Thanks guys. I bumped on the answer here:
http://electron9.phys.utk.edu/phys135d/modules/m6/Work.htm
and I did not know that many have asked this similar question before. The thread below seems to have a good discussion on that matter:
https://www.physicsforums.com/showthread.php?t=482584
Suppose I have an object moving in a free space at initial velocity of v (no friction). Then two forces of equal magnitude but opposite directions are applied on the moving object of which one of them is in the same direction as the moving object. Is there work done by that force?
This is not a homework. I am just pretty confused about a problem (with solution!) I saw in the internet found here:
http://www.transtutors.com/physics-homework-help/electromagnetism/Electromagnetic-induction.aspx
The part I am confused is the mutual inductance part. Here I repeat the question...
Imagine charging up two conducting plates, placed nowhere near each other in an open air, to several hundred thousand kiloVolts. Can the positive plate strips off electrons from the air molecules and thus ionizing them without any event of electrical sparks between the two plates?
Let v = \alpha_1 x_1 + \alpha_2 x_2 + \cdots , where \alpha_i \geq 0, \forall i are real numbers in which \alpha_1 + \alpha_2 + \cdots = 1.
Then
\alpha_1 A x_1 + \alpha_2 A x_2 + \cdots \leq \alpha_1 b + \alpha_2 b + \cdots.
Hence
Av \leq b.
Note also that v \geq 0.
Right you have the point there:there might not be any point (u - t_i \hat{g}_i) \in K such that it satisfies \hat{g}^\top d = t_i < 0. This means the book cannot also claim that the point (u - s_i \hat{g}_i) \in K satisfyng \hat{g}^\top d = s_i > 0 always exist.
You mean in the definition of K? But you can't change that.
Yes, I agree with you that nothing states about the implication but
since K is a cone which is closed and convex, u \in K exists and must be a unique point.
I encountered a problem in a book with a proof given. But I am a bit skeptic about it. I hope someone can help shed some light.
Let \{g_{i}\} be a set of vectors and imagine a cone defined as K = \left\{v \,\bigg|\, v =-\sum_{i}\lambda_{i}g_{i}, \textup{ where }\lambda_{i}\geq 0 \ , \forall i...
Let me clarify. An open cover for a set A is a collection of open sets whose union contains A. Similar definition goes for closed cover, as noted by HallsofIvy. If you stick to this definition, then it resolves your question b).
With that in mind, and to avoid confusion, don't use...
It was my fault, HallsofIvy, for being ignorant. I could have pointed out his mistake about "open subcover". I took that as a typo, as I did not wish to argue much on that matter (I wished to go straight to the main point). Anyway I did mention the union of arbitrary open sets is open in passing...
I am not surprise (in fact, none of us is!) with your observation -- it is trivially true! You must have missed the most crucial point about Heine-Borel theorem that many had tried to get to you. The theorem does not argue anything about there does not exist a cover that has finite (open/closed)...
No such theorem exists. If it does, it is probably useless. If we replaced open subcovers with closed subcovers, then even closed and bounded set can possesses such cover with no finite (closed) subcovers. For example:
Define closed interval I_n = [1/n, 2] for all n \in \mathbb{N}. Then the...
Sorry for not explaining my problem clearer.
That was what my problem was. Many thanks, Hurkyl! :)
Does this mean \lim_{k \to \infty} f(x_k) is the limit over all sequences as well?
Also, using the given example, can I write 0 \leq \lim_{x \to 0} f(x) \leq 1? or that \lim_{x \to 0} f(x)...
I have questions regarding this subject.
By definition, \limsup_{k \to \infty} f(x_k) \equiv \lim_{k \to \infty} \sup_{n \geq k} f(x_n) and \liminf_{k \to \infty} f(x_k) \equiv \lim_{k \to \infty} \inf_{n \geq k} f(x_n). Say a sequence \{x_k\} converging to 0 from the left in the following...
HallsofIvy: I find your method quite elegant.
Benorin: Thank you for an alternative way for showing C_1 \cap C_2 \subseteq \textup{cl}(C_1) \cap \textup{cl}(C_2).
Ah! I see it now (after going over it a couple of times). :redface: Thank you.
Just after I've posted my last reply, I came...