# All Questions

**-1**

votes

**0**answers

36 views

### Euler equation formula [on hold]

when I am using Euler equation for Fourier transform integrals of type $
\int_{-\infty}^{\infty} dx f(x) exp[ikx] $ I am getting following integrals:
$\int_{-\infty}^{\infty} dx f(x) cos(kx)$ (for ...

**2**

votes

**0**answers

82 views

### Twisting by a multiplicative Character in Katz, Perversity and Exponential sums

Let $C(x_1,\ldots,x_n)$ be a nonsigular cubic form with integral coefficients.
In his Proof that $C$ fulfills the Hasse-Principle, if $n\geq 9$, Hooley used the following estimate that was provided ...

**6**

votes

**0**answers

75 views

### $F[[T]] \times F[[1/T]]$ fundamental domain, show compactness

Let $p$ be a prime number. What is the easiest way to see that $(\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T)))/\mathbb{F}_p[T, 1/T]$ is compact? Here $\mathbb{F}_p[T, 1/T]$ is embedded in ...

**3**

votes

**2**answers

149 views

### Jacobson-Morozov theorem

Jacobson-Morozov theorem for a semisimple algebraic group $G$ (presumably I am working over algebraically closed field) states that: given a unipotent u, there exists a homomorphism $\phi$ from $SL_2$ ...

**0**

votes

**1**answer

50 views

### Comparison of Lp norm of matrix and its entry wise norm. [on hold]

I need to know the relation between operator norm of a matrix i.e. $ \Vert A\Vert_p$ for case of p=1 and 2 and its entry wise Frobenius norm $ \Vert A\Vert_F$.

**-2**

votes

**0**answers

125 views

### What is the symmetry of SU(3) - when seen as a group / algebra? [on hold]

Simply asked: is it more correct to state that the symmetry of the SU(3) manifold is $Z_3$ or $S_3$? Or neither of the two?
SU(3) has a kind of threefold symmetry; but which one exactly? When ...

**4**

votes

**0**answers

56 views

### Prove or disprove a claim about covering a polytope by convex polytopes in a certain way

Here is the claim:
Given a polytope $K$ in a unit ball in $\mathbb{R}^d$, there exists a universal constant $C(d)>0$ depending only on $d$ and a countable collection of convex polytopes ...

**0**

votes

**0**answers

65 views

### How do i show that the fixed points of this dynamics $ x_{n+1}=x_{n}^2-x_{n-1}^2 $ are stable? [on hold]

Is there somone who can show me how do i show that the fixe point of this
dynamics $$ x_{n+1}=x_{n}^2-x_{n-1}^2 $$ are stable ?
$x_{0}+x_{1}>0 $,$x_{0}=0,x_{1}=\frac{1}{2}$
*My attempt only I ...

**-1**

votes

**0**answers

49 views

### Normal Sub-groupoid [on hold]

Is there a definition of normal groupoid?
For normal sub-quasi-group I found two:
The first one: a sub-quasi-group $H$ is called normal if there exists a normal congruence $\theta$ such that $H$ ...

**3**

votes

**0**answers

42 views

### Expected size of determinant of $AA^T$ for non-square random Toeplitz $A$

If $A$ is chosen uniformly at random over all possible $m$ by $n$ Toeplitz (0,1)-matrices, what is the expected size of the value of the determinant of $AA^T$? We can assume $m \leq n$ and all ...

**2**

votes

**2**answers

91 views

### Identities involving sums of Catalan numbers

The $n$-th Catalan number is defined as $C_n:=\frac{1}{n+1}\binom{2n}{n}=\frac{1}{n}\binom{2n}{n+1}$.
I have found the following two identities involving Catalan numbers, and my question is if ...

**2**

votes

**2**answers

102 views

### Is the boundary of an open, regular, bounded, path-connected, and simply connected set a Jordan curve

Trying to find weakest condition on an open bounded set to apply Carathéodory's theorem.
My bounded open sets can be assumed to be pretty well-behaved, but I wonder if the above conditions are ...

**2**

votes

**0**answers

49 views

### TTF triples are recollements

The notion of recollement
$$
\mathcal{A}'
\stackrel{\overset{i^*}{\longleftarrow}}{\stackrel{\overset{i_*}{\longrightarrow}}{\underset{i^!}{\longleftarrow}}}
...

**1**

vote

**0**answers

68 views

### $n$-recollements and perverse t-structures

A recent preprint on arXiv brought my attention on the notion of $n$-recollement (def. 2) a generalization of the notion of recollement among three abelian or triangulated categories behaving like a ...

**1**

vote

**0**answers

46 views

### Beauville's Integrable System with singular spectral curves

Let us consider Beauville's Integrable System. So, we live on $\mathbb{P}^1$. There is the moduli space of matrices $M_r(d)/\mathrm{PGL}(r)$ with polynomial entries of degree less than or equal $d$. ...

**0**

votes

**0**answers

69 views

### Normal subgroupoid? [on hold]

Is there a definition of normal groupoid?
For normal sub-quasi-group I found two:
The first one: a sub-quasi-group $H$ is called normal if there exists a normal congruence $\theta$ such that $H$ ...

