Willie Wong
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97/100 score
 1d awarded Yearling Aug 26 comment Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not? @JiK: you are right. But outside of semantics the two are asking about the same thing, so I ask you to forgive me for being lazy and not editing the two to perfectly align. Aug 21 comment Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not? @user2520938: the quoted theorem from the other post says that if both $f$ and $f^{-1}$ are connected then $f$ is continuous. It does not, as you seem to think, say that $f^{-1}$ connected $\implies$ $f$ is continuous. (The connectedness of the forward map is used implicitly in the contradiction step.) Aug 13 comment lim sup and lim inf of sequence of sets. @JackBauer: While I don't understand why you are asking about what undergraduate or graduate students can or cannot understand, I do expect a beginning graduate student to understand what a complete lattice is and how it applies to the discussion at hand after learning the definition from, say, Wikipedia. Aug 12 awarded Good Answer Jun 29 comment Hausdorff-Young / Restriction Inequality To make it even more explicit: take $f_i$ to be "frequency translations" of one fixed $f_0$ which has compact support in frequency space. Jun 29 comment Hausdorff-Young / Restriction Inequality Unfortunately, I have neither the time nor energy to hold your hand through every single step of the argument. (I really suggest you discuss this with your tutor or professor, since from your comments you are missing some very basic techniques in analysis.) So one last-ditch effort: to prove an inequality $\|Tf\|_{L^q} \leq C \|f\|_{L^p}$ is false, the usual way is to construct a sequence $f_i$ such that $\|f_i\|_{L^p} = 1$ and $\|Tf_i\|_{L^q} \nearrow \infty$. So see if you can construct such an $f_i$ using exactly the ingredients that I gave above. Jun 26 comment Is :$\frac{\Bbb d}{\Bbb d x}$ a chaotic operator in infinite-dimensional Hilbert space? You have given us an abstract Hilbert space, but you have not told us how $d/dx$ acts on elements of the Hilbert space. Are you thinking about $H = L^2(\mathbb{R})$ or something? In which case $d/dx \not\in B(H)$ so is trivially not a chaotic operator by your definition. Jun 25 comment Conditions under which a function vanishing on the boundary belongs to $H_0^1$ $H^1_0$ is usually defined as the closure of $C^\infty_0$ (or equivalently $C^1_0$) in the $H^1$ norm. Functions in $C^\infty_0$ or $C^1_0$ have compact support. Jun 25 comment Hausdorff-Young / Restriction Inequality 1. Let $K$ be any compact set, on the complement $\mathbb{R}^d \setminus K$, there are not mutual inclusions between $L^p$ and $L^q$ for $p\neq q$. 2. Let $h:\mathbb{R}^d \setminus K \to \mathbb{R}$ be a function with $\inf h > M > 0$. Then the mapping $f\mapsto hf$ for $f\in L^p$ is either unbounded, or has operator norm at least $M$. 3. Try to prove the statement for the case the density of $\mu$ is $(1 + |x|^2)^\sigma$ for $\sigma > 0$. The general case follows not too more difficultly. Jun 25 awarded Nice Answer Jun 24 answered Two different trigonometric identities giving two different solutions Jun 24 comment “Commutation” of parallel transport with covariant derivative and Riemann curvature tensor Note that for the second question, there is no need for $X,Y,Z$ to actually be vector fields along $c$. Your question as posed only depends on the pointwise values of $X,Y,Z$, so we can consider the equivalent problem by extending those vectors to vector fields along $c$ by parallel transport. Jun 24 comment “Commutation” of parallel transport with covariant derivative and Riemann curvature tensor For 2, the answer is obviously no. You are equivalently asking the following question: if $X, Y, Z$ are vector fields parallel transported along $c$, is $R(X,Y)Z$ a vector field that is parallel transported along $c$? Taking the covariant derivative with respect to $c$ you see that this is only the case if $\nabla_{c'} R = 0$. Jun 24 comment “Commutation” of parallel transport with covariant derivative and Riemann curvature tensor The RHS of your first equation makes no sense. $P_{t_1,t_2}(Y(t_1))$ is a single vector in $T_{c(t_2)}M$, and not a vector field along $c$, so it is meaningless to take the $c'$ covariant derivative of it. Jun 23 comment Conditions under which a function vanishing on the boundary belongs to $H_0^1$ $u_\delta$ is certainly not necessarily smooth. It equals $u$ when $|u(x)| > 2$ so if $u$ is not smooth there neither will $u_\delta$. The key thing about the hint is not that $u_\delta$ is smooth, but that $u_\delta$ has compact support in $\Omega$. Jun 23 comment Conditions under which a function vanishing on the boundary belongs to $H_0^1$ Side comment: it is generally a good idea to ask two tangentially related questions as separate ones on this website, rather than as one question with two distinct parts. Jun 23 comment Hausdorff-Young / Restriction Inequality The sum comes from the example in the back of my mind (sorry for not being clear) when $\mu$ has a density that can be decomposed as the sum of a $L^r$ function and a $L^\infty$ function. (If the density is not bounded near $\infty$ then you expect to lose derivatives on the physical side and cannot generally expect the sort of restriction inequality you want to hold; so when I see your question I think of the case where the density is bounded outside a compact set and within the compact set is only $L^r$.) Jun 22 comment Is $(((\sqrt{2})^ \sqrt{2})^ \sqrt{2})^{\cdots}$ an irrational number? Voting to re-open based on @rajb245's comment Jun 22 comment Overview of nonlinear analysis, differential equations (ODE and PDE), dynamical systems, and mathematical physics, and their relationships Can't help you with that. Any narrowing will be a different question from what you have asked, and I can't read your mind (sorry) well enough to guess where you would prefer the emphasis be placed.