Good list of exercises in Hatcher’s book algebraic topology
I’m doing self study of Hatcher’s book on algebraic topology. There are too many exercises after each section and try to solve all the exercises needs a lot of work and definitely not an efficient and...
View ArticleReference request: ring and geometry of mixed characteristics
It seems that to study p-adic numbers or ring of mixed characteristics, one has to start from a number theory perspective. I wonder if there is a good textbook about ring or geometry of mixed...
View ArticleWhat does "within the same order of magnitude" convey?
This question originates from a quandary about the meaning of the statement that two values are within the same order of magnitude. I wonder whether there is an established usage, of (rather more...
View ArticleFeelings of inadequacy as a PhD mathematician [closed]
I've recently started a PhD in pure mathematics, and I'm about to finish my first year. One thing that I have become acutely aware of during this time is that my understanding of much of the...
View ArticleWhy use geometric algebra and not differential forms?
This is somewhat similar to Are Clifford algebras and differential forms equivalent frameworks for differential geometry?, but I want to restrict discussion to $\mathbb{R}^n$, not arbitrary...
View Article"Negative" versus "Minus" when referring to Real numbers
As a math educator, do you think it is appropriate to insist that students say "negative $0.8$" andnot"minus $0.8$" to denote $-0.8$?The so called "textbook answer" regarding this question reads:A...
View Articleis it true that mathematicians mostly have very good memory? [closed]
I did not find a good tag so I chose some, I hope thats okay.I recently talked to someone who knows a fields medalist and he said that his memory is incredible, without going into details. Of course...
View ArticleHow do these two category-theoretic descriptions of set difference differ?
Most of the set operations are easy to transcribe into the category-theoretic language. Formally, let $A$ be a set, then:A subset $B \subset A$ is a category-theoretic subobject, whose accompanied...
View ArticleWhat mathematical consequences might there be if Euler Mascheroni constant is...
So far as I know, no one has proved the irrationality of Euler Mascheroni constant. There are discussions about the difficulty of proving the irrationality of this constant.Since we cannot prove that...
View ArticleInfinity question [closed]
Numbers have to stop at one point, right? So, can you ever really have an "infinite" number of things, or is infinity just a concept?
View ArticleIs there a group theoretic proof that $(\mathbf Z/(p))^\times$ is cyclic?
Theorem: The group $(\mathbf Z/(p))^\times$ is cyclic for any prime $p$.Most proofs make use of the fact that for $r\geq 1$, there are at most $r$ solutions to the equation $x^r=1$ in $\mathbf Z/(p)$,...
View ArticleModern references for Field Theory [duplicate]
I am currently taking a class on Groups and Rings. I have been reading a text called Topics in Field Theory by Gregory Karpilovsky in my free time. I started reading this book mainly to motivate my...
View ArticleFormal definition of plane
The formal definition of plane says that:A plane is a set of points such that if any two points are taken on it, all the points lying on the line joining these two points also lie on the plane.The...
View ArticleIs it reasonable to assert that "strong duality" is useless in practice...
I find the concept of strong duality hard to appreciate.In optimization, our ultimate objective is to find the optimizer $x^*$ that lives in the constraint set of the optimization problem$$\min f(x)...
View ArticleA possible explicit formula for the odd "Collatz numbers"
I have a possible interesting question about the Collatz Conjecture, that I think it is "answerable".Let me state my question as follows.We are only interested in odd natural numbers under the Collatz...
View ArticleWhat problems will arise if we define matrix multiplication using zero-padding?
Let $ a \in \mathbb{R}^n $ with $ a = (a_1, a_2, \dots, a_n) $. We define this to be equivalent to$$ (a_1, a_2, \dots, a_n, 0, 0, \dots, 0) $$where a finite number of zeros are appended. Using this...
View ArticleCan I use Ravi Vakil's way of learning for elementary subjects?
I mean you can't learn math in a linear order. Can I just read a paper first on a subject I haven't studied and just work backwards? For example, I have never studied combinatorics but I sort of have a...
View ArticleInterpretation of homology of $K(A,n)$
It is well known that the singular (co)homology of the space $K(G,1)$ is one possible definition of group homology. Now for non-abelian groups, we cannot expect to extract anymore obvious group...
View ArticleBook Recommendations for Learning Python for Mathematics.
Lately, I've been finding that I often need to compute various things and graph some pretty complicated functions. I've realized that learning to program, especially in Python, could be really helpful...
View ArticleSoft question - Uncommon examples of linear operators
Common examples of linear operators areMatrices in finite dimensional spaces.Differentiation and definite and indefinite integration in appropriately regular functional spaces.I remember that the...
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