Articles of big list

Important applications of the Uniform Boundedness Principle

There’s like three applications of the uniform boundedness principle in wikipedia: 1) If a sequence of bounded operators converges pointwise to an operator, then the limit operator is also bounded, and the convergence is uniform on compact sets. 2) Any weakly bounded subset of a normed space is bounded. 3) A result in pointwise convergence […]

Properties of the number 50

I will shortly be engaging with my 50th (!) birthday. 50 = 1+49 = 25+25 can perhaps be described as a “sub-Ramanujan” number. I’m trying to put together a quiz including some mathematical content. Contributions most welcome. What does 50 mean to you?

Proof of Pythagorean theorem without using geometry for a high school student?

There are some proofs of Pythagoras theorem which don’t even require high school maths to understand, but they all are using shapes to prove of the theorem. However, I am trying to find some proofs of Pythagoras theorem that don’t use shapes in their proofs, for example a purely algebraic proof. Besides, they would still […]

What are some examples of principal, proper ideals that have height at least $2$?

Krull’s principal ideal theorem states that in a Noetherian ring $R$, any principal proper ideal $I$ has height at most $1$. Presumably the Noetherian hypothesis is required, so what are some (preferably simple) examples where the result of the theorem doesn’t hold? A search for similar examples turned up this question, but perhaps not requiring […]

List of equivalents to Law of the Excluded Middle under intuitionistic logic

What are commonly-known equivalents of LEM among logicians, assuming intuitionistic axioms? I’ll list a few to begin with. Below, $\varphi$ and $\psi$ denote arbitrary sentences. (DNE) Double negation elimination: $\neg \neg \varphi \vdash \varphi$ (RAA) Reductio ad absurdum: $\neg \varphi \to \psi, \neg \varphi \to \neg \psi \vdash \varphi$ (P) Peirce’s Law: $\vdash ((\varphi \to […]

I'm searching for some books with guidance into mathematical study.

Yesterday, I’ve found this. It’s a PDF file with this purpose, from Oxford. Some weeks ago I’ve also found two books tha seems to fill this purpose: Prelude to Mathematics; I Want to Be a Mathematician. I’m not searching for something specific as I still have no idea on what I want to do with […]

Funny translations of mathematical words

As already noticed in this question there are some mathematical words that literally translated from a language to english (or from english to this language) means something totally different. A few examples: “Eigenwert” (= “eigenvalue” in german) is translated “intrinsic value” by google translate. “coprs” (= “division ring” in french) is translated “body” or “corps” […]

Extending the ordered sequence of 'three-number means' beyond AM, GM and HM

I want to create an ordered sequence of various ‘three-number means‘ with as many different elements in it as possible. So far I’ve got $12$ ($8$ unusual ones are highlighted): $$\sqrt{\frac{x^2+y^2+z^2}{3}} \geq \color{blue}{ \frac{\sqrt{x^2+y^2}+\sqrt{y^2+z^2}+\sqrt{z^2+x^2}}{3 \sqrt{2}}} \geq $$ $$\geq \color{blue}{\frac{\sqrt{(x+y)^2+(y+z)^2+(z+x)^2}}{2 \sqrt{3}} } \geq \frac{x+y+z}{3} \geq $$ $$ \geq \color{blue}{ \frac{\sqrt[3]{(x+y)(y+z)(z+x)}}{2} } \geq \color{blue}{\sqrt{\frac{xy+yz+zx}{3}}} \geq $$ $$\geq […]

Example of non-noetherian algebras which are tensor products of noetherian algebras

We suppose all rings are commutative with unity. I am looking for examples of a tensor product $B\otimes_A C$ which is not noetherian, where $A$ is a noetherian ring and $B, C$ are noetherian $A$-algebras. The more examples the better. In other words, I’m asking a big list of examples.

Group theory applications along with a solved example

As I asked in previous question, I am very curious about applying Group theory. Still I have doubts about how I can apply group theory. I know about formal definitions and I can able to solve and prove problems related to Group theory. But when comes to applications, I don’t know where to start. I […]