# All Questions

805 views

### Analytic Geometry: Point coordinates, same distance from two points

Given are two points, $P1(x_1, y_2)$ and $P2(x_2, y_2)$, and distance $a$. Now I want to find the two points $T1$ and $T2$. $$d(P1,T1) = d(P2, T1) = a = d(P1,T2) = d(P2, T2)$$ Eg: (T1 and T2 are my ...
783 views

### Intuition for Little Picard's theorem

Little Picard's theorem is the following: Suppose $f:\mathbb{C}\rightarrow \mathbb{C}$ is entire. Then either 1) $f$ is constant 2) $f$ is surjective or 3) $f$ is onto $\mathbb{C}-\{p\}$ for ...
784 views

### Proving the countability of algebraic numbers

I am trying to prove that algebraic numbers are countably infinite, and I have a hint to use: after fixing the degree of the polynomial, consider summing the absolute values of its integer ...
77 views

### Formula for $\sum_{i = 1}^{n} \frac{i}{K - i}$

How can I find a formula for $\sum_{i = 1}^{n} \frac{i}{K - i}$, $K \in \mathbb{Z}$, $K > n$?
251 views

585 views

### Continuous representatives in Sobolev Spaces

My question arise from the study of the possible extensions of Rademacher's Theorem to the Sobolev Space $W^{1,p}(\Omega)$, with $\Omega\subset \mathbb{R}^n$. In specific I'm studying the proof of the ...
1k views

### particular solution of $4y''-y= \sin(x)\cdot \cos(x/2)$

So I'm working with a nonhomogeneous second order differential equation: $$4y''-y=\sin(x)\cos(x/2).$$ I know that the general solution, $y$, equals $y_c + y_p$ where $y_c$ is the general ...
668 views

### Definition of definition

I was wondering if there is a good way to "define" what definition means exactly in mathematics. Since the answers may be subjective or philosophical, I want to ask only for references on this topic. ...
1k views

### The last number of infinity [closed]

First of I am not a math guru, so it might be a dumb question, but today in the metro I was bored, and started thinking what is the last number of the infinity, and came up with this theory, that the ...
516 views

### Geometry/ Similar Triangles Problem

Consider the trangle shown below with vertices A, B, C where point D lies on the side AB, point E lies on the side BC and point F lies on the side AC and the three lines AE, BF, and CD intersect at a ...
361 views

518 views

### What it takes to a mathematician [closed]

I am a junior in Caltech's mathematics department. I keep being harassed by some non-mathematicians and fellow students that I do not have what it takes to do mathematics. They say that one must ...
181 views

### When are two modular functions equal?

Suppose $f, g \in M_{k}(\Gamma)$ for some $\Gamma$ a congruence subgroup of $PSL_{2}(\mathbb{Z})$. If $k \geq 1$ then it is a finite computation to determine whether or not $f = g$. What if $k = 0$, ...
416 views

### Help with integrating 101: $\int y \ln{y}\,dy$

Would appreciate it if someone would please help me solve this $$\int y\;\ln y\, dy$$ taking time to explain reason for each step taken. Thanks in advance!
176 views

### Does solution of $x=\sum_{n=0}^\infty e^{-A_n/x}$ exist?

Usually when working with indefinite sums, I want to work out the sum or whether it convergence. But now I encountered a problem they other way around and I'm clueless... Is there even a general ...
128 views

### Help me formalize this calculation

I needed to find the number of five digits numbers that are made of numbers from $0,1,2,3,4,5$ and are divisble by 3. One of the proper methods can be, that $0+1+2+3+4+5 = 15$ So we can pick out ...
316 views

### Calculate measurements for a diagonal fence beam

Given the width W and the height H of a rectangle, and the thickness T of a beam extending exactly from the upper left corner to the lower right corner as shown, how do I solve for length X and angle ...
209 views

### fundamental representation of $\ sl(l+1,F)$

This problem concerns the topic representation theory of Lie Algebras. The main purpose of the exercise is to study the form of the fundamental dominant weights of a Lie Algebra. I would be very ...
294 views

### Unbiased estimator and Variance

I am having a hard time trying to solve this problem. I don't know how to start it. Any help would be greatly appreciated. Let T be any unbiased estimator of $\tau(\theta),$ and let $W$ be a ...
192 views

### Approximation of a real number as a linear combination of two reals with coprime integral coefficients

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, give me an algorithm that will provide coprime integers $a$ ...
521 views

### Finding a Möbius transformation

Let $f$ be a holomorphic mapping from {$z:\Re(z)>0$} into itself. Let $1$ be a fixed point of $f$. In addition suppose that $\left|\frac{f(2)-1}{f(2)+1}\right|=\frac13$. I want to show that ...
874 views

### Is the empty graph connected?

Is the empty graph always connected ? I've looked through some sources (for example Diestels "Graph theory") and this special case seems to be ommited. What is the general opinion for this case ? As ...
363 views

372 views

### Indefinite integral $\int^{\infty}_{0}\frac{x}{x^4+1}dx$ via residues

I want to compute $\displaystyle \int^{\infty}_{0}\frac{x}{x^4+1}dx$ using the residue theorem. The poles in the upper half plane are: location: $\large e^{\frac{\pi i}4}$, order: 1, residue: ...
2k views

### Proof of Euler's Theorem without abstract algebra?

Every proof I've seen of Euler's Theorem (that $\gcd(a,m) = 1 \implies a^{\phi(m)} \equiv 1 \pmod m$) involves the fact that the units of $\mathbb{Z}/m\mathbb{Z}$ form a group of order $\phi(m)$. ...
733 views

### Questions on perfect squares

I recently attended a test which ask me two question based on perfect squares; here they are: $1.$ How many even perfect squares between $1000$ and $5000$ are divisible by both $5$ and $9$? $2.$ Can ...
222 views

158 views

### Combinations Help

I have an application where I iterate through all k-combinations of a set of size n. For example here I have listed all k-combinations for when n is 4. Also I have separated each list of combinations ...
### Simple estimation $e^{a\sqrt{r}} > r$
I want to prove a simple theorem about contour integration via residues and I need the following estimation: $e^{a\sqrt{r}} > r$ for any real a > 0 and r >> 0. Is this true? If so, what is an ...