# All Questions

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### Journals dedicated towards work exploring the development of toy systems of axioms and their (subsequent?) theories?

Due to certain aspects of my current thesis (computational biology, not mathematics), I find myself needing to learn about the development of a toy system of axioms and the theories that subsequently ...

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**1**answer

20 views

### What are the properties of a matrix A so that A*A=A?

I am trying to find the general properties needed for a m*m matrix A, so that A*A=A will hold.
(other than the identity matrix, ...

**3**

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**1**answer

36 views

### Subgroups from which all class functions extend to class functions on the ambient group

Until someone suggests better terminology, let me call a subgroup H of a finite group G segregated if every class function on H can be extended to a class function on G. Equivalently, H should have ...

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**1**answer

46 views

### blow-up of $\mathbb{P}^5$ as a projective bundle

I am wondering if the blow-up of $\mathbb{P}^5$ along three disjoints $\mathbb{P}^1$ (say in generic position) can be understood as a projective bundle over some nice (Fano?) variety.
If one ...

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28 views

### Triangle (constrained number, rather than shape) packing?

Are there any interesting results on optimal packings in the plane using a fixed number of triangles (without a fixed size or shape constraint)?
For instance, what's the maximum area packing of the ...

**3**

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28 views

### Are Schubert varieties for Kac-Moody groups cut out by linear equations?

Let $G$ be a reductive group, and let $X$ be a partial flag variety for $G$. Then it is known that for any projective embedding of $X$, that the equations scheme-theoretically cutting out a Schubert ...

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### The proximality of low rank function approximation

The paper "Best $n$-Dimensional approximation to sets of functions" by A. L. Brown in 1964 gave a negative answer to the following question:
Q1: Is there for a given integer $n$ always a best ...

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**1**answer

36 views

### On the moments of Lévy processes

For a Brownian motion $B_t$, the evolution of the moments with $t$ obeys the simple rule:
$$\mathbb{E}[|B_t|^p] = \kappa_p |t|^{p/2},$$
with $\kappa_p<\infty$. The proof only requires to remark ...

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**1**answer

52 views

### Physical and real life interpretation of the concept of regularity used in differential equations?

I guess the title kind of speaks for my questions: I'm curious to know what could be the physical interpretation or real life application of the concept of regularity that arises in PDE: take for ...

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7 views

### How can one use stability analaysis of finite differences methods in linear Schrodinger to the NLS?

Specifically, I've seen a lot of analysis of grid stability for solving Linear Schrodinger with Forward Euler, Backward Euler and Crank-Nicolson. However, most of the usages I've seen for the same ...

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### Quantifiers and predicate logic [on hold]

Please help me know are these answers correct. I have solved this.
Question 1
(i) Every student passes at least one assignment. (ii) Some students think they know more than some lecturers and some ...

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32 views

### Galois field theory [migrated]

$\mathrm{GF}( 29^2)$ is created by adjoining the root of the irreducible quadratic $p= x^2 + 7x +15$ to the field $\mathrm{GF}(29)$ . The cubic polynomial $q = Y^3 + (26x + 26)Y^2 + (8x+22)Y +13x+23$ ...

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136 views

### Is Hardy-Littlewood k-tuple conjecture known to imply Goldbach's conjecture?

The question is in the title: is Hardy-Littlewood k-tuple conjecture known to imply Goldbach's conjecture? I tried to give a heuristics in Upper bound for $r_{0}(n)$ through probabilities that seems ...

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**1**answer

60 views

### extension of Riemannian metric on real affine variety

Given a Riemannian metric $g$ on the real part $X_R$ of a real affine variety $X$,
is there a "natural" way to extend $g$ to be a Riemannian metric on $X$?

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31 views

### Intersections of open sets and $\alpha$-favorable spaces

I would like to ask about some classes of topological spaces whether they have been studied, what are they called (if they have a name) and whether they have some interesting properties. For the ...

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15 views

### Time decay for Hartree equation with Coulomb potential

Are there any time-decay results for the solution of the Hartree equation
\begin{equation}\frac{1}{i}\partial_t\phi-\Delta\phi=-(|x|^{-1}\ast|\phi|^2)\phi,\quad x\in\mathbb{R}^3\end{equation} in ...

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**2**answers

237 views

### Have axioms / axiom schemata of this flavour been proposed or otherwise considered?

With the exception of a few miscellaneous cases, the axioms (and/or schemeta) of ZFC can roughly be divided into two kinds:
Those that guarantee the existence of more complicated sets, given that ...

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48 views

### how to construct 3D curve in highway geometric design

give you some control points ,also give you the initial point and final point ,their curvature ,torsion and coordinate are kown.How to construct a three-dimensional space curve under the constaint of ...

