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 Jul 2 awarded Curious Jun 12 awarded Nice Question Jun 5 asked Prove that $\lceil \frac{\sqrt{n^2+1+\sqrt{n^2}}}{\sqrt{n^2+3+\sqrt{n^2+2}}-\sqrt{n^2+2+\sqrt{n^2+1}}}\rceil = 2n^2+n+3$ May 9 awarded Yearling May 9 awarded Nice Question May 9 asked Why is analysis called “analysis”? Apr 6 comment Are there numbers that if proven rational (or irrational) will have important consequences to mathematics? Well, we could consider both irrationality and transcendence. I'm just wondering if we're trying to prove numbers irrational (and transcendent) just for the fun of it, or are there practical applications for such results. Apr 6 comment Are there numbers that if proven rational (or irrational) will have important consequences to mathematics? I'm thinking of something like is there an instance of the following kind of implication: "x: rational $\Rightarrow$ 'Very-important-theorem-A' holds" Apr 6 asked Are there numbers that if proven rational (or irrational) will have important consequences to mathematics? Nov 26 asked Evaluate(?) $\int_{-2}^2 \sqrt{1-z^2} dz$ Sep 24 asked Why do we believe the equation $ax+by+c=0$ represents a line? Feb 6 awarded Tumbleweed 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! ^_^ Jan 29 accepted Given $F:X \to X$ on $X$:Hilbert space satisfying some properties, prove that $F$ is surjective. 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. 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 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 $P_{1,0}(A)=1$? 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.) 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 $g(\bar{x})=\bar{x}$? Jan 28 asked Given $F:X \to X$ on $X$:Hilbert space satisfying some properties, prove that $F$ is surjective.