All Questions

270 views

Calulus of variations Euler Lagrange equation

How can I find an extremal for the following functional: $$J[y] = \int_0^1 e^y (y')^2 dx$$ with the boundary conditions $y(0)=1, y(1)=\log(4)$? I have tried computing the Euler-Lagrange equation but ...
180 views

166 views

237 views

How can a large matrix be compressed?

I have a large matrix, around 10x10. Each individual element in the matrix itself is also a very large number, roughly of the order 10^30. I know that matrices can be used to solve linear equations. ...
56 views

About the boundedness of the derivative of a function which is in special function space.

If $f \in C^1 ([0,T] , L^2) \cap C^0 ([0,T] , W^{1,2} )$, then how can I conclude that $$\left \| \frac{\partial f}{\partial t} \right \|_{L^\infty([0,T] \times \Bbb R^n )} < \infty ?$$ Here $f$ ...
954 views

Prove: If $A \subseteq C$ and $B \subseteq D$, then $A \cap B \subseteq C \cap D$

Is the form and correctness of my elementwise proof of this correct? I don't have any other way of getting feedback for my proofs and I want to improve. Proof. Suppose $A, B, C, D$ are sets such ...
2k views

How to prove volume and surface area of sphere [duplicate]

Possible Duplicate: Why is the volume of a sphere $\frac{4}{3}\pi r^3$? We know that the surface area of a sphere is $4\pi r^2$ where $r$ is the radius of the sphere, and the volume is ...
46 views

Example of no inclusion in set of closures

Let $(X,d)$ be a metric space and $A_j \subseteq X$ for $j = 1,2,...$ . Let $B = \bigcup_{j=1}^\infty A_j$. Find an example for which $\bigcup_{j=1}^\infty \bar{A_j} \neq \bar{B}$. Also, in general ...
142 views

Is $O(n^2) = O(n^3)$? Prove your answer.

I am not sure how to go about doing this, I know that: $$O(g(n))=\{f : \exists \ c \ \in \Bbb R_+, \ \exists \ n_0 \in \Bbb N, \ \forall \ n\geq n_0 :f(n) \le c·g(n)\},$$ but how do I go about using ...
638 views

Stratified random sampling

Which of the following is NOT a characteristic of stratified random sampling? (A) Random sampling is part of the sampling procedure. (B) The population is divided into groups of units that are ...
92 views

Lebesgue measure: show that $\mu^*(k\,A)=k \mu^*(A)$

For $k\gt 0$ and a subset $A$ of $\mathbb{R}$, let $k\,A=\{kx\mid x∈A\}$. Show that $$\mu^*(k\,A)=k \mu^*(A)$$ and that $A$ is measurable if and only if $k\,A$ is measurable.
253 views

Use $\epsilon - \delta$ definition to prove $\lim\limits_{(x,y)\rightarrow (0,0)} \frac{(xy)^4}{ (x^2 + y^4)^3}$ exists.

As the topic, Use $\epsilon - \delta$ definition to prove $\lim\limits_{(x,y)\rightarrow (0,0)} \frac{(xy)^4}{ (x^2 + y^4)^3}$ exists. I tried to use the inequalities $|x+y|>|xy|$ and ...
502 views

Math contest: Find number of roots of $F(x)=\frac{n}{2}$ involving a strange integral.

Edit summary: A good answer appeared. CW full answer added, based on given answers. Removing my ugly-looking attempts, as they still remain in the rev. history. Here's a final-round calculus ...
3k views

Relations between p norms

The p-norm is given by $||x||_{p} = (\sum_{n=1}^{\infty }|x_{n}|^{p})^{1/p}$. For $0 < p < q$, it can be shown that $||x||_{p} \geq ||x||_{q}$ (1, 2). It sppears that in $R^{n}$ a number of ...
91 views

functional analysis-normed linear space

Can somebody please help me to find the answer for this problem... Let $V$ be a norm linear space and let $x\in V\setminus\{0\}$. Also let $W$ be a linear subspace of $V$. Show that if there is ...
119 views

Coordinates of point inside of circle

I'm sorry if what I'm asking has already been answered or is really easy but I struggled a little and haven't been able to come up with an idea. TThe context is as follows: a robot is being placed ...
277 views

How do I prove the field $F[x]/(p(x))$ is an extension of a field $F$

Let $F$ a field and $p(x)$ an irreducible polynomial in $F[x]$. I'm trying to prove that $F[x]/(p(x))$ is an extension of $F$. I know there are two approaches. Either we can prove that $F$ is a ...
133 views

Number of powers of $2$ having leading digit $1$

How many of the numbers $2^m$ (where $0\le m\le M)$ have leading digit $1$? My trial - Since leading digit $=1$, whenever $2^m$ reaches or just crosses a $10^x$ and is less than $2 \cdot 10 ^ x$, ...
318 views

Cauchy Riemann equations for real functions.

Let $f(x+iy) = x^2 + i.0$. Then $u(x,y) = x^2$ and $v(x,y) = 0.$ Hence $u_x = 2x$, $v_y = 0$, $u_y = 0$, and $v_x = 0.$ Clearly this doesn't satisfy Cauchy Riemann equations, and hence is not ...
135 views

The number of curves of given genus over a field

Let $k$ be a field. Let $g\geq 0$ be an integer. I have an elementary question. Let $N$ be the "number" of $k$-isomorphism classes of smooth projective geometrically connected curves over $k$ of ...
273 views

Find the possible values of $a$, $b$ and $c$?

Given $(a,\space b,\space c)\in \mathbb Z^3$ and that $$\sqrt[3]{\sqrt{a}+\sqrt{b}} + \sqrt[3]{\sqrt{a}-\sqrt{b}} = c$$ Find the possible values of $a$, $b$, and $c$.
problem understanding $a\times((b\cdot b)a-(b\cdot a)b)=-a\cdot ba\times b$
$a\times(b\times(a\times b))=a\times((b\cdot b)a-(b\cdot a)b)=-a\cdot ba\times b$ can anyone expand on how the final answer is derived? I try to expand but ended up scratching my head. the best I can ...