0
votes
0answers
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
0answers
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
1answer
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
0answers
54 views

Is countably complete lattice bounded? [on hold]

I wonder if countably complete lattice is bounded and, if it is why ?
3
votes
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
3answers
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
0answers
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
2answers
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
0answers
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
1answer
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
3answers
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
3answers
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
0answers
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
3answers
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
0answers
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
8answers
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
2answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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 ...

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