# All Questions

**2**

votes

**0**answers

14 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 ...

**2**

votes

**0**answers

53 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 ...

**-4**

votes

**0**answers

40 views

### I'm PHD student (finance) [on hold]

I want to apply CIR model for a chive asset price.
can I do this if Feller condition is considering?
please help me

**1**

vote

**0**answers

30 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 ...

**3**

votes

**0**answers

39 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 ...

**4**

votes

**1**answer

83 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 ...

**-2**

votes

**0**answers

14 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.

**4**

votes

**0**answers

46 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 ...

**-4**

votes

**0**answers

44 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.

**3**

votes

**0**answers

37 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,\ ...

**0**

votes

**0**answers

95 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 ...

**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 ...

**6**

votes

**1**answer

80 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 ...

**0**

votes

**1**answer

22 views

### Linear programming - Simplex tool

is there any tool which helps with solving linear optimisation problems with the Simplex algorithem?
The tool should have some API or (better) receive a problem (text file) and return the solution.
...

**-3**

votes

**1**answer

105 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 ...

**4**

votes

**0**answers

48 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 ...

**12**

votes

**1**answer

318 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

45 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 ...

**5**

votes

**0**answers

74 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
...

**7**

votes

**0**answers

67 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?
...

**-2**

votes

**0**answers

41 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

64 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

105 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

**0**answers

97 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

65 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

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

**4**

votes

**2**answers

244 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

77 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 ...

**9**

votes

**1**answer

257 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 ...

**-2**

votes

**1**answer

62 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 ...

**0**

votes

**0**answers

76 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 ...

**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 ...

**0**

votes

**1**answer

237 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**

vote

**0**answers

47 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 ...

**0**

votes

**1**answer

86 views

### Osculating ellipsoids

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

**4**

votes

**0**answers

73 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 ...

**1**

vote

**0**answers

28 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 ...

**1**

vote

**1**answer

177 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, ...

**3**

votes

**1**answer

135 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 ...

**4**

votes

**1**answer

292 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,\; ...

**0**

votes

**0**answers

55 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 ...

**2**

votes

**1**answer

168 views

### Does this PDE only have the trivial solution?

Let $(M,g)$ be a closed Einstein manifold of dimension $m>2$ and
$$
\mathrm{Ricc}(g)=\lambda g,
$$
$h$ a symmetric $2$-covariant tensor, $\Delta=\nabla^*\nabla$ the Laplacian on functions as well ...

**1**

vote

**1**answer

52 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 ...

**4**

votes

**1**answer

158 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

**0**answers

44 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 ...

**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

78 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 ...

**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 ...

**10**

votes

**2**answers

500 views

### Does every (co)homology functor (in particular, stable homotopy) factor through chain complexes?

Ordinary homology and cohomology factor through chain complexes via singular homology and cohomology. What about other (co)homology theories?
That is, for each spectrum $E$, do we have a lift in the ...

**7**

votes

**0**answers

69 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 ...