All Questions

303 views

Understanding tensor divergence notation in an integral

Given a smooth tensor valued function $\sigma:R^2\rightarrow R^{2\times2}$, I'm trying to show that $\int_\Omega \nabla\cdot\sigma=\int_{\partial\Omega}\sigma n$, where $\Omega$ is a connected ...
206 views

Evaluate the following improper integral with bounds.

I need ideas for solve this improper integral, i know is hard and is a bonus for my analysis course, so i would really appreciate your help, thanks $$\int_{1}^{\infty}\dfrac{x\sin(2x)}{x^2+3}dx$$ ...
249 views

Simple example of a short exact sequence of groups

I am new to this and would like to understand $$0 \overset{a0}{\to} B \overset{a1}{\to} A \overset{a2}{\to} A/B \overset{a3}{\to} 0,$$ where $B \subset A$ and they are both Abelian groups. Also maybe ...
75 views

Proof or disproof a problem about critical points

There is a question im trying to solve, but im not sure im doing the right thing. "Let $p(x) = \sum_{j=0}^na_jx^j$ a polynomial without multiple roots, then all critical points of ...
140 views

Let $U\subset \mathbb{R}^n$ (open set) and $f:U\longrightarrow V$ a homeomorphism then we can say that $V$ is a open set in $\mathbb{R}^n\,?$

Let $U\subset \mathbb{R}^n$ (open set) , $V\subset \mathbb{R}^n$ and $f:U\longrightarrow V$ a homeomorphism then we can say that $V$ is a open set in $\mathbb{R}^n$ ? Any hints would be appreciated. ...
361 views

Can the Identity Map be a repeated composition one other function?

Consider the mapping $f:x\to\frac{1}{x}, (x\ne0)$. It is trivial to see that $f(f(x))=x$. My question is whether or not there exists a continuous map $g$ such that $g(g(g(x)))\equiv g^{3}(x)=x$? ...
115 views

Set of Bounded linear Operators on $l_2$ is dense on the set of bounded operators on $l_2$?

Let $l_2^{+}$ be the Hilbert space of all square summable sequences $\{x_n\}, n \in \mathbb{N}$ under some definition of inner-product $\langle,\rangle_l$. Define $B[l_2^{+}]$ as the set of all ...
781 views

homeomorphism question relating to the topological 3-sphere

I have a question concerning an exercises from a text call Topology and Groupoid authored by Ronald Brown The question is as follows: Let $E^2 = \{(x, y) \in \mathbb R^2 : x^2 + y^2 \leq 1\}$. The ...
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Summation of Modulo Sequences

I came across through simulation that multiplying and adding certain modulo sequences yield equal results. Consider the following two sequences \begin{align} g_0[k] &= \sum_{n=0}^{N-1} \left< ...
288 views

Inequality. $a\sqrt{2b+c^2}+b\sqrt{2c+a^2}+c\sqrt{2a+b^2} \leq 3\sqrt{3}$

Could you help me please with the following inequality $a,b,c$ non-negative numbers such that: $a+b+c=3.$ Prove that: $$a\sqrt{2b+c^2}+b\sqrt{2c+a^2}+c\sqrt{2a+b^2} \leq 3\sqrt{3}.$$
95 views

Is the product of generators equal to the generator of the product?

Let $(X, \mathcal{J})$ and $(Y, \mathcal{F})$ be two measure spaces. Let us assume that $J$ is a collection of subsets of $X$ which generates $\mathcal{J}$, i.e. $\sigma(J)=\mathcal{J}$. Similarly, ...
75 views

Lowest possible price before any discount

I am having difficulty solving the following problem A toy store regularly sells all stocks at a discount price of 20% to 40%. If an additional 25% were deducted from the discount price what would ...
396 views

Is the set of integer coefficient polynomials countable? [duplicate]

Possible Duplicate: Is the set of polynomial with coefficients on $\mathbb{Q}$ enumerable? The set of integer coefficient polynomials are countable, when the cardinality of each set of ...
77 views

Searching for a multiplicative function to separate complex numbers

Let $g$ be a complex number, where $a$, $b$, $c$, $d$ are real numbers, and $i = \sqrt{-1}$. $g = \frac{{(a + bi)\exp (ci)}}{{{\rm{abs}}(a + bi)}}$ Since the absolute value (i.e. modulus) of a ...
187 views

Quickest way to find a point on a circumference

Given the image below, A is the centre of the circle, B is a point on the circumference and AC and DB lie on parallel lines. Knowing A, C, D and the radius of the circumference, I need to find the ...
291 views

In Need of Ideas for a Small Fractal Program

I am a freshman in high school who needs a math related project, so I decided on the topic of fractals. Being an avid developer, I thought it would be awesome to write a Ruby program that can ...
215 views

Nilpotent Matrices

Let $\mathbb K$ be a field and $A, B\in M_n(\mathbb K)$ be nilpotent matrices. Suppose that $nullity(A)\cap nullity(B)\geq 1$. Can we find a regular matrix $T$ such that the first columns of the two ...
303 views

Martingales: Stopping Time

On page 217 of the book Probability Essentials By Jean Jacob and Philip Protter found an issue that can not do, is about martingales and stopping times. If anyone has any tips on how to do, I'd ...
151 views

Fiber Bundle: Hairbrush

I am trying to understand the hairbrush example of a fiber bundle from the Wikipedia article on fiber bundles. If I am understanding this, in the hairbrush example E is the hairbrush, ie. all the ...
117 views

Which is the probability to a random line to be parallel to a specific other line?

In my perception, using the common sense, is less common, or less probable, to a random line be parallel that not to be, because to be parallel a line needs obey a restrictive rule. But anyone can, ...
133 views

Subset of a function

Suppose we have a function $f:X \rightarrow Y$. Now, consider the function $g:X'\rightarrow Y$ where $X'\subset X$. I'd like to say the $g$ is a "subset" of $f$ ; is there a correct term for ...
How come that $$\left(1-\frac{1}{x}\right)^x \approx e^{-1}\ ?$$ Is there a proof or something to understand this?