# All Questions

**0**

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16 views

### Analogues of the Lagrange inversion theorem.

Does anyone know if there exists other theorems similar to the Lagrange Inversion Theorem. I'm interested in collecting methods for determining the asymptotic behaviour of implicitly defined ...

**2**

votes

**2**answers

53 views

### Is it consistent that $\frak{d} < 2^{\aleph_0}$?

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. We write $f <^* g$ if there is $N\in\omega$ such that $f(n) < g(n)$ for all $n>N$. A set $D\subseteq \omega^\omega$ is ...

**-4**

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**0**answers

21 views

### Probably some naive question on conditional probability [on hold]

As known, three variables x_1, x_2 and y, if x_1 and x_2 are conditional independent given y, we have p(x_1, x_2|y) = p(x_1|y)p(x_2|y).
I was wondering about p(y|x_1, x_2), is that possible to get ...

**0**

votes

**1**answer

22 views

### Powers in compact coset spaces

Let $G$ be a topological group, let $K$ be a closed cocompact subgroup (i.e. the coset space $G/K$ is compact in the quotient topology) and let $g \in G$. Is there a sequence of positive powers ...

**0**

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35 views

### Two (strictly related) proofs by induction of inequalities

This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...

**-3**

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**0**answers

38 views

### Proving integration techniques [on hold]

I've recently competed my A levels and now that I'm in the university I finally found the time to understand calculus on a intuitive level. So I've been reading up on books such as "Calculus with ...

**1**

vote

**0**answers

30 views

### Inequivalent definitions of Cartan subalgebra

As far as I can tell, there exists no acknowledgment on the internet of the fact (or maybe it's not a fact) that inequivalent definitions of "Cartan subalgebra" of a real Lie algebra exist in the ...

**2**

votes

**1**answer

174 views

### Anti-compactness

Let $(X,\tau)$ be a topological space such that $\tau$ is a proper superset of $\{\emptyset, X\}$. We call an open cover $\mathcal{U}$ of $(X,\tau)$ proper if $X\notin \mathcal{U}$. Moreover we say ...

**5**

votes

**3**answers

196 views

### Sierpinski's construction of a non-measurable set

In the early 20th century there was a lot of fuss over the axiom of choice implying that there are Lebesgue non-measurable sets of reals. In his book about The Axiom of Choice, Gregory Moore points to ...

**-4**

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**0**answers

59 views

### Two problems in functional analysis [on hold]

Let $f$ be linear functional on Banach space $B$ and $ker f$ is closed subspace of $B$, prove that $f$ is a bounded linear functional.
Let $\{e_n\}$ be an orthonormal basis of Hilbert space H. T is ...

**5**

votes

**1**answer

174 views

### Is there a “good” reason that the universal central extension of $SL(2,\mathbb Z)$ is $Br_3$?

Let
$$ Br_3 := \langle \tau_1,\tau_2\ :\ \tau_1 \tau_2 \tau_1 = \tau_2 \tau_1 \tau_2 \rangle $$
be the braid group on three strands, and consider the surjection
$$\phi : Br_3 \twoheadrightarrow ...

**9**

votes

**0**answers

138 views

### Applications of $p$-adic Hodge theory

I am trying to learn $p$-adic Hodge theory. I found some materials explaining main theorems (or aspects) of the theory. However, I could not find references which explaining how to use the theory. ...

**0**

votes

**1**answer

87 views

### Bombieri-Vinogradov in short intervals

In 1985 Perelli, Pintz & Salerno proved a short-interval form of the Bombieri-Vinogradov theorem with $\theta \in (7/12, 1]$. Have there been any improvements on this, in particular with the ...

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votes

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23 views

### Kan extension pseudo-2-functor

Consider the 2-category $[S, H] $ of 2-functors $S\to H $ (in which, obviously, $S$ and $H$ are 2-categories). And consider a (possibly fully faithful) functor $T: S\to Z $
For simplicity, let's ...

**0**

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49 views

### Space of positive matrices of a form

$\mathsf{Sym}^+_n$ be the space of symmetric matrices with entries in $\Bbb R_+\cup\{0\}$.
$\sum_{i=1}^{k}a_ia_i'$ where $a_i\in\Bbb R_{\geq 0}^n$ from $i=1,\dots, k\leq n$ characterizes all the ...

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70 views

### Futaki invariant on $X=Bl_p(\mathbb CP^2)$ for different line bundles

Let $X$ be a projective variaty which blow up at a point $p$ , i.e, $X=Bl_p(\mathbb CP^2)$, then for the Line bundle $L=-K_X$, we have for Futaki invariant $Fut_L\neq 0$, I want to see what about ...

**-1**

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**0**answers

75 views

### Maximum connected components $0-1$ matrix

Let the notion of connected matrix be as in here Connected components $0-1$ matrices
Here a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by ...

