# All Questions

**1**

vote

**1**answer

52 views

### Did Erdős prove there are two primes $4a+1, 4b+3$ between between $n$ and $2n$?

http://mathworld.wolfram.com/ChoquetTheory.html
Is the claim in the link true? Here's the reference given there:
https://www.renyi.hu/~p_erdos/1934-01.pdf
Erdős proved that there exist at least ...

**0**

votes

**0**answers

23 views

### A question on the integrability of eigenfunctions of the Laplacian

Let $(M,g)$ be a closed Riemannian manifold. Let $\lambda$ and $u$ be (the $k$-th) eigenvalue and eigenfunction,
$$\Delta u=-\lambda u.$$
I was wondering under what condition (for example, spaces ...

**0**

votes

**0**answers

21 views

### Showing a permutation module is reducible

For a permutation module $V$, which is a permutation module if it has the basis $B = \{v_1,...,v_n\}$ such that the matrix of every $g \in G$ with respect to this basis is a permutation matrix. I need ...

**1**

vote

**1**answer

23 views

### Veronese embeddings and locally free resolutions

Let $i : \mathbf P^1 \to \mathbf P^2$ be the second Veronese embedding. Clearly, $i_\star \mathcal O_{\mathbf P^1}$ has a locally free resolution of the form
$$
0 \to \mathcal O_{\mathbf P^2} (-2) ...

**-4**

votes

**0**answers

18 views

### Problem on Shell Method [on hold]

I've been on this problem for awhile and have no idea how to figure it out. Some help would be greatly appreciated.
Thank you! http://i.stack.imgur.com/I4dP7.png

**0**

votes

**0**answers

11 views

### $d\bar d$-lemma on pair $(X,D)$

Let $X$ be a Kahler manifold with a simple normal crossing divisor $D$, i.e., pair $(X,D)$. Let $\omega$ and $\omega'$ be two Kahler forms in the same Kahler class then have we $d\bar d$-lemma on pair ...

**2**

votes

**2**answers

86 views

### Finite orbits on an elliptic curve with two generic involutions

Let $C$ be a (very) general genus 1 curve embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a (2,2)-divisor.
Each projection defines $C$ as a double cover of $\mathbb{CP}^1$ and induces an ...

**3**

votes

**0**answers

48 views

### Plurisubharmonic functions on Kähler manifolds, intuition?

As the question suggests, what is the intuition for working with plurisubharmonic functions on Kähler manifolds?

**-4**

votes

**0**answers

55 views

### Lexicographic rank of an odd/even permutation [on hold]

Let $S_n$ be the set of all permutations of integers from $1$ to $n$. Let $P_1$ and $P_2$ be two partitions defined on $S_n$ as follows.
$P_1$ is the set of all those permutations which have even ...

**13**

votes

**1**answer

162 views

### Are there irreducible polynomials with all zeros on two concentric circles?

This is somewhat similar to this recent question, but extending in a different direction.
Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a bicycle ...

**1**

vote

**0**answers

74 views

### Bertini's theorem in positive characteristic for smooth morphisms

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

**2**

votes

**1**answer

239 views

### What is the Euler characteristic of a mapping space?

Suppose that $A$ and $B$ are topological spaces homotopy equivalent to finite cell complexes, and let $B^A = \mathrm{maps}(A,B)$ denote the space of maps from $A$ to $B$. Is it there a formula for ...

**3**

votes

**1**answer

289 views

### Is a manifold generically real analytic (with generic real analytic metric)?

I have heard it said in some differential geometry talks that "the generic situation in such and such case is real analytic". My question is, is the generic smooth manifold also real analytic in some ...

**2**

votes

**1**answer

85 views

### A question about simple closed curves in 3-dimensional Euclidean space

Let E(3) be 3-dimensional Euclidean space. I have submitted the following question to Mathstackexchange and other mathematical websites, but have never received any responses-not even rejections on ...

**0**

votes

**0**answers

37 views

### Which self-reference restrictions can be weakened in probabilstic logic?

This work suggests that there is some generalization of Truth in terms of probability, which can be definable within the logic itself.
Is where any other thorems on self-reference restrictions, which ...

**-3**

votes

**0**answers

116 views

### Has this special functor a left and a right adjoint? [on hold]

I would like to know if there exists the left and the right adjoint functors of the functor : $ X \to \displaystyle \bigoplus_{ n \geq 0 } H^n ( X , \mathbb{Q} ) = \displaystyle \bigoplus_{ n \geq 0 } ...

**2**

votes

**0**answers

74 views

### Do the zeros of the sum/difference of these series all reside on the line $\Re(s)=\frac12$?

The following series seems convergent for all $s\in \mathbb{C}$:
$$\displaystyle f(s):=\sum_{n=1}^\infty \frac{(-1)^n}{(n+s)^{n+s}}$$
The function itself does not appear to have any real or complex ...

