# 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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### 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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### 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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### 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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### 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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### 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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### 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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### Why are asymptotically one half of the integer compositions gap-free?

This is a longish post about something that has been haunting me for a while about a kind of restricted composition, namely gap-free and complete compositions. First, I will define the terms that are ...
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### Is the sequence $(a_n)$ defined by $a_n=\tan{a_{n-1}}$, $a_0=1$, dense in $\Bbb{R}$?

Let $a_0=1,a_n=\tan{a_{n-1}}$. Then is $\{a_n\}_{n=0}^\infty$ dense in $\Bbb{R}$? I've drawn a map of this dynamical system and it seems that the sequence is dense on $\Bbb{R}$.
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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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### Extending the result $\int_{0}^{\infty} \left( ( 1 - 2C(x))^{2} + (1-2S(x))^{2} \right) \, dx = \frac{4}{\pi}$

While generalizing this result, I succeeded in proving that for $\alpha > 0$, $\beta < 1$ and $1 < 2\alpha + \beta < 3$, we have \begin{align*} &\int_{0}^{\infty} \left[ \left( ...
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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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### Applying Freyd-Mitchell's embedding theorem on large categories

One commonly reads that the Freyd-Mitchell's embedding theorem allows proof by diagram chasing in any abelian category. This is not immediately clear, since only small abelian categories can be ...
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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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### 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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### What is the Picard group of $z^3=y(y^2-x^2)(x-1)$?

I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the ...
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### Computing the Chern-Simons invariant of SO(3)

I am an undergraduate learning about gauge theory and I have been tasked with working through the two examples given on pages 65 and 66 of "Characteristic forms and geometric invariants" by Chern and ...
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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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### 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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### Limit of sequence of growing matrices

Let $$H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right),$$ ...
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### 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 ...
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### 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 ...
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### Does this sequence have any mathematical significance?

Take the sequence 001 and repeatedly append its second half to itself, using the larger half if the length is odd. This gives you ...
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### 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 $Y'$ ...
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### Application of Hilbert's basis theorem in representation theory

In Smalo: Degenerations of Representations of Associative Algebras, Milan J. Math., 2008 there is an application of Hilbert's basis theorem that I don't understand: Two orders are defined on the set ...
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### 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. ...
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### Making Friends around a Circular Table

I have $n$ people seated around a circular table, initially in random order. At each step, I choose two people and switch their seats. What is the minimum number of steps required such that every ...
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### 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. ...
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### 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 ...
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### 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": $S = \{(x,y) : |x|^p + |y|^p = 1\}$, ...
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### 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 $F$: ...
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### 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 ...
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### 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 ...
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### Subgroups as isotropy subgroups and regular orbits on tuples

Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω? Motivation: In ...
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### 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 ...
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### References on the theory of $2$-groups.

Many theorems about odd order $p$-groups fail miserably for $2$-groups. These can range from simple $2$-group exceptions (e.g. Frobenius complements can be either cyclic or generalized quaternion) to ...
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### Is every model of modular arithmetic either even or odd?

Modular Arithmetic (MA) has the same axioms as first order Peano Axioms (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$. ...
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 ...
### What does $H^0(Y',f^*N_Y)$ measure?
Let $X$ be a smooth variety and let $Y\subset X$ be a smooth subvariety. Let $f:Y'\to Y$ be a (say, finite surjective) morphism. When $f$ is the identity, the cohomology group $H^0(Y',f^*N_Y)$ ...