**1**

vote

**1**answer

30 views

### Sum of Stirling numbers with exponents

I have a trouble with the following sum
$\sum_{i=0}^n\binom{n}{i}S(i,m)3^i$, where $S(i,m)$ is the Stirling number of the second kind (the number of all partitions of $i$ elements into $m$ nonempty ...

**2**

votes

**0**answers

36 views

### Which weighted projective spaces (and their finite quotients) are local complete intersections?

Let $G$ be a finite subgroup of $Gl_{n+1}(k)$ (for $k$ being an algebraically closed field). My question is: do there exist examples of $G$ such that the corresponding quotient $P$ of $\mathbb{P}^n$ ...

**0**

votes

**0**answers

25 views

### Anisotropic limit of a Dirac delta function

I hope this is the right place for this question, sorry if it is not. As part of a fairly long equation which I will not bore you with I have a Dirac delta function $\delta({k}-{k}_1-{k}_2)$ where the ...

**3**

votes

**0**answers

89 views

### fundamental groups of $SO(n)$ and $Sp(2n)$

Let $k$ be an algebraically closed field of characteristic zero. and let $$\sigma: SL_n(k)\rightarrow SL_n(k)$$
be an involution.
My questions are:
How could one calculate the fundamental group of ...

**2**

votes

**0**answers

22 views

### good choice of extension of equivariant map

Let $K$ be a compact Lie group. Let $C_K$ denote the category whose objects are the compact lie groups containing $K$ and whose morphisms are inclusion of the groups. Let $Y$ be a $K-$space such that ...

**0**

votes

**0**answers

19 views

### A question regarding forcing in $NGBC^{-f}$+$BAFA$

Suppose one has a model $M$$\vDash$$NGBC^{-f}$+BAFA. Does there exist a (class) forcing extension $M[G]$$\vDash$$NGBC^{-f}$+$BAFA$ that has a submodel $N$$\vDash$$NGB^{-f}$+$BAFA$+$\lnot$$AC$? Can ...

**0**

votes

**0**answers

28 views

### Non-normality of limit of random variables

I have encounter the following difficulty in the study of limits of random variables. Assume that $\{X_n\}_{n\geq 1}$ is a sequence of real-valued random variables such that
...

**0**

votes

**0**answers

16 views

### A general theory for boundary value problems

One can study the characterization of a linear differential operator $T$ from scalar product $(f,g)=\int_{a}^{b}f(t)g(t)dt$ and the theory of adjoint operators solving $Tf=g$ by finding a right ...

**1**

vote

**0**answers

17 views

### Power-spectrum of quasi-periodic functions

From Scholarpedia:
Quasiperiodic oscillation is an oscillation that can be described by a quasiperiodic function, i.e., a function $F$ of real variable $t$ such that ...

**5**

votes

**1**answer

47 views

### references for faithful orthogonal (or unitary) representation of symmetric groups

Let $S_n$ be the symmetric group of $n$ points. I want to find references (or proofs) for the following statement (1).
(1). There does not exist any faithful orthogonal representation
$$
...

**0**

votes

**0**answers

30 views

### Schur polynomial, change of variable

Let $k=(k_1,k_2,k_3,k_4)\in \mathbb{N}^4$ and let $s_k(x_1,x_2,x_3,x_4)$ be the Schur polynomial on $GL_4$.
Question 1: If I replace $x_3$ with $x_1$ and $x_4$ with $x_2$, can $s_k(x_1,x_2,x_1,x_2)$ ...

**0**

votes

**1**answer

40 views

### Convex cones: strict separation

Consider two closed convex cones $A$ and $B$ in $\mathbb{R}^3$. Assume that they are convex even without zero vector, i.e. $A \setminus \{0\}$ and $B \setminus \{0\}$ are also convex (it helps to ...

