# All Questions

**-2**

votes

**0**answers

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

**1**answer

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} ...

**27**

votes

**4**answers

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

**0**answers

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

**0**answers

121 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

**1**answer

76 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

**1**answer

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

**0**answers

87 views

### Hausdorff topologies on Q

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

**0**

votes

**0**answers

135 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

**0**answers

148 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

**1**answer

113 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

**0**answers

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

**0**answers

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

**2**answers

381 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

**1**answer

74 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

**0**answers

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

**1**answer

70 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

**0**answers

41 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

**0**answers

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

**0**answers

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

**0**answers

107 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

**0**answers

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

**2**answers

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

**0**answers

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 $f_i$.
...

**0**

votes

**0**answers

17 views

### 4th order statistics of Circularly Symmetric Complex Normal random vector? [on hold]

Assume that ${\bf z} \in C^{n×1}$ is a CSCG random vector denoted with $C (μ,Σ)$ where $μ$ and $Σ$ are mean and contrivance matrix, respectively, and defined as
$μ=E({\bf z})$, $Σ=E({\bf z}{\bf ...

**-2**

votes

**0**answers

29 views

### maximization of products of two trace function [on hold]

consider the following optimization problem:
\begin{array}{l}
\mathop {\max }\limits_{\bf{X}} \,\,\,\operatorname{trace}\left( {{\bf{XA}}} \right)\operatorname{trace}\left( {{\bf{XB}}} \right)\\
...

**4**

votes

**0**answers

168 views

### Etale Slice Theorem

I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful?
This is Luna's Slice theorem from a ...

**0**

votes

**0**answers

23 views

### Is triple point intersection 'generic' in Teichmuller space?

Let $S$ be a hyperbolic surface of finite type and $\alpha,\beta$ be two closed curves. Consider $X$ to be the set of all those points $\chi$ in the Teichmuller space $\mathcal{T}(S)$ of $S$ such that ...

**1**

vote

**0**answers

54 views

### Kontsevich integral for 2-bridge knots

Are there any articles that explain a formula for Kontsevich integral of 2-bridge knots?

**1**

vote

**0**answers

33 views

### Cell(J) vs Cof(J) in $\text{sSet}_{\text{Quillen}}$

consider sSet equipped with its Quillen model structure $\text{sSet}_{\text{Quillen}}$, we know that a trivial cofibration is a retract of a transfinite composition of pushouts of horn inclusions. I ...

**-1**

votes

**0**answers

66 views

### Help me to proof Mobius-Euler equation [on hold]

Can you help me to proof that
$$
\sum_{d | n}^{\, } \left ( \mu \left ( d \right ) \times \varphi \left ( d \right ) \right ) = 0\: for\: \mathbf{n}\geq 2, \mathbf{n}\: is\: even
$$
where ...

**0**

votes

**0**answers

29 views

### Poisson bivector on the product of two manifolds [migrated]

Let $X, Y$ be two manifolds. Let $(U, x_1, \ldots, x_n)$ and $(V, y_1, \ldots, y_m)$ local coordinates of $X, Y$ respectively. A Poisson bivector on $X$ is defined by
\begin{align}
\pi_X = \sum_{i,j} ...

**0**

votes

**0**answers

30 views

### A question related to Bernoulli trial [on hold]

I'm thinking a Bernoulli process $X_1, X_2, X_3, ...$ that stops when $n\left( X=0 \right)+2n\left( X=1 \right)\ge A$, where $n(X=0)$ and $n(X=1)$ are the number of 0 and 1 in the sequence ...

**18**

votes

**4**answers

2k views

### Massive cancellations

Let $A=\{a_1,\ldots,a_k\}$ be a fixed, finite set of reals. Let $S_A(n)$ be the set of all reals that are expressible as the sum of at most $2^n$ terms, where each term is a product of at most $n$ ...

**0**

votes

**0**answers

34 views

### Bound on change of function given bound on Hessian

Suppose I have very some smooth function $F(x)$, and let $x_0 = \text{argmin}_x F(x)$. I would like to bound $F(x) - F(x_0)$ from above, in terms of the gradient $\nabla f(x)$ and the Hessian matrix ...

**0**

votes

**1**answer

93 views

### Normals along a Sphere [on hold]

Let $M \subset \mathbb{R}^d$ be a smooth 2-manifold that is homeomorphic to a sphere or a connected sum or tori. Does there always exists two points $x,y \in M$ such that the normals $\angle(n_x, n_y) ...