**-2**

votes

**0**answers

54 views

### Are the following interpretations elementarily equivalent? [on hold]

Are the following interpretations elementarily equivalent?
$$ < \mathbb N, \le > \text{ and }<\mathbb N + \mathbb Z, \le> $$
If so, prove it. Else make a formula that distinguishes them.
...

**2**

votes

**2**answers

142 views

### Removing subtrees

Let $T$ be a complete infinite rooted binary tree. Is it possible to remove (infinitely many) subtrees of $T$ and get a subgraph $G$ such that:
$G$ has no complete subtrees (the graph below any ...

**1**

vote

**1**answer

69 views

### Combinations Question about the construction of some special sets

Let n and k be two given numbers. The goal is to choose n subsets from {1,2,…,n} such that the union of any k of these subsets is the set {1,2,…,n} and the union of any m < k of ...

**1**

vote

**1**answer

64 views

### Deterministic finite-state automaton driven by a Markov chain

I've stumbled on some problem, and I have the feeling that this is closed to something well-studied in dynamical systems. The problem is the following. Consider a finite-state automaton with state ...

**2**

votes

**1**answer

57 views

### Measuring products of finitely generated subgroups of free groups

Let $F$ be a finitely generated free group, $H_1, \dots, H_n$ finitely generated subgroups of infinite index in $F$, and $\epsilon > 0$. Must there be an epimorphism to a finite group $\phi \colon ...

**2**

votes

**1**answer

113 views

### Is there any elementary proof of No wandering domain for polynomials

It seems that it is almost impossible to give a elementary proof of Sullivan's no wandering domain for rational map or even more general class of maps.
I think it is interesting to ask whether we ...

**8**

votes

**2**answers

191 views

### Can a positive measure subset of a free group be nowhere dense?

Let $F$ be a finitely generated free group and let $S \subseteq F$ be a subset for which there is some $\epsilon > 0$ such that for any epimorphism to a finite group $\phi \colon F \to G$ we have ...

**2**

votes

**2**answers

129 views

### Tensor product over a monoid in a monoidal category

nLab article on tensor product says:
"Given two objects in a monoidal category $(C,\otimes)$ with a right and left action, respectively, of some monoid $A$, their tensor product over $A$ is the ...

**3**

votes

**1**answer

52 views

### Extension of an involutive automorphism

Suppose that $g$ is a complex semi-simple Lie algebra and $g'$ its reductive subalgebra.
If $\tau$ is an involutive automorphism of $g'$, can $\tau$ be extended to an involutive automorphism of $g$ ...

**21**

votes

**1**answer

341 views

### Cantor's theorem for presheaves?

Some years back (before MathOverflow was born), Tom Leinster asked an interesting question at the $n$-Category Café which I don't recall ever seeing an answer for:
Does there exist a ...

**6**

votes

**1**answer

162 views

### Blow-ups of $\mathbb{P}^{n-3}$ and $(\mathbb{P}^1)^{n-3}$

Let us consider the points
$$p_1=[1:0:...:0],p_2 = [0:1:...:0],...,p_{n-2} =[0:...:0:1],\\
p_{n-1}=[1:1:...:1]\in\mathbb{P}^{n-3}$$
and the blow-up $X = Bl_{p_1,...,p_{n-1}}\mathbb{P}^{n-3}$.
...

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votes

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28 views

### Caffarelli-Silvestre extension definition of fractional Laplace-Beltrami on hypersurface

Let $\Gamma \subset \mathbb{R}^{n+1}$ be a $n$-dimensional (closed) $C^k$-hypersurface. Consider the problem
$$\nabla_\Gamma \cdot (y^{1-2s}\nabla_\Gamma u)(x,y) =0 \quad \text{for $(x,y) \in \Gamma ...

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41 views

### The hypertriangular function of $n$

I'm looking for papers or recent results on the hypertriangular function of $n$:
$$H_t(n)= \displaystyle\sum\limits_{k=1}^{n} k^k$$
This is A001923 in the OEIS.
I don't have much experience with ...

**1**

vote

**0**answers

71 views

### The definition of $SK_1$ for an arbitrary ring

Let $R$ be a unitary associative ring. If $R$ is commutative, then one defines $SK_1(R)$ as the quotient $$SK_1(R)=SL(R)/E(R)$$ (Definition 2.8 of ...

**-1**

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49 views

### $V(A)\cong {\mathbb N}\cup\{0\}$ and $$V(A_+)\cong\{(m,n)\in {\mathbb Z}^2 \mid m,n \geq 0, \hbox{ $m+n$ even}\}. $$ [on hold]

In Professor Blackadar's book "K theory for operator algebras", there is an example in Chapter 3, $K_0$-theory and order:
Let
$$ A=\{f :[0,1]\to M_2 \mid f(0)={\rm diag}(x,0), f(1)={\rm ...

**1**

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61 views

### How would you call a variety that is locally a complete intersection up to defect c?

Let $X$ be an equidimensional variety of dimension $n$ over a field that can be covered by open subvarieties of certain intersections of $N-n$ hypersurfaces in $P^N$ (for a large enough $N$; we ...

