# All Questions

**0**

votes

**0**answers

63 views

### Lefschetz hyperplane theorem for Neron-Severi

Suppose that $X$ is a smooth projective variety of dimension at least $3$, and that $D$ is a smooth ample divisor. I am wondering to about the status of the Lefschetz hyperplane theorem for the map ...

**0**

votes

**0**answers

13 views

### Probability distribution of the distances between N mobile nodes in a square plain of length l [on hold]

For N randomly moving nodes enclosed in a square plain of length l. The (Nchoose2)W samples of the distances between the nodes are collected over a window of length W. We can assume W is large, what ...

**2**

votes

**1**answer

25 views

### Analytical value for the first eigenvalue of a certain spherical triangle

I am testing some numerical algorithms for computing the Laplace-Beltrami eigenvalues on the sphere. One thing that came up was computing the first eigenvalue of the "equilateral" spherical triangle ...

**-1**

votes

**0**answers

54 views

### Is countably complete lattice bounded? [on hold]

I wonder if countably complete lattice is bounded and, if it is why ?

**3**

votes

**1**answer

86 views

### Proof for additivity of cumulants

If one does not define cumulants via the cumulant generating function (cgf), e.g. because the cgf does not exist, then an alternative way is to use the recusion
\begin{align*}
...

**-6**

votes

**0**answers

37 views

### Help with derivative rules [on hold]

Hi I need som help with some rules Y to Y'.
1: Y = A(^2)/B
2: Y = E(^2)/1
3: Y = E^(X+X)
4: Y = sin(X(^3))/cos(X(^2))
5: Y = -X(^B)cos(B(^X))
What is the Y' of all functions?

**1**

vote

**1**answer

138 views

### On Q-Cartier Divisors

I have my question on Q-Cartier Weil divisor.
People say $D$ is Q-Cartier divisor if $nD$ is Cartier for some $n \geq 1$. Especially for $n > 1$, I have never seen the `rigorous' definition of ...

**1**

vote

**1**answer

256 views

### Numbers $n$ such that the sum of the divisors of $n$ is a nontrivial power

Let $\sigma (n)$ be the sum-of-divisors function. For example, $\sigma(7)=1+7=2^3$.
I know some results about triplets of positive integers $(n,a,b)$ where $a,b\ge 2$ such that $\sigma (n)=a^b$, but ...

**0**

votes

**0**answers

46 views

### What can we say about variational energies here?

Let $V_{ij}^{lk}$ be any $nm \times nm$ real symmetric matrix, $\forall i,j,k,l$
\begin{equation}
V_{ij}^{kl}=V_{ji}^{lk}
\end{equation}
(So for the indices we have $1 \leq k,l \leq m$ and $1 \leq i,j ...

**-1**

votes

**0**answers

42 views

### The product of the power and the natural number in the short interval [on hold]

It is obvious that if $a,b,x\in\mathbb{N}$ and $a^n\leq 2x+1$ then there exists $b\in\mathbb{N}$ such that $a^nb\in\left[x^2,(x+1)^2\right]$. For example for $n=3$, $a=2$ and $x=4$ we have $b=2$ and ...

**4**

votes

**3**answers

294 views

### Automatically generate BibTeX item from arxiv [on hold]

I'm looking for a tool which generates a BibTeX item for a given arxiv id. I only found http://www.crcg.de/arXivToBibTeX/ using Google but this tool always tells me that the arxiv ids I enter don't ...

**-2**

votes

**0**answers

38 views

### periodic function satisfy the condition f’’(x)f(x)>0 at -inf < x < +inf? [on hold]

Can a periodic function f(x) satisfy the condition f’’(x)f(x)>0 at -inf < x < +inf?

**11**

votes

**2**answers

572 views

### Which way for reading the proofs?

I am a master student in mathematics. For me a large part of doing mathematics is thinking about, reading and verifying the proof of theorems that I find them in my field of study. I can do this ...

**4**

votes

**0**answers

66 views

### Interesting triple integral

Some time ago I stumbled on an alleged identity
$$\int\limits_0^\infty \frac{dx}{x} \int\limits_0^x \frac{dy}{y}
\int\limits_0^y \frac{dz}{z} [\sin{x}+\sin{(x-y)}-\sin{(x-z)}-\sin{(x-y+z)}]=
...

**5**

votes

**1**answer

177 views

### Strange problem about triplets of differential forms

Suppose we have the following map:
$$(\Omega^1(\mathbb{R}^n))^3\longrightarrow(\Omega^2(\mathbb{R}^n))^3$$
...

**19**

votes

**3**answers

577 views

### Connectedness in the language of path-connectedness

Is there a topological space $(C,\tau_C)$ and two points $c_0\neq c_1\in C$ such that the following holds?
A space $(X,\tau)$ is connected if and only if for all $x,y\in X$ there is a ...

**3**

votes

**3**answers

164 views

### how to find explicitly given component in a regular representation

Given a finite group $G$ and its irreducible representation $\pi$ I want to find explicit elements of the group algebra $\mathbb{C}[G]$ lying in components of the left regular representation ...

