# All Questions

**1**

vote

**2**answers

496 views

### Are hyperreal numbers isomorphic to formal power series?

I wonder whether hyperreal numbers isomorphic with formal Laurent series?
It seems that any hyperreal number can be represented in the form of Laurent series over $\omega$. For instance,
...

**7**

votes

**1**answer

142 views

### A question on the twistor space of a manifold

Let $M$ be either (a) self-dual conformal 4-manifold, or (b) hypercomplex $4n$-manifold.
In either case one can construct the twistor space $Z$ (in the case (b) $Z=\mathbb{C}\mathbb{P}^1\times M$ as a ...

**0**

votes

**1**answer

71 views

### What is a one-parameter Newton's method?

The Newton's method that I know is defined as follows:
$$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$
However, I've recently encountered a paper that talks about a one-parameter family of Newton's method ...

**2**

votes

**0**answers

81 views

### Isomorphic subcategories of directed graphs and presets

For the purposes of this post, a digraph (directed graph) has neither loops nor multiple parallel edges, and a preset is an ordered pair consisting of a set $S$ and a preorder (viz., a reflexive and ...

**2**

votes

**1**answer

114 views

### the number of indecomposable modules of finite groups over finite fields of a fixed dimension

I am interested in determining the the number of indecomposable modules of finite groups over finite fields of a fixed dimension. Specifically, I have the following conjecture:
Conjecture. Suppose we ...

**1**

vote

**0**answers

35 views

### optimal strategies for 2-player zero-sum games of perfect information

I asked essentially this on math.SE slightly more than
3 days ago, and it hasn't received any answer there.
Do finite-state 2-player zero-sum games of perfect information with only win-draw-loss ...

**1**

vote

**0**answers

136 views

+50

### Killing vector fields on sphere and nodal sets

Let $u$ be a smooth function on $S^2$ such that $\int_{S^2} ux_j =0$, for $j=1,2,3$. Does there exist a killing vector field $V$ on $S^2$ such that the nodal line of $\varphi=V(u)$ ($\{x\in S^2: ...

**-3**

votes

**0**answers

162 views

### Why is SO(3) not $S^1 \times S^2$? (Where is the mistake?) [migrated]

I was trying to calculate the fundamental group of SO(3). In order to represent the group I reasoned the following way:
In order to build the 3X3 orthogonal matrix I need an orthonormal positive ...

**-4**

votes

**0**answers

84 views

### paradox about the Axiom of Choice? [on hold]

I want to know the paradox about the axiom of choice and how to illustrate the paradoxes.Thank you!

**2**

votes

**1**answer

75 views

### Counting linearly ordered subsets of maximal length in partially ordered $d$-tuples of nonnegative integers

Given $d\in \mathbb{N}$, let $X_d:= \{(\ell_1, \ldots , \ell_d): 0 \le \ell_1 \le \ldots \le \ell_d \le d\}\subset \mathbb{Z}^d$, and endow $X_d$ with the (usual) partial order, namely, $x\le y$ if ...

**9**

votes

**2**answers

316 views

### Can a graph be reconstructed from its cycle lengths?

All graphs discussed are finite and simple. The cycle sequence of a graph $G$, denoted $C(G)$, is the nondecreasing sequence of the lengths of all of the cycles in $G$, where cycles are distinguished ...

**2**

votes

**1**answer

46 views

### Local rotations to world rotations [on hold]

I am using a digital gyroscope and I am getting very good results with it, only problem is the local angle does not match the world angle (seen by the world).
Red = local X-axle
Green = local ...

**11**

votes

**1**answer

411 views

### Elliptic curves and connected components

Are there elliptic curves of positive rank with two real connected components
in which all the rational points lie only on one component?
Concrete examples are really appreciated.