**0**

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17 views

### Maximizing a certain concave function over a non-convex set

I have been working on a problem which involves maximizing a concave function over a non convex constraint set. The problem is the following $$max. \frac{1}{2}x^TBx\\ s.t.\ x_i(1-x_i)=0,\ \ ...

**-2**

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**0**answers

31 views

### Finding ellips equation by focuses and tangent line [migrated]

The Ellips which has focuses in $(±3,0)$ and a tangent line $x+y-5=0$.
I need to find ellips equation.
I've founded these equations
$\frac{x_{0}}{a^2} = \frac{1}{5}, \frac{y_{0}}{b^2} = \frac{1}{5}$
...

**4**

votes

**3**answers

271 views

### Do geodesics in SL2R map to geodesics in the hyperbolic plane?

I am looking for a reference/proof/disproof of the following statement.
Equip the Lie group $SL_2(\mathbb{R})$ with the left-invariant Riemannian metric, whichis given on the Lie algebra by $\langle ...

**-2**

votes

**0**answers

45 views

### Blood type frequency given probability [on hold]

I have calculated the probability that any child will have a particular blood type from both the genotype level and the phenotype level assuming the human ABO Rh system is followed.
Here are the ...

**0**

votes

**1**answer

87 views

### Connected curve

Assume we have a normal,connected quasi projective scheme $Y:=X\backslash D$ where $X$ is a quasi projective scheme over field $k$, not necessarily char zero and also $D$ is a simple divisor, not ...

**0**

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26 views

### on the existence of holomorphic coordinates under bounded curvature

Let $(X,g)$ be a complete Kahler manifold with the Riemann curvature and its derivatives bounded. Suppose also the injectivity radius of this manifold is bounded below by a constant $i_0>0$. My ...

**3**

votes

**2**answers

112 views

### Can the pre-image of the real points in the complex upper-half plane of a modular elliptic curve under the modular parametrization be identified?

Consider an elliptic curve $E/\Bbb Q$ and let $\Phi:\Gamma_0(N)\backslash\overline{\mathfrak{H}}\rightarrow E(\Bbb C)$ be the analytical description of its modular parametrization. We know that this ...

**4**

votes

**1**answer

217 views

### Open problems in compressed sensing

What are the main open problems in compressed sensing?
I am interested in theoretical as well as in numerical point of view.

**-4**

votes

**0**answers

45 views

### How can i be distinguished from -i? [migrated]

Mathematicians designate one solution to x^2 = -1 as i and the other as -i. Would anybody notice if we switched their identities? Any polynomial p(x) with a complex root will also have its conjugate ...

**3**

votes

**2**answers

187 views

### Sets of squares representing all squares up to $n^2$

Let $S_n=\{1,2,\ldots,n\}$ be natural numbers up to $n$.
Say that a subset $S \subseteq S_n$
square-represents $S_n^2$ if every
square $1^2,2^2,\ldots,n^2$ can be represented by adding or subtracting
...

**2**

votes

**1**answer

194 views

### Primes $p=x^2+27y^2$ and Ramanujan's $x_1^{1/3} + x_2^{1/3} + x_3^{1/3}$

I was trying to generalize,
...

**3**

votes

**1**answer

107 views

### Prime labelling of graphs

A prime labeling of a graph is an injective function $f: V(G) \to \{1, 2, ..., |V(G)|\}$
such that for every pair of adjacent vertices $u$ and $v$, $\text{gcd}(f(u), f(v)) = 1$ (labels of any two ...

**-3**

votes

**0**answers

24 views

### Model of function of 2 random variables [on hold]

In my model W = f(E, K). f is a complex function (several operations on E and K).
for any W, infinity pairs of (E, K) exist that satisfy f.
E and K are between [0, +oo]
I have observations for W ...

**3**

votes

**0**answers

25 views

### On a continuous extension of a linear 2nd order PDE

Consider an elliptic (hyperbolic) equation
$A(x,y) u_{xx} + 2B(x,y) u_{xy} + C(x,y) u_{yy} = 0$
in a bounded open plane set $D$, with real-valued functions $A$, $B$, and $C$. Is it true that at least ...

**1**

vote

**0**answers

37 views

### How the modular theory of von Neumann algebras, deal with generating C*-algebras?

Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$. Suppose the existence of a bicyclic ...

**1**

vote

**0**answers

44 views

### Uniqueness of an embedding theorem for Real differential fields

I will follow a preliminary exposition for the problem in question, which will essentially follow the format on http://www4.ncsu.edu/~singer/papers/model_diff_fields.pdf [pg. 87]:
Let $K$ be a real ...

**0**

votes

**1**answer

104 views

### Murray–von Neumann equivalence on C$^*$-algebra and von Neumann algebra

Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$.
Let $p,q \in M_{\infty}(A)$ be ...