**3**

votes

**1**answer

55 views

### Conformal changes of metric and geodesics

Suppose $(M,g)$ is a Riemannian manifold. Let us assume that $X$ denotes a vector field in this manifold and consider the integral curves of this vector field.
Does there exist a conformal factor $c$ ...

**-2**

votes

**0**answers

34 views

### Rearrangement of difficult algebraic equations [on hold]

I have always had difficulty when rearranging large equations of functions to find the roots and also the turning points(using derivatives) and was hoping some people could give me some tips when it ...

**0**

votes

**0**answers

24 views

### Embedding Riemmanian Manifold Linearly

Given a Hilbert Manifold $M$ does there exist a smooth map into some very large Hilbert space taking geodesics to straight lines?

**5**

votes

**1**answer

66 views

### Is an elliptic curve that is isomorphic to its Frobenius conjugate defined over $\mathbb{F}_p$?

Let $p$ be prime and $q = p^n$. Let $E$ be an elliptic curve over $\mathbb{F}_q$, and let $E^{(p)}$ be the pullback of $E$ by the $p$-power Frobenius of $\mathbb{F}_q$. If $E$ is isomorphic (over ...

**2**

votes

**0**answers

27 views

### Construct a sequence of i.i.d random variables with a given distribution function, diagonalization? [on hold]

Assume we have a sequence of i.i.d. random variables $X_1, X_2, \dots,$ on a probability space $(\Omega, \mathcal{F}, P)$ with$$P(X_n = 1) = P(X_n = -1) = {1\over2}.$$Given a distribution function ...

**9**

votes

**1**answer

412 views

### Are the following identies well known?

$$
x \cdot y = \frac{1}{2 \cdot 2 !} \left( (x + y)^2 - (x - y)^2 \right)
$$
$$
\begin{eqnarray}
x \cdot y \cdot z &=& \frac{1}{2^2 \cdot 3 !} ((x + y + z)^3 - (x + y - z)^3 \nonumber \\
...

**0**

votes

**0**answers

58 views

### Investigate the $\inf$ of the sequence $(\sqrt[n]{|\sin{n}|})_{n=1}^{\infty}$ [on hold]

According to this MathStackExchange post years ago we have $\displaystyle\lim_{n\to\infty}(\sqrt[n]{|\sin{n}|})=1$. So the $\limsup$, the $\liminf$, and the $\sup$ of this sequence are clearly 1.
But ...

**-4**

votes

**0**answers

29 views

### For v = (x, y, z) let a, b, c denote the angles between v and the respective x, y, z axes. Show that cos^2(a) + cos^2(b) + cos^2(c) = 1 [on hold]

I am unsure how to approach this problem, as I have not yet learned many of the trig identities for working in 3 dimensions. The only thing I can think of is if A, B, and C have to add up to 180 (I am ...

**0**

votes

**0**answers

15 views

### Bounding function of norms in constrained vector space

$v$ is a vector of length $n$, where $v_1 = 1$ and very element $v_i \in [0,1]$
$w = \| v \|_1^1$
$x = \| v \|_2^2$
$y = \| v \|_3^3$
$z = \| v \|_4^4$
Can you recommend a strategy for achieving a ...

**13**

votes

**1**answer

119 views

### Non-negative polynomials on $[0,1]$ with small integral

Let $P_n$ be the set of degree $n$ polynomials that pass through $(0,1)$ and $(1,1)$ and are non-negative on the interval $[0,1]$ (but may be negative elsewhere).
Let $a_n = \min_{p\in P_n} \int_0^1 ...

**5**

votes

**0**answers

51 views

### Differentiation of Lie $\infty$-groupoids

I've been trying to understand how to differentiate Lie $\infty$-groupoids to get a Lie $\infty$-algebroid. First of all, I will state the definitions that I'm assuming.
A Lie $\infty$-groupoid is a ...

