# All Questions

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### A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
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### Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...
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### The Ring Game on $K[x,y,z]$

I recently read about the Ring Game on Mathoverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative ...
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### Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}},$$ where ...
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### A variation of Fermat's little theorem

Fermat's little theorem states that for $n$ prime, $$a^n \equiv a \pmod{n}.$$ The values of $n$ for which this holds are the primes and the Carmichael numbers. If we modify the congruence ...
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### What properties of busy beaver numbers are computable?

The busy beaver function $\text{BB}(n)$ describes the maximum number of steps that an $n$-state Turing machine can execute before it halts (assuming it halts at all). It is not a computable function ...
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### Grothendieck 's question - any update?

I was reading Barry Mazur's biography and come across this part: Grothendieck was exceptionally patient with me, for when we first met I knew next to nothing about algebra. In one of his first ...
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### Differential forms on fuzzy manifolds

This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me. Question In the usual setting of open subsets of ...
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### Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
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### Is this similarity to the Fourier transform of the von Mangoldt function real?

Mathematica knows that the logarithm of $n$ is: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ The von Mangoldt function should then be: ...
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### Continuous projections in $\ell_1$ with norm $>1$

I was trying to find papers and articles about non-contractive continuous projections in $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case ...
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### Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
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### Is $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ algebraic or transcendental?

It's easy to show that $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ is irrational. However, can it be shown whether it is algebraic or transcendental? My hunch is that it's transcendental but I don't know ...
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### Is there a homology theory that counts connected components of a space?

It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$. I have recently learned that for Čech homology the ...
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### Geometric interpretation of the Riemann-Roch for curves

Let $X$ be a smooth projective curve of genus $g\geq2$ over an algebraically closed field $k$ and denote by $K$ a canonical divisor. I have some clues about the geometrical interpretation of the ...
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### Regular way to fill a $1\times1$ square with $\frac{1}{n}\times\frac{1}{n+1}$ rectangles?

The series $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=1$$ suggests it might be possible to tile a $1\times1$ square with nonrepeated rectangles of the form $\frac{1}{n}\times\frac{1}{n+1}$. Is there a ...
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### Solving Special Function Equations Using Lie Symmetries

