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| bio | website | |
|---|---|---|
| location | ||
| age | ||
| visits | member for | 4 months |
| seen | Jun 9 at 3:03 | |
| stats | profile views | 5 |
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Feb 6 |
awarded | Tumbleweed |
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Jan 30 |
asked | Given $F(u)=u''+\lambda_n u$ on $X=\{u \in C^2 | u'(0)=u'(l)=0\}$, prove Im$(F) = \{ \theta | \langle \theta, \phi_n \rangle = 0 \}$ |
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Jan 29 |
comment |
Given $F:X \to X$ on $X$:Hilbert space satisfying some properties, prove that $F$ is surjective. I dived into my lecture notes and found a theorem in topological degree theory that is applicable (I think it's the homotopy invariance of the Leray-Schauder degree.) Thank you very much for the solutions and clarifications! ^_^ |
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Jan 29 |
accepted | Given $F:X \to X$ on $X$:Hilbert space satisfying some properties, prove that $F$ is surjective. |
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Jan 29 |
comment |
Given $F:X \to X$ on $X$:Hilbert space satisfying some properties, prove that $F$ is surjective. Thank you very much for this counterexample, clearing the ambiguity in the problem. The statement of the problem doesn't explicitly say anything about linearity, and in class we seemed to have no agreement on assuming linearity as a default. |
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Jan 28 |
accepted | Absolute continuity of the distribution of $X_t=aB_t+bt$, $Y_t=a(t)B_t$ with respect to the Wiener measure |
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Jan 28 |
comment |
Absolute continuity of the distribution of $X_t=aB_t+bt$, $Y_t=a(t)B_t$ with respect to the Wiener measure Sorry for answering so late, I forgot to check the notifications that someone kindly answered. I understand why $P_{a,b}(A)=0$ but may I ask why do we know $P_{1,0}(A)=1$? |
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Jan 28 |
comment |
Given $F:X \to X$ on $X$:Hilbert space satisfying some properties, prove that $F$ is surjective. (As stated above. Sorry I mistakenly hit the "add comment" button before I finished writing the questions, so I edited it again.) |
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Jan 28 |
comment |
Given $F:X \to X$ on $X$:Hilbert space satisfying some properties, prove that $F$ is surjective. Thank you very much for the answer. I've been reading your solution carefully for a while, but there are things I still couldn't figure out, and hope you could kindly explain. (1) Why does the existence of a $k$ such that $\|x-Tx\| \geq k$ dist$(x,K)$ implies $\text{Im} (1-T)$ is closed? (2) Doesn't the closed graph theorem requires $F$ to be linear? (3) Which fixed point theorem are you referring to in the last sentence? How can I choose an $\bar{x}$ that gives $g(\bar{x})=0$ (not $g(\bar{x})=\bar{x}$? |
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Jan 28 |
asked | Given $F:X \to X$ on $X$:Hilbert space satisfying some properties, prove that $F$ is surjective. |
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Jan 23 |
awarded | Teacher |
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Jan 23 |
answered | Proof that a certain number is disivible by 6 |
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Jan 23 |
asked | Absolute continuity of the distribution of $X_t=aB_t+bt$, $Y_t=a(t)B_t$ with respect to the Wiener measure |
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Jan 22 |
awarded | Supporter |
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Jan 22 |
comment |
Given the SDE: $dX_t=dB_t+b(X_t) dt$ with $(x,b(x)) \leq 0, \forall x \in \mathbb{R}^n$, prove that $E[|X_t|^2] \leq nt+E[|X_0|^2]$ My apologies. I was copying from my notes and there I'm just writing $i$'s instead of $(i)$'s (and also the "$b(X_t^{(i)})$" thing too, typing too fast and carelessly.) Thank you very much for the solution. |
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Jan 22 |
awarded | Scholar |
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Jan 22 |
accepted | Given the SDE: $dX_t=dB_t+b(X_t) dt$ with $(x,b(x)) \leq 0, \forall x \in \mathbb{R}^n$, prove that $E[|X_t|^2] \leq nt+E[|X_0|^2]$ |
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Jan 22 |
awarded | Student |
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Jan 22 |
asked | Given the SDE: $dX_t=dB_t+b(X_t) dt$ with $(x,b(x)) \leq 0, \forall x \in \mathbb{R}^n$, prove that $E[|X_t|^2] \leq nt+E[|X_0|^2]$ |
