# All Questions

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### Young-Fibonacci lattice and purely periodic continued fractions

The Fibonacci lattice $\mathcal{F}$ is the poset of all finite words consisting of 1's and 2's where a word $v$ covers a word $u$ if $v$ is obtained from $v$ by either (a) inserting a 1 in $u$ prior ...
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### Maximally nonplanar graphs

Is there any way to characterize maximally non-planar graphs? For example, given a random collection of k-regular graphs, there is a clear distinction between the planar and non-planar graphs by ...
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### From interpolation to separation

Lusin's separation theorem states that, if $A$ and $B$ are disjoint analytic subsets of a Polish space, then there is a Borel set $X$ separating them ($A\subseteq X$, $B\cap X=\emptyset$). Craig's ...
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### What are retracts of polynomial rings?

Is there a known example of a ring endomorphism $f: \mathbb{Z}[x_1, \ldots, x_n] \to \mathbb{Z}[x_1, \ldots, x_n]$ such that $f \circ f = f$ but whose image is not isomorphic to a polynomial ring? ...
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### What useful admissible rules does ZFC have beyond the deduction theorem?

I'm interested in formal proof verification, and one of the surprisingly difficult parts of this is dealing with proofs by contradiction. The issue is that the final step of such proofs is typically ...
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### Real analyticity of solution of heat equation

Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...
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### A question about equivariant sheaves

Suppose we have an G-equivariant sheav $\mathcal F$ on a smooth variety $X$. Can we split $\mathcal F$ as sum of eigensheaves? (I have seen this for structure sheaf but not sure if we can do it for ...
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### Inverse Galois problem for simple Lie type groups

Progress towards the Inverse Galois problem over $\mathbb{Q}$ is very well documented for sporadic groups ($M_{23}$ is the only case open) and for $PSL_n(q)$ (a lot of cases known, but wide open in ...
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### Tautological line bundle after blow-up

Let $X$ be a projective manifold, and $Z$ be a submanifold of $X$ with codimension at least 2. Let $Y$ be the blow-up of $X$ along $Z$ with the exceptional divisor $E$. Question: what is the relation ...
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### Equivariant Fredholm operators classify equivariant K-theory

Let $\mathcal{F}$ be the space of Fredholm operators on a separable Hilbert space $H$ with the topology induced by the operator norm. If $X$ is compact, Atiyah-Jänich proved that ...
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### On compact operators with domain $c_{0}$

Let $X$ be a Banach space and $T$ be a compact operator from $c_{0}$ into $X^{*}$. Let $\epsilon>0$. Is there a compact operator $S$ from $X$ into $l_{1}$ such that $S^{*}|_{c_{0}}=T$ and ...
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### Are all differential forms with integral periods closed?

Are all differential forms with integral periods necessarily closed? The idea being that the integral of such a form $A$ over any nullhomotopic cycle would have to be zero, and by contractibility and ...
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### Minimal immersions of the 2-sphere

Following the ideas of S.-S. Chern, J. L. M. Barbosa associated a holomorphic curve in $\mathbb{C}P^m$ to a minimal immersion of the 2-sphere into the $2m$-sphere in his 1972 paper. However, his ...
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### Does every connected component of a covering space over a connected base intersect all its fibers?

This statement is used without explanation in the proof of a Corollary to Proposition 4.3.5, pages 210-211 of the book "Algèbre et théories galoisiennes" by R. and A. Douady, second edition, Cassini, ...
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### A matching that covers vertices with maximum degree

We have a graph G with maximum degree $\Delta$. The induced subgraph on vertices with degree equal to $\Delta$ is a bipartite graph (while the original graph is not). Prove that G has a matching that ...
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### Shortest path problem

Say that $G'$ is a graph re-weighted from $G$ using the rule: $w' (u, v) = w(u, v) − f(u) + f(v)$, where $f$ always produce the positive results for any nodes. Can we prove that the shortest path ...
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### Why is taking the inverse Laplace transform valid in this case?

Assume $F \in L^2([0,\infty))$, so that the Laplace-transform $L[F]$ is well-defined. Assume furthermore, that $$y \mapsto \frac{L[F](iy)}{1+L[F](iy)}$$ is in $L^2(\mathbb{R})\cap L^1(\mathbb{R})$, ...
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### Modulus of continuity of analytic functions [on hold]

Let $H$ be the class of all analytic functions of the unit disk onto itself. For $r\in (0,2)$ let $$h(r)=\sup\{|f(z)-f(w)|: f\in H, |z-w|\le r\}.$$ How to determine $h$ explicitly? Schwarz lemma ...
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### Exotic 2-adic lifts of mod $2$ Steinberg idempotent

Denote $B_n$ the Borel subgroup of $Gl_n(Z/2)$, i.e., the subgroup of upper triangular matrices, $\Sigma _n$ the subgroup of permutation matrices. The (conjugate) Steinberg idempotent is defined to be ...
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### Geometric interpretation of the conjugation operation in the dual Steenrod algebra

As the dual mod 2 Steenrod algebra, $A$, is a Hopf algebra, it has the conjugation operation, $\chi:A\to A$. Milnor also gives a formula for this. I wonder if there is any source telling about a ...
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### Cannot multivectors be classified more easily than general tensors?