**3**

votes

**1**answer

115 views

### Do the following two filtrations of the affine Grassmannian agree?

Let $H = L^{2}(S^{1},\mathbb{C}^{n})$, $H_{0}\subseteq H$ the subset of maps that extend holomorphically to the unit disc, and $H_{m} = z^{m}H_{0}$. Consider the affine Grassmannian for $GL_{n}$ in ...

**0**

votes

**0**answers

54 views

### Bruhat order in homogeneous spaces

Let $R$ be a root system with simple roots $\Delta$. For all $\alpha\in\Delta$ let $\omega_\alpha$ be the fundamental weight associated to $\alpha$. Is then the length of $\alpha$ equal to the length ...

**3**

votes

**1**answer

158 views

### adjoint of this closed (?) operator

I am currently dealing with an unbounded operator
$T:\{f \in L^2(-2\pi,2\pi); f \in AC((-2\pi,2\pi)), T(f) \in L^2, \lim_{x \rightarrow \pm 2 \pi} f(x)g(x)=0\} \subset L^2(-2\pi,2\pi)\rightarrow ...

**1**

vote

**1**answer

55 views

### Conversion between condtional expection conditioned on $\sigma$-algebra and on r.v

Let $(\Omega, \mathcal F, P)$ be a probability space, and let $\mathcal G \subseteq \mathcal F$ be a sub-$\sigma$-algebra of $\mathcal F$ and $X : \Omega \to \mathbb R$ a random variable. Then the ...

**-4**

votes

**0**answers

50 views

### What are “small” finite groups with “exponentially” large expansion? [on hold]

Given an integer $k$ suppose one wants the group to be polynomial in size in $k$ but the expansion to be exponential in size in $k$. I am not sure I am asking a precise question. Kindly add in ...

**0**

votes

**0**answers

24 views

### Integral over conditioning variable of a Gaussian

The marginal of a multivariate Gaussian can be computed in closed form, i.e.,
$p(x) = \int_y \mathcal{N}((x,y);\mu,\Sigma)\ dy$
is simple. But what I need is
$L(x) = \int_y \mathcal{N}((x\mid y); ...

**4**

votes

**0**answers

58 views

### Multiplicity of $Ext^{d-t}(M,\omega_R)$, ($d=\dim R, t=\dim M$)

Let $R=\bigoplus_{i \geq 0} R_i$ be a Cohen-Macaulay graded ring ($R_0$ is a field and $R$ is generated by $R_1$) of dimension $d$ with canonical module $\omega_R$, and $M$ a graded Cohen-Macaulay ...

**0**

votes

**0**answers

67 views

### What does the equation $\tau \tau^* = \sigma^* \sigma$ represent in the ADHM construction of vector bundles?

I'm looking at the explicit construction of vector bundles with Anti-Self-Dual (ASD) connections on them via the ADHM construction of instantons. At the heart of this is the complex
$$
V ...

**6**

votes

**0**answers

109 views

### Connection between quasifibrations and homotopy cartesian squares

Let me first fix the definitions.
A map $p\colon E\rightarrow B$ is called a quasi-fibration, iff the canonical inclusion $p^{-1}(b)\rightarrow hofib_b(p)$ is a weak equivalence for all for all $b\in ...

**0**

votes

**1**answer

128 views

### Book on Convergence Concepts in Probability without Measure Theory [on hold]

I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...

**1**

vote

**1**answer

75 views

### On whether a formula of KP is $\Pi_3$

In the context of KP, is the formula $\forall w(w\in x \leftrightarrow\forall y\exists z F(w,y,z))$ $\Pi_3$ when $F(w,y,x)$ is $\Delta_0$?

**2**

votes

**1**answer

74 views

### If $u_n \to u$ in $H^{\frac 12}(\partial\Omega)$, does $f(u_n) \to f(u)$ in $H^{\frac 12}(\partial\Omega)$ for $f$ Lipschitz?

Let $f:\mathbb{R} \to \mathbb{R}$ be a smooth function with $|f'(x)| \leq C$ for all $x$ and $f(0)=0$.
Suppose $u_n \to u$ in $H^{\frac 12}(\partial\Omega)$, where $\Omega$ is a bounded domain of ...

