0
votes
0answers
19 views

Question on Stationary & Cointegration Test (Augmented Dickey Fuller & Engle Granger test) [on hold]

I'm performing the stationary and cointegration test on stock prices. What I'm confused is 1) the difference between ADF stationary test and ADF cointegration test. 2) Also, in ADF stationary test, ...
-2
votes
0answers
27 views

Mean time for the renewal process [on hold]

The system is as below. Energy keeps coming at a node with a constant rate $\rho$. Node has files of size exponential($\lambda$) to be transmitted. At time zero, say the energy at the node be zero. ...
-1
votes
0answers
79 views

Degrees of multilinear polynomials satisfying some constraints [on hold]

Let $t<\sqrt{n}$. $\Bbb Z^t[x_1,\dots,x_n]=\{f\in\Bbb Z[x_1,\dots,x_n]: deg(f)\leq t$ and $f$ is multilinear$\}$. Fix an ordering of $S=\{0,1\}^n.$ If $f\in\Bbb Z[x_1,\dots,x_n]$, let $f(S)$ be ...
0
votes
0answers
69 views

Derivability of a function defined on the tangent bundle. Foundations of Finsler metrics

My question is linked to the foundations of Finsler metrics (with weak derivability assumptions). Let $M$ be a manifold of dimension $n$, and $F$ is a function from the tangent bundle $TM$ to ...
1
vote
1answer
96 views

ideal of maximal minors is cohen-macaulay?

Let $k$ be an algebraically closed field. Let $A$ be an $m \times n$ matrix with linear forms $a_{ij} \in k[x_1, \ldots, x_p]_1$ as entries. Let $I$ be the ideal generated by the maximal minors of ...
3
votes
1answer
121 views

On the size of centralizers in a non-abelian finite simple group

It is known that for a finite non-abelian simple group $G$ we have $|G|<|C_G(x)|^3$ for some involution $x$. Is there a better bound for the order of centralizer of a nontrivial element of $G$ (not ...
2
votes
0answers
104 views

Example of symplectic 4-manifolds with no Lefschetz fibration structure?

I just read about Donaldson's result on existence of Lefschetz pencil structure on symplectic manifolds (Donaldson 1999). However, one has to blow up the base locus to get a Lefschetz fibration ...
0
votes
0answers
61 views

Asymptotic analysis of a sum of complex summands using integral

I'm trying to find the exact asymptotics of a sum: $$A = \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^{i} y^{2n-i} $$ as $n\rightarrow\infty$. Here $x,y$ are complex numbers, $|x|\leq1, ...
2
votes
0answers
85 views

“simulteneous eigenvectors” under the full set of weighted Laplacians on a $g$-fold product of the Poincare half plane

This question is closely related to the following MO question Characterizing the real analytic Eisenstein series Let $\mathfrak{h}=\{z=x+iy\in\mathbf{C}\}$ be the Poincare upper half plane endowed ...
1
vote
1answer
103 views

Absolutely continuous functions

it is well known that if a function $f:[0,T]\to\mathbb{R}$ satisfies the inequality $$\vert f(t)-f(s)\vert\leq \int_s^t{m(r) dr},$$ for $s<t$ and some $m\in L^1([0,T])$ then $f$ is absolutely ...
1
vote
1answer
125 views

abelian p- subgroups of E_6(q)

Is there any result about maximal abelian p-subgroups of the exceptional group E_6(q), where q=p^a is prime power?
8
votes
1answer
175 views

Non-abelian freeness of SU_2

The distribution of the trace of a random element of $SU_2$ is the Sato-Tate distribution. The analogue of the Gaussian distribution in free probability theory is the Wigner semicircle distribution. ...
3
votes
1answer
86 views

Dehn twist about an arbitrary curve

I need to know if there is an algortihm to write down a Dehn twist about an arbitrary curve on an orientable surface $S$ , as product of a set of generators of $MCG(S)$. Since we have the conjugacy ...
-4
votes
0answers
42 views

finding points on elliptic curve over finite field [on hold]

Find the points on the elliptic curve y^2 = x^3 + 2x + 2 in F17 (field of prime 17). Do I have to guess a first point and then use an algorithm to spit out all other points?
29
votes
3answers
2k views

A game of stones

How I arrived at this question is a rather long story having to do with the honors calculus class I am teaching. At this point it's sheer curiosity on my part. Here is the game. ...
1
vote
0answers
100 views

Proper monomorphisms in complex analytic spaces

In the algebraic geometry of schemes, we know that monomorphisms which are universally closed (= every pullback is a closed map) and of finite type are closed immersions. See Gortz, Wedhorn "Algebraic ...
3
votes
1answer
175 views

Example of proof using the generic matrix

There is a really nice proof of the Cayley-Hamilton Theorem using the generic matrix. I expose it briefly. One defines the generic matrix $G:=(X_{ij})_{ij} ...
0
votes
0answers
41 views

Bounding a ratio by its complement [on hold]

Given some integers $n > \beta + \alpha, \beta > \alpha$ and $\alpha > 0$, is there a real number $\delta$ for which $\frac{n-\alpha}{n-\beta} \geq (\frac{\beta}{\alpha})^{\delta}$, where ...
1
vote
0answers
92 views

How to see that this pairing of line bundles is multiplicative?

