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I am a graduate mathematics student, and you are embarrasingly wrong. First, the term kernel certainly does have meaning in this context. The kernel of a mapp

I am a graduate mathematics student, and you are embarrasingly wrong. First, the term kernel certainly does have meaning in this context. The kernel of a mapping is the set of arguments it sends to the identity element (zero here), so in this case the kernel is just {1111, 2222, 3333, ... , 9999}. Like you, I thought the author should have used the term "fixed point," but there are situations where his way would be better. Details if anyone asks. Now, why could anyone possibly gain from studying permutations on the digits of integers represented in base ten? EDIT: the short answer is "cryptography." I'm cutting this post in two. For the longer answer, see below.

mnemonicsloth 18y ago
Here's the long answer to the question: what could anyone learn from rearranging the digits of an integer? First off, it's helpful to know that the integers (denoted {Z}) are the canonical example of what's called ring structure, which basically means + and * work normally, but you aren't guaranteed clean division. Since they're simple, mathematicians sometimes use {Z} as a stand-in for more persnickety structures in the same way they use n-dimensional Euclidean space as a proxy for thinking about infinite-dimensional vector spaces. Articles like this one usually get written when an algebraist or number theorist publishes an epic proof, and the only way to make it comprehensible to a lay audience is to project it down into toy examples in a simple space. That's kind of a "trust me" answer by itself, of course. So without further ado, the math: There are two notable things about rings. You can make polynomials with + and * in them, and division generally leaves you with remainders. Remainders are okay, though, because they let you define new rings. {Z mod 12 Z}, for example, is shorthand for "integers mod 12," or more formally, "map the integers to a new space where all the multiples of 12 get sent to zero," so {1, 2, ..., 11 } act normal, 12 is actually zero, 13=12+1=0+1=1 and so on. It turns out that this is all you need to represent numbers in place-value notation. A number N in base r has a representation of n digits: N = (A_n)(r^n) + ... + (A_2)(r^2) + (A_1)r + A_0 which is just a polynomial whose coefficients are members of the ring {Z mod r*Z}. If you map that back to the integers, plug in the actual value of r, and crunch the numbers, you get your original value of N back. So, "rearranging the digits of an integer" is actually "permuting the coefficients of polynomials with coefficients mod r." Now, suppose you have an email message that's 12000 characters long. You can think of that message as the polynomial representation, modulo 256, of a really big number. Mapping that number into different modular spaces, or permuting the coefficients of its various representations so that you can get it back later ("encrypting the message") forms the basis of the field of cryptography, which in turn is closely related to information theory and the theory of ergodic processes, etc, etc, etc. For more details, you might start with: http://www.amazon.com/Applied-Cryptography-Protocols-Algorithms-Source/dp/0471117099 http://www.amazon.com/Applied-Cryptography-Protocols-Algorit... http://www.inference.phy.cam.ac.uk/mackay/itila/book.html http://www.inference.phy.cam.ac.uk/mackay/itila/book.html
jdoliner 18y ago
I agree that if we decide that 0 is the identity element then the concept of the kernel does have meaning. Well except that the article says the map isn't defined over {1111, 2222, 3333, ... , 9999} However 0 is just the additive identity and there's really no reason to select addition. We could just as easily select the multiplicative identity (1). So yes if you arbitrarily select 0 and then redefine function you can form a concept of the kernel (it's pretty boring as far as I can see). However the article does specifically say that 6174 is the kernel. Now for that to be correct under your definition it would need to be true that: 7641 - 1467 = 0 Which we all know to be false. Your post seems to suggest that you think this use of kernel was a poor word choice instead of completely incorrect. Perhaps you could send me the details that your post promises.

mnemonicsloth 18y ago

Here's the long answer to the question: what could anyone learn from rearranging the digits of an integer? First off, it's helpful to know that the integers (denoted {Z}) are the canonical example of what's called ring structure, which basically means + and * work normally, but you aren't guaranteed clean division. Since they're simple, mathematicians sometimes use {Z} as a stand-in for more persnickety structures in the same way they use n-dimensional Euclidean space as a proxy for thinking about infinite-dimensional vector spaces. Articles like this one usually get written when an algebraist or number theorist publishes an epic proof, and the only way to make it comprehensible to a lay audience is to project it down into toy examples in a simple space. That's kind of a "trust me" answer by itself, of course. So without further ado, the math: There are two notable things about rings. You can make polynomials with + and * in them, and division generally leaves you with remainders. Remainders are okay, though, because they let you define new rings. {Z mod 12 Z}, for example, is shorthand for "integers mod 12," or more formally, "map the integers to a new space where all the multiples of 12 get sent to zero," so {1, 2, ..., 11 } act normal, 12 is actually zero, 13=12+1=0+1=1 and so on. It turns out that this is all you need to represent numbers in place-value notation. A number N in base r has a representation of n digits: N = (A_n)(r^n) + ... + (A_2)(r^2) + (A_1)r + A_0 which is just a polynomial whose coefficients are members of the ring {Z mod r*Z}. If you map that back to the integers, plug in the actual value of r, and crunch the numbers, you get your original value of N back. So, "rearranging the digits of an integer" is actually "permuting the coefficients of polynomials with coefficients mod r." Now, suppose you have an email message that's 12000 characters long. You can think of that message as the polynomial representation, modulo 256, of a really big number. Mapping that number into different modular spaces, or permuting the coefficients of its various representations so that you can get it back later ("encrypting the message") forms the basis of the field of cryptography, which in turn is closely related to information theory and the theory of ergodic processes, etc, etc, etc. For more details, you might start with: http://www.amazon.com/Applied-Cryptography-Protocols-Algorithms-Source/dp/0471117099 http://www.amazon.com/Applied-Cryptography-Protocols-Algorit... http://www.inference.phy.cam.ac.uk/mackay/itila/book.html http://www.inference.phy.cam.ac.uk/mackay/itila/book.html

jdoliner 18y ago

I agree that if we decide that 0 is the identity element then the concept of the kernel does have meaning. Well except that the article says the map isn't defined over {1111, 2222, 3333, ... , 9999} However 0 is just the additive identity and there's really no reason to select addition. We could just as easily select the multiplicative identity (1). So yes if you arbitrarily select 0 and then redefine function you can form a concept of the kernel (it's pretty boring as far as I can see). However the article does specifically say that 6174 is the kernel. Now for that to be correct under your definition it would need to be true that: 7641 - 1467 = 0 Which we all know to be false. Your post seems to suggest that you think this use of kernel was a poor word choice instead of completely incorrect. Perhaps you could send me the details that your post promises.