# All Questions

**0**

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4 views

### Advanced use of commutation matrices

I am aware of matrix operators vec and kronecker product, commutation matrices and various related identities like stated in ...

**0**

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**1**answer

8 views

### Maximal chromatic number with a fixed number of edges

Given a graph $G$ with $m$ edges, what is the maximum chromatic number $\chi(G)$ that the graph can have?
My guess is that $\chi(G) \leq r(m)$ where $r(m) := \max\{k\in \mathbb{N}:
\frac{k(k-1)}{2} ...

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14 views

### Was “arithmetical translation” (coding in the Goedel sense) ever a part of Hilbert's Program?

Was "arithmetical translation" (that is, coding in the Goedel sense) ever a part of Hilbert's Program? I ask this question for several reasons:
i) it gives the numerals |, ||, |||,.... an ersatz ...

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4 views

### conformal deformation with fixed boundaries

For a flat plane with certain boundary, e.g., a rectangular patch, is it possible to conformally displace or deform such patch to a curved bump with exact same boundary? In this thesis, Dr. Keenan ...

**2**

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27 views

### Definition of a normed ring

A normed ring "should" be a monoid object in the monoidal category of normed abelian groups. There are (at least) two choices of morphisms of normed groups, namely bounded or short homomorphisms, ...

**-6**

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10 views

### Why End to End Testing is Required and How to Perform It?

Nobody wishes to be known for their negligence and mistakes, and there same is the case with the Testers. When Testers are assigned any project to test, then from that moment, they take the burden and ...

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21 views

### Shape-related vector fields

Assume that $M$ is a surface in $\mathbb{R}^{3}$. We denote its shape operator by $S$. A vector field $X$ is shape related to $Y$ if $S(X)=Y$.
(of course it is not an equivalent relation).
...

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8 views

### Convex Optimization in an Ellipsoid

Suppose we want to minimize a linear objective inside an ellipsoid that is,
$\min _x l^Tx$
such that $(x - \mu)^TA(x - \mu) \leq \beta ^2$.
Here, A is PSD and $\mu$ is a fixed vector. Can this be ...

**-1**

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36 views

### Alternative Geometries

In our world, the distance between two points (in 2d) is defined as $\sqrt{(\Delta x)^2 + (\Delta y)^2}$. Suppose that in an alternative geometry, it was defined as $\sqrt[p]{|\Delta x|^p + |\Delta ...

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35 views

### Algebraic independence criterion

Is there any criterion for checking algebraic independence of a set of polynomials in $n$ variables in terms of the leading monomials with respect to some monomial order ? The Jacobian criterion is ...

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33 views

### Tight binomial left tail bound

Let $X \sim \text{Bin}(n,p)$. Wikipedia claims
$$\mathbf P[X \leq (p-\epsilon)n ] \leq e^{ - 2 \epsilon^2 n}.$$
This follows from Hoeffding's inequality ...

**1**

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50 views

### Is the positive existential theory undecidable?

Could you tell if the positive existential theory of $\mathbb{C}[e^{\mu x} \mid \mu \in \mathbb{C}]$ is undecidable in the language $\{+, \cdot , \frac{d}{dx} , 0, 1, e^x\}$ ?
How can we prove the ...

**4**

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**0**answers

96 views

### If $k$ is an algebraically closed field of any characteristic, then the fundamental group of $A$ is abelian

This is a followup to my earlier question, see here.
I am curious as to whether or not we can, in a similar way, show that if $k$ is an algebraically closed field (of any characteristic) then the ...

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29 views

### Simplicial approximation diagram

Let $K$ and $L$ be simplicial complexes as given, and let $\phi:|K|\to|L|$ be the continuous map, where $A=\phi(a)$, $B=\phi(b)$, and so on.
Check whether the map $\phi$ has a simplicial ...

**5**

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106 views

### all subsets borel

Assume Martin's axiom plus $\neg CH$. It is well known, via almost disjoint forcing, that every set of reals of size less than continuum is an example of a metric space whose subsets are all ...

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17 views

### F-points of product of closed subgroups vs. product of F-points, F a local field, reference?

Let $F$ be a finite extension of $\mathbb Q_p$, where p is an odd prime. Let $G$ be a connected reductive group defined over $F$. Let $M, H$ be closed $F$-subgroups of $G$ (in particular, I'm ...

**5**

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**2**answers

107 views

### What characterizations of relative information are known?

Given two probability distributions $p,q$ on a finite set $X$, the quantity variously known as relative information, relative entropy, information gain or Kullback–Leibler divergence is defined ...

**5**

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71 views

### Improving Baumgartner's result?

Q1: Is it consistent with the failure of CH to have an $\aleph_1$-dense subset $A \subseteq \mathbb{R}$ such that for every $X \subseteq \mathbb{R}$ of size $\aleph_1$, there is a $C^{\infty}$ map $F: ...