**0**

votes

**1**answer

52 views

### Are Carnot groups (as Carnot Caratheodory metric spaces) doubling?

I need to use the Lebesgue differentiation theorem for doubling metric measure spaces and was wondering if Carnot groups are doubling. If yes, is there any reference you can point me to? Thank you.

**9**

votes

**2**answers

594 views

### nonstandard models and mathematical theorems

Is there a first order statement about the natural numbers (not nonstandard analysis) such that the truth of the statement is easier to see in a nonstandard model? In other words, do nonstandard ...

**-1**

votes

**0**answers

154 views

### Question about Lusternik-Schnirelmann Category?

I have this sets: $\Omega\subset \mathbb{R}^N, N\geq 3$ a smooth bounded domain
$\Omega_{r}^+=\{x\in \mathbb{R}^N, d(x,\Omega)\leq r\}$ and ${\Omega}^-_{r}=\{x\in \Omega, d(x,\partial\Omega)\geq ...

**2**

votes

**2**answers

136 views

### Given functors $F$ and $G$, does $\mathrm{Res}_F \cong \mathrm{Res}_G$ imply $F \cong G$?

Assume $F, G : \mathbf C \to \mathbf D$ be functors. Denote by $\widehat{\mathbf C} = \mathrm{Fun}(\mathbf{C}^{\mathrm{op}}, \mathbf{Set})$ the category of presheaves of sets on $\mathbf C$. Then, $F$ ...

**4**

votes

**0**answers

50 views

### On Minkowski sum of two independent Poisson point processes

Suppose that $\Phi_1$ and $\Phi_2$ represents two independent Poisson point processes respectively with intensity $\lambda_1$ and $\lambda_2$ (therefore). We know very well different operations on ...

**1**

vote

**0**answers

40 views

### Nuclearity properties of Integral operators with Schwartz type kernels

Consider an integral operator $$T:L^2({\mathbb R}^n)\to L^2({\mathbb R}^n),\qquad (Tf)(x)=\int_{\mathbb R^n} dy\,K(x,y)f(y),$$ with Schwartz type kernel $K\in{\mathscr S}({\mathbb R}^{2n})$ (smooth ...

**1**

vote

**0**answers

29 views

### Recent Survey on Dynamics of Linear Operator

I'm studying Linear Dynamics using the textbook Linear Chaos by grosse erdmann. I'm looking for a recent encyclopaedic article/survey which gives me a big picture of the area.
It seems erdmann and ...

**0**

votes

**4**answers

162 views

### Approximating an arbitrary $\sigma$-algebra by simpler $\sigma$-algebras

A $\sigma$-algebra $\mathcal F$ over $\Omega$ is generated by an countable partition if there exits a countable partition $\mathcal B = \{ B_i \}$ of $\Omega$ such that $\mathcal F = \sigma(\mathcal ...

**-5**

votes

**0**answers

129 views

### perfect numbers and their properties [on hold]

Yesterday I asked a question about perfect numbers. After thinking about the answer and comments I received, I now conjecture that the cube of any perfect number can be written in the form of the sum ...

**1**

vote

**0**answers

67 views

### Uniqueness of the solution of a nonlinear PDE [on hold]

Given the nonlinear PDE
$$
\partial^2\phi+\phi^3=0
$$
I consider the characteristics $\xi=\kappa\cdot x$. Then I look for a solution in the form
$$
\phi(x)=a\cdot\chi(\xi).
$$
Then, provided ...

**1**

vote

**0**answers

37 views

### Question on the local existence theory for the classical solution for the incompressible fluid dynamics equation

For the incompressible fluid dynamics system
\begin{equation}
\begin{split}
&\nabla\cdot v=0,\\
&\dfrac{\partial v}{\partial t}+v\cdot \nabla v+\nabla p=A(S,D)v,
\end{split}
\end{equation}
...