**5**

votes

**2**answers

174 views

### mod p cohomology ring of alternating groups

Let $A_n$ be the alternating group of $\{1,2,\cdots,n\}$.
(1). What is the cohomology ring
$$
H^*(A_4;\mathbb{Z}/3)
$$
and its Steenrod operation $P^i$'s?
(2). Are there general results about the ...

**29**

votes

**25**answers

6k views

### Most intriguing mathematical epigraphs

Good epigraphs may attract more readers. Sometimes it is necessary.
Usually epigraphs are interesting but not intriguing.
To pick up an epigraph is some kind of nearly mathematical problem: it ...

**5**

votes

**0**answers

98 views

### Traces of operators in nuclear spaces

I am currently reading up on nuclear spaces in Yarchow, "Locally Convex Spaces", but I got confused and don't seem to find my mistake. In said book, theorem 21.5.9 states:
Let $F$ be a nuclear ...

**-4**

votes

**1**answer

56 views

### Proof equation is of O(log(n)) [on hold]

I am following a course of CS and we are getting Big Oh Notation ( discrete math)
We have to proof certain equations are of O(n^2) etc
I can solve easy equations like 3N + 4 and (n +1)^2 = o(n^2).
...

**1**

vote

**0**answers

118 views

### the first chern class of complex vector bundles

Let $\xi^\mathbb{C}$ be a complex vector bundle over a manifold $M$ (or $CW$-complex $B$).
Case~1: $\xi^\mathbb{C}$ is a complex line bundle. Then the first Chern class
$c_1(\xi^\mathbb{C})$ is zero ...

**51**

votes

**10**answers

3k views

### Proposals for polymath projects

Background
Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. This is a concept introduced in 2009 by ...

**7**

votes

**0**answers

94 views

### A conjecture of Lubotzky on ranks of subgroups of special linear groups over the integers

In a 1985 paper named "Dimension function for discrete groups" Lubotzky conjectured that:
For any integer $n \geq 3$ the group $\mathrm{SL}_n(\mathbb{Z})$
contains infinitely many finite index ...

**2**

votes

**1**answer

85 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 ...

**4**

votes

**1**answer

123 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 ...

**-2**

votes

**1**answer

113 views

### Diophantine equations over natural numbers [on hold]

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 ...

**1**

vote

**1**answer

291 views

### When does $R [x]/I $ has 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 idempotents and an unextended ideal $I$ in $R[x]$ such that $R[x]/I$ has ...

**7**

votes

**0**answers

81 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 ...

**8**

votes

**0**answers

107 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?
...

**4**

votes

**2**answers

172 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 ...

**-3**

votes

**0**answers

35 views

### Similarity estimation [on hold]

http://www.diku.dk/summer-school-2014/course-material/mikkel-thorup/bottomk-exercise.pdf Can somebody help with exercise 4 in chapter 2.2? Any hints would be highly appreciated.

**0**

votes

**0**answers

18 views

### Bayesian inference on gamma distribution

The likelihood of an observation $x$ under a gamma distribution is
$$L(x | \alpha, \beta) \propto \beta^\alpha x^{\alpha-1} \frac {\exp(-x\beta)} {\Gamma(\alpha)}$$
Suppose I have some observations ...

**1**

vote

**0**answers

61 views

### When does $R [x]/I $ have infinitely many idempotents in special case?

At < When does $R [x]/I $ have infinitely many idempotents? >, Er_Ro asked the following question.
Let $R $ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking ...

**-4**

votes

**0**answers

92 views

### Find an example of functions f, g such that limx→0 f(x) and limx→0 g(x) both do not exist, but limx→0 f(x) + g(x) = 1 [on hold]

Find an example of functions f, g such that limx→0 f(x) and limx→0 g(x) both do not exist, but limx→0 f(x) + g(x) = 1.

**1**

vote

**1**answer

183 views

### Bertini-type theorem in positive characteristic

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

**0**

votes

**0**answers

124 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 $E$. Then $\pi_*:T_Y\rightarrow ...

**3**

votes

**0**answers

51 views

### Cardinal of a set cinsist of product of two sets?

Let
$$
A=\{1,2,\ldots,p-1\},\qquad B=\{1,2,\ldots,q-1\}
$$
where $p,q$ are primes not necessarily distinct.
Is there any elementary way to find the cardinal of the following set
$$
AB=\{ab:\ a\in A,\ ...

**6**

votes

**1**answer

133 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 ...

**12**

votes

**1**answer

370 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) = ...

**4**

votes

**0**answers

73 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 ...

**7**

votes

**0**answers

110 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
...

**4**

votes

**2**answers

260 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 ...

**-2**

votes

**0**answers

46 views

### Maximally nonplanar graphs [on hold]

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 ...

**2**

votes

**1**answer

119 views

### Does every connected component of a covering space over a connected base intersect all the fibers of the covering space?

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

**1**answer

152 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 ...

**2**

votes

**0**answers

109 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

**0**answers

76 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 ...

**4**

votes

**1**answer

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

**2**

votes

**0**answers

86 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 ...