**3**

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**2**answers

176 views

### Can all the sporadic groups be expressed as permutation groups based on a single big cycle?

Working on M11, I came up with that it can be generated using the following permutations:
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
[[2, 0, 1, 7], [3, 4, 5, 6]]
[[4, 0, 6, 7], [2, 3, 1, 5]]
[[0, 7], [4, 6], ...

**9**

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**2**answers

128 views

### Factorization of a matrix as a product of a symmetric and a skew-symmetric matrix

When can an $n\times n$ matrix $M$ be written as a product $M=AB$, where $A^T=A$ and $B^T=-B$?
For example, a necessary condition is that the trace of $M$ vanishes. In this case, it is easy ...

**-1**

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108 views

### The difference between Hilbert Scheme and Chow Scheme

I am confused by Hilbert Scheme and Chow Scheme. Whenever you have a point in hilbert scheme, take its fiber in the universal family and take its fumdamental class, we get a point in Chow Scheme; and ...

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**1**answer

66 views

### Time Hierarchy Theorem and P vs NP

One obvious strategy for proving P not equal to NP would be to show that there is some problem in NP which is hard for a time class strictly containing P (the origin of this question is the recent ...

**5**

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**1**answer

165 views

### Grothendieck's paper on principal bundles, reduction to a torus step

In Grothendieck's paper "Sur la Classification des Fibres Holomorphes sur la Sphere de Riemann", there is a step I don't understand in section 4, where he proves reduction to a torus. He states (lemma ...

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43 views

### When is $\left[\begin{smallmatrix} D_1 & B \\\\ -B^T & D_2 \end{smallmatrix} \right]$ $\mathbb{R}$-diagonalizable?

Is there some block-wise characterization of $\mathbb{R}$-diagonalizability (by similarities) of
$$\begin{bmatrix} D_1 & B \\\\ -B^T & D_2 \end{bmatrix},$$
where $D_1$ and $D_2$ are real ...

**4**

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43 views

### Equalizing Geometric means of Graph Cycles

Consider a strongly connected directed graph $G$. I have been stuck on the following question: can you assign real numbers in $[0,1]$ to each edge of $G$ so that the geometric mean of all cycles are ...

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17 views

### Fractal Dimension Estimate from Hurst Paramater using Higuchi's Algorithm [on hold]

I am using R Stats package to calculate the Hurst parameter of my Center of Pressure Data set. I am using Higuchi's Algorithm to calculate it. I went and read his paper and saw that "index H is ...

**4**

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97 views

### What is known about the large cardinal strength of Shelah's categoricity conjecture?

Shelah's categoricity conjecture states that for every Abstract Elementary Class $\mathcal{K}$ there is a cardinal $\mu$ depending only on $\operatorname{LS}(K)$ (i.e. the Löwenheim–Skolem number of ...

**-1**

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64 views

### A statistical conundrum [on hold]

I have a statistical problem in a domain that I can not talk about due to a NDA. However, I have worked out how to describe an exactly analogous problem as follows.
You are in charge of record ...

**1**

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64 views

### Does $\mathfrak{P}(A)\cong\mathfrak{P}(B)$ imply $A\cong B$? [duplicate]

Does $\mathfrak{P}(A)\cong\mathfrak{P}(B)\implies A\cong B$ hold for arbitrary sets $A, B$?
Notation:
We write $\mathfrak{P}(M)$ to denote the power set of a set $M$.
$M_1\cong M_2$ is just an ...

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9 views

### Proofing Analytic continuation and stationary increments of an exponential Family

In U.Küchler "Exponential Families of Stochastic Processes" 1997 Theorem 4.2.1 we have the following setup.
Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq0})$ be a filtered measurable space. Let ...

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**0**answers

15 views

### regularization and conversion of sgn(x) to difference of convex [on hold]

sgn(x) or sign function has discontinuity in 0 which make it nonconvex function.
however i have tried to represet sgn(x) as a limit for a sequence of converging function which are smooth and ...

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69 views

### Prove that k ≤ log2N [on hold]

I have the following problem and I don't know where to star:
Let n ≥ 2 and let n = p1p2...pk be its prime factorization, where the primes are not necessarily distinct. Prove that k ≤ log2N (hint: ...

**2**

votes

**1**answer

45 views

### Complexity of Deciding Feasibility of a system of linear inequalities over restricted variables

I am working out an interesting problem and would like some help with this particular sub problem:
Suppose we have a matrix $ M =\left\lbrace a_{ij}\right\rbrace $ of size $n\times m$ where $ ...

**1**

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**0**answers

18 views

### Relation between Aitchison Distance on a Simplex and Geodesic distance on the multinomial manifold

I am trying to understand the difference/relation between the Aitchison distance on a simplex
$$\left[ \sum^D_{k=1} (\log{\frac{x_{ik}}{g(\mathbf{x}_i)}} - \log{\frac{x_{jk}}{g(\mathbf{x}_j)}})^2 ...

