# All Questions

**0**

votes

**1**answer

43 views

### How can two random variables are continuous infers that their jointly random variable is continuous [on hold]

We assume that $\forall a,b$ suchthat $a^2+b^2>0$, $aX+bY$ is continuous random variable.
But we don't assume that $X$ and $Y$ are independent.
My question is the following:
Is it true that the ...

**1**

vote

**0**answers

47 views

### Cramer-Rao type bound for absolute estimation error

Let $\{X_1, X_2, \ldots, X_n\}$ be independent and identically distributed (i.i.d.) random variables sampled from a common distribution with density $f_{\theta}(x)$, where $\theta$ is an unknown ...

**5**

votes

**1**answer

208 views

### When are configuration spaces aspherical?

It is a theorem of Fox and Neuwirth that the space $C_k \mathbb R^2$ of unordered configurations of $k$ points in $\mathbb R^2$ is apsherical, i.e. has trivial higher homotopy groups.
This has some ...

**5**

votes

**1**answer

167 views

### real representation of a product group

Let $G_1$ and $G_2$ be compact Lie groups. We know that each finite-dimensional complex irreducible representation of $G_1\times G_2$ is the tensor product of an irreducible representation of $G_1$ ...

**1**

vote

**1**answer

31 views

### Splines linearly independent

Let $N_1:=\chi_{[0,1]}$ be defined as this characteristic function and $N_n:=N_{n-1}*N_1$ then this leads to polynomials with support $[0,n]$. These splines are well-studied click for wikipedia My ...

**2**

votes

**1**answer

126 views

### Random walk on a sphere along latitude-longitude grid

Suppose a sphere is partitioned by a latitude-longitude grid, with
grid quadrilaterals $\Delta \times \Delta$.
All grid nodes have degree $4$, while the North & South poles have
degree $2 \pi / ...

**1**

vote

**1**answer

56 views

### Is the product of two supermodular functions supermodular?

The definition of Supermodularity is that for every $x′>x$ and $y′>y$, $f(x′,y′)+f(x,y)>f(x′,y)+f(x,y′)$.
Suppose $f$ and $g$ are supermodular, non-negative and increasing in both arguments. ...

**1**

vote

**1**answer

66 views

### Coxeter Isometry groups whose center has torsion

I'm looking for examples of Riemannian manifolds M of dimension $\geq 2$ such that the isometry group $Isom(M)$ contains as subgroup a finite Coxeter group $G$ such that $Tor(Z(G))$, the torsion group ...

**8**

votes

**2**answers

454 views

### Bit String Bijection

I am searching for a bijection between two types of bit strings (strings of 0's and 1's) both of even length (2n).
The restriction on the first type of bit string is that they must have the same ...

**-5**

votes

**0**answers

35 views

### Formula of infinite sum [on hold]

Hi im looking for the formula
$$\sum_{n=0}^\infty na^n $$
Can someone give it to me ? I couldn't find it on the net.

**0**

votes

**0**answers

66 views

### Counting ways to Arrange Variable Sized Objects into Fixed Number of Spaces [on hold]

I scoured the web for a generalized notion of balls and bin that encompasses the needs of my questions. Essentially, I'm looking for a way count the number of ways to fit i different types of objects, ...

**0**

votes

**0**answers

41 views

### what is the space $T_Y^*X$ if $Y$ is a complex analytic submanifold of $X$? [on hold]

Help with notation! I'm a physicist, but I've come across the following notation, $T_Y^* X$ where $Y$ is a complex analytic submanifold of $X$. A phrase I've heard used is "conormal bundle" but is ...

**9**

votes

**0**answers

214 views

### Proof of Cauchy's Theorem from Group Theory - Generalizable?

There are many proofs for Cauchy's Theorem from group theory, which states that if a prime $p$ divides the order of a finite group $G$, then $\exists g\in G$ of order $p$.
Recently I've encountered ...

**-3**

votes

**0**answers

25 views

### A book on numerical optimization [on hold]

I have a background in mathematics, and I'm looking for a good book in numerical optimization. I know a lot of them: Bertsekas, Bonnans, Fletcher, Nocedal. Those are all very good books.
I'm looking ...

**2**

votes

**0**answers

30 views

### Smooth bivariate functions identifiable under permutations

Consider a "smooth" $f : [0,1]^2 \to [0,1]$ and let $\pi : [n] \to [n]$ be a permutation of $[n] =\{1,\dots,n\}$. Let us define
$$
A^{f,\pi} := \Big( f\Big(\frac{\pi(i)}{n},\frac{\pi(j)}{n}\Big) ...

**0**

votes

**0**answers

36 views

### projective module over C*-algebra [on hold]

Suppose we have $V$ a projective module over C*-algebra $A$. Suppose we define another projective module $W$ which is same as $V$ but the action is given by: $v.a := v.\alpha(a)$, where $\alpha$ is ...

**1**

vote

**0**answers

53 views

### Symmetric and antisymmetric powers of SU(2) representations [on hold]

Recently, I took a course in representation theory at Imperial College, and on the first homework the questions were about certain sneaky relationships when it came to representations of SU(2).
...