**1**

vote

**0**answers

196 views

### An example of threefold

Its description is a little bit complicated but it would be great if anyone can give an example.
I tried to construct it as a toric variety (See the previous question) but did not succeed.
I am ...

**4**

votes

**3**answers

302 views

### reflexive banach space

I want to ask this non-expert question:
What does it mean geometrically for a Banach space to be reflexive?
Well, we could say a Banach space is reflexive iff unit ball is weakly compact. Or some ...

**-1**

votes

**0**answers

27 views

### Global existence of power-type nonlinear Schrodinger equations on compact manifolds [on hold]

Consider the nonlinear Schrodinger equation
$$i\partial_t u + \Delta u = K|u|^ru$$
on a compact manifold, may be with boundary (with Dirichlet boundary conditions). It is known that on $\mathbb{R}^n$, ...

**27**

votes

**8**answers

4k views

### Uninteresting questions with interesting answers [on hold]

What are best examples of questions in mathematics that are not interesting until one knows the answers, whose answers themselves are what is interesting?
The thing that prompts me to post this is ...

**3**

votes

**2**answers

107 views

### distributional Hessian for semiconvex functions on non-smooth manifolds

Let $f:R^n \to R$ be convex. Then there exist signed Radon measures $\mu^{ij}=\mu^{ji}$ such that
$$
\int_{R^n} f \frac{\partial^2 \varphi}{\partial x_i \partial x_j} dx= \int_{R^n} \varphi d\mu^{ij} ...

**1**

vote

**0**answers

46 views

### Subclass of semimartingales for which all characteristics can be estimated?

I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great.
An Ito semimartingale is a martingale for which the ...

**4**

votes

**0**answers

108 views

### Does this inequality always hold?

Denote the adjacency matrix of a given undirected graph by $g$. It is an $n$-by-$n$ symmetric Boolean matrix with elements on the diagonal to be zero ($n\geq 3$). Let $g_{12}=g_{21}=g_{13}=g_{31}=1$ ...

**-4**

votes

**0**answers

34 views

### Solving a tough a PDE shifting data [on hold]

How would I solve this one:
$u_t-\nabla^2u = f(r,\theta, t) \quad r<a, t>0$
$u(r,\theta, 0)=\phi(r,\theta) \quad r<a$
$u=h(\theta) \quad r=a$
So I guess I need to make the BC's ...

**5**

votes

**1**answer

158 views

### Evaluation maps for moduli of stable maps

Let $\overline{M}_{0,n}(\mathbb{P}^N,d)$ be the moduli space of stable maps of degree $d$ from curves of genus zero with $n$-marked points to $\mathbb{P}^N$.
Consider the product of the evaluation ...

**1**

vote

**1**answer

90 views

### Is there any simpler form of this function

Assume that $n$ is a positive integer. Is there any simple form of this hypergeometric value $$_2\mathrm{F}_1\left[\frac{1}{2},1,\frac{3+n}{2},-1\right]?$$

**-3**

votes

**0**answers

44 views

### On ranks of matrix products [on hold]

Tensor product of two matrices increases simultaneously sizes of product matrix, size of rank multiplicatively.
Is there a function on two matrices which increases size multiplicatively while rank ...

**2**

votes

**0**answers

54 views

### completion of non-finitely generated ideal

Let consider $A=k[x_{1},x_{2}...]$, the polynomial ring with countably many indeterminates.
Then we can consider the completion ...

**0**

votes

**0**answers

99 views

### Solving the transcendental equation $Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$

I need to solve the following equation:
$Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$
for $x\in\mathbb{R}^{\ast}$ and where $k\in\mathbb{R}^{+}$. Here $Li_{3}$ and $Li_{2}$ are the third and ...

**-1**

votes

**0**answers

51 views

### Clarification on notation of “left invariant fields” (Lie groups) [migrated]

In these notes in Definition 1.4 we learn that
A vector field $X$ on a Lie group $G$ is called left invariant if $d(L_g)_h(X(h))=X(g(h))$ for all $g,h \in G$, or for short $(L_g)_*(X)=X$.
where ...

**6**

votes

**1**answer

151 views

### Isometries of some simple Cayley graphs

Consider a Cayley graph of a group $G$ with respect to a symmetric finite generating set $S$.
There are some obvious candidates to isometries of this graph - for example, translation by elements of ...

**0**

votes

**0**answers

11 views

### Boundary Condition for LevelSet Reinitialization

Recently, I'm curious about the boundary condition of Levelset Reinitialization. Generally, when we try to express the interface, we use the levelset advection
$$\phi_t + (V\cdot\nabla)\phi = 0$$
But ...

