Generalization of winding number for surfaces of the form $\mathbb{R}^n...
IdeaI have an idea, that it is possible to generalize the winding number for surfaces of the form $f: \mathbb{R}^n \rightarrow \mathbb{R}^{n+1}$The winding number for $n=1$ is $w_{\gamma}(x) =...
View ArticleWhy do we write $ghg^{-1}$ (in algebra) and $PDP^{-1}$ (in linear algebra)...
Why do we write $ghg^{-1}$ (in algebra) and $PDP^{-1}$ (in linear algebra) and not $g^{-1}hg$ and $P^{-1}DP$?Conjugation by $g$ is always written $ghg^{-1}$. We could just have easily defined it as...
View ArticleSubjects of study that involve some or all of: analysis and low-dimensional...
According to my studies, I see my interests in the following subjects:Analysis and low-dimensional topologyAlgebraGraph theory and combinatoricsI am looking for a subject in mathematics that is somehow...
View ArticleWhat are the topics in mathematics I need to study to become a cryptographer
What are the topics in mathematics I need to study to become a cryptographer?And what all are the books I am to read?
View ArticleExamples of the Pigeonhole Principle
As most of you might know, the Pigeonhole Principle basically states that If $n$ items are put into $m$ containers, with $n>m$, then at least one container must contain more than one itemIt always...
View ArticleTips on proving that a set has an algebraic structure
If we read the article about ring on wikipedia, we can see that the properties of ring are as follows.A ring is a set $R$ equipped with two binary operations $+$ (addition) and $â‹…$ (multiplication)...
View ArticleIs there a word that encompasses both "definitions" and "theorems"? [closed]
I sometimes write sentences like this:To prove the desired result, the following notions will be handy.What follows is a usually a sequence of definitions, theorems and proofs. Therefore, I don't think...
View ArticleDifferent ways to find the inverse of a function
I recently started to look into analog circuit implementation of mathematical operations on signals, so while computing integration and differentiation, it was straight-forward with the use of...
View ArticleHelp me decipher this linear algebra proof
I am studying for a linear algebra exam using notes provided by my professor, who has a somewhat idiosyncratic approach to notation, showing work, and omission of detail. While I understand most of the...
View ArticleInteresting math-facts that are visually attractive
To give a talk to 17-18 years old (who have a knack for mathematics) about how interesting mathematics (and more specifically pure mathematics) can be, I wanted to use nice facts accompanied by nice...
View ArticleStruggling to retain mathematical knowledge as a self-learner: How can I...
I’m a self-learner who studies mathematics because of a deep passion for the subject. However, I’ve been facing a persistent issue that has started to seriously affect both my progress and motivation....
View ArticleApplications of information geometry to the natural sciences
I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has...
View ArticleShowing the $\mathcal{P}$ subalgebra generated by a subset $S$ is the...
This question is intended to be as general as possible when defining "subalgebra". Hence the universal-algebra tag and the category-theory tag.The Question:Show that for an algebra $A$, the $\mathcal...
View ArticleIs there an abstract reason why alot of calculus tools that exist on Banach...
As far as I can tell, which doesnt have to mean much, Frechet spaces are basicly the "closest possible" generalisation of Banach spaces, since every Frechet space is a sequential limit of Banach spaces...
View ArticleTheorems with an extraordinary exception or a small number of sporadic...
The Whitney graph isomorphism theorem gives an example of an extraordinary exception: a very general statement holds except for one very specific case.Another example is the classification theorem for...
View ArticleIs $\tan\theta\cos\theta=\sin\theta$ an identity?
A friend of mine, who is a high school teacher, called me today and asked the question above in the title. In an abstract setting, this boils down to asking whether an expression like "$f=g$" is...
View ArticlePhilosophy of Mathematical Foundations [closed]
I am only an amateur mathematician, and I have recently been interested in the foundations of mathematics. Upon reading about predicate logic and finding language defined in terms of sets and functions...
View ArticleBook series like AMS' Student Mathematical Library?
I had the joy of discovering AMS'Student Mathematical Library book series today, and I have been pleasantly surprised by how enticing some of the titles seem: exciting and expositionary, a perfect...
View ArticleWhat does it mean by killing a subgroup in quotient process?
We often use the word killing$H$ while describing the quotient $G/H$,where $H$ is a normal subgroup of $G$.In this process what we actually do is we consider all the cosets induced by $H$ and consider...
View ArticleHow to develop patience in mathematics?
How to develop the patience in mathematics?When you read the books like Rudin, Conway, etc., it takes a lot of time for one problem; sometimes $10$ hours each.I usually devote one problem for $30$...
View Article