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May
13
comment Are there any nontrivial examples of contradictions arising in non-foundational or applied math due to naive set theory?
Fair enough, I see your point.
May
13
comment Are there any nontrivial examples of contradictions arising in non-foundational or applied math due to naive set theory?
The only difference is that it is differential equations instead of number theory. But other than that, it is identical.
May
13
comment Are there any nontrivial examples of contradictions arising in non-foundational or applied math due to naive set theory?
You can construct a differential equation whose solution you know, but which you cannot prove. without assuming naive set theory is consistent. That is, you write a computer program that checks proofs in naive set theory, and then give it the Goedel statement "this statement cannot be proved" as the initial condition.
May
13
answered Are there any nontrivial examples of contradictions arising in non-foundational or applied math due to naive set theory?
May
12
revised On random rotational fluctuations in $\mathbb{R}^n$
Forgot the 1 in the expansion of exp.
May
12
answered On random rotational fluctuations in $\mathbb{R}^n$
May
11
comment On random rotational fluctuations in $\mathbb{R}^n$
I have an answer, but if you don't know stochastic differential equations, it is unlikely to mean much to you. You at least need to understand the Ito integral from a non-rigorous point of view.
May
11
comment On random rotational fluctuations in $\mathbb{R}^n$
Do you know the language of stochastic differential equations?
May
11
comment Existence of the Brownian Motion using the Kolmogorov extension theorem
The only hard part is showing that the Brownian Motion is continuous almost surely. Everything else is just applying Kolmogorov's extension theorem. Showing Brownian motion is continuous requires quite a bit more work.
Apr
27
comment Plate trick demonstrating SO(3) not simply connected.
Sounds right to me.
Apr
27
comment Plate trick demonstrating SO(3) not simply connected.
Think of your arm representing a map from $p:[0,1] \to SO_3$, where $t$ is how far along the arm we are (in units so that the length of the whole arm is $1$), and $p(t)$ is the orientation of that piece of the arm. Then any homotopy which takes $p$ to the trivial homotopy should, in principle, convert to a way to twist your arm so that it becomes untwisted. (Fixing the endpoints corresponds to not allowing one to change the orientation of the shoulder or the hand.)
Apr
27
comment Plate trick demonstrating SO(3) not simply connected.
I am not sure what you are asking. If you are asking whether "if it can be proved that the arm cannot be untwisted, then it can be proved that the path is noncontractible?" then I think the answer is "yes." If you are asking "can we provide an easy to understand proof that the arm cannot be untwisted?" then the answer is "not to my knowledge."
Apr
19
awarded  Informed
Apr
14
comment Is there a relationship between trigonometric functions and their “co” functions?
Follow up question? How do they decide which is the original function and which is the co-function? My theory is that the original function (sin, tan, sec) is increasing on $[0,\pi/2)$. Otherwise, why is secant the reciprocal of cosine?
Apr
9
comment Prove or disprove a claim related to $L^p$ space
I am really busy for the next few days, so I won't be adding much detail for a while. I do want to provide more details for why $f$ is in the Morrey space. I also got caught on the fussy details several times when I tried to make this work. I think sleeping on it was what helped the most. I find my mind does a lot of unconscious thinking while I sleep or daydream.
Apr
9
comment Prove or disprove a claim related to $L^p$ space
The proof that $\int_a^b f(x) \, dx \le \sqrt{b-a}$ has quite a few details which I omitted, because I am rushed for time right now. Also thinking about it, I might have to change it to $\int_a^b f(x) \, dx \le 3\sqrt{b-a}$.
Apr
9
revised Prove or disprove a claim related to $L^p$ space
Corrected upper bound for E_N
Apr
9
answered Prove or disprove a claim related to $L^p$ space
Apr
8
comment Prove or disprove a claim related to $L^p$ space
This is a really fascinating problem. I am sure the conjecture is false, but I am struggling to find the counterexample.
Mar
16
comment $X = ABA^T$, $X$ is PSD and $B$ is Symmetric. Does $B$ have to be PSD to satisfy this equation?
Take $B$ to be any symmetric matrix, some of whose eigenvalues are positive, and some of whose eigenvalues are negative. Let $A$ be the orthogonal projection onto the space spanned by eigenvectors corresponding to eigenvalues that are positive. So, for example, start with $B = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$. Then $A = 2^{-1/2} \begin{bmatrix} 1 & -1\\-1 & 1\end{bmatrix}$ will work (and you don't need the $2^{-1/2}$ - that's just to make it a projection).