# All Questions

0answers
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### 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, ...
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. ...
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 ...
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 ...
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 ...
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 ...
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 ...
0answers
61 views

0answers
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### 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 ...
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 ...
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 ...
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 ...
0answers
211 views
+50

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 $\forall ... 0answers 51 views ### matrix theory understand the notion of transpose [on hold] Can you explain me, please, what does it mean the transpose of a matrix ? I know the definition in the context of matrix theory and its generalization to adjoint operators (transpose of a linear ... 0answers 138 views ### Whiskering a monad In "The Geometry of Iterated Loop Spaces", May shows that any monad that is coming from an operad may be "whiskered", so that the unit map becomes a closed cofibration. The ability to do this is vital ... 1answer 357 views ### A funny factorization of the Jacobian coming from the lines on the Fermat cubic Here is something which came up in my algebraic geometry class, and I'm wondering if it has a deeper explanation. Let$F(w,x,y,z) = w^3+x^3+y^3+z^3$and let$X$be the cubic surface in$\mathbb{P}^3$... 1answer 107 views ### Powers of orthogonal matrices is closed This might be a basic question, nonetheless I cannot give a proof. Given an orthogonal matrix$A$with eigendecomposition$A = Q \Lambda Q^{-1}$with only non-real eigenvalues. Given also a diagonal ... 0answers 45 views ### Weak topology and topology by semi-norm [on hold] Wikipédia: -The weak topology on X is the initial topology with respect to X* (let's note it T') -If the field K has an absolute value , then the weak topology σ(X,F) is induced by the family of ... 1answer 39 views ### Distribution of the$\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 $$\mathbf{U}=\begin{pmatrix} 1 & 0 \\ 0 & \mathrm{e}^{\mathrm{j}\beta_1}\end{pmatrix} \begin{pmatrix} ... 4answers 2k views ### Fermat's last theorem over larger fields Fermat's last theorem implies that the number of solutions of x^5 + y^5 = 1 over \mathbb{Q} is finite. Is the number of solutions of x^5 + y^5 = 1 over \mathbb{Q}^{\text{ab}} finite? Here ... 0answers 23 views ### Properties of unit quaternion transformation [on hold] It is commonly known that unit quaternions can be used to represent spatial rotations. The usual interpretation is as follows:$$ \tilde{q} = \cos{\alpha \over 2}+(a\cdot i+b\cdot j+c\cdot ... 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. ... 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 ... 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 ... 0answers 86 views ### Hausdorff topologies on Q Is there any description known of the Hausdorff topologies on$\mathbb{Q}$compatible with the group operations? 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. ... 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 ... 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 ... 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 $$\nabla ... 0answers 68 views ### Strong solution to u_t - \Delta_p u = f For p > 1, consider the equation$$\langle u_t, v \rangle + \int_\Omega |\nabla u|^{p-2}\nabla u \nabla v = \langle f, v \rangleu(0) = u_0u|_{\partial\Omega} =0$$for all v \in ... 2answers 378 views ### Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated? Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated? Or are there some tables or handbooks which list some common calculated results of ... 1answer 73 views ### Are the natural numbers a disjoint union of infinite sets of zero asymptotic density? [on hold] Suppose \mathbb{N}=\bigsqcup_{i\in\mathbb{N}}E_i with \#E_i=\infty for each i. Is it possible that \limsup_{N\to\infty}\frac{1}{N}\#(E_i\cap\{1,\ldots,N\})=0 for all i, which would mean ... 0answers 66 views ### Adelic integral factorization In order to calculate Tamagawa numbers, I need to justify that for a nice (say Schwartz-Bruhat) function, the following identity holds :$$\int_{\mathbf{A}^2} f(x)dx = \int_{SL_2(\mathbf{A})/SL_2(K)} ... 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 ... 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: ... 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 ... 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$j\in \mathbb N$... 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 ... 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$B_{\alpha_1}\subset B_{\alpha_2}$... 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 ... 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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