**8**

votes

**1**answer

192 views

### When does the continuous Galois(=etale) cohomology of fields coincide with the naive one? Often true by the Bloch-Kato conjecture?

For a field $F$ I am interested in its $l$-adic (Galois=\'etale) cohomology; here $l$ is a prime distinct from the characteristic of $F$ (for simplicity one may assume that the latter is $0$).
For ...

**0**

votes

**0**answers

44 views

### Linear codes in MAGMA [on hold]

How can I compute the socle of a linear code in MAGMA? Indeed I need MAGMA to view my code as a $G$-Module over $GF(p)$ not just a subspace.

**-3**

votes

**0**answers

71 views

### IA automorphisms of group of order pq [on hold]

Let $G$ be a group with order $pq$ then is it necessary that the order of corresponding group of IA automorphisms is also $pq$?

**5**

votes

**1**answer

94 views

### integral p-adic Hodge theory and de Rham representations

$p$-adic Hodge theory gives us some comparison theorems between several cohomology theories.
It also provides a hierarchy in the category of $p$-adic representations of the absolute Galois group of a ...

**1**

vote

**1**answer

73 views

### Semi-riemannian hypersurfaces

Let us consider the pseudo-euclidean space $\mathbb{R}^3_1$; that is, the space $\mathbb{R}^3$ endowed with the metric
$$\langle x, y\rangle = x_1y_1+x_2y_2-x_3y_3.$$
I see in O'Neill's book that ...

**0**

votes

**0**answers

26 views

### finding the formula to a given table of values [on hold]

I created a spreadsheet that i filled with values i got from a game.
The values may be rounded, but they were calculated, so there has to be a formula behind.
...

**16**

votes

**0**answers

421 views

### What are the products $\prod_{A\subset{\mathbb F}_p\colon |A|=n} \sum_{a\in A} \zeta^a$ equal to?

This is a somewhat more explicit version of a question I have recently asked.
Let $p$ be an odd prime, and write $\zeta:=\exp(2\pi i/p)$ (any other primitive $p$th root of unity will do as well). For ...

**-2**

votes

**0**answers

86 views

### Exercise from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris [on hold]

Let π:C′→C an unramified double cover of a complex Riemann surface C of genus g. With the symbol Nmπ we mean the norm application that takes a meromorphic function on C′ named h and produce a ...

**-2**

votes

**0**answers

67 views

### involution of a riemann surface [on hold]

I've have found in the book GEOMETRY OF ALGEBRAIC CURVES by Arbarello, Cornalba, Griffiths that if π:C′→C it's a double unramified cover of a complex riemann surface named C that we can define the ...

**8**

votes

**1**answer

361 views

### Open access journals in number theory

I'm a phd student in number theory, and I'm required (by the funding council) to publish any article I write in open access journals. The problem is that all journals I can find are either out of my ...

**5**

votes

**2**answers

101 views

### Image of poset with Hausdorff interval topology

Given a poset $(P,\leq)$ the interval topology $\tau_{\text{int}}(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = ...

**0**

votes

**0**answers

46 views

### Linking theorem, elliptic pde

I am trying to solve some linear system of the form
$$ \Delta \phi +p v(r)^{p-1} \psi = -f, \qquad \Delta \psi + qu(r)^{q-1} \phi =g $$ in $ \mathbb{R}^N$ where $ f,g$ are given and $ \phi,\psi$ are ...

**3**

votes

**1**answer

46 views

### Product of posets with Hausdorff interval topology

Given a poset $(P,\leq)$ the interval topology on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and ...

**5**

votes

**1**answer

222 views

### Infinite graphs isomorphic to their line graph

The only finite connected graphs $G$ that are isomorphic to their line graph $L(G)$ are the cycle graphs $C_n$ (see this link for example).
There are connected countable graphs that are isomorphic to ...