The lie group + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's ...
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In Mathworld's "Pi Approximations", (line 58), Weisstein mentions one by the mathematician Daniel Shanks that differs by a mere $10^{-82}$, $$\pi \approx ... 0answers 439 views ### Is Erdős' lemma on intersection graphs a special case of Yoneda's lemma? Under which name is the following proposition filed actually: Every poset P embeds fully and faithfully in the powerset of P, ordered by subset inclusion. Let me call it Dedekind's lemma. ... 0answers 431 views ### Dedekind Sum Congruences For a,b,c \in \mathbb{N}, let a^{\prime} = \gcd(b,c), b^{\prime} = \gcd(a,c), c^{\prime} = \gcd(a,b) and d = a^{\prime} b^{\prime} c^{\prime}. Define \mathfrak{S}(a,b,c) = a^{\prime} ... 0answers 232 views ### When are two proofs “the same”? Often, we find different proofs for certain theorems that, on the surface, seem to be very different but actually use the same fundamental ideas. For example, the topological proof of the infinitude ... 0answers 452 views ### Find all functions f such that if a+b is a square, then f(a)+f(b) is a square Question: For any a,b\in \mathbb{N}^{+}, if a+b is a square number, then f(a)+f(b) is also a square number. Find all such functions. My try: It is clear that the function$$f(x)=x$$... 0answers 670 views ### Rotman's exercise 2.8 “S_n cannot be imbedded in A_{n+1}” This question is about the (in)famous Rotman's exercise 2.8 in "An Introduction to the Theory of Groups." I've searched and found similar questions here and in MO, but none of them contains a valid ... 0answers 554 views ### Definitive source about Dirichlet finally proving the Unit Theorem in the Sistine Chapel There is a remark one can find in various books or survey articles (e.g., page 49 of Helmut Koch's "Number Theory: Algebraic Numbers and Algebraic Functions") saying Dirichlet figured out a proof of ... 0answers 419 views ### Tate conjecture for Fermat varieties I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture ... 0answers 272 views ### Projective profinite groups I'm reading the first chapter of Serre's Galois Cohomology. On p. 58, He gives two examples of projective profinite groups: the profinite completion of free (discrete) groups; the cartesian product ... 0answers 2k views ### Pullback and Pushforward Isomorphism of Sheaves Suppose we have two schemes X, Y and a map f: X\to Y. Then we know that \operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F}), where ... 0answers 605 views ### How many points of intersection between an ellipse and an L_p-circle? Consider an ellipse E in the plane, centered at the origin. (In my case, the minor axis points into the nonnegative quadrant.) Let S be an "L_p-circle": S = \{(x,y) : |x|^p + |y|^p = 1\}, ... 0answers 389 views ### Classification of local Artin (commutative) rings which are finite over an algebraically closed field. A result in deformation theory states that if every morphism Y=\operatorname{Spec}(A)\rightarrow X where A is a local Artin ring finite over k can be extended to every Y'\supset Y where Y' ... 0answers 305 views ### Existence of a certain functor F:\mathrm{Grpd}\rightarrow\mathrm{Grp} Let \mathrm{Grpd} denote the category of all groupoids. Let \mathrm{Grp} denote the category of all groups. Are there functors F\colon\mathrm{Grpd}\rightarrow \mathrm{Grp}, ... 0answers 172 views ### Geometric & Intuitive Meaning of SL(2,R), SU(2), etc… & Representation Theory of Special Functions Many special functions of mathematical physics can be understood from the point of view of the representation theory of lie groups. An example of the power of this viewpoint is given in my question ... 0answers 503 views ### On the problem of polynomial bijection from \mathbb Q\times\mathbb Q to \mathbb Q The question titled "Polynomial bijection from \mathbb Q\times\mathbb Q to \mathbb Q" which was posed on MathOverflow attracted quite a lot of attention (and may be the question with most wrong ... 0answers 221 views ### Strengthening the intermediate value theorem to an “intermediate component theorem” Let f be a continuous function on a closed Euclidean ball (in dimension \ge 2) that is negative in the center of the ball and positive on its boundary. On any path from the center of the ball to ... 0answers 414 views ### Generating function solution to previous question a_{n}=a_{\lfloor n/2\rfloor}+a_{\lfloor n/3 \rfloor}+a_{\lfloor n/6\rfloor} In attempting to answer this question, I reduced it to a seemingly simple generating functions question, but after days of work was unable to construct a proof. Since I do not have experience trying ... 0answers 462 views ### This one weird thing that bugs me about summation and the like Most of us know$$\sum_{n=a}^b c_n=c_a+c_{a+1}...+c_{b-1}+c_b$$Some of us know$$\prod_{n=a}^b c_n=c_a \cdot c_{a+1}...c_{b-1} \cdot c_{b}$$A few of us know ... 0answers 876 views ### A very nice divisibility problem A very hard problem, here it is: Prove that, if 2^{2^j} a + 1 divides c^{2^j}+1 for fixed integers a,c and all nonnegative integers j, then a=1 and c=2^l for some odd positive integer ... 0answers 382 views ### If \beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q, then \alpha=0.a_1a_2a_3\cdots\in\mathbb Q? Question : For every even k\ge 4, is the following (\star) true?$$\begin{align}\text{If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb ... 0answers 2k views +200 ### What are some strong algebraic number theory PhD programs? I am currently applying for PhD programs in the US. My main interests are number theory and algebra. More specifically, I am interested in algebraic number theory (number fields, Galois groups, ... 0answers 214 views ### Geometric way to view the truncated braid groups? This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question. I also asked a related question on MO, although ... 0answers 433 views ### When are nonintersecting finite degree field extensions linearly disjoint? Let$F$be a field, and let$K,L$be finite degree field extensions of$F$inside a common algebraic closure. Consider the following two properties: (i)$K$and$L$are linearly disjoint over$F$: ... 0answers 384 views ### Which number fields can appear as subfields of a finite-dimensional division algebra over Q with center Q? I have some idle questions about what's known about finite-dimensional division algebras over$\mathbb{Q}\$ (thought of as "noncommutative number fields"). To keep the discussion focused, let's ...

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