# All Questions

**1**

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22 views

### 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 ...

**-1**

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**0**answers

18 views

### 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 ...

**1**

vote

**0**answers

17 views

### 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 ...

**3**

votes

**0**answers

25 views

### 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?
...

**0**

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**0**answers

46 views

### Bertini type theorem in positive characteristic

Let $f:X \to Y$ be a morphism of finite type over an algebraically closed field of characteristic $0$. Assume that $Y$ is irreducible and non-singular. Let $x \in X$ be a closed point and $T_xf:T_xX ...

**-2**

votes

**1**answer

54 views

### Diophantine equations over natural numbers

Are there versatile techniques that are applicable to deciding if a system of multivariate quadratic diophantine equations with one determinantal restriction has solutions over natural numbers?
In ...

**10**

votes

**1**answer

253 views

### Erdös-Turán via Hardy-Littlewood circle method?

For any set $B\subseteq \mathbb{N}$ one can associate the formal series
$$f_B(z) = \sum_{b\in B}z^b$$
and obtain
$$f_B(z)^k = \sum_{n\geqslant 0} r_{B,k}(n)z^n,$$
where $r_{B,k}(n) = ...

**3**

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**0**answers

51 views

### 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 ...

**5**

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**0**answers

50 views

### 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 ...

**2**

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**0**answers

73 views

### 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 ...

**4**

votes

**2**answers

211 views

### 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 ...

**0**

votes

**0**answers

69 views

### 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 ...

**3**

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**0**answers

45 views

### 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
...

**1**

vote

**0**answers

23 views

### 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 ...

**-3**

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**0**answers

63 views

### 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 ...

**1**

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**0**answers

45 views

### 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 ...

**0**

votes

**1**answer

66 views

### 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, ...

**2**

votes

**0**answers

38 views

### 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 ...

**-2**

votes

**1**answer

57 views

### 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 ...

**3**

votes

**1**answer

64 views

### 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 $L^2(\mathbb{R})\cap L^1(\mathbb{R})$, ...

**0**

votes

**0**answers

52 views

### 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 ...

**3**

votes

**0**answers

69 views

### 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 ...

**3**

votes

**1**answer

132 views

### 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 ...

**2**

votes

**0**answers

51 views

### 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 ...

**2**

votes

**0**answers

42 views

### 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 ...

**1**

vote

**1**answer

49 views

### 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 ...

**1**

vote

**1**answer

170 views

### 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, ...

**1**

vote

**0**answers

51 views

### 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 ...

**9**

votes

**1**answer

236 views

### 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 ...

**7**

votes

**0**answers

66 views

### 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 ...

**2**

votes

**1**answer

80 views

### 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 ...

**-4**

votes

**0**answers

28 views

### 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?
...

**-4**

votes

**0**answers

43 views

### 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?

**-4**

votes

**0**answers

29 views

### 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 ...

**4**

votes

**1**answer

277 views

### 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,\; ...

**4**

votes

**1**answer

232 views

### 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?
...

**1**

vote

**0**answers

101 views

### 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 ...

**0**

votes

**1**answer

209 views

### 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 ...

**1**

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**0**answers

50 views

### 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. ...

**0**

votes

**1**answer

66 views

### Is a continuous family of contractible spaces simultaneously contractible?

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 ...

**8**

votes

**0**answers

46 views

### A kaleidoscopic coloring of the plane

Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap ...

**0**

votes

**0**answers

83 views

### Constructing special holomorphic functions

I would appreciate any help with this question as I am not sure how I should approach it.
Suppose $ D$ is the unit disk and that $A(x)$ is a real valued smooth function on $D$.
Does there exist a ...

**4**

votes

**1**answer

154 views

### Generalized density functions on the natural numbers

If $a_1,a_2,\dots$ are IID random bits (correction as per Anthony Quas: these "bits" are $+1$ and $-1$ with equal probability), then with probability 1, the set of natural numbers $n$ such that ...

**2**

votes

**1**answer

72 views

### k-fellow traveler property and automatic structur

Let P be a permutation group with some generating set S and let W be the word acceptor automaton of P, if I know the value of k (k-fellow-traveller property of CayleyGraph CG(P,S)).
I realized that ...

**0**

votes

**1**answer

63 views

### Is the restricted root system of a simple real Lie group irreducible?

As the title asks, is the restricted root system of a simple real Lie group irreducible?
I believe this is true but I need a reference to cite.

**2**

votes

**1**answer

280 views

### Does Borel's proof for existence of normal numbers make an essential use of axiom of choice?

A normal number is a real number whose infinite sequence of digits in every base $b$ is distributed uniformly in the sense that each of the $b$ digit values has the same natural density $\frac{1}{b}$, ...

**0**

votes

**0**answers

57 views

### Eigenvalues of a partitioned self-adjoint matrix

This is a repost of the same question on MSE (with no reply/comment):
http://math.stackexchange.com/questions/1454314/eigenvalues-of-a-partitioned-self-adjoint-matrix
I would be grateful just for a ...

**1**

vote

**1**answer

97 views

### Galois group - unknown word

I found in some articles a definition of the Galois group of a differential module. In the definition appeared the expression
$(Repr(Gal(M,\nabla),Forget)$. According to the the articles, this pair ...

**0**

votes

**0**answers

94 views

### Harmonic function with Dirichlet boundary condition

Consider the domain $D = \{(x_1, x_2,.., x_n) \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$. Let $D$ be divided into two parts $D_1$ and $D_2$ by the hyperplane $H = \{x_1 = \frac{1}{2}\}$. My question is: ...

**0**

votes

**0**answers

68 views

### Osculating ellipsoids

Let $K$ be a given smooth, origin-symmetric convex body in $n$ dimensional euclidean space. At each point $x$ on the boundary of $K$ there exists an origin-symmetric ellipsoid that touches $x$ of ...