122
votes
2answers
8k views

Evaluate $ \int_{0}^{\frac{\pi}{2}}\frac{1}{(1+x^2)(1+\tan x)}\,\mathrm dx$

Evaluate the following integral$$\int_{0}^{\Large\frac{\pi}{2}}\frac{1}{(1+x^2)(1+\tan x)}\,\mathrm dx.$$ My Attempt: Let $$\tag1 I = \int_{0}^{\Large\frac{\pi}{2}}\frac{1}{(1+x^2)(1+\tan ...
1
vote
0answers
7 views

Let $s$ be a simple function st. $s\le g+h$, show there exists simple functions $u\le g$, $v\le h$ st. $s=u+v$

Let $g,h$ be two functions and let $s$ be any simple function such that $s \le g+h$. Show there exists simple functions $u\le g$ and $v\le h$ such that $s=u+v$. I have been thinking about this ...
0
votes
0answers
5 views

Angle between two parabolas

I'm a little confused about a problem that asks me to find the angle between the two parabolas $$y^2=2px-p^2$$ and $$y^2=p^2-2px$$ at their intersection. I used implicit differentiation to find the ...
4
votes
2answers
36 views

When does this equation $\cos(\alpha + \beta) = \cos(\alpha) + \cos(\beta)$ hold?

I come across this problem in an advanced maths textbook for grade 11 in my country. And it's marked a star, which means that it's a difficult exercise, and so, no solution for this problem is given. ...
5
votes
1answer
86 views

Find the sum of the series below

Find the sum $$(1\cdot2)+(1\cdot3)+(1\cdot4)+\cdots+(1\cdot2015)+(2\cdot3)+(2\cdot4)+\cdots+(2\cdot2015)+\cdots+(2014\cdot2015)$$ What I have tried... We are looking for ...
0
votes
1answer
12 views

Diffeomorphism between Euclidean space

How does one show that if $f:U\rightarrow V$ is a diffeomorphism between open sets $U\subset\mathbb{R}^m$ and $V\subset\mathbb{R}^n$ then $m=n$? Here is some working: For $u\in U$ let $v=f(u)\in V$. ...
3
votes
0answers
11 views

Rearrangements that never change the value of a sum

Which bijections $f:\{1,2,3,\ldots\}\to\{1,2,3,\ldots\}$ have the property that for every sequence $\{a_n\}_{n=1}^\infty$, $$ \lim_{n\to\infty} \sum_{k=1}^n a_n = \lim_{n\to\infty} \sum_{k=1}^n ...
0
votes
0answers
10 views

Sums of partially dependent Bernoulli random variables

I am looking for any kind of Chernoff type large deviation bound for the following random variable: $$X = \sum_{i=1}^NX_i$$ where each $X_i$ is an identically distributed Bernoulli random variable ...
9
votes
16answers
971 views

What is $0\div0\cdot0$?

We all know that multiplication is the inverse of division, and therefore $x\div{x}\cdot{x}=x$ But what if $x=0$? $0\div0$ is undefined so $0\div0\cdot0$ should be too, but whatever happens when we ...
0
votes
1answer
300 views

Induction to prove regular expression

Prove that is if S and T are any regular expressions over the one-letter alphabet, (for example: Σ = {a}), and if n is any natural, then the languages (ST)^n and (S^n)(T^n) are equal. I have to use ...
0
votes
1answer
20 views

Show that the smallest positive value of $ax+by$ is equal to $gcd(a, b)$.

I already read the proof of the following theorem (Abstract Algebra by T. W. Judson): Let $a$ and $b$ be nonzero integers. Then there exist integers $r$ and $s$ such that $gcd(a,b)=ar+bs$. ...
3
votes
0answers
9 views

Number of left cosets equals number of right cosets?

So I've been working on abstract algebra out of John B. Fraleigh's 3rd edition text. In the exercises of chapter 11, I came upon a question which I cannot even begin to solve. "Show that there are ...
1
vote
2answers
42 views

Fourier transform to find an harmonic function (Strauss)

I am trying to solve one of the problems of section 12.4 of the book "Partial Differential Equations" by Strauss. The problem says: Use the Fourier transfor in the $x$ variable to find the harmonic ...
0
votes
1answer
22 views

Open sets in $\mathbb{R}$ are countable unions of disjoint intervals--not a duplicate

I have seen many proofs of this and they all seemed somewhat technical to me. I'd like to argue as follows, and would appreciate your comments: We use only these two easy facts: $(1)$: The union of ...
0
votes
1answer
11 views

What is the difference between orthogonal and orthonormal in terms of vectors and vector space?

I am beginner to linear algebra. I want to know detailed explanation of what is the difference between these two and geometrically how these two are interpreted?
0
votes
3answers
22 views
0
votes
0answers
13 views

Is $\sum\limits_{{\rm{i}} = 1}^{n - k} {{b_i}} \times {b_{n-2}} + b_{k}$ correct to express this pattern in base n?

having seen the pattern be,ow i have tried to express it in base n, of course there should be few constraints added to parameters. $$\begin{array}{l} 1 \times 8 + 1 = 9\\ 12 \times 8 + 2 = 98\\ 123 ...
5
votes
6answers
5k views

If I buy 2 lottery tickets do I double my chance of winning?