**1**

vote

**0**answers

74 views

### On moduli space of torsion free semi-stable sheaves on nodal curves

Let $X$ be a projective irreducible nodal curve of genus at least $2$. Denote by $U(r,d)$ the moduli space of semi-stable torsion free sheaves on $X$ of rank $r$ and degree $d$. There are several ...

**0**

votes

**0**answers

37 views

### Auslender-Teboulle asymptotic cones Theorem 6.4.1

Is there any typo in proof of theorem 6.4.1 of Auslender-Teboulle asymptotic cones book.
In this theorem the author claims that if $\text{rge}\,A\subset\text{aff}\,C$ then for $\epsilon>0$ we have
...

**-1**

votes

**0**answers

67 views

### Is there a name for these prime and composite numbers? [on hold]

For example 67 is the 19th prime number and the 19th composite number is 30.
The 37th prime is 157 and the 37th composite is 54.
The 329th prime is 2207 and the 329th composite is 410.
I need a word ...

**-4**

votes

**0**answers

43 views

### How to solve Ax=b using matlab [on hold]

What is the best way to solve XA=B in matlab, where A and B are 3*4 matrices and X is 3*3 matrix.

**2**

votes

**0**answers

85 views

### Generating free groups by small subgroups and an element

Let $F$ be a free group of countably infinite rank, and let $L \leq F$ be a finite index subgroup. For some prime $p$ and $k \in \mathbb{N}$, let $\sigma \colon L \to ...

**0**

votes

**0**answers

30 views

### Powers of compact operators [migrated]

Consider a Hilbert space $H$ and a compact self-adjoint operator $T : H \to H$. I want to prove that all positive powers (especially fractional powers) of $T$ are compact. From the spectral theorem, I ...

**1**

vote

**0**answers

87 views

### A question about intersection theory over non algebraically closed field

Suppose we have a bilinear map $f\colon k^m\times k^n\to k^l$ such that $f(x,y)=0$ implies $x=0$ or $y=0$. We want to show $l\geq m+n-1$ for $k$ alg closed. (There is a result of Hopf for ...

**4**

votes

**1**answer

108 views

### K-fellow traveler property and automatic structure

I have been reading several articles about automatic groups and metric spaces of negative curvature. However it is not clear for me the relationship between automatic groups, hyperbolcity and the ...

**8**

votes

**0**answers

129 views

### No limit shape for random Young diagrams under z-measure?

In their paper Random partitions and the Gamma kernel (Advances in Mathematics 194 (2005) 141–202), Borodin and Olshanski state that:
An important difference between the Plancherel measures and ...

**2**

votes

**0**answers

43 views

### Pro-G_p topology

Let $G_p$ be pseudovariety of all finite $p$-group. The pro-$p$ topology on a group $G$ is the unique group topology such that the set of normal subgroups $N$ with $G/N$ in $G_p$ is a fundamental ...

**3**

votes

**1**answer

306 views

### Is this version of van der Waerden's Theorem consistent with ZFC?

One way to phrase van der Waerden's Theorem is:
For every finite coloring of $\mathbb N$ and every finite $F \subseteq \mathbb N$, there exist $a,b \in \mathbb N$ such that $a + b \cdot F$ is ...

**10**

votes

**2**answers

482 views

### Does van der Waerden's Theorem hold for $\omega_1$?

One way to phrase van der Waerden's Theorem is:
For every finite coloring of $\mathbb N$ and every finite $F \subseteq \mathbb N$, there exist $a,b \in \mathbb N$ such that $a + b \cdot F$ is ...

**-1**

votes

**0**answers

49 views

### relation between characters [on hold]

My assumption: $ H $ is a subgroup with index $ m $ in the finite group $ G $ & $ F $ is an algebraic closed field of characteristic zero & $ \chi $ is an irreducible $ F $-character of $ G $ ...

**11**

votes

**2**answers

894 views

### What is prime power of this equation of p?

Let $p$ be a prime number, I think when $p^2+p+1=q^a$, where $q$ is a prime number, then $a=1$. But I can't prove it. Is it true?

**7**

votes

**0**answers

154 views

### Homotopy transfer in the opposite direction

Let $X\rightleftarrows Y\circlearrowleft$ be a strong deformation retraction of chain complexes (a.k.a. contraction), i.e. $X\rightarrow Y\rightarrow X$ is the identity, $Y\rightarrow Y$ is a homotopy ...

**3**

votes

**0**answers

67 views

### Problem with a proof in Wellfounded trees in categories

I'm reading the paper Wellfounded trees in categories by Moerdijk and Palmgren. I'm having trouble understanding the proof of Theorem 7.2 (page 216), i.e. that in $\mathbf{ML}_{< \omega} ...

**-1**

votes

**0**answers

31 views

### $ \mathbb{C} $-character table of $ D_{14} $ [on hold]

Is there any reference where I can find the $ \mathbb{C} $-character table of the dihedral group $ D_{14} $?