**-5**

votes

**0**answers

45 views

### Calculate the probability that the sum of the square of two integers selected random are divisible by 2 [on hold]

P((X,Y)|X^2+Y^2 is divisible by 2)X,Y are positive integers greater than 4 possibly equals

**1**

vote

**0**answers

45 views

### Of the standard distance metrics, which ones can/cannot be embedded in Euclidean space?

Given the discussion from:
Representability of finite metric spaces
it appears that a 1974 paper by Morgan gives the criteria for when a distance metric can be embedded in Euclidean space. My first ...

**4**

votes

**1**answer

110 views

### $n$ groups of $n$ queens on a toroidal chessboard

An interesting question came up in the Puzzling Stack Exchange a few days ago about "queen-connected sets". When trying to solve this problem, I came across an arrangement of five colours of queens ...

**-3**

votes

**0**answers

29 views

### A question on subordinate matrix norm

Suppose:
$B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$
${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...

**7**

votes

**0**answers

124 views

### Calculation-free proof of the Weyl Integral formula for U(n)

The Weyl integral formula states that if f is a class function on U(n), T is the torus of diagonal matrices in U(n), and dU(n) and dT are the standard Haar measures on U(n) and T, then
$$\int_{U(n)} ...

**0**

votes

**0**answers

41 views

### A Question about compactness of an embedding into $L^p$ spaces

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says
For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have
$$ \Lambda \int_{\Omega} ...

**4**

votes

**1**answer

139 views

### Plethysm of $S^3(S^2V)$ as $\mathfrak{sl}_3(\mathbb{C})$-module

I have asked this question in MSE before, but have not got any answer. So here I am asking it again with some more detail.
I believe that the following sequence of ...

**0**

votes

**0**answers

32 views

### Full row rank of a specific matrix

Let $A \in \mathbb R^{n \times n}$ and $b \in \mathbb R^n$ and consider the following row-infinite matrix
\begin{align*}
\begin{pmatrix}
\mu_0 b & \mu_1 Ab & \mu_2 A^2b & \mu_3 A^3b & ...

**3**

votes

**0**answers

74 views

### Is $C^\infty(M,\mathbb{R})$ an ind-(smooth manifold)?

Let $\mathrm{Man}$ be the category of smooth manifolds (2nd countable, Hausdorff, no boundary, not necessarily compact) and smooth maps, and let $M$ be an object thereof. Is the presheaf $S \mapsto ...

**3**

votes

**0**answers

50 views

### Is the signature of inverse images of diffeomorphic submanifolds (along a homotopy equivalence) the same?

Suppose it is given an orientation preserving homotopy equivalence $h:N→M$ between closed oriented connected manifolds. Let $X,Y\subset M$ be diffeomorphic submanifolds, and assume $h$ to be ...

**6**

votes

**2**answers

252 views

### Algebraic proof without using comparison theorem for étale cohomology

Let $X$ be some smooth scheme over $\mathbf C$ equipped with an action of $\mu_n$ (the group of $n$th roots of unity).
The étale cohomology groups of X are therefore equipped with an action of ...

**1**

vote

**0**answers

54 views

### What are the E7(7) invariants in the adjoint representation?

Take a real vector space $R$ transforming in the adjoint representation of
the ${\rm E}_7(7)$ Lie group as $R \rightarrow G R G^{-1}$. One can define
invariants using traces of products of $R$ as ...

**1**

vote

**1**answer

106 views

### limit and combinatorics

Given $x \in (0,\frac{1}{2})$ and $y \in (0,\frac{1}{2}]$, what is the value of the following limit:
$\lim_{n\rightarrow \infty}\sum_{k=0}^{n}{n \choose k}|x^{n-k}(1-x)^{k}-y^{n-k}(1-y)^{k}|?$
When ...

**15**

votes

**2**answers

163 views

### Which partitions of $[0,1]$ are collection of level sets of a real continuous function?

Let $f:[0,1]\to[0,1]$ be given. The level sets of $f$ (ie the collection of all sets of the form $\{x\in[0,1]:f(x)=y\}$, for each fixed $y\in[0,1]$) partition the domain of $f$. I am curious for set ...