**-2**

votes

**0**answers

26 views

### Ezcontour in Matlab [on hold]

I am using ezcontour to plot an ellipse in matlab, but I would like to get only level 1 contour.
How can I specify that I only want level 1 contour? I can't find anything about this in the ...

**0**

votes

**0**answers

61 views

### Is there an improvement for the Schur-Horn inequalities for positive semi-definite matrices?

By the Schur-Horn inequality I am thinking of the statement that for any Hermitian matrix $H$ its diagonal n-tuple $(H_{11},H_{22},..,H_{nn})$ for any choice of basis lies in the convex hull of the ...

**0**

votes

**0**answers

121 views

+50

### Eigenvalue of a linear map over finite field

Let $ F_q $ be a finite field with $ q $ elements.
Let $ g $ be a multiplicative generator of $ F_{q^2}^* $.
It implies that
$ <g^{q+1}> = F_q^* $.
Let $ l $ be a prime greater than $ q^2-1 ...

**2**

votes

**0**answers

56 views

### “Semiclassical approximation” in random matrix theory

I am reading Planar Diagrams by Brezin, Parisi, Itzykson and Zuber. If you specialize the discussion in section 5 there we seem to derive the eigenvalue distribution of $N \times N$ random Hermitian ...

**-4**

votes

**0**answers

52 views

### Let Z be the set of integers, and consider the function f : Z → Z defined f(x) = 2x. Which of the following is correct? [on hold]

a) f is invertible
b) f is injective
c) f is bijective
d) f is surjective
I put b, though apparently that's wrong.
Thanks :)

**1**

vote

**0**answers

61 views

### translation invariance of the Laughlin wave function

This is a translation into math of the following question, posted on PhysicsOverflow.
Let $H:=L^2(\mathbb C)$.
For every $N$, let $\psi_N\in\Lambda^N H\cong (L^2(\mathbb C^N))^{S_N}$ be the function
...

**0**

votes

**0**answers

31 views

### Calculating Swaps and Swaptions [on hold]

Hi all I have a problem when I have to calculate swaps/swaptions.
n=10-period binomial model for the short-rate, ri,j. The lattice parameters are: r0,0=5%, u=1.1, d=0.9 and q=1−q=1/2.
1.Compute the ...

**8**

votes

**3**answers

272 views

### Reference for a strong intermediate value theorem for measures

Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states ...

**-1**

votes

**0**answers

58 views

### On multilinear linear combinations [on hold]

If $K$ is algebraically closed field, then consider $m$ multilinear polynomials $f_i$ for $i=\{1,\dots,m\}$ in $K[x_1,\dots,x_n]$ of degree $d_i$ each with no common root. We know there exists $g_i$ ...

**4**

votes

**1**answer

275 views

### Free action of $\mathbb{Z}(2^{\infty})$ on a compact space

Assume that $X$ is a Hausdorff compact space such that $\forall n\in \mathbb{N}$, we have a free action of $\mathbb{Z/{2^{n}}\mathbb{Z}}$ on $X$. Must $\mathbb{Z}(2^{\infty})$ act freely on ...

**0**

votes

**0**answers

61 views

### Number of graphs with M edges that does not contain K-clique [on hold]

If we consider the space of graphs $G(n,M)$ where $M$ denotes the number of edges. Is there any known way of calculating the number of graphs within this space that does not contain any k-cliques? Can ...

**4**

votes

**1**answer

140 views

### Nuclearity noncommutative torus

I read that the Noncommutative torus (rotation algebra) is nuclear when $\theta\in\mathbb{R}\setminus\mathbb{Q}$. Unfortunately, I haven't found a proof. Could someone give me a reference and/or an ...

**-2**

votes

**2**answers

40 views

### Systems of ODEs that fulfill a matrix relationship at steady state [on hold]

It is well known that for a system of linear ODE $$x'(t) = A(t) \cdot x(t) + b(t)$$
with initial condition $x(t_0) = x_0$, that for a solution at any other time point $t_1$, $x(t_1) = (z_1, \ldots, ...

**6**

votes

**2**answers

133 views

### Exactness of an additive left Kan extension

Let $\phi:R\to S$ be a flat ring homomorphism and consider the induced adjoint pair
$$\phi_!:R-Mod\rightleftarrows S-Mod:\phi^*,$$
where $\phi_!=(S\otimes_R -)$. The right adjoint $\phi^*$ is easily ...