3
votes
1answer
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 ...
4
votes
1answer
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 $$\int_{1}^{\infty}\dfrac{x\sin(2x)}{x^2+3}dx$$ ...
0
votes
1answer
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 ...
1
vote
1answer
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 ...
1
vote
1answer
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. ...
6
votes
4answers
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 $g(g(g(x)))\equiv g^{3}(x)=x$? ...
0
votes
0answers
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 ...
2
votes
2answers
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 ...
1
vote
1answer
59 views

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< ...
0
votes
1answer
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: $$a\sqrt{2b+c^2}+b\sqrt{2c+a^2}+c\sqrt{2a+b^2} \leq 3\sqrt{3}.$$
2
votes
1answer
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, ...
0
votes
2answers
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 ...
0
votes
1answer
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 ...
0
votes
1answer
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 ...
2
votes
4answers
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 ...
2
votes
4answers
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 ...
0
votes
1answer
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 ...
2
votes
1answer
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 ...
2
votes
0answers
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 ...
2
votes
3answers
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, ...
2
votes
2answers
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 ...
4
votes
2answers
104 views

Approximation with 1-exponential

How come that $$\left(1-\frac{1}{x}\right)^x \approx e^{-1}\ ?$$ Is there a proof or something to understand this?
5
votes
1answer
535 views

When does the summation of a quotient equal the quotient of summations?

That is, under what conditions would $$ \sum_{i = 1}^n \frac{a_i}{b_i}= \frac{\sum_{i = 1}^n a_i}{\sum_{i = 1}^n b_i} $$ be true? What about for infinite summations, i.e. when $n \rightarrow ...
2
votes
3answers
169 views

Average of random variable - can't find its value!

I have $P(t) = 1-(1-R^{-t})^N$ which is the probability that I need more than t operations in my algorithm, t is a random variable then, I can assign a probability to it. Why is it that the sum of ...
2
votes
1answer
160 views

Double Inequality. $ab+bc+ca \leq \frac{1}{8}\sum{\sqrt{(1-ab)(1-bc)}} \leq a^2+b^2+c^2$

Let $a,b,c$ be non-negative numbers such that $a+b+c=1.$ Prove that : $$ab+bc+ca \leq \frac{1}{8}\sum_{cyc}{\sqrt{(1-ab)(1-bc)}} \leq a^2+b^2+c^2.$$ Thanks:)
5
votes
3answers
87 views

$(1-t)^{-d}= \Sigma_{k=0}^\infty {d+k-1 \choose d-1} t^k$?

I'm trying to see why the equation $(1-t)^{-d}= \Sigma_{k=0}^\infty {d+k-1 \choose d-1} t^k$ holds in the power series ring $\mathbb{Z}[[t]]$. I assume it's a counting argument about the number of ...
2
votes
1answer
248 views

How to solve a set of cosine equations?

suppose I have an equations of the following with two unknowns $A$ and $\theta$ $A\sin(x+\theta)=D$ I have two points $(E,F) (G,H)$ how do I go about solving this equation analytically. I can solve ...
-1
votes
5answers
416 views

When to use transfinite induction?

How do we know when we are allowed to use transfinite induction in a proof ? Edit : considering the replies i should say the following Consider an infinite sum of fractions. By induction we can ...
4
votes
2answers
63 views

If $f \colon [-1,1] \to \mathbb R$ satisfies $\vert f'' \vert \le k$ then $\vert f \vert \le \frac{k}{2}$

Let $f \colon [-1,1] \to \mathbb R$ be a twice differentiable function s.t. $f(-1)=f(1)=0$ and there exists $k>0$ s.t. $\vert f''(x) \vert \le k$ for every $x \in [-1,1]$. Show that $$ ...
1
vote
1answer
116 views

Lebesgue measure and matrix notation problem

I have trouble with understanding following from my text book in Measures and Integral theory. Let T be an orthogonal $n\times n$ matrix. If $\lambda^{n}$ is the Lebesgue measure then we have: ...
1
vote
2answers
849 views

How do I solve this integral? $\int e^{-t/2}\sin(3t) dt$? [duplicate]

Possible Duplicate: Name of this identity? $\int e^{\alpha x}\cos(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \cos(\beta x)+\beta \sin(\beta x))}{\alpha^2+\beta^2}$ I might have missed ...
2
votes
4answers
988 views

Limit of the difference quotient of $f(x) = \frac{2}{x^2}$, as $x\rightarrow x_0$

Could someone please show me how to derive the limit of the difference quotient of $f(x) = \frac{2}{x^2}$, as $x\rightarrow x_0$ The difference quotient is just the expression: $(f(x+h)-f(x))/h$ So, ...
2
votes
2answers
96 views

Are fractional exponents considered logarithms?

