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I am studying "FAMILIES OF SETS" in the book "Real analysis for graduate
students(v 3.1)" by Richard F. Bass and I could not figure out this example.
The example
Definition of an algebra and sigma-algebra
They are stating that
Verifying parts (1) and (2) of the definition is easy.
This is exactly the part I do not understand.
I do not understand how we define the complement for a set {0,1,2}. The set {0,1,2} should be in D, as it is countable, but what is its complement? It seems that it is {...,-3-2-1} union {3,4,5,...}. Are these sets both countable?
And what about the set {1.1, 2.5, 3.4}, how do we define the complement of such a set? (and how do we show that it is in fact in D?)
P.S.
I do not know how to write formulas so I'm sorry for the ugly mathematical writing
The complement of {0,1,2} in R is every real number except those three. It's also in the algebra because that was the definition, you defined an algebra of all countable subsets or the complements of countable subsets.
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In formal, does there exist such that for all ?
No, $\pi$ and thus $\pi/2$ are irrational, thus the (additive) equivalence classes of the integers modulo $2\pi$ are dense in $\Bbb R$ and thus approach infinitesimally, but never reach $\pi/2$.
The fundamental fact is that for any given number x the set of numbers {mx+n : m,n integer} is either
an arithmetic sequence {mr : r integer} which implies and is equivalent to x as a multiple of r being rational, or
dense in the real numbers, which by the first case happens for all irrational x.
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I am reading a review to compare Mathematica to APL/J. One question raised in the article seems very interesting to me:
Is Mathematica really the way to go to express our creative thoughts –
viz back to a 17th century notation designed for parchment instead of
forward to a twentieth-century one designed for computers?
Can one share examples of Iverson's notation vs traditional math notation to demonstrate the edge of APL/J on expressing and solving math problems? This would be greatly helpful for new comers.
One example: Alternating series.
Alternating sum is very common in mathematics. But it is cumbersome to put the sign before each term:
in APL and J, because of the order of operations, it is
-/a
I recommend reading Iverson's paper Notation as a Tool of Thought, kindly provided by the J folks. It deals precisely with this issue.
In it you'll find many Math proofs derived using APL instead of the classical notation, along with accompanying commentary. Here's a redacted example, proving Gauss's formula for the arithmetic series:
+/⍳n
+/⌽⍳n ⍝ as + is associative and commutative
((+/⍳n)+(+/⌽⍳n))÷2 ⍝ as x=(x+x)÷2
(+/(⍳n)+(⌽⍳n))÷2 ⍝ as + is associative and commutative
(+/(n/n+1))÷2 ⍝ summing each respective x∊⍳n and y∊⌽⍳n, y=n+1-x → (x+y)=n+1
(n×n+1)÷2 ⍝ per definition of × (times)
Other articles by Iverson, Hui and friends are also illuminating. Again, the J folks provide a notable library.
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For my theory of computation class, we are supposed to do some review/practice problems to work off the rust and make sure we are ready for the course. Some of the problems are induction proofs. I did this at one time, but apparently it has completely escaped me. I've watched a couple tutorials, but still can't do problem 'a'. If anyone can walk me through the first problem I'm pretty sure I could figure out the second one on my own. Any help would be appreciated!
First verify it holds for n = 1.
Then assume it is true for n = x ( the sum of the first x squares ) and then try to compute the sum of the the first x + 1 squares. You know the result for the first x, you just add the last square to that sum. From there it should be easy.
And you posted on the wrong site.
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Maybe my question will fail to be specific but when fitting a glme model (using lme4 package in R) I get for one of the parameters SE=1000, with the estimated parameter as high as 16. The variable is a dichotomous variable. My question is if there might be an explanation for such a result, considering that the other parameters have parameters and SE that seem ok
That's a sign that you have complete separation. You should re-run the model without that covariate. Since its an ME model you may need to do a tabulation of outcome by covariate by levels to see what is happening. More details would allow greater specificity in our answers.
This is a link to a posting by Jarrod Hadfield, one of the guRus on the R mixed model mailing list. It demonstrates how complete separation leads to the Hauck-Donner effect, and it offers some further approaches to attempt dealing with it.
You may be seeing a case of the Hauck-Donner effect. Here is one post that discusses it, you can read the original paper or search the web for additional discussions.
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I'm programming some program which calculates the limit of markov chain.
if the markov matrix diverges, I should transform it into the form
dA + (1-d)E, where both A and E are n * n matrix, and all of the elements of E are 1/n.
But if I apply that transformation when the input converges, the wrong value comes out.
Is there any easy way to check if the markov matrix converges?
I'm not going to go into detail, because it's an entire field unto itself. Although the general convergence theorem states that any finite Markov chain that is aperiodic and irreducible converges (to its stationary distribution). Irreducibility is simple to check (it's equivalent to connectedness in graphs), and periodicity is also easy to check (the definition of both is found in the first chapter of the book below, and the convergence theorem is proved in chapter 4 of the book).
It's worth noting that if there isn't irreducibility that can be easily solved in the symmetrical case by splitting the state space into "connected components", and considering each one separately. While periodicity can be patched by doing something similar to what you're suggesting. It's called creating the lazy Markov chain. If you want to understand the whole topic a little better (Mixing times for example will be very helpful in your convergence algorithm), this is an excellent book (available for free):
http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf