Highest scored questions
159,035 questions
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Topology of the moduli space of a 2-dim closed surface
Consider the moduli space $\cal{M}_{\Sigma_g}$ of a 2-dim closed surface $\Sigma_g$ of genus $g$. What is the topology of such a moduli space $\cal{M}_{\Sigma_g}$?
For example, what is $\pi_n ( \cal{M}...
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1
answer
1k
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Multiplication of a symmetric matrices [closed]
I'm wonder if the next claim is true or not:
If A,B is a symmetric matrices over the real numbers,
and A is PSD , B is PD implies than AB is PSD.
PD - positive definite
PSD - positive semidefinite
If ...
- 1
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votes
1
answer
101
views
Asking for reference about a relation related to Fourier transform [closed]
Sorry for the not-perfect question. I am asking for a reference for the following relation:
$$\int f . g. h ...= \int_{\xi_1 +\xi_2 +...=0} \hat{f}(\xi_1) \hat{g}(\xi_2)... d\xi_1 d\xi_2...$$
Could ...
- 159
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votes
1
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1k
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Pull back a vector field [closed]
In Voisin's book Hodge theory and complex algebraic geometry, I Section 9.1.2, p.223, the author writes:
Let $\phi:\mathcal X\to B$ be a family fo complex manifolds. The differential $\phi_*$ is a ...
- 471
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votes
1
answer
302
views
Is the super square root of $2$ irrational? [closed]
The super square root of $n$ is the solution/solutions to $x^x=n$. Is the super square root of $2$ irrational?
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votes
1
answer
361
views
Basis for space of continuous, surjective monotone functions on $\mathbb{R}$ [closed]
$\DeclareMathOperator\CM{CM}$
I recently came across Okhezin - Study of families of monotone continuous functions on Tychonoff spaces describing monotone functions on general topological spaces and I ...
- 5,405
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votes
2
answers
294
views
Are there (in)finitely many primes of this form? [closed]
Are there (in)finitely many primes $p$ such that $1+kp$ is a prime for some positive integer $k$ ?
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votes
2
answers
425
views
Coordinate free proof of Gauss-Bonnet theorem [closed]
Can the theorem be proved invariantly, without any reference to charts,frames, basis vectors or coordinates?
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votes
1
answer
250
views
Twin prime based Dirichlet series
Assuming there are infinitely many twin primes, one can consider a Dirichlet series $ \sum_{n>0}a_{n}{n^{-s}} $ and replace the sequence of positive integers with the sequence of twin primes. That ...
- 6,982
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votes
1
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125
views
Does the Hadwiger-Nelson graph have a perfect matching?
The Hadwiger-Nelson graph on $\mathbb{R}^n$ is defined to be $(\mathbb{R}^n,E_n)$ where $$E_n = \big\{\{x,y\}: x,y\in \mathbb{R}^n \text{ and } |x-y|=1\big\},$$ where $|\cdot|$ denotes the Euclidean ...
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165
views
Does this particular L-series built from L-functions of prime degree define an L-function?
Throughout this question, I call 'L-function' any automorphic L-function belonging to the Selberg class.
Suppose $ (F_i)_{(i>0)} $ is a sequence of L-functions with $ F_i $ of degree $ p_i $ ...
- 6,982
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votes
1
answer
70
views
Hamiltonian path in countable connected graph such that $\text{deg}(v)=\omega$ for all $v$ [closed]
Is there a countable connected graph $G=(\omega, E)$ such that $\text{deg}(v)=\omega$ for all $v\in\omega$, but there is no Hamiltonian path in $G$?
- 48.9k
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votes
1
answer
580
views
How does deletion-contraction affect chromatic number? Can it increase chromatic number? [closed]
Question: In graph theory, contracting an edge or deleting an edge are basic operations in many topics such as graph minors or Wagner's theorem on planar graphs. And I'm interested in how these ...
- 419
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votes
1
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534
views
Proof of im/possibility of constructing any fractal by iterated function systems? [closed]
Well the question is as simple as that and what I really want to see is if there is a mathematical proof that can tell whether every fractal(object of any non-integer dimension) can be constructed by ...
- 143
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votes
2
answers
122
views
a inequality of $L^p$ [closed]
I want to prove the inequality
$$
|x-y|^p \le \frac{p}{2}\big|x-y\big|\;\big(x^{p-1}+y^{p-1}\big)
$$
if $p \ge 1$. For the case $p$ is an integer, it's easy to do, but I have no idea when $p$ is not ...
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1
answer
267
views
A generalization of an old group problem [closed]
Here is an old exercise in group theory: (1) If $G$ is a group of order $2n$ with $n$ odd then $G$ is not simple and in fact $G$ has a normal subgroup of order $n$. I am going for one straight ...
user39125
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votes
1
answer
619
views
a question on Siegel Upper Half Space [closed]
Is it known whether siegel upper half plane is dense in the space of nonsingular matrices of same dimension .$.http://en.wikipedia.org/wiki/Siegel_upper_half-space.
Actually the question i have in my ...
- 2,106
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1
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411
views
Can Hartogs' extension theorem be used to prove there's no naked singularity?
Maybe for the first time my question doesn't deal with number theory. Tonight a friend of mine told me about Hartogs' extension theorem and said his work in analytic microlocal analysis was somehow ...
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1
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1k
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Why does the Lefschetz Operator not Square to Zero? [closed]
I'm trying to learn about the Lefschetz decomposition but am having a very basic problem: For the fundamental form $K$ of a Kahler metric on a complex manifold $M$, the corresponding Lefschetz ...
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votes
2
answers
768
views
Question on Linear Operators
Let $V$ be a normed infinite dimensional vector space. Let $L: V \longrightarrow V$ be a bounded linear operator. Moreover assume that $L$ is 'locally nilpotent' that is:
$$ \forall v \in V \quad \...
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votes
2
answers
1k
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Finite versus infinite on non-Hausdorff topologies [closed]
Question: Does there exist some real-valued function $f(x)$ where $f: \mathbb{R} \to \mathbb{R}$, for which $\lim_{x \to \infty}$ converges on a non-Hausdorff topology but does not converge on a ...
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358
views
Ordering of tuples equivalent to mapping to R?
Suppose we have a non-strict total ordering on tuples of real numbers (the ordering includes tuples of differing lengths). Any tuple, t2, generated from another tuple, t1, by increasing one or more ...
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1
answer
169
views
Can there be an effectively generated consistent theory that extends PA and be consistently extended by its own completeness and consistency?
[The question has been edited]
Can we have an effectively generated consistent theory $T$, that extends $\sf PA$, such that:
$T + \\ \forall \mathcal S \, [\exists x: \operatorname {Proof}_T(x, \...
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votes
1
answer
77
views
Sobolev embedding [closed]
I was trying to understand Sobolev embedding, some results about this topic are not clear to me.
My question is the following:
what are the condition on $p_1 , \alpha_1, p_2 $and $\alpha_2$ for
$W^{...
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1
answer
350
views
Can we find a closed form formula for this function?
I'm interested in this function
$$
h(m,n) = \operatorname{sgn}\Bigl(\lim_{w \to \infty}{\sum_{x_1=1}^{w}\dotsi\sum_{x_u=1}^{w}\dfrac{1}{1 + wp^2(m,n,x_1,\dotsc,x_u)}}\Bigr)
$$
where $p$ is a ...
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1
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177
views
Prove that the equation $2^a - 2^b - 1=3^c$ has no integral solution with $a,b\geq 3$
I was looking for a natural power of 3 that could be written like
Binary format:
11..(N times)..11011..(M times)..11
Example: 1111110111111111111111 (...isn't a power of 3)
Or could also be written ...
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1
answer
125
views
Basis of Euclidean topology on $\mathbb{R}$ such that no element is contained in another [closed]
What is an example of a topological base ${\cal B}$ for $\mathbb{R}$ with the Euclidean topology such that for every $B_1\neq B_2 \in {\cal B}$ we have $B_1\not\subseteq B_2$?
- 48.9k
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votes
1
answer
124
views
Periodical functions with $f^{(n)} = f$, but $f^{(k)} \neq f$ for $k\in \{1,\ldots,n-1\}$ [closed]
For any infinitely differentiable function $f: \mathbb{R}\to \mathbb{R}$ and positive integer $k\in\mathbb{N}$, let $f^{(k)}$ denote the $k$-th derivative of $f$.
For which $n\in\mathbb{N}$, $n>1$, ...
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1
answer
140
views
Doubt about lemma for polynomial equivalence [closed]
Multivariate polynomials $f,g$ are equivalent if there exists
invertible linear transformation $A$ such that $f(X)=g(A\cdot X)$
From paper p.1:
Lemma 1.1. (Structure of quadratic polynomials). Let $F$...
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votes
1
answer
382
views
Is the 2-dimensional Gauss-Bonnet theorem applicable in higher dimensions? [closed]
This is a cross-post of this MSE post that users commented that it is appropriate for MO.
I want to know
Question: Is the 2-dimensional Gauss-Bonnet theorem applicable (any topological or ...
- 4,195
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votes
1
answer
380
views
References of research papers which lead to starting of Sieve Theory
Question - I am thinking to present one or two papers on Sieve Theory in my masters thesis. I will also present 3 other papers on Riemann Zeta Function which I have studied earlier . But I have no ...
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1
answer
317
views
Factor group isomorphic with Klein four-group [closed]
Let $G$ be a finite solvable group and $N$ be a normal subgroup of it which is an elementary abelian 2-group. Suppose that $G/N\cong \mathbb{Z}_2\times \mathbb{Z}_2$ and $|C_G(x)|=16$ for any $x\in G-...
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1
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156
views
The existence of an interval $I\subset (0.8856,+\infty)$ such that the derivative $(\Gamma^{-1})’(x)$ is greater than $1$ [closed]
The principal inverse function of the gamma function is denoted by $\Gamma^{-1}$. See the paper: Uchiyama - The principal inverse of the gamma function.
