-1
votes
0answers
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
0answers
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
0answers
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
2answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
2answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
2answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
6answers
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
0answers
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
4answers
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
0answers
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
2answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
0answers
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 ...

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