**0**

votes

**0**answers

14 views

### K nearest neighbors estimation with a kernel

If I have a bunch of data points $x_1,\dots,x_n$, I can build a density function $f(x)$ based on these data points by defining $f(x) = c/d_k(x)$ for an appropriate constant $c$, where $d_k(x)$ is the ...

**1**

vote

**0**answers

64 views

### All relations among degree n monomials in n variables

In the course of my work, I have run into the problem of finding exactly all relations among degree $n$ monomials in $k[x_1,\dotsc,x_n]$, with specific interest in the case $n=3$ (e.g. $x_1^2 x_2 ...

**4**

votes

**2**answers

482 views

### Idea of using etale site

I have just read an article which mentions that, when Grothendieck considered using etale morphism, he did borrow the idea from Riemann that multivalued function on an open subset of complex plane ...

**0**

votes

**1**answer

31 views

### Convergence to equilibrium via gradient descent

J. B. Rosen proved that in concave games of n players (which assumes that Cartesian product of strategy profiles is convex) if the game satisfies the condition of diagonally strictly concave then ...

**0**

votes

**0**answers

54 views

### Proof of formula for dimension of moduli of stable vector bundles on smooth curves [migrated]

Let $C$ be a smooth curve of genus $g \ge 2$ over an algebraically closed field of positive characteristic. If I understand correctly, the dimension of the moduli space of vector bundles on $C$ of ...

**0**

votes

**1**answer

126 views

### Model over DVR for smooth projective curves

Let $C$ be a smooth, projective, geometrically irreducible curve of genus at least $2$ over a complete discrete valued field $F$ of characteristic zero (not necessarily algebraically closed). Let $R$ ...

**0**

votes

**0**answers

175 views

### Getting back into advanced mathematics [on hold]

I hope that this is the correct forum to post my question on. I'm a 37 year old IT professional working in the Banking Sector who previously studied for a PhD in Computattional Fluid dynamics back in ...

**3**

votes

**0**answers

52 views

### Relation between linear independence of lattice vectors and the toric variety defined by that lattice

I have been reading some basic and elementary work on toric varieties, but even though people assured me that toric varieties are very well understood, several questions remained.
Setup. Let ...

**-1**

votes

**0**answers

13 views

### lower incomplete gamma function Holomorphic extension [on hold]

How to use repeated application of the recurrence relation for the lower incomplete gamma function to lead to the power series expansion?

**5**

votes

**0**answers

52 views

### $C^1$ regularity of harmonic functions on Riemannian manifolds

Consider a smooth, connected and complete Riemannian manifold $M$. It is well known that harmonic functions defined on some open subset of $M$ are $C^\infty$.
I'm interested in knowing whether there ...

**0**

votes

**0**answers

36 views

### 3D matching modification

Consider all instances of the $3D$ matching problem where all edges that intersect-intersect in "exactly" one vertex (1-edge intersection).
Consider all instances of the $3D$ matching problem where ...

**1**

vote

**0**answers

95 views

### Proof of Arnold-Liouville theorem in classical mechanics [on hold]

I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point.
Especially, I am talking about the part that ...

**0**

votes

**0**answers

9 views

### The inter-request time distribution after aggregating some arrivals in the renewal process

This is a follow-up question of the question "Aggregate arrivals from a Poisson Process"
The inter-arrival time of a renewal process, $t$, conforms to a general distribution, denoted by PDF ...

**0**

votes

**0**answers

32 views

### Algebraic independence in normed spaces

A set of $n$ points in $\mathbb{R}^2$ is algebraically independent over $\mathbb{Q}$ if there is no polynomial dependency among the $2n$ coordinates.
A result (Lemma 3.3) from "Globally linked pairs ...

**1**

vote

**1**answer

90 views

### Expected size of determinant of $AA^T$ for non-square random $A$

If $A$ is chosen uniformly at random over all possible $m$ by $n$ (0,1)-matrices, what is the expected size of the absolute value of the determinant of $AA^T$. We can assume $m < n$ and all ...

**-2**

votes

**0**answers

167 views

### Question about Fermat's Last Theorem [on hold]

Is there a way to prove that having $x \gt 0, z \gt 0, n \gt 2$ with $x, z, n \in \mathbb{Z}$,
$$
\sum_{k = 0}^{n - 1}{\binom{n}{k} x ^k} = z ^ n
$$
have no solution without using Fermat's Last ...

**0**

votes

**0**answers

26 views

### A general method to integrate rational functions [on hold]

$\int\frac {x^3}{1+x^5}$
ATTEMPT:
I did the following substitution:
Let $x=\frac{1}{t}.$
$dx=\frac{-1}{t^2}dt.$
substituting back:
$I=\int\frac{-1}{1+t^5}dt$ which doesn't seems a simpler ...

**0**

votes

**0**answers

16 views

### On important functions relflecting spectral properties of Jacobi operators [migrated]

The spectral analysis of Jacobi (semi-infinite, tridiagonal) operators acting on $\ell^{2}(\mathbb{N})$ is deeply investigated. A crucial role is played by function $m$ which is usually known as Weyl ...

