Do exists an inverse Cayley–Dickson construction for deducing...
Is there a known inverse or reverse Cayley–Dickson construction that enables deduction of numbers in the reverse order, from higher-dimensional to lower-dimensional sets?For example, starting from...
View ArticleUnexpected examples of natural logarithm
Quite often, mathematics students become surprised by the fact that for a mathematician, the term “logarithm” and the expression $\log$ nearly always mean natural logarithm instead of the common...
View ArticlePutnam 2008 B2 -- Soft Question
The 2008 Putnam exam has the following question for B2:Let $F_0(x) = \ln x.$ For $n \geq 0$ and $x>0,$ let $F_{n+1} (x) = \int _0 ^xF_n(t) \, dt$. Evaluate $$\lim _{n \rightarrow \infty} \frac{n!...
View ArticleIs it a bit circular to say that every thing is equal to exactly one thing?
This is a theorem of first-order logic with equality: $(\forall x)(\exists! y)x=y$, where the $\exists!$ means the unique existential quantifier. However, this theorem seems a bit circular, or at least...
View ArticleSDEs and irreversibility in statistical mechanics
In statistical mechanics, the equations governing particle behavior are typically deterministic ordinary differential equations (ODEs). However, in real-life particle systems, these systems are never...
View ArticleReal analysis (series) practice questions.
I have an analysis 1 exam coming up this week, and I am still not particularly comfortable proving series convergence / divergence as well as continuity using the epsilon delta definition. I’m looking...
View ArticleRelation of twisted graded modules and twisted sheaves
I came across the definitions of graded twisted modules while reading about the syzygy theorem. In the meantime I also attended an algebraic geometry course where twisted sheaves occured. For both...
View ArticleShall proofs be externalised?
I might have a reached a point in a proof I’m writing in which I can take two different directions:I can complete my proof by relying on someone else’s theorem;I can complete my proof without relying...
View ArticleWhat are some conjectures of your own?
Background: Although this site is most-often used for specific one-off questions, many of the highest scored questions (also on MathOverflow), which gather a lot of attention to the site are about...
View ArticleConnection between twisted graded modules and twisted sheaves
I came across the definitions of graded twisted modules while reading about the syzygy theorem. In the meantime I also attended an algebraic geometry course where twisted sheaves occured. For both...
View Article(When) are recursive "definitions" definitions?
This is a "soft" question, but I'm greatly interested in canvassing opinions on it. I don't know whether there is anything like a consensus on the answer. Under what conditions (if any) are recursive...
View ArticleCommutative algebra from Hungerford’s algebra [closed]
I just finished a course in abstract algebra (group theory, ring and module theory, field and Galois theory) from Hungerford’s algebra GTM. I want to study algebraic geometry, and commutative algebra...
View ArticleHow can this open map characterization be explained or interpreted?
I hope you're having a good day.I'm an undergrad mathematics student, I took a general topology course a few months ago, and I'm now reviewing topological spaces to prepare for functional analysis.I...
View ArticleThe Relationship Between Field Extension Tower and Tensor Product
Let $F$ be a field.Consider field extension $ F \subset L \subset K $. Let $ \alpha_1, \dots , \alpha_m $ be an $F$-basis of $ L $, $\beta_1, \dots, \beta_n$ an $L$-basis of $ K $. By the theory of...
View ArticleWhat makes a math problem important? [closed]
Given that pure mathematics is, by definition, not concerned with applications, how does one decide that one problem is more valuable than another? Is it just a matter of certain topics becoming...
View ArticleIs it right to think of the grothendieck group $K(A)$ as the categorification...
Grothendieck $K$-groups make sense for exact categories I think, but let $k$ be a field and let's assume $A$ is a $k$-linear abelian category, which are supposed to 'categorify' vector spaces over $k$....
View ArticleCan I simplify multiple inequality cases into a single absolute inequality?
I'm not an expert and I haven't delt with these kinds of inequalities for a while, so I'm hoping someone can explain if my thought process is reliable and will suffice to prove the results I'm looking...
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Möbius band alike shape found trying to understand the Bloch-sphere
Colored/Labeled finite set of natural numbers describing a curve in space resembling a Möbius strip found by playing with the idea of the bloch-sphere.The title is too long but the question is...
View ArticleInterpretation of algebraic-geometric language into differential-geometric...
I want to know how to interpret a proposition stated by algebraic geometry language in a differential geometry way.For example, roughly speaking, I think schemes corresponds to complex manifolds,...
View ArticleWhat are the arguments of the mathematicians who objected against the...
Q:What are the arguments of the mathematicians who objected against the ontological argument/proof Gödel offered?$$\begin{array}{rl} \text{Ax. 1.} & \left\{P(\varphi) \wedge \Box \; \forall...
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