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**1**answer

119 views

### Question about conjugate points

If there exist two geodesics from $p$ to $q$ that are not only different from each other but also infinitesimally close to each other, then it implies that $q$ is conjugate to $p$.
Can anyone give an ...

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**1**answer

32 views

### Concurrency related problems in $n$ independent, parallel $M/M/1$ queues

Model:
Consider $n$ independent, parallel $M/M/1$ queues with identical arrival rates $\lambda$ and service rates $\mu$. For each $M/M/1$ queue, we use the FCFS (First Come First Served) discipline ...

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### Equivalence of Ramanujan's complete series with modular forms

In his notebooks Ramanujan mentions something called a "complete series" which is some power series $\sum_{n = 0}^{\infty}a_{n}q^{n}$ in terms of $q = e^{-y}$ with $y = \pi K'/K$ and $z = 2K/\pi$ such ...

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### Find an analytic function [migrated]

Is it possible to find an analytic expression for a smooth, continuous, single-variable function $y=f(x)$ such that:
1) $f(0) = 0$
2) $f(x_1) = y_1$
3) $f(x) > 0, \forall x>0$
4) ...

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**1**answer

106 views

### What are the solutions for discrete integers b, d to a^b≡c^d (mod p) where p is a large prime number?

Is there a way to efficiently discover or choose the integers b,d for the congruence relationship below where p is a large prime number? Is there a name for this relationship?
$$
a^{b} = c^{d} (mod ...

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47 views

### Do we have $\widetilde{K_0}(\mathbb{Z}[G])=Wh_{0}(G)$ for the general group

We clearly have that $\widetilde{K_0}(\mathbb{Z}[G])\subseteq Wh_{0}(G)$, and if can be seen from a theorem of Swan, we have that for a finite group, this is true. In the commutative case, this should ...

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83 views

### Ramanujan's series for $(1/\pi)$ and modular equation of degree $29$

In his famous paper "Modular Equations and Approximations to $\pi$", Ramanujan gives the following famous series for $1/\pi$: $$\frac{1}{2\pi\sqrt{2}} = \frac{1103}{99^{2}} +
...

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**1**answer

123 views

### If there exists a nontrivial vector field $V$ such that $\nabla_{X}V=0$ for any vector field $X$, the manifold must be flat?

If there exists a nontrivial vector field $V\not=0$ in Riemannian manifold $M$ and an open set $U\subset M$ such that $\nabla_{X}V=0$ in $U$ for any vector field $X$ in $M$, then dose $U$ have to be ...

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58 views

### primitive polynomial on $\mathbb{F}_2[x]$

For some reason I need some primitive polynomial $f$ on $\mathbb{F}_2[x]$ where $\deg f \in [1,10^4]$. (Especially for $\deg f = 10\pm \epsilon, 10^2\pm \epsilon, 10^3\pm \epsilon, 10^4 \pm ...

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119 views

### Find the Range of Function

What is the range of the map $\mathbb{C}^m\to\mathbb{C}^m$,
$$(z_1,\ldots,z_m)\mapsto (b_1,\ldots,b_m),$$
where $b_k=\prod_{j\neq k}(z-z_j)$ for $1\leq k\leq m$ ?

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29 views

### Factor of 2 In the Definition of Metric Contact Structure

In Blair's book and many many literatures, I see definition of a contact metric manifold which involves a relation
\begin{equation}
d\kappa \left( {X,Y} \right) = g\left( {X,\Phi Y} \right)
...

**21**

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**3**answers

727 views

### Are the higher homotopy groups of the Hawaiian earring trivial?

The fundamental group of the Hawaiian earring is very complicated, but since it's "1-dimensional" one might guess that the higher homotopy groups vanish. Do they? Since the Hawaiian earring does not ...

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### Integral straight-line embeddings of planar graphs

Wikipedia says (in the article on Fáry's theorem),
"Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The ...

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**1**answer

229 views

### Congruence for the number of points in the elliptic curve $y^2 = x^3+b \pmod{p}$

Let $E$ be the elliptic curve $y^2=x^3+1$ and $p \equiv 1 \pmod{3}$ a prime. Computing the number of points mod $p$ of $E$ using the naive method gives:
$$ \#E(\mathbb F_p) = 1+ \sum_{x=0}^{p-1} ...

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154 views

### A convergence issue

Disclaimer: This could be a stupid question and could have a very simple answer which I am unable to see.
Edited
Let $\{x_n\}_{n=1}^\infty$ be a sequence of vectors in a Hilbert space ...

