Recent content by kalish1

I need to show that the following expression, $$a^3b-a^3c+a^3z+a^3x+a^3y-a^2bx+a^2by+a^2cx-a^2cy-a^2zx+a^2zy-a^2x^2+a^2y^2-abcz-abcx-aczx-acx^2+b^2c^2+2bc^2x+c^2x^2-b^2c-2bcx-cx^2,$$ is positive given that: $1.$ $\ a,b,c,x,y,z$ are positive real numbers $2. \ \ a>b+x$ $3. \ \ c<b+y$ I...

A friend and I have been working on this problem for hours - how can the following expression be simplified analytically? It equals $\frac{1}{2},$ and we have tried the following to no avail: 1. Substitution of $x = \sqrt{5}$ 2. Substitution of $x = 2\sqrt{5}$ 3. Substitution of $x =...

Let $[a,b]$ be a closed real interval. Let $f:[a,b] \to \mathbb{C}$ be a continuous complex-valued function. Then $$\bigg|\int_{b}^{a} f(t)dt \ \bigg| \leq \int_{b}^{a} \bigg|f(t)\bigg| dt,$$ where the first integral is a complex integral, and the second integral is a definite real integral...

Problem: Given $W = \{z: z=x+iy, \ y>0\}$ and $g(z) = e^{2 \pi i z},$ what does the set $g(W)$ look like, and is it simply connected? Attempt: $W$ represents the upper-half complex plane. And $$g(z) = e^{2 \pi i (x+iy)} = \cdots = e^{-2\pi y}(\cos (2 \pi x) + i \sin (2 \pi x)).$$ (Am I on the...

How can I derive a contradiction from the following nasty statement: Assume $\sqrt{5} = a + b\sqrt[4]{2} + c\sqrt[4]{4} + d\sqrt[4]{8},$ with $a,b,c,d \in \mathbb{Q}$? *This is the last piece of an effort to prove that the polynomial $x^4-2$ is irreducible over $\mathbb{Q}(\sqrt{5}).$*

I started by setting $\alpha= e^{2\pi i/3} + \sqrt[3]{2}.$ Then I obtained $f(x) = x^9 - 9x^6 - 27x^3 - 27$ has $\alpha$ as a root. How can I proceed to find the minimal polynomial of $\alpha$ over $\mathbb{Q},$ and identify its other roots?

I am reading the following proof of a proposition from Royden+Fitzpatrick, 4th edition, and need help in understanding the last half of the proof. (My comments in italics.)----------Proposition: Let $A$ be a countable subset of the open interval $(a,b).$ Then there is an increasing function on...

Here is the modified problem: Let $x \in R - \{0\},$ where $R$ is a domain. Define $T_x(M) = \{m \in M \ | \ x^n m=0 \ \ \mathrm{for \ some} \ n \in \mathbb{N}\}$ as the $x$-torsion of $M.$ I need to show that $T_x(M \oplus N) = T_x(M) \oplus T_x(N)$ for $R$-modules $M,N$ or show that there...

Let $x \in R - \{0\},$ where $R$ is a domain. Define $T_x(M) = \{m \in M \ | \ x^n m=0 \ \ \mathrm{for \ some} \ n \in \mathbb{N}\}$ as the $x$-torsion of $M.$ I know that $T_x(M \oplus N) = T_x(M) \oplus T_x(N)$ for $R$-modules $M,N$ only if $R$ is a PID. But I can't think of a...

Letting $X$ be a ring and $K$ be an $X$-module, I need to show that **if** $K \cong A \times B$ for some $X$-modules $A,B$, **then** $\exists$ submodules $M'$ and $N'$ of $K$ such that: $K=M' \oplus N'$ $M' \cong A$ $N' \cong B.$----------I understand the concepts of internal and external...

This problem has been on my mind for a while. ---------- **Problem:** Show that **if** \begin{equation} |z_1+z_2+\dots+z_n| = |z_1| + |z_2| + \dots + |z_n| \end{equation} **then** $z_k/z_{\ell} \ge 0$ for any integers $k$ and $\ell$, $1 \leq k, \ell \leq n,$ for which $z_{\ell} \ne 0.$...

I need to prove that the complex power series $\sum\limits_{n=0}^{\infty}a_nz^n$ converges uniformly on the compact disc $|z| \leq r|z_0|,$ assuming that the series converges for some $z_0 \neq 0.$ *I know that the series converges absolutely for every $z,$ such that $|z|<|z_0|.$ Since...

I'm having some trouble showing that a Mobius transformation $F$ maps $0$ to $\infty$ and $\infty$ to $0$ iff $F(z)=dz^{-1}$ for some $d \in \mathbb{C}.$ Mainly with the "only if" part. Do I need to use pictures? This is Exercise $23$ in Section $3.3$ of Conway's *Functions of One Complex...

**Motivation:** I am studying for an exam over Chapters $1-3$ of *Real Analysis* by Royden and Fitzpatrick, 4th edition. I am stuck on understanding some of Proposition $11$, which I have reproduced below: **Proposition 11:** Let $f$ be a simple function defined on $E.$ Then for each...

So I was bored in math class and came up with this series of related questions, that I cannot answer: Is there a clean expression for $f'(x),$ where $$f(x)=\prod_{i=1}^{n}\dfrac{(x-i)}{(x+i)}?$$ What about for $f''(x)?$ Or for $$f(x)=\prod_{i=1}^{n}\dfrac{(x^2-i)}{(x^2+i)}?$$