**0**

votes

**0**answers

2 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

6 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 ...

**0**

votes

**0**answers

5 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 ...

**-1**

votes

**0**answers

13 views

### Common Area of study of Analytical Number theory and Real analysis. ?

I am planning to do a Study Project in Analytical Number theory and My Professor's research interest is in Analysis. So I want to know the common areas of both the fields which i can study.
Thanks ...

**2**

votes

**0**answers

11 views

### 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

8 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

**0**answers

13 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 ...

**2**

votes

**0**answers

18 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

26 views

### Rearrangement of difficult algebraic equations

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

14 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?

**3**

votes

**1**answer

44 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

20 views

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

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 ...

**6**

votes

**1**answer

163 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

41 views

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

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

28 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

12 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 ...

**11**

votes

**1**answer

105 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

48 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

43 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

42 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

103 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

28 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

118 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

36 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

125 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

67 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

41 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

243 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

47 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

103 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 ...

**13**

votes

**1**answer

139 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

81 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

72 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

54 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

22 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

67 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

47 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

52 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

50 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

17 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

138 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

109 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 ...

**-5**

votes

**0**answers

28 views

### Elementary Expected Value Question [on hold]

Let $N$ be a positive integer. A soon to be bankrupt
casino lets you play the game $G(N)$. In the game $G(N)$, you roll a
typical, fair, six-sided die, with faces labeled 1 through 6, up to $N$
times ...

**2**

votes

**0**answers

48 views

### space of stability condition for an elliptic curve

Let $E$ be an elliptic curve. I want to understand why $\mathrm{Stab}(E)/\mathrm{Aut}(D^b(E))$ is a $\mathbb{C}^\times$-bundle over the moduli space $\mathbb{H}/\mathrm{SL}(2,\mathbb{Z})$ of elliptic ...

**3**

votes

**0**answers

93 views

### Langlands-Shahidi method carried out in the simplest case?

I would like to read the simplest examples of Langlands-Shahidi method carried out to prove the functional equation of $L$-function. Could the constant term of GL(2)-Eisenstein series to prove any ...

**7**

votes

**2**answers

231 views

### Average digit sum in different bases

Given two natural numbers $n\geq 1$ and $b\geq 2$, denote by $S_b(n)$ the sum of the digit of $n$ in its representation in base $b$. Clearly $S_b(n)$ varies from 1 (when $n$ is a power of $b$) to ...

**3**

votes

**1**answer

75 views

### Is this notion of 'closed subset' of a semigroup action something people have thought of?

Suppose $S$ is a semigroup (or a monoid, or a category), and $X$ is an $S$-set -- that is, a set with an action of $S$. Say that a sub-$S$-set $Y$ is "downward closed" (or maybe "well-generated") if ...