There's a lottery. There are 6 balls chosen randomly from 49 and you have to match all the balls to win. I buy one ticket. If I buy two tickets with different numbers for the same draw, do I ...
1
vote
0answers
10 views

Proof that Sum of $n$ Squared Errors ~ Chi Square with $n$ $df$

There is a youtube video dealing with the proof that the sums of the squares of normally distributed $n$ random errors, each one distributed as $\sim \chi^2(1\text{ df})$ follows a chi square ...
0
votes
1answer
51 views

Set partitioning in ZFC

Does $\sf ZFC$ allow the partitioning of a set by claiming that $a$ and $b$ are in the same subset if $f(a,b)$? Cause I've once seen this technique being used in a proof but I can't see how it is ...
3
votes
1answer
60 views

Help with a proof regarding simple functions.

The question is If $f>g≥0$, then there exists non-negative measurable simple functions $f_k↗f$ st. $f_k≥g$ for all $k$. My attempt. Using a theorem in my text book For every non-negative ...
1
vote
1answer
21 views

How to factor a third degree polynomial?

How do I factor $2x^3-6x$? I've looked it up, but I haven't seen advice that helps with my exact problem.
0
votes
0answers
13 views

Prime counting function

How much of an impact would the discovery of an exact formula that is equivalent to the prime counting function have on the mathematics community and acedemia as a whole?
3
votes
2answers
51 views

Sum of Squares in terms of Sum of Integers

We know that the sum of squares can be expressed as a multiple of the sum of integers as follows: $$\begin{align} \sum_{r=1}^n r^2 &=\frac 16 n(n+1)(2n+1)\\ &=\frac {2n+1}3\cdot \frac ...
1
vote
2answers
25 views

Differentiate vector function wrt vector

I have a function $\frac{df(\mathbf{y})}{d\mathbf{y}}=\mathbf{y}g(\kappa)$ where $\kappa=||\mathbf{y}||_2$ and $g(\cdot)$ is a scalar function. Thing is when I differentiate this function I get a ...
0
votes
0answers
9 views

Additional assumptions on function to ensure uniform convergence

Given a sequence $u=(u_n)_{n\geq1}$ converging to $1$, I would like to prove uniform convergence of the sequence of functions $f_n$ defined by $f_n(x)=f(u_n x)$ for $x\in\mathbb{R}_+$ to the function ...
0
votes
0answers
7 views

Positive hyperplane? Is there a name for these type of hyperplanes?

Consider an affine hyperplane $\{ x : \langle x,v\rangle=a \}$ where $a\ge 0$ and $v\in\mathbb{R}^n_{+}$. That is, both the level $a$ and the vector $v$ are non-negative. Is there any special name ...
1
vote
4answers
80 views

The simplification of divided difference of cosine function

What is the following limit? $$\lim_{h \to 0}\frac{\cos(\pi/2+h)-\cos(\pi/2)}{h}$$ Why when simplified do you get $(-\sin(h))/h)$?
4
votes
3answers
87 views

Geometry of images of maps $f: \mathbb R \to \mathbb C$?

I am having trouble seeing what a continuous map $f: \mathbb R \to \mathbb C$ might look like. If it was linear it would look like a line but it's not clear to me what happens if it's any map. I ...
0
votes
0answers
20 views

A Quotient of the Euclidean Group

$\newcommand{\euc}{\mathscr I}\newcommand{\R}{\mathbf R}$ Let $\euc(n)$ denote the the Euclidean group $\R^n\rtimes O_n(\R)$. Recall that $\euc(n)$ acts on $\R^n$ as $(\mathbf x, T)\cdot \mathbf ...
0
votes
2answers
28 views

Get the distribution of $X|Y=y$ given this joint probability density function

Given the joint probability density function $f(x,y) = \lambda^2 \exp(-\lambda y)$ with $0 < x < y.$ How do I get the distribution of $X|Y=y$ ? Thanks in advance!
0
votes
2answers
20 views

Inequality involving floor

Let $x$ be randomly chosen from $\{1,...n\}$. Define $X_{p}$ such that \begin{equation} X_p= \begin{cases} 1, & \text{if}\ p|x, \\ 0, & \text{otherwise.} ...
0
votes
0answers
16 views

Is this proof correct? Show that the residues are equal

I think I solved the following question, but was unsure about a couple of details. Let $f$ be meromorphic in a neighborhood of $0$, $\phi$ analytic in a neighborhood of $0$, such that $\phi(0) = ...
2
votes
5answers
25 views

Is it possible to factor a quadratic equation when $a$, $b$, and $c$ are all equal?

I have the equation $4x^2+4x+4$ to factor. I know that need to start with $$(2x \quad )(2x \quad )$$ to make $4^2$, but I can't seem to factor the rest of the way. What should I do?
2
votes
1answer
45 views

How do you evaluate elements of the Drinfeld double $D(H)$ against elements of $H^\ast$ or $H$?