Given a projective flat morphism $p: X \rightarrow Y$ of integral noetherian schemes of relative dimension one. For a coherent sheaf $F$ on $Y$ we can define a line bundle $det(F)$ on $Y$ and for a ...
3
votes
0answers
62 views

Worst-Case Solution to (Stochastic) Matrix Inequality

EDIT: Some specific conjectures added. This problem comes with an associated stochastic process, but I phrase everything as linear algebra in case somebody from a non-probability community has seen ...
1
vote
0answers
32 views

Reference for a special case of the Hanson-Wright inequality

I would like find tail bounds for the expression $$ \begin{align*} \left|\left\langle a,\phi\right\rangle \left\langle \phi,b\right\rangle -\left\langle a,b\right\rangle\right|, \end{align*} $$ where ...
12
votes
0answers
211 views
+50

Algebraic dependency over $\mathbb{F}_{2}$

Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$ such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall ...
-4
votes
0answers
51 views

matrix theory understand the notion of transpose [on hold]

Can you explain me, please, what does it mean the transpose of a matrix ? I know the definition in the context of matrix theory and its generalization to adjoint operators (transpose of a linear ...
7
votes
0answers
138 views

Whiskering a monad

In "The Geometry of Iterated Loop Spaces", May shows that any monad that is coming from an operad may be "whiskered", so that the unit map becomes a closed cofibration. The ability to do this is vital ...
12
votes
1answer
357 views

A funny factorization of the Jacobian coming from the lines on the Fermat cubic

Here is something which came up in my algebraic geometry class, and I'm wondering if it has a deeper explanation. Let $F(w,x,y,z) = w^3+x^3+y^3+z^3$ and let $X$ be the cubic surface in $\mathbb{P}^3$ ...
-1
votes
1answer
107 views

Powers of orthogonal matrices is closed

This might be a basic question, nonetheless I cannot give a proof. Given an orthogonal matrix $A$ with eigendecomposition $A = Q \Lambda Q^{-1}$ with only non-real eigenvalues. Given also a diagonal ...
-2
votes
0answers
45 views

Weak topology and topology by semi-norm [on hold]

Wikipédia: -The weak topology on X is the initial topology with respect to X* (let's note it T') -If the field K has an absolute value , then the weak topology σ(X,F) is induced by the family of ...
0
votes
1answer
39 views

Distribution of the $\alpha$-parameter of a $2\times 2$ Haar-distributed, unitary matrix

It is well known that any $2\times 2$ unitary matrix $\mathbf{U}$ can be parametrized as $$\mathbf{U}=\begin{pmatrix} 1 & 0 \\ 0 & \mathrm{e}^{\mathrm{j}\beta_1}\end{pmatrix} \begin{pmatrix} ...
26
votes
4answers
2k views

Fermat's last theorem over larger fields

Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite. Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite? Here ...
-1
votes
0answers
23 views

Properties of unit quaternion transformation [on hold]

It is commonly known that unit quaternions can be used to represent spatial rotations. The usual interpretation is as follows: $$ \tilde{q} = \cos{\alpha \over 2}+(a\cdot i+b\cdot j+c\cdot ...
2
votes
0answers
120 views

State of the art in the theory of integer sequences

I was going through N.J.A. Sloane's 'Encyclopedia of Integer Sequences'. In it are discussed many tricks that are used to determine the recursive definition or explicit formula for a given sequence. ...
3
votes
1answer
75 views

Strongly asymmetric graphs

Asymmetric graphs are graphs that have a trivial automorphism group $\textrm{Aut}(G)$, i.e. the only graph isomorphism from $G$ to itself is the identity. Let's call a graph $G$ strongly asymmetric ...
0
votes
1answer
178 views

Continuity of a Functional

A certain functional $T$ is defined as: $$T(F)=\int_{(0,1)}F^{-1}(s)M(ds)$$ where $M$ is a probability measure with support $[\alpha,1-\alpha]$,for $\alpha>0$. The result that above functional is ...
1
vote
0answers
86 views

Hausdorff topologies on Q

Is there any description known of the Hausdorff topologies on $\mathbb{Q}$ compatible with the group operations?
0
votes
0answers
133 views