**-1**

votes

**1**answer

105 views

### How compute combinatorial expression [on hold]

How compute
$\sum_{j=1}^k \binom{x}{j}\binom{k-1}{j-1}\alpha^j, \quad x, \alpha\in\mathbb{R}$

**2**

votes

**1**answer

157 views

### Mal'cev “rational equivalence” and model theory

In Universal Algebra, it is possible to say that two presentations denote the "same" kind of algebraic structures, if the two corresponding varieties are "rationally equivalent" (Mal'cev 1958).
In ...

**1**

vote

**0**answers

103 views

### Dual space of $l^p(\mathbb{Z},X)$

Let $X$ be a Banach space, $p \in [1,\infty)$ and $l^p(\mathbb{Z},X)$ the usual sequence space taking values in $X$. Is it always true that $(l^p(\mathbb{Z},X))^* = l^q(\mathbb{Z},X^*)$ and ...

**0**

votes

**1**answer

88 views

### Existence of a special Kahler structure on the contangent bundle

It is known that the total space of the cotangent bundle $T^{\ast}M\to M$ of any manifold $M$ can be equipped with a Kahler structure. My question is, it is known when $T^{\ast}M$ admits a Special ...

**2**

votes

**1**answer

119 views

### How to choose a random proper coloring

I am studying proper colorings of complete bipartite graphs and I'd like to be able to pick a random proper coloring and the compute some things about it.
Recall that a proper coloring of a complete ...

**7**

votes

**1**answer

205 views

### Can you decide whether the commutator subgroup of a f.p. group is f.g?

Is the following algorithmic problem known to be decidable/undecidable?
Input: a finite group presentation $P$.
Decide: is the commutator subgroup of the group presented by $P$ finitely generated?

**-3**

votes

**1**answer

72 views

### Can functional invariants of dynamical systems be used in data science (or parameter identification)? [on hold]

Given a functional of the form:
$$
F[x]=\int^{t}_{0} \mathcal{L}(x^{(n)},x^{(n-1)},...,x,\tau)\,\text{d}\tau+g(x(t))
$$
Where $x$ is in $\mathbb{R}^{m}$ and is in ...

**16**

votes

**1**answer

221 views

### On a drawing in Dixmier's Enveloping Algebras

This image
comes from Dixmier's book, 'Enveloping Algebras' ('Algèbres enveloppantes').
Dixmier writes that
The curves shown on p. XIV have their origin in the study of U(sl(3)).
They are ...

**6**

votes

**0**answers

111 views

### Singularities of an analytic function over a non-archimedean field

What do we know about the types of singularities that a convergent power series over a non-archimedean field can have?
More specifically:
i) What types of essential singularities can occur?
ii) Are ...

**5**

votes

**1**answer

474 views

### What is the value of the infinite product: $(1+ \frac{1}{1^1}) (1+ \frac{1}{2^2}) (1+ \frac{1}{3^3}) \cdots $? [on hold]

What is the value of the following infinite product?
$$\left(1+ \frac{1}{1^1}\right) \left(1+ \frac{1}{2^2}\right) \left(1+ \frac{1}{3^3}\right) \cdots $$
Is the value known?

**5**

votes

**0**answers

114 views

### Factorization of antiderivative of minimal polynomials

In any totally real number field, is there an element whose minimal polynomial has the property that its antiderivative factors completely over the rationals?
(I’ll let you choose whichever constant ...

**2**

votes

**2**answers

148 views

### Connections having the same holonomy along loops at a point

I find myself stuck with the following question, which seems very classical but for which I have not been able to find a reference.
Consider a smooth vector bundle $E$ of rank $r$ over a compact ...

**5**

votes

**0**answers

68 views

### Real Zeros - tail estimate

Given a random polynomial with Gaussian coefficients, the Kac-Rice formula tells us what the expected number of real zeros is (for more on this, see the excellent paper of Edelman and Kostlan in the ...

**0**

votes

**0**answers

52 views

### Three-wise deformation [on hold]

Let $A$ be an associative algebra. A deformation is a "new product" $\cdot$ given by:
$$
a\cdot b := ab +f(a,b),
$$
where $f:A\times A\to A$ satisfies a suitable cocycle condition.
Suppose that ...

**0**

votes

**0**answers

63 views

### Lattice cocycle for an extension of line bundle on a complex tori

Let $X=V/\Lambda$ be a complex torus. Suppose there is an extension of vector bundles on $X$
$$
0 \to \mathcal{L} \to \mathcal{F} \to \mathcal{L} \to 0,
$$
where $\mathcal{L}$ is line bundle. Assume ...

**-3**

votes

**0**answers

56 views

### Permutations with fixd points [on hold]

I am trying to write a java program that counts permutations of a string, I would like to check my results by hand, but I can remember (or find) the formula to count the number of permutations, and ...

**2**

votes

**1**answer

96 views

### Prove or disprove an inequality concerning zeros of a polynomial

If a polynomial $p(z)$ of degree $n$ with zeros $z_1,z_2,\cdots,z_n$ assumes maximum at $w$ on $|z|=1.$ Prove or disprove that the Harmonic mean of $|z_k-w|,$ $k=1,2,\cdots,n$ is greater or equal to ...