**2**

votes

**1**answer

150 views

### Torsion theory for quasi-coherent sheaves?

In a category $\mathcal C$, we will say that $(\mathcal T,\mathcal F)$ is a torsion theory if it satisfies:
(1) $Hom(T,F)=0$ for all $T\in \mathcal T$ and $F\in \mathcal F$.
(2) If $Hom(T,F)=0$ for ...

**1**

vote

**1**answer

136 views

### Enumerating Lattice points

Let $A \in \mathbb{R}^{d\times d}$ be an invertible matrix. Consider the set
$$P_d := A\mathbb{Z}^d = \{A x| x \in \mathbb{Z}^d \} \subset \mathbb{R}^d$$.
and
$$ Q_d := [-1,1]^d.$$
I am interest in ...

**4**

votes

**1**answer

131 views

### Upper bound of the waiting time of a sum process

Let $n \in \mathbb{N}$, $x_1, \ldots, x_n \in (0,1)$ fix but arbitrary, s.t. $\sum_{i=1}^n x_i = 1$. Let $X_i \sim \operatorname{Unif}(\{x_1, \ldots, x_n\})$ i.i.d., and $T_n = \min\{t \in \mathbb{N} ...

**-1**

votes

**0**answers

100 views

### Order of element in algebraic group [migrated]

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...

**4**

votes

**1**answer

139 views

### Order dimension of $\omega^\omega/(fin)$

Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we say $f\simeq g$ if and only if $\exists N \in \omega$ such that $f(n) = g(n)$ for all $n\geq N$.
...

**-4**

votes

**0**answers

34 views

### Find point of triangle [on hold]

There is the triangle (http://www.mathportal.org/calculators/plane-geometry-calculators/triangleRightAngle.gif) where we know:
coordinates of points: A, C and all sides: a, b, c
Angles are 45 and 90
...

**3**

votes

**1**answer

158 views

### Generalization of Borsuk-Ulam to arbitrary ratio

Let $g: S^n \to R^n$ be a continuous odd function (i.e. $g(-x)=-g(x)$ for all $x$). The Borsuk-Ulam theorem implies that $g$ has a zero, i.e. there is an $x$ such that $g(x)=(0,0,...,0)$.
Suppose $g$ ...

**1**

vote

**0**answers

46 views

### Presentation of the Rybnikov matroid

In this well celebrated work Gregory Rybnikov exhibit an example of two arrangements with the same underlying matroid, but with fundamental groups which are not isomorphic. This is a key ...

**-4**

votes

**0**answers

32 views

### I have to show any non-invertible matrix is a reducible matrix [on hold]

Suppose that $A \in M_n(D)$ and $D$ be a division ring. An $n × n$ matrix $A = (a_{ij} )$ is called reducible if $A$
has a non-trivial invariant subspace in $D^n$. I have to show any non-invertible ...

**6**

votes

**2**answers

178 views

### Extending the Abel-Jacobi map over the DM-compactification $\overline{\mathcal{M}}_2$?

Let $\mathcal{M}_2$ be the moduli space of genus two curves and $\mathcal{A}_2$ the moduli space of principally polarized abelian surfaces. Then the Abel-Jacobi map gives an open embedding ...

**4**

votes

**1**answer

164 views

### Macaulay's example of prime ideals in $\mathbb C[X_1,X_2,X_3]$ having large number of generators

There is a famous example of Macaulay which shows that there are prime ideals of height two in $\mathbb C[X_1,X_2,X_3]$ having at least $l$ generators for any $l\ge 3$.
In Macaulay's words, the ...

**3**

votes

**2**answers

117 views

### ${\frak b}$ and ${\frak d}$ defined with $\leq$ instead of $\leq^*$

Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we define
$f\leq g$ if $f(n)\leq g(n)$ for all $n\in\omega$;
$f\leq^* g$ if there is $N\in\omega$ ...

**-2**

votes

**0**answers

37 views

### $L_{\infty}$-norm of a $\delta(t)$-“function”? [on hold]

In different contexts the $L_\infty$- norm may sometimes be defined as the essential least upper bound $\|\cdot\|_\infty=\operatorname{ess} \sup_t |\cdot |$ and sometimes as just the least upper bound ...

**3**

votes

**1**answer

131 views

### Fixed point property for intersection of spaces which are homeomorphic to a disk

The following question is question 9.8 from Miller's paper ``Some interesting problems
'':
Question Suppose $D_n$ a subset of the plane is homeomorphic to a disk and for every
$n\in \omega, ...

**0**

votes

**0**answers

35 views

### Logic and Metamath book recommendation [migrated]

Recently, I got interested in Mathematical Logic and now I am looking for good introductory books on Mathematical Logic for beginners. In fact, I plan to read some good books on Metamathematics also. ...

**0**

votes

**1**answer

42 views

### Centralizer of the derived algebra in a non-perfect Lie algebra

Is there a non-perfect Lie algebra for which the centralizer of the derived algebra is trivial?