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Hi, I am trying to figure out the topology on that space. If I understand correctly' the clopen sets in this topology are A={F | A is in F}. My question is this: suppose that a point x is in A in the original topology of the space X. Does the clopen set A we defined on the Stone-Cech Compactification perform an open neighborhood for the principal filter of x? Is this the idea of this compactification? And if so, then what is the meaning of convergence in this space? Thanx! —Preceding unsigned comment added by Topologia clalit ( talk • contribs) 09:13, 3 August 2010 (UTC)
Thanks this is very helpfull. From what you have said, I see now, that for every infinite sequence of ultrafilters there exists an open set that contains an infinite number of ultrafilters from that sequence, since for every A in the original topological space X, either A is in infinite number of ultrafilters or the complement of A is in infinite number of ultrafilters. But, How can I show that every sequence of ultrafilters has a convergence subsequence in this Stone-Cech topology. In other words, I guess, how can I show or be convinced that the Stone-Cach compactification is indeed compact? Thanx! —Preceding unsigned comment added by Topologia clalit ( talk • contribs) 10:54, 3 August 2010 (UTC)
If A is a set with 10 elements how many total order binary relations are possible on A. Thanks - Shahab ( talk) 13:56, 3 August 2010 (UTC)
Here is the discussion moved from above:
Thanks for the responses, guys. I have some follow-up questions, but unfortunately I also have other priorities, so it'll take me a while to compile them together. Feel free to add any further contributions in the meantime. -- COVIZAPIBETEFOKY ( talk) 14:37, 3 August 2010 (UTC)
Correct me if I'm wrong, but I'm pretty sure both of the examples that 67.122.211.208 gave are not quite what I'm asking for; they are people who believe that number theory is either outright inconsistent or that it is nonsense not worth studying. My question refers to people who believe in the inherent concept of number theory, but don't believe that there is only one platonic ideal of what a model of number theory should look like exactly. Of course, this is well outside the bounds of mathematical discourse and quite blatantly into philosophy, so possibly it is a question not worth discussing too much. I'm not entirely sure how to justify my own belief that number theory ought to only have one reasonable model, except to say that it's obvious, because in number theory, we can name every object, at least in theory (starting with 0, then S0, SS0, SSS0, and so on).
203.97.79.114's point seems silly to me, since having a reasonable model of set theory is clearly quite a stronger statement than having a reasonable model of number theory (in the sense that it must include a reasonable minimal inductive set containing the empty set, serving as our set of natural numbers, and much more). Maybe there's some merit to it that I don't see, though. -- COVIZAPIBETEFOKY ( talk) 11:43, 6 August 2010 (UTC)
I saw this problem in a computer programming competition and while I found it trivial to write a program to perform the task, I would like to know if this is a named problem in mathematics so I can see if there are better well-known algorithms. The problem:
I did it two ways. The first, which I know will not work in special cases, is to remove the lowest number in any row/column with more than one value until there is no row/column with more than one value. That passed the test in the competition. I wrote another version which attempted removing each number, one at a time, in a breadth-first search until it tried all combinations of removals and picked one of the maximized solutions. That works, but is obviously very cost-intensive. Therefore, I am interested in reading about better solutions. -- kainaw ™ 14:53, 3 August 2010 (UTC)
Ok the definition of "linear in each variable" is a little dodgy when we talk about infinite tensor products. How about this:
Given a collection Mi of R modules, call a map L defined on their direct product linear in i (index/variable) if for every two tuple x,y such that x(j)=x(j) for every j not equal to i, then L(z)=L(x)+L(y), where z(j)=x(j) for j different from i and z(i)=x(i)+y(i). And for every tuple x and scalar a in R, L(z)=aL(x), where z(i)=ax(i) and same as x everywhere else. L is called multilinear if it's linear in every index.
