
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In reply to this post by Tom Johnson
Tom,
Perhaps one of the most common usages of the Kronecker delta, and a usage we skirted in the discussion, is to establish biorthogonality between a vector space and its dual space. The Kronecker delta arises when given an indexed basis and its indexed dual set (which may or may not span the dual space), we take inner products of vectors in the first with vectors in the second. Because of linear independence in both sets, the inner product will take the value 1 when the vectors correspond and 0 otherwise. The Kronecker delta, in this case, is a manifestation of inner products and duality. Jon -- Sent from: http://friam.471366.n2.nabble.com/ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/ |
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Regardless of its use, the concept is equally simple. The definition of "=" is the only difference. Frank --- Frank C. Wimberly 140 Calle Ojo Feliz, Santa Fe, NM 87505 505 670-9918 Santa Fe, NM On Sat, Jun 13, 2020, 12:23 PM Jon Zingale <[hidden email]> wrote: Tom, - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/ |
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