Rational curves and Serre's construction

Why rational curves, used in Serre's construction of vector bundles, usually corresponds to unstable bundle? I saw this affirmation in Richard Thomas's paper on an obstructed bundle on a CY threefold. ...
-4
votes
0answers
147 views

Examples of weird 'modular like mathematical space' that behaves like as if it is infinite until a threshold value is reached? [on hold]

(Might be a bit layman because I don't have rigorous math term to describe the concept) Generalize it to mathematical spaces, are there spaces which are sort of like a. Consists of multiple ...
-3
votes
1answer
111 views

Decidable theorem or result that is not weaker than Tarski's theorem

I am wondering what other decidable theorem or results that is not weaker or stronger than Tarski's theorem. Could any one give reference or a simple introduction about such result known in their ...
-1
votes
0answers
23 views

If the linearization of a section is surjective on a slice, is the image under a submersion also a smooth manifold?

Let $V \rightarrow M \times N_1 $ be a vector bundle, where $M$ and $N_1$ are smooth manifolds and $s: M \times N_1 \rightarrow V$ a smooth section such that whenever $s(p,q) =0$ then $$ \nabla ...
2
votes
0answers
68 views

Strong solution to $u_t - \Delta_p u = f$

For $p > 1$, consider the equation $$\langle u_t, v \rangle + \int_\Omega |\nabla u|^{p-2}\nabla u \nabla v = \langle f, v \rangle$$ $$u(0) = u_0$$ $$u|_{\partial\Omega} =0$$ for all $v \in ...
8
votes
2answers
378 views

Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated?

Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated? Or are there some tables or handbooks which list some common calculated results of ...
0
votes
1answer
73 views

Are the natural numbers a disjoint union of infinite sets of zero asymptotic density? [on hold]

Suppose $\mathbb{N}=\bigsqcup_{i\in\mathbb{N}}E_i$ with $\#E_i=\infty$ for each $i$. Is it possible that $\limsup_{N\to\infty}\frac{1}{N}\#(E_i\cap\{1,\ldots,N\})=0$ for all $i$, which would mean ...
0
votes
0answers
66 views

Adelic integral factorization

In order to calculate Tamagawa numbers, I need to justify that for a nice (say Schwartz-Bruhat) function, the following identity holds : $$\int_{\mathbf{A}^2} f(x)dx = \int_{SL_2(\mathbf{A})/SL_2(K)} ...
0
votes
1answer
69 views

Schur's lemma for antiunitary operators on complex Hilbert spaces

Suppose to have a linear irreducible unitary representation $\rho:G\rightarrow U(H)$ on a complex Hilbert space $H$ with $G$ a generic group. Let $A$ be an $\textit{anti}$-linear operator such that ...
-2
votes
0answers
40 views

Expected value of minimum of an exponential function [on hold]

Find expected value of minimum of n random variables: x = (x1,x2,x3,..,xn) The distribution is an exponential function: ...
-1
votes
0answers
55 views

Continuous versions of tensors/ Tensors with infinite indices?

In linear algebra and general relativity, we knew that vectors can be represented by a linear combination of components and a basis $$\mathbf{V}=\sum_{i=1}^n A_i\mathbf{e_i}$$ Or in Einstein ...
1
vote
0answers
37 views

decomposition of tempered distributions by entire analytic functions

Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1~~\text{if}~|\xi|\leq 1\}$$ Let $j\in \mathbb N$ ...
-1
votes
0answers
105 views

infinitesimally commutative diagram [on hold]

Consider $f:X\rightarrow Y$, $g:Y\rightarrow Z$, $h:Y\rightarrow Z$ morphisms of intregal and separated $k$-schemes of finite type. We assume that at at point $x\in X$, $h(x)=g(f(x))$ the level of ...
-2
votes
0answers
44 views

Why is the finite extension field of the p-adic numbers $\mathbb{Q}_p$ spherically complete? [on hold]

Here by spherical completeness it is meant that given a non-empty nest of closed balls $\{B_\alpha|\alpha\in I\}$, that is, $\forall \alpha_1,\alpha_2\in I$ either $B_{\alpha_1}\subset B_{\alpha_2}$ ...
6
votes
2answers
231 views

On a minimal algebraic number field which satisfies the principal ideal theorem

By an algebraic number field, we mean a finite extension field of the field of rational numbers. Let $k$ be an algebraic number field, we denote by $\mathcal{O}_k$ the ring of algebraic integers in ...
-1
votes
0answers
39 views

singular point of a complete intersection surface [migrated]

Let $S:= H_1\bigcap H_2\bigcap \cdots \bigcap H_N \subset\mathbb{P} _{\mathbb{C}}^{N+2}$ be a complete intersection surface, where each $H_i$ is a hypersurface defined by a homogeneous equation $f_i$. ...

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