updates
Solve $x^2 + x = y^4 + y^3 + y^2 + y$ over integers.
I am trying to solve solve $x^2 + x = y^4 + y^3 + y^2 + y$ over the integers. So far I have decomposed it into $x(x + 1) = y(y + 1)(y^2 + 1)$ and noticed ...
Apr 13, 2026
Archive: Updates
updates
The Minkowski sum of two convex sets is convex
Let $A$ and $B$ be two convex subsets in $\mathbb{R}^n$. Define a set $C$ given by $$C = A + B = \{a + b : a \in A \mbox{ and } b \in B\}.$$ Is $C$ a conv...
Apr 13, 2026
updates
Find non-trivial solution and then find all solutions
Find the value of b for which the following system has a non-trivial solution and find all the solutions in this case $$2x + 6z = 0$$ $$4x + y + bz = 0$$ ...
Apr 13, 2026
updates
Definition of Topological Embedding
Definition. Let $X$ and $Y$ be topological spaces. Suppose $f:X\to Y$ is an injective continuous map. If the function $f':X\to\ f(X)$ obtained by res...
Apr 13, 2026
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Converging Sequences Definition
Q. Explain exactly what it means for $\{a_n\}$ $n\in\mathbb N$ to converge to $L ∈ R.$ I wrote that for $\{a_n\}$ to converge to $L ∈ R$ means that the in...
Apr 13, 2026
updates
$dy/dx=e^{-y}$ differential equation
I am trying to solve the following differential-equation: $$dy/dx=e^{-y}$$ This is what I tried: However that is not a correct solution. How can I derrive...
Apr 13, 2026
![Prove that $n<2^n$ for every positive integer $n$ [duplicate]](https://news.conceivablytech.com/uploads/prove-that-n-2-n-for-every-positive-integer-n-duplicate_720.jpg)
updates
Prove that $n<2^n$ for every positive integer $n$ [duplicate]
Prove that $n<2^n$ for every positive integer $n$. $P(1): 1 < 2^1$ $n+1<2^n+1$ for induction hypothesis $n+1<2^n+2^n$ I can't undertstand...
Apr 13, 2026
updates
How to solve $n < 2^{n/8}$ for $n$?
This is from an exercise (1.2.2) in introduction to algorithms that I'm working on privately. To find at what point a $n \lg n$ function will run fas...
Apr 13, 2026
updates
Real Analysis, Folland Exercise 1.5.33 Borel measures on the real line
Exercise 33 (Real Analysis - Folland): "There exists a Borel set $A \subset [0,1]$ such that $0 < m(A \cap I) < m(I)$ for every subinterval $I$ of $...
Apr 13, 2026
updates
Why is it that when proving trig identities, one must work both sides independently?
Suppose that you have to prove the trig identity: $$\frac{\sin\theta - \sin^3\theta}{\cos^2\theta}=\sin\theta$$ I have always been told that I should mani...
Apr 13, 2026
