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 Yearling
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Aug
25
comment Intuition of a norm vector space/ infinite dimensional vector space
At this point, you might have to build more "algebraic" intuition. At first, it helps to think of an infinite dimensional vector space as a finite dimensional space with some exceptions. The number of exceptions you need to keep in mind depends on the "additional structures" your vector space has. For example, if it is an inner product space, then you can still keep orthogonality. One property that is quite commonly mentioned is that a closed and bounded set is not necessarily compact in an infinite dimensional Hilbert space.
Aug
11
comment Wave Equation - like 4th Order PDE
@Lurco There are many things that won't work. The most obvious reason is probably that when the number of solutions is uncountable, summation does not work. People have tried to use integration instead, and that yields two common alternatives to Fourier series: Laplace transform (half real line) and Fourier transform (whole real line). Of course there are many other alternatives (even for the finite case), but I think you get the picture.
Aug
10
comment Wave Equation - like 4th Order PDE
When the domain is finite, it may be possible to find a countable complete orthogonal basis that consists of functions that satisfy given homogeneous boundary conditions.
Jun
26
awarded  Yearling
Jun
12
comment What is the best algorithm for finding a $g \in S_n$ which $a^g=b$ for given $a, b \in S_n$
Find the disjoint cycles in $a$ and $b$. Since you assume that they have the same cycle type, you can just match the cardinality of each cycle in $a$ and $b$.
Jun
11
revised Find the distribution of some random variable connected to Wiener Process. Please, check my solution.
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Jun
11
comment Find the distribution of some random variable connected to Wiener Process. Please, check my solution.
I think you might want to think of $D^2(W_t)$ as $\text{cov}(W_t, W_t)$ so you won't confuse yourself with the signs. Or better yet, think of $\text{cov}$ as the dot product. $D^2$ is just the square of the norm of the vector.
Jun
11
answered Find the distribution of some random variable connected to Wiener Process. Please, check my solution.
Jun
11
comment Relationship between sine and cosine in a circle
I would have thought this picture had been drawn on the board in his class already...
Jun
11
comment Find the distribution of some random variable connected to Wiener Process. Please, check my solution.
$W_7 - W_3$ has the same distribution as $W_4$, but it is not equal to $W_4$. (Otherwise $W_7$ would be completely determined by $W_3$ and $W_4$, which is NOT the case.)
Jun
11
comment Relationship between sine and cosine in a circle
I don't know if "in a circle" has any significance. May I guess $\sin(x)^2 + \cos(x)^2 = 1$?
Jun
11
comment Relationship between sine and cosine in a circle
There are a many answers to this. The simplest one I can think of right now is $\sin(x) = \cos\left(\frac{\pi}2 - x\right)$.
Jun
11
comment Will someone please explain multivariate normal distributions with a real-life example?
I can try to concoct one example. The GPS data at each time is a 2D random variable, which is assumed by many apps to be normally distributed.
Jun
11
answered Bernouli Trial Probability of Stopping After X Trials
Jun
11
revised Geometric Interpretation of Determinant of Transpose
added 406 characters in body
Jun
11
revised Geometric Interpretation of Determinant of Transpose
added 406 characters in body
Jun
10
asked Geometric Interpretation of Determinant of Transpose
Jun
10
revised How to encode matrices uniquely
added 54 characters in body
May
25
comment Prove that a function $f:X \to Y $ is injective if and only if $\forall x_1, x_2 \in X$ where $f(x_1) = f(x_2)$ implies that $x_1 = x_2$
Am I right to assume that "$x + 1$" is supposed to be "$x_1$"?
May
9
comment Function whose limit does not exist at all points
@moni94 Your function definition is not valid because the "last digit of the decimal representation" is not defined for every real number. (Take $1/7$ for example.)