0
votes
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
2answers
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
2answers
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
1answer
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
2answers
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
0answers
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
1answer
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
0answers
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
1answer
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?

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