**1**

vote

**1**answer

86 views

### On conflicting descriptions for tor of a local cohomology group

Let $X$ be a smooth projective surface and $C$ a Cartier divisor on $X$. Denote by $\mathcal{H}^1_C(\mathcal{O}_X)$ the sheaf associated to the presheaf $U \mapsto H^1_{C \cap U}(\mathcal{O}_X|_U)$. ...

**4**

votes

**1**answer

46 views

### Bounding the number of points at integral distance from vertices of a triangle

Can the number of points at integral distance to all three points of a non-degenerate triangle of area $A$ be bounded by $1+cA$ for some suitable constant $c$?
Remark: Since it is easy to bound this ...

**-1**

votes

**0**answers

88 views

### How does this small change in the Pollard Rho method affect its complexity?

In finding the smaller factor $p$ of an input number $n$, the Pollard Rho method takes time bounded by a function in $O(\sqrt{p})$. (Did I get that right?)
Now let's say I tweak the method just a ...

**0**

votes

**0**answers

67 views

### Is the localization sequence exact in the middle mod. algebraic equivalence?

Let $X$ be a smooth projective $k$-variety ($k=\bar k$) and $U\subset X$ an nonempty open subset. Is it true that a cycle algebraically equivalent to zero in $U$ comes from a cycle of $X\backslash U$ ...

**2**

votes

**0**answers

23 views

### Is the unit bundle of a Finsler vector bundle a sphere bundle?

I asked this at mathstackexchange but got no answer, so I am trying here.
Let $E$ be a Finsler vector bundle* of rank $k$ over a manifold $M$. Does the unit "bundle" $UE$ admits a structure of a ...

**2**

votes

**0**answers

73 views

### Operads and the Stable Module Category

I am seeking references to places where operads and their algebras have been studied for the stable module category. Colored operads are fine too.
Let $k$ be a field and $R$ a $k$-algebra. The stable ...

**1**

vote

**1**answer

56 views

### Generalized Theorem of Laguerre

There is known theorem of Laguerre, that every linear ordinary differential equation of second order
$$y''+A(t)y'+B(t)y=0$$
by point transformation could be mapped into
$$y'' = 0,$$ that in few words ...

**1**

vote

**0**answers

55 views

### Where does the algebraic closure enter into Block's Theorem?

When applying Block's Theorem on the structure of differentiably simple rings to Lie algebras most authors require an algebraically closed field, but I can see no reference to algebraic closure in ...

**2**

votes

**0**answers

55 views

### Stopping time sigma-fields

Let $(F_n)$ be a discrete Filtration and $S_n,S$ (not necessarily finite) stopping times with $S_n\uparrow S$ (increasing convergence).
Is it true that the associated sigma-fields satisfy ...

**1**

vote

**0**answers

18 views

### Constrained absolute orientation of 3D point sets

Let us assume we have two 3D point sets, $P=\{p_i\}$ and $Q=\{q_i\}$, and that we need to recover the transformation that takes $P$ as close to $Q$ as possible. In particular, I am interested in ...

**1**

vote

**1**answer

55 views

### an inequality about kronecker product with eigenvalues question

Recently i'm reading a paper,there is a inequality that confuse me. L is a symmetric,irreducible and semi-positive definite matrix with eigenvalues of ...

**3**

votes

**1**answer

141 views

### Examples when one can use the the symmetric power $L$-functions to study topics related to the number theory

"The symmetric power $L$-functions are a powerful tool for studying algebraic or geometric objects through analytic methods." I read this sentence in the introduction of a Master thesis. I want to ...

**8**

votes

**0**answers

119 views

### $\pi_1$ of the moduli of G-bundles on elliptic curves and the double affine braid group

For a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, the fundamental group of $\mathfrak{h}_\text{reg}/W$ (where $\mathfrak{h}$ is the Cartan subalgebra, $\mathfrak{h}_\text{reg}$ is the subset ...