Proposition 8.2 of Majid's primer on quantum groups says that if $H$ is a finite dimensional Hopf algebra with quantum double $D(H)$, then this is a factorizable Hopf algebra with quasi-triangular ...
0
votes
1answer
22 views

If I have a function that's continuous and it's limits at $\pm \infty$ are $\pm \infty$ is it surjective?

I was trying out some problems where I needed to prove that a function was surjective, and I thought I could do this, is this true? Intuitively, it seems so. If I have a function that's continuous ...
0
votes
0answers
9 views

For a root system, why does $\beta\in\Delta_+\setminus\{\alpha_i\}$ imply $(\beta+\mathbb{Z}\alpha_i)\cap\Delta\subset\Delta_+$?

Let $\mathfrak{g}(A)$ be a Kac-Moody algebra for a matrix $A$, with root basis $\{\alpha_1,\dots,\alpha_n\}$. There is a remark on the bottom of page 6 of Kac's Infinite Dimensional Lie Algebras ...
14
votes
4answers
537 views

Visualizing the factorial

Often in basic mathematics, we can visualize things very easily, which I believe helps understanding (instead of just working out a number theoretical proof). For example: $$(n+1)^2 - n^2 = (n+1) +n$$ ...
2
votes
1answer
24 views

For any $N$ and $B$, is there always a $B$-smooth relation $x + y \equiv 0 \pmod{N}$?

Let $N$ be any integer and $B \geq 2$ be a smoothness bound. Does there always exist $B$-smooth integers $x,y$ such that: $$x + y \equiv 0 \pmod{N}\text{ ?}$$ My only progress is that I know the ...
2
votes
1answer
19 views

Question about fundamental solution to the biharmonic operator $\Delta^2=\Delta(\Delta)$?

Show that when $n=2$, the function $u(x)=-\frac{1}{8\pi}|x|^2log|x|$ is a fundamental solution to the biharmonic operator $\Delta^2=\Delta(\Delta)$. That is, show that $$\varphi(0)=\int_{\Bbb ...
1
vote
1answer
16 views

Integral related to Poisson kernel

$\textbf{Problem}$: Find the value of the integral $$I=\int_{-\infty}^0 P.V.\frac{1}{\pi}\int_{-\infty}^{\infty} \frac{\partial f}{\partial y} \frac{(x-y)}{(x-y)^2+z^2} \ dy \ dz,$$ with $f$ a ...
0
votes
1answer
19 views

Prove whether the linear equations are solvable or not?

I am beginner to linear algebra. I am confused for finding the solution for following question. There are set of linear equation(m equations and n unknown) represented in the form of matrices. ...
4
votes
0answers
13 views

Why would we a priori expect $V(I)$ to satisfy axioms to define the closed sets for a topology on $\text{Proj}(S)$?

The topological space $\text{Proj}(S)$ has the underlying set$$\text{Proj}(S) = \{\mathfrak{p} \text{ a homogeneous prime such that }S_+ \not\subseteq \mathfrak{p}\},$$and the closed sets are the loci ...
14
votes
1answer
214 views

There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x^b_{1}+x^b_{2}+\cdots+x^b_{k}|\ge 1$

This problem is a 2014 Sydney mathematics competition problem (11 grade). It seems difficult to solve. (I previously posted the n=2 case for which André Nicolas and Dan Robertson proposed solutions) ...
2
votes
0answers
70 views

Of strings and substrings: A problem of probability

Problem Let $\Sigma=\{a, b\}$. Let $\Sigma^*$ denote the Kleene star of $\Sigma$: \begin{equation*} \Sigma^* = \{\varepsilon, a, b, aa, ab, ba, bb, aaa, aab, \ldots\} \end{equation*} where ...
0
votes
2answers
36 views

What is the motivation behind this (tested, working) 2d coordinate transform?

I am working with programming and 2d geometry and need to transform between two different coordinate systems. I have two different representations of the same world, where I can sample any point and ...
1
vote
0answers
11 views

Partial D.E. Change of Variables

Can anyone tell me how to derive the value for alpha and beta, I'm guessing 6 and 1 in one order or another - "via quadratic". In transforming the equation, could someone show also how the operator ...
0
votes
1answer
22 views

How to show that action of an algebra $H$ on a vector space is the same as the coaction of $H^*$ on $V$?

Let $H$ be a Hopf algebra and $V$ a finite dimensional $H$-module. How to show that action of an algebra $H$ on a vector space is the same as the coaction of $H^*$ on $V$? Thank you very much.
-5
votes
0answers
21 views

Three problems of discrete math [on hold]

Show that if $A,B,C,$ and $D$ are sets with $\lvert A\rvert=\lvert B\rvert$ and $\lvert C\rvert=\lvert D\rvert$, then $\lvert A\times B\rvert=\lvert B\times D\rvert$. Prove that a natural ...
1
vote
0answers
26 views

Application of Stoke's Theorem

Edit: I think I misunderstood the problem. Upon reading my textbook again, I think what they mean by $F(x,y,z)=<yz,2xz,e^{xy}>$ ; C is the circle $x^2+y^2=16, z=5$ is just literally a ...

15 30 50 per page