# 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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### 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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### How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
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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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### Is There An Injective Cubic Polynomial $\mathbb Z^2 \rightarrow \mathbb Z$?

Earlier, I was curious about whether a polynomial mapping $\mathbb Z^2\rightarrow\mathbb Z$ could be injective, and if so, what the minimum degree of such a polynomial could be. I've managed to ...
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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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### Is there a categorical definition of submetry?

(Updated to include effective epimorphism.) This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use ...
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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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### 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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### 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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### 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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### 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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### 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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### Generalization of Liouville's theorem

As proposed in this answer, I wonder if the answer to following question is known. Let $E = E_0$ be the set of elementary functions. For each $i > 0$, inductively define $E_i$ to be the closure ...
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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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### Denesting radicals like $\sqrt[3]{\sqrt[3]{2} - 1}$

The following result discussed by Ramanujan is very famous: $$\sqrt[3]{\sqrt[3]{2} - 1} = \sqrt[3]{\frac{1}{9}} - \sqrt[3]{\frac{2}{9}} + \sqrt[3]{\frac{4}{9}}\tag {1}$$ and can be easily proved by ...
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### Limit of sequence of growing matrices

Let $$H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right),$$ ...
I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like $$\infty !=\mathop{\hat{\prod}}_{k=1}^\infty k = \sqrt{2\pi}$$ and \infty ... 0answers 498 views ### In n>5, topology = algebra During the study of the surgery theory I faced following sentence: Surgery theory works best for n > 5, when "topology = algebra". I don't know what is the meaning of topology=algebra. ... 0answers 396 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 252 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 400 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 119 views ### 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 ... 0answers 378 views ### Irrationality of \sum_{p\in\mathbb{P}} \frac{1}{2^{p}} Let \mathbb{P} be the set of prime numbers, and consider m=\displaystyle\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}. Is m irrational? In the following paper, the author recalls several sufficient ... 0answers 470 views ### Is there an exposition of complex analysis firmly separating the algebra, analysis, and topology? Complex analysis seems to work because of the interplay between algebraic geometry over \mathbb{C}, and analysis and topology exploiting the fact that \mathbb{C}/\mathbb{R} happens to be a ... 0answers 555 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 369 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 440 views ### Who was Hermann Künneth? Question as in the title: Who was Hermann Künneth? Where can I find some biographical information beyond what is available on Wikipedia? The well-known Künneth formula, for example in the form of ... 0answers 523 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 306 views ### Combinatorial Interpretation of a Certain Product of Factorials Let \mu denote the Moebius function. What is a combinatorial interpretation of the following integer, \begin{align} \prod_{d \mid n} d!^{\,\mu(n/d)}, \end{align} where the product is taken over ... 0answers 420 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 400 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 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 412 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 606 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 362 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 $\alpha=0.a_1a_2a_3\cdots\in\mathbb ... 0answers 563 views ### Rotman's exercise 2.8 “$S_n$cannot be imbedded in$A_{n+1}$” This question is about the (in)famous Rotman's exercise 2.8 in "An Introduction to the Theory of Groups." I've searched and found similar questions here and in MO, but none of them contains a valid ... 0answers 193 views ### Geometric way to view the truncated braid groups? This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question. I also asked a related question on MO, although ... 0answers 258 views ### Existence of a certain functor$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}, ...
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 ...