**1**

vote

**0**answers

64 views

### Volume of arithmetic quotients of symmetric spaces

Now let $\textbf{G}$ be some connected semisimple linear algebraic group over a number field $F$. Let $G_{\infty}$ be $\textbf{G}(\mathbb{R}\otimes_{\mathbb{Q}} F)$. Let $K_{\infty}$ be a maximal ...

**3**

votes

**1**answer

82 views

### Differential Operators On A Curve And On Osculating Circle

Given a 1D Riemannian manifold $\Gamma$ embedded in 2D Euclidean space (e.g. a parametric curve on a plane $\mathbb{R}^{2}$ ), and point $x_{0}\in \Gamma$, we denote $S^{1}(x_{0})$ the circle ...

**1**

vote

**2**answers

124 views

### Subsets of $\mathbb{N}$ whose lower density respects complements

The lower density of $A\subseteq\mathbb{N}$ is defined to be $\lambda(A)=\lim\text{inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$. We set $${\cal C} = \{A\subseteq \mathbb{N}: ...

**8**

votes

**6**answers

685 views

### Algorithms for calculating R(5,5) and R(6,6)

Calculating the Ramsey numbers R(5,5) and R(6,6) is a notoriously difficult problem. Indeed Erdős once said:
Suppose aliens invade the earth and threaten to obliterate it in a year's time unless ...

**0**

votes

**0**answers

32 views

### Logic resolution and logic consequence [on hold]

Which of this are false?
a) If some formula H results from premises D, then H could be derived from D with using resolution (reapetedly) rule.
b) If some formula H results from premises D, then we ...

**11**

votes

**4**answers

639 views

### Determining if a matrix is orthogonal

Let g be an element of $GL_n(\mathbb C)$. We know that there are orthogonal groups $O(\beta)=\{X\in GL_n(\mathbb C) \mid X^t\beta X=\beta\}$ for any $\beta$, invertible symmetric matrix. Though these ...

**0**

votes

**0**answers

113 views

### If C is a closed, unbounded subset of Ord, are the sets generated by the new class of ordinals Ord restricted to C, a Universe?

Given any universe, and some closed unbounded subset of Ord, is it possible toy generate a universe in the following way:
Restrict Ord to a target club. Then generate all look the sets necessary to ...

**18**

votes

**2**answers

797 views

### History of Geometric Analogies in Number Theory

My question, put simply, is: When did mathematicians/number theorists begin viewing questions in number theory through a geometric lens?
For example, was it before Grothendieck introduced schemes to ...

**-1**

votes

**0**answers

24 views

### Fast Algorithm to compute the Discrete Fourier Transform with a constraint on the summation index

I really appreciate if anyone can help me with this problem.
Problem:
Let $W_n=e^{\frac{2\pi i}{N}}$ which is the $N$th root of unity. The backward Discrete Fourier Transform of a complex vector ...

**0**

votes

**1**answer

79 views

### A question about the Vandermonde determinant

We know that the Vandermonde determinant of order $n$ is the determinant defined as follows:
$$\begin{vmatrix}
1&x_1&x_1^2&\dots&x_1^{n-1}\\
...

**-3**

votes

**0**answers

38 views

### Mutual Information: How these two equations are equal? [on hold]

I'm a biologist trying to apply the Mutual Information (MI) to some RNA secondary structure. I know that there exists two MI equation that, mathematically, are equal:
$I(X,Y) = \sum_{x,y} p(x,y) ...

**-4**

votes

**0**answers

38 views

### What does b^(3x+1)×b^(2x−5) equal? [on hold]

I am taking a grade 12 math course and this question is really confusing me b^(3x+1)×b^(2x−5). The answer is b^(6x^2−13x−5). However since both the bases ("b") are the same, and they are being ...

**1**

vote

**0**answers

143 views

### Functors similar to $H^i(\cdot)$

Suppose $T$ is a contravariant functor from the category of pointed topological spaces to the category of abelian groups, then we have homomorphisms $\alpha\colon T(X)\times T(Y)\to T(X\times Y)$ and ...

**3**

votes

**1**answer

98 views

### group completion theorem by using homology fibrations

In the paper Homology fibrations and group completion theorem, McDuff-Segal (www.maths.ed.ac.uk/~aar/papers/mcdsegal.pdf), page 281:
Let $M$ be a topological monoid such that $\pi_0M$ is generated by ...

**2**

votes

**2**answers

185 views

### binomial/factorial identity mod p

In trying to determine the spectrum of a well-known ergodic transformation, I came up with the following useful (for me) result.
Let $p$ be a prime and $a$ a positive integer. Then for $M$ a positive ...

**0**

votes

**0**answers

20 views

### Weighted Perturbation Bound for Polar Decomposition

Setup: Let $X\in\mathbb{R}^{n\times r}$ be a matrix with orthogonal columns, with $\Sigma = X^TX$, and assume that $\Sigma$ is invertible (note, $\Sigma$ is not necessarily the identity).
Suppose we ...