Recent content by drawar

Yeah, that also. Thanks for being so patient with me. I really appreciate your instructive help and guidance.

##E_1(F_n \circ S)=\left\{w \in V|F_n(S(w))=w\right\}##. ##W \subset E_1(F_n \circ S)## is because of the fact that ##S## is orthogonal. It remains to find a vector that ##\not\in W## but belongs to ##E_1(F_n \circ S)##, which is nothing but ##w##.

No, unfortunately. Would appreciate further hints if possible!

##F_n^2=I##, so the equation in (a) reduces to ##S(w)=F_n(w)## (- I guess this requires ##F_n## to be bijective or at least injective but I don't know how to prove it). From here ##n## can be found by subtracting ##F_n(w)## (or equivalently ##S(w)##) from ##w## followed by a normalization. Does...

Homework Statement Let ##n## be a unit vector in ##V## . Define a linear operator ##F_n## on ##V## such that $$F_n(u) = u-2\langle u, n \rangle n \; \mathrm{for} \; u \in V.$$ ##F_n## is called the reflection on ##V## along the direction of ##n##. Let ##S## be an orthogonal linear operator on...

Thank you all, I think I've got what you said, for (1) since it was given that x_1,x_2 \geq 0, 2x_1+x_2 \geq 0 hence the constraint is equivalent to 2x_1+x_2 \leq 3. for (2) whatever the larger is, the constraint makes no sense because 6 > 3, so the problem is infeasible. If 'max' were changed...

conditions (1) should read "the larger of (2x_1+x_2) and 0 is smaller than 3", that is, if 0 \leq 2x_1+x_2 then we also have 0 \leq 2x_1+x_2 \leq 3, otherwise, if 2x_1+x_2 \leq 0 then 2x_1+x_2 \leq 0.

Homework Statement Consider the LP: max \, -3x_1-x_2 \,\,s.t. \,\,\,\, 2x_1+x_2 \leq 3 \quad \quad \ -x_1+x_2 \geq 1 \quad \quad \quad \quad \ x_1,x_2 \geq 0Suppose I have solved the above problem for the optimal solution. (I used dual simplex and get (0,1) as the optimal solution.) Now...

Yeah, it's a real time-saver. Btw, I think a simple counter-example will do but here's how the proof goes: (Please tell me if it needs any correction!) $$f(\lambda(x_1,x_2)+(1-\lambda)(y_1,y_2))=f(\lambda x_1+(1-\lambda) y_1,\lambda x_1+(1-\lambda) y_1)\\ =\mathrm{min}(\lambda x_1+(1-\lambda)...

Thanks, actually I had tried to plot f(x1,x2) in the first place but with no success, mainly due to the fact that f involves both x1 and x2. I know, however, how to plot f when it involves only x1 (or x2), for example f(x1)=max(2x1,-x1). Btw, I've just given it another try after seeing your...

Homework Statement Given 2 problems: (P1) min min(##x_1,x_2##) s.t ##x_1, x_2 \geq 0## (P2) min t s.t ##t \leq x_1## ##t \leq x_2## ##x_1, x_2 \geq 0## (i) Is the mapping f(##x_1,x_2##)=min(##x_1,x_2##) convex? (ii) What are the objectives of (P1) and (P2)? Homework Equations The...

Oh that's enlightening, thank you!

Thank you for the help, but doesn't sin(1/x) have infinitely many zeros in [ε, ∞)?

Homework Statement Given $$y''+e^{-x}y=0. \qquad (*)$$ Let ##y(x)## be any nontrivial solution of ##(*)##, show that y has finitely many positive zeros. Hint: Consider ##z''+\frac{C}{x^4}z=0## where ##C>0## is sufficiently large, which has a solution ##z(x)=x\sin \frac{\sqrt C}{x}##. Homework...

Thanks for the idea, but I can't seem to arrive at anything useful. Here's what I have done $$f(t) = \sum_{k=0}^{99} [k \cos(2\pi k t) - k \sin(2\pi k t)]=\frac{1}{2\pi}\frac{\partial}{\partial t}\sum_{k=0}^{99} [\sin(2\pi k t) + \cos (2\pi k t)].$$ $$\sum_{k=0}^{99}\sin(2\pi k...