The relations to make the tensor product in the free module generated by the direct product can be done in similar ways. Am I correct? I find "linear in each variable" in the usual sense quite hard to interpret when the direct product is infinite Money is tight ( talk) 14:56, 3 August 2010 (UTC)
In terms of side length, what's the hypervolume of each of the regular polychora? -- 138.110.25.31 ( talk) 19:51, 3 August 2010 (UTC)
Should we add the volumes to the data in tables and boxes for these six "Platonic bodies", and for the three corresponding larger dimensional families? JoergenB ( talk) 17:41, 8 August 2010 (UTC)
Okay, what I mean is I want to be able to type a sum with two lines of description, such as for an Eisenstein series
I used \stackrel there but note that the top line is smaller. I see in books often where they do it and the two lines are same size font. Thanks. StatisticsMan ( talk) 21:28, 3 August 2010 (UTC)
Mathematics desk | ||
---|---|---|
< August 2 | << Jul | August | Sep >> | August 4 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Hi, I am trying to figure out the topology on that space. If I understand correctly' the clopen sets in this topology are A={F | A is in F}. My question is this: suppose that a point x is in A in the original topology of the space X. Does the clopen set A we defined on the Stone-Cech Compactification perform an open neighborhood for the principal filter of x? Is this the idea of this compactification? And if so, then what is the meaning of convergence in this space? Thanx! —Preceding unsigned comment added by Topologia clalit ( talk • contribs) 09:13, 3 August 2010 (UTC)
Thanks this is very helpfull. From what you have said, I see now, that for every infinite sequence of ultrafilters there exists an open set that contains an infinite number of ultrafilters from that sequence, since for every A in the original topological space X, either A is in infinite number of ultrafilters or the complement of A is in infinite number of ultrafilters. But, How can I show that every sequence of ultrafilters has a convergence subsequence in this Stone-Cech topology. In other words, I guess, how can I show or be convinced that the Stone-Cach compactification is indeed compact? Thanx! —Preceding unsigned comment added by Topologia clalit ( talk • contribs) 10:54, 3 August 2010 (UTC)
If A is a set with 10 elements how many total order binary relations are possible on A. Thanks - Shahab ( talk) 13:56, 3 August 2010 (UTC)
Here is the discussion moved from above:
Thanks for the responses, guys. I have some follow-up questions, but unfortunately I also have other priorities, so it'll take me a while to compile them together. Feel free to add any further contributions in the meantime. -- COVIZAPIBETEFOKY ( talk) 14:37, 3 August 2010 (UTC)
Correct me if I'm wrong, but I'm pretty sure both of the examples that 67.122.211.208 gave are not quite what I'm asking for; they are people who believe that number theory is either outright inconsistent or that it is nonsense not worth studying. My question refers to people who believe in the inherent concept of number theory, but don't believe that there is only one platonic ideal of what a model of number theory should look like exactly. Of course, this is well outside the bounds of mathematical discourse and quite blatantly into philosophy, so possibly it is a question not worth discussing too much. I'm not entirely sure how to justify my own belief that number theory ought to only have one reasonable model, except to say that it's obvious, because in number theory, we can name every object, at least in theory (starting with 0, then S0, SS0, SSS0, and so on).
203.97.79.114's point seems silly to me, since having a reasonable model of set theory is clearly quite a stronger statement than having a reasonable model of number theory (in the sense that it must include a reasonable minimal inductive set containing the empty set, serving as our set of natural numbers, and much more). Maybe there's some merit to it that I don't see, though. -- COVIZAPIBETEFOKY ( talk) 11:43, 6 August 2010 (UTC)
I saw this problem in a computer programming competition and while I found it trivial to write a program to perform the task, I would like to know if this is a named problem in mathematics so I can see if there are better well-known algorithms. The problem:
I did it two ways. The first, which I know will not work in special cases, is to remove the lowest number in any row/column with more than one value until there is no row/column with more than one value. That passed the test in the competition. I wrote another version which attempted removing each number, one at a time, in a breadth-first search until it tried all combinations of removals and picked one of the maximized solutions. That works, but is obviously very cost-intensive. Therefore, I am interested in reading about better solutions. -- kainaw ™ 14:53, 3 August 2010 (UTC)
Ok the definition of "linear in each variable" is a little dodgy when we talk about infinite tensor products. How about this:
Given a collection Mi of R modules, call a map L defined on their direct product linear in i (index/variable) if for every two tuple x,y such that x(j)=x(j) for every j not equal to i, then L(z)=L(x)+L(y), where z(j)=x(j) for j different from i and z(i)=x(i)+y(i). And for every tuple x and scalar a in R, L(z)=aL(x), where z(i)=ax(i) and same as x everywhere else. L is called multilinear if it's linear in every index.
The relations to make the tensor product in the free module generated by the direct product can be done in similar ways. Am I correct? I find "linear in each variable" in the usual sense quite hard to interpret when the direct product is infinite Money is tight ( talk) 14:56, 3 August 2010 (UTC)
In terms of side length, what's the hypervolume of each of the regular polychora? -- 138.110.25.31 ( talk) 19:51, 3 August 2010 (UTC)
Should we add the volumes to the data in tables and boxes for these six "Platonic bodies", and for the three corresponding larger dimensional families? JoergenB ( talk) 17:41, 8 August 2010 (UTC)
Okay, what I mean is I want to be able to type a sum with two lines of description, such as for an Eisenstein series
I used \stackrel there but note that the top line is smaller. I see in books often where they do it and the two lines are same size font. Thanks. StatisticsMan ( talk) 21:28, 3 August 2010 (UTC)