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207 views

### Why is the kth cohomology group of the DM-compactification of the moduli space of curves pure of weight k?

I'm trying to understand the paper
Arbarello, Enrico, Cornalba, Maurizio,
Calculating cohomology groups of moduli spaces of curves via algebraic geometry.
Inst. Hautes Études Sci. Publ. Math. No. 88 ...

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61 views

### Profinite Local Ring inside Polynomial Ring

This is a "technical" question that I came across in my research.
Let $A = \textbf{Z}_{p}[\![t_1, \cdots, t_a ]\!]<z_1, \cdots, z_b>$ be the $(p, t_1, \cdots, t_a)$-adic completion of the ...

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94 views

### Proof of existence of recursively inaccessible and Mahlo ordinals

As in title - I'm looking for a proof of the existence of a countable recursively inaccessible or recursively Mahlo ordinals, especially the first one. When looking for it in all the papers I stumbled ...

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### Show that there is a matris only $ A $ such that $ \varphi (t) = e ^ {tA} $. [on hold]

Let $ \varphi(t)$ of a matrix $n \times n$ functions $C^1$. If $\varphi(0)=I$ (identity) and $\varphi(t + s) = \varphi (t) + \varphi (s)$ for all $ t, s \in \Re $, show that there is a matris only $ A ...

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### Solution of ODE where A has not eigenvalue [on hold]

Suppose $\mu$ is not an eigenvalue of A. Show that the equation $x'= Ax + e^{\mu t}b$ has a solution of the form $\varphi(t) = ve^{\mu t}$.

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### Why is it true that write $\bar{G}=\bar{H}\bar{K}$ [on hold]

Let $G=HK$ and $N$ be a normal subgroup of $G$. Also let $\bar{G}=G/N$. Why is it true that write $\bar{G}=\bar{H}\bar{K}$

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### Unfoldings of trajectories on the Veech triangle $V_4$

Let $V_4$ be the isosceles triangle with base angle $\pi/8$. $V_4$ is a Veech triangle, so the dynamics of billiards on it are very well understood.
Above is the unfolding of $V_4$, with edge ...

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48 views

### Hessian Matrix and Kronecker Product

Given the following equation,
$\Delta Y=J\Delta X+\frac{1}{2}H \Delta X \otimes \Delta X$
where $\Delta Y, \Delta X \in \mathbb{R}^{n}$, $J \in \mathbb{R}^{n \times n}$ is the Jacobian and $H \in ...

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66 views

### Probability of differently loaded dice summing to a value

I have a real world problem that boils down to the following:
I'm playing dice. I have $n \approx o(10)$ differently biased die. The probability of the $i^{th}$ die showing $x_i$ is given by ...

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43 views

### How to find the inverse function of a function like f: N x N -> N [on hold]

I need help on how to find the inverse of a function N x N -> N
For example, if anybody could give me a step by step explanation how to find the inverse function of
f(x,y)=3x-2y
I would be very ...

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306 views

### Is it meaningful to work on convergencies, integration, etc. on the Zariski topology? [on hold]

Since I have studied analysis as well as algebra recently, I am familiar to work on integrablities, and such concepts when I look at topologies. Currently, I am studying algebraic geometry, and I want ...

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25 views

### Depth of multigraded modules

Can any one please give me some references on depth of multigraded module over a standard multigraded ring.

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88 views

### Besides F_q, for which rings R is K_i(R) completely known?

With the exception of finite fields and "trivial examples", which rings $R$ are such that Quillen's algebraic $K$ groups $K_i(R)$ are completely known for all $i\geq 0$?
Here, by "trivial examples" ...

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28 views

### Measurability of functions with multiple parameters

For a formalisation of the Giry monad in a theorem prover, I think I require some notion of measurability of “curried” functions. I.e. I have measure spaces $A$, $B$, and $C$ and a function $f: A ...

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**1**answer

96 views

### Which complete lattices arise as images of the Galois connections induced by binary relations?

Any binary relation $R\subseteq X\times Y$ gives rise to a Galois connection between the powersets of $X$ and $Y$ in a well known way (on MO you can see it e. g. in this answer; in fact, such Galois ...

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**1**answer

57 views

### Iwasawa decomposition of the pseudo-orthogonal group

This is a soft-question, but I haven't found an answer anywhere: do the factors of the Iwasawa decomposition of the pseudo-orthogonal group SO(p, q) have a simple form, in the same way that the ...

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**1**answer

31 views

### Any generic way to move a psd matrix to its neighbors?

Given a two positive matrices $A,B$. For simplicity, let's assume that $Tr A=Tr B=1$. Assume that $\|A-B\|_1\leq\varepsilon$, for some small $\varepsilon>0$, where $\|\cdot\|_1$ is the $l_1$-norm, ...