This is sort of a spinoff of Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces? - seems to be almost hopeless, but maybe some partial ...
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### To calculate $Tor_1^G(\mathbb{Z},N_{ab})$ and $Tor_1^Q(\mathbb{Z},N_{ab})$

Let $G$ be a finite $p$-group and $N$ be a normal subgroup of $G$. I wish to calculate $Tor_1^G(\mathbb{Z},N_{ab})$ and $Tor_1^Q(\mathbb{Z},N_{ab})$, where $Q=G/N$. I could not found any lecture notes ...
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### Existence of free operators, independent and with given distributions

Excuse me if the question is not appropriate for Mathoverflow. I havs asked it in math.stackexchange, but did not get any response. And so, I dared to put it here. I am trying to learn free ...
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### Best way to find recent papers in a special field of mathematics?

My subjects of interest are Geometry of Banach spaces, renorming theory and fixed point theory. When I want to find recent papers in these fields of mathematics, mostly, I search name of paper, say, ...
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### What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite

When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism ...
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### positions of a methane molecule with carbon atom at the origin

Let $\text{CH}_4$ be the molecule of Methane: The four hydrogen atoms form vertices of a regular tetrahedron with the carbon atom in the center of the regular tetrahedron. Here we regard all atoms ...
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### Angular distribution of zero sets of sparse polynomials

Consider a sequence of complex polynomials $f \in \mathbb{C}[z]$, $f(0) \neq 0$, that are composed of a negligible fraction $o(\deg{f})$ of monomials. Are the zeros of such polynomials necessarily ...
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### What function is a Gaussian integral

Let $g(u,\delta)=E[f(x)]$ where the expectation is over $N(u,\delta^2)$. Is there a characterization what function $g(u,\delta)$ can be produced this way? Is there a procedure solve the inverse ...
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### Proof for sequent (Logic) [on hold]

I'm currently trying to prove the following: http://i.imgur.com/dACOdOS.png And this is what I've come up with so far: And now I'm stuck. Perhaps the next step might be to eliminate for all x? ...
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### True for any expression of integer in the form of a sum of products, quotients of expressions of integers, could be an answer to a counting problem? [on hold]

For example, the super catalan number is in this form. There is no combinatorial interpretation found yet for the super catalan number but how could you tell whether there exists some?
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### How to computer the integral associated with Gamma function? [on hold]

I do know how to computer the following integral. Who can help me? Thanks! $$\int_R\frac{1-cos(tu)}{u^{2H+1}}du=-2|t|^{2H}\cos(\pi H)\Gamma(-2H),$$ where $H\in(0,1)$ and $\Gamma$ denotes the Gamma ...
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### Intuition behind salient numbers in number of h-cobordism classes of smooth homotopy n-spheres

The Wikipedia article on Exotic Sphere displays the sequence of numbers (see also OEIS A001676 and the Milnor link therein) for the order of the classse as 1, \;1, \;1,\; 1,\; 1, \;1, \;28,\; 2,\; ...
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### Listing all solutions to $n = x^2 + y^2 + z^2$ with integers

I would like to list all ways of writing $n$ as the sum of 3 squares. This is slightly different from finding just one: Is there an algorithm for writing a number as a sum of three squares? ...
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### A basic question on local cohomology

I had posted this question on stackexchange but did not get any response, hence putting it up on mathoverflow. Let $X$ be a smooth, projective variety, $i:X \hookrightarrow \mathbb{P}^n$ a closed ...
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### When does $R [x]/I$ have infinitely many idempotents?

Let $R$ be a commutative ring with identity and $R[x]$ its polynomial ring. I am looking for a ring with finitely many idempotent and an unextended ideal $I$ in $R [x]$ such that $R[x]/I$ has ...
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### How Universal is the Topological $\mathbb K$-algebra $C(\Omega, \mathbb K)$?

For $\Omega$ an arbitrary set the family $C(\Omega, \mathbb K)$ of all functions $\Omega \to \mathbb K$ becomes a complete topological $\mathbb K$-algebra under the topology of uniform convergence. ...
Let's say I have a surjective, continuous map $f: X \to Y$, and there is a deformation retract of the fibers $f^{-1}(y)$ to a point for every $y \in Y$. Is it always the case that there is a ...