$\Gamma^{-1}$ is an increasing and concave ...
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1
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227
views
Equation $\ \binom xn'\ =\ \log(n)$ [closed]
Problem: Solve equation
$$ \binom xn'\ =\ \log(n) $$
Here prime stands for the derivative with respect to $x$.
Observe that:
$\quad$ for integer $n$ large,
the approximate solution is $\ ...
- 4,786
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votes
1
answer
117
views
Product of critical graphs and Hedetniemi's conjecture
If $G, H$ are finite, simple, undirected graphs, their categorical product $G\times H$ is defined by $V(G\times H) = V(G)\times V(H)$ and $$E(G\times H) = \big\{\{(v_1, w_1),(v_2,w_2)\}: v_i \in V(G)\...
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1
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191
views
The usual topologies [closed]
My questions are :
Why do we commonly use certain usual topologies rather than others ? For example the usual topology on the real numbers, the topology of
uniform convergence, the compact-...
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votes
1
answer
173
views
Even-odd partitioned groups! [closed]
Let $G$ be a group with the property $G=G_e\dot{\cup} G_o$ with
$G_oG_o\subseteq G_e\leq G$.
($\dot{\cup}$ denotes disjoint union, $\leq$ is subgroup notation, and $G_o^{-1}=\{x^{-1}: x\in G_o\}$)
...
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1
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303
views
Quotient of a Banach algebra [closed]
Let $A$ be a Banach algebra. Is there a Banach algebra $B$ and a non-trivial closed ideal $I$ of $B$ such that $\frac{B}{I}\cong A$?
- 565
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votes
3
answers
439
views
if V(f) is irreducible, then how to show that the polynomial f itself is irreducible? [closed]
V(f) is the zero locus of the polynomial f in the polynomial ring k[x1, x2, ..., xn] with k an algebraically closed field.
If V(f) is irreducible, then how to show that 'f' is irreducible?
- 3,477
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votes
1
answer
158
views
Randomness about coefficients of series
$B\subset \mathbb{N}\bigcup \{0\}$ is finite and not empty, infinite series:$$f(x)=\sum_{i=1}^{\infty}a_i x^i,a_i \in B$$ Now $f(x)$ is rational or has a natural boundary.
Now,the question :if $f(x)$...
- 3,104
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1
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230
views
$L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)$?
Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at $\...
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votes
1
answer
1k
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Chow groups and short exact sequence
Let $X$ and $Y$ be subvarieties of a smooth projective variety $M$ such that $M=X \bigcup Y$. Can you explain to me why $ A_k ( X \bigcap Y ) \to A_k ( X ) \oplus A_k ( Y ) \to A_k ( X \bigcup Y ) \to ...
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votes
1
answer
1k
views
Continuous surjection $S^n\to S^m$, $n<m$ [closed]
I had an exam last week on algebraic topology.
"Suppose $f:S^n\to S^m$ is continuous, where $n<m$. Prove that $f$ is homotopic to a constant mapping. The fact that $S^n$ minus one point is ...
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votes
1
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490
views
How to prove that homometric sets lead to same result in this problem? [closed]
First let me define Difference multiset for a set of integers
$$P=\{p_1,p_2, \dots,p_K\} ,\quad p_i \in\{1,2,\dots,N\},\quad p_i\ne p_j
$$
as below:
$$
D = \{p_i-p_j \mod N ,\quad i \ne j\}
$$
I ...
- 123
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votes
1
answer
634
views
compactly supported harmonic functions [closed]
Do a significant class of compactly supported smooth functions u on Ω⊂Rn such that Δu≥0 exist?
Thanks!
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votes
1
answer
332
views
Convergence Question [closed]
If $\alpha _{n}\rightarrow \alpha$, then how does one show that for any j=1,2,... and $\epsilon> 0$, if $sup\int \left | x \right |^{j+\epsilon }d\alpha _{n}<\infty$, then $\int x^{j}d\alpha _{n}...
- 1
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votes
1
answer
682
views
Gamma(e)=Pi/2,Zeta(e)=4/Pi ? [closed]
I find that Gamma(e) is close to Pi/2 and Zeta(e) is close to 4/Pi. So I have a question:
$\Gamma (e) = \pi /2$
$\zeta (e) = 4/\pi $
Is it true in fact?
- 217
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votes
3
answers
628
views
Roulette probability [closed]
I'm looking for some knowledge on probability, I've scoured the net but I can't really grasp the answer.
I was having a discussion with a co-worker about roulette probability. He says that at any ...
-3
votes
2
answers
14k
views
Roots of a polynomial in several variables [closed]
Hi,
I'm new here, and not a mathematician at all :-(
I am looking for an algorithm to find the roots (in the complex domain) of a polynom of several variables.
Thanks for any light you could bring ...
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votes
1
answer
344
views
A problem seeking for algorithm [closed]
Now there are n independent projects, each project is composed by several steps, each step is labeled, there are k workers are going to work on these projects.
Assume that each person can do only one ...
- 455