Say I have a number with a fractional exponent, $10^{\frac{1}{3}}$. Could this number be considered a logarithm, even though it is not written as $10^{0.\overline{3}}$?
3
votes
2answers
288 views

Compute expectation of certain $N$-th largest element of uniform sample

A premier B-school has 2009 students.The dean,a math enthusiast,asks each student to submit a randomly chosen number between 0 and 1.She then ranks these numbers in a list of decreasing order and ...
1
vote
0answers
91 views

Left-modules over a bialgebra form a monoidal category

Let $B = (B, \nabla, \eta, \Delta, \epsilon )$ be a bialgebra over a commutative ring $k$. Let $M$ and $N$ be two left $B$-modules. Then the tensor product $M \otimes_k N$ becomes a left $B$-module ...
7
votes
2answers
441 views

An analytic function with a simple pole

Let $f(z)$ be analytic in the disk $|z|<R \ \ \ (R>1)$ except for a simple pole at a point $z_0$, $|z_0|=1$. Consider the expansion $f(z)=a_0+a_1 z+ \cdots$, and show that $$\lim_{n \to \infty} ...
4
votes
1answer
122 views

Does Bayesian probability have a different interpretation of a random variable?

Bayesian probability interprets the meaning of the probability of a random variable as some degree of belief. But does this result in any difference in the interpretation of a random variable itself? ...
3
votes
2answers
305 views

Inequality. $a^7b^2+b^7c^2+c^7a^2 \leq 3 $

Let $a,b,c$ be positive real numbers such that $a^6+b^6+c^6=3$. Prove that $$a^7b^2+b^7c^2+c^7a^2 \leq 3 .$$
1
vote
2answers
85 views

Prove $(1+z)^a \geq 1+az$, for $z>-1, a>1$, using mean value theorem

Prove $(1+z)^a \geq 1+az$, for $z>-1, a>1$, using the mean value theorem Hint says try using: $f(z)=(1+z)^a$ I've tried messing around with this, but I can't seem to get 1 + az on the RHS. ...
1
vote
1answer
86 views

Inequality. $\frac{a}{(b+c)^4}+\frac{b}{(c+a)^4}+\frac{c}{(a+b)^4} \geq \frac{3}{2(a+b)(b+c)(c+a)}$

For $a,b,c >0$ prove that : $$\frac{a}{(b+c)^4}+\frac{b}{(c+a)^4}+\frac{c}{(a+b)^4} \geq \frac{3}{2(a+b)(b+c)(c+a)}.$$ I don't know how should I start. It is difficult for me because the ...
-2
votes
1answer
318 views

Proving a set closed under a binary operation is associative [closed]

Please help me this is my honework problem
2
votes
2answers
125 views

Integrating $\int_0^{500}e^{\frac{(x-a)^2}{b^2}} dx$

Well the title says it all: Can someone please help me integrate: $$\int_0^{500} e^{\frac{-(x-a)^2}{b^2}} \, dx$$ I need to understand how $e$ can be integrated when it is to the power of a ...
3
votes
3answers
159 views

Constructing a basis for finite abelian groups

Let $G$ be a finite abelian group, and $g_1, \ldots, g_k$ a set of "linearly independent elements", namely such that $\langle g_1 \rangle \oplus \ldots \oplus\langle g_k \rangle$. I would like to ...
1
vote
1answer
64 views

Inequalities of summations

I am thinking if the following condition is in general true: $\frac{n}{m} \leq \frac{\sum_{i = 1} ^ {n} a_i}{\sum_{i = 1} ^ {m} b_i}$, when $n \leq m$ and $a_i \geq 0$, $b_i \geq 0$ but i cannot find ...
2
votes
1answer
109 views

Formation of a positive-definite matrix via a positive-semidefinite one

Let the square symmetric matrix $L\in\mathbb{R}^{n\times n}$ be positive semi-definite with vector $1_n$ spanning its null-space (i.e., vector $1_n$ is the eigenvector of $L$ corresponding to the ...
0
votes
0answers
109 views

sequences of poles and zeros

Let $f$ be a meromorphic function in a domain $D$. The set of zeros $Z_f$ and the set of poles $P_f$ are both discrete in $D$; it means that doesn't exist a sequence of zeros (risp. sequence of poles) ...
8
votes
2answers
113 views

Evaluation of probability related integral

I have encountered the following integral in my research which does not give-in to my attempts: $$ \int_\mathbb{R} x \left( \frac{1}{\sigma_1} \phi\left(\frac{x}{\sigma_1}\right) ...
3
votes
3answers
288 views

How to get the N-th word in a sequence

Suppose I have an alphabet (e.g. consisting of ABCDEF) and a lexicographic order is defined i.e. A -> B ... -> F -> AA -> AB .. -> AF -> BA -> BB -> ... -> BF ... -> FF -> AAA -> ... Is there a ...
11
votes
3answers
1k views

Keep the value of an 8-sided die roll, or gamble by taking a 12-sided die roll. What's the best strategy?

Consider a dice game in which you try to obtain the largest number (e.g. you win a prize based on the final dice roll). You roll an 8-sided die, with numbers 1–8 on the sides. You may either ...
3
votes
1answer
611 views

Finding the virtual center of a cloud of points.

Given: (latitude, longitude) points $P_1, P_2,\ldots, P_n$. Presumably, all the points should form a dense cloud. However, noise is possible. Needed: The virtual center of the points. For ...

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