**4**

votes

**1**answer

44 views

### Proving moduli of uniform continuity in RCA_0

Simpson's Subsystems of Second Order Arithmetic (pp. 134ff.) uses RCA$_0$ to prove various theorems of analysis for all continuous functions with a suitable modulus of uniform continuity. And he ...

**4**

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**0**answers

51 views

### Strongly real elements of odd order in sporadic finite simple groups

Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution.
Question: Is it true ...

**-2**

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**0**answers

123 views

### Linear forms that avoid numbers with lot of factors

Is following true?
For every given $c>0$ there is an $n_c>0$ such that for every $n>n_c$ there are integers $n<a,b<2n$ such that there are two positive integers $\frac{n}{2(\log ...

**3**

votes

**1**answer

59 views

### Horn's spectrum problem with random Hermitian matrices

An important problem in matrix analysis, completely solved in the early 2000's by A. Knutson & T. Tao (The honeycomb model of GLn(C) tensor products. I. Proof of the
saturation conjecture. J. ...

**0**

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**0**answers

167 views

### relation between algebraic geometry and complex geometry

As a complex manifold $\mathbb{P}^n$ is locally the euclidean space $\mathbb{C}^n$, as a projective variety it is locally $\mathbb{C}^n$ with the Zariski topology, as a scheme it is locally ...

**5**

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36 views

### Compute the index of the Dirac operator on $C_0(R^2)$ to obtain Bott element in $K_0$

I am studying the paper of Baum-Connes-Higson to understand the Connes-Kasparov conjecture. In example 4.23, they discuss the case $G=\mathbb{R}^2$. I have constructed the Dirac operator, but I’m ...

**5**

votes

**1**answer

238 views

### P.G.Goerss, J.F.Jardine, “Simplicial Homotopy Theory” prerequisites

I know that such questions may be better suited for math.stackexchange, but I believe that that the topic of simplicial homotopy theory is advanced enough for mathoverflow.
Besides, I know that there ...

**1**

vote

**1**answer

106 views

### Jensen formula in $\mathbb{C}^n$?

Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function with zero set $X\subset \mathbb{C}$. Jensen's formula reads
$$
\log(|f(0)|)+\int_0^R\frac{|X\cap B_t(0)|}{t}dt = ...

**1**

vote

**0**answers

72 views

### Equivariant form of Nagata's compactification theorem?

Given a finite group $G$ acting on an algebraic variety $X$ (let's say over $\mathbb C$, if that helps), is there always a proper variety $\bar X$ with a $G$ action such that $X \to \bar X$ is a ...

**6**

votes

**2**answers

95 views

### Simple question: different definitions of Bousfield localization

I am not an expert on model categories and I am getting lost with two different definitions I have found on Bousfield localizations. I don't see the link between them.
First definition: Let ...

**0**

votes

**2**answers

38 views

### Prove or disprove a monotonicity property of a certain convex optimization problem

Let $R = (r_{ij})$ be an $n\times k$ real matrix with only positive entries,
and consider the convex optimization problem
$\max f(x) = \sum_{i=1}^n \log \sum_{j=1}^k r_{ij} x_j$
such that ...

**1**

vote

**0**answers

43 views

### Has every Lusin vector space a stronger Polish vector space topology?

Let $X$ be a topological vector space or even a locally convex space such that its (vector space) topology is Lusin, i.e. there is some stronger Polish topology. Does there also exist a stronger ...

**0**

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57 views

### Why are these two gamma functions equal? [on hold]

$\gamma_{1}(s)=\pi^{\frac{1}{2}-s}\frac{\Gamma(\frac{s}{2})}{\Gamma(\frac{1}{2}(1-s))}$
$\gamma_{2}(s)=(\frac{b}{2\pi})^{\frac{1}{2}-a} e^{-2i\theta(b)}$
,where $s=a+bi$
I calculated ...

**13**

votes

**1**answer

279 views

### Okounkov-Vershik approach to representation theory of $S_n$

This is a rather soft question. I was wondering if someone could explain on a fundamental and intuitive level, what the Okounkov-Vershik approach to representation theory of $S_n$ is all about. It's ...

**0**

votes

**0**answers

28 views

### Implicit feature space of Power Kernel

For the polynomial kernel, $K(x,y) = (x^Ty+c)^d$, the implicit feature space $\phi$ for which $K(x,y) = \phi(x)^T \phi(y)$ is of finite dimension and well known [1][2].
It is also well known that the ...

**8**

votes

**2**answers

518 views

### A new result on the Diophantine equation $x^3 + y^3 +z^3 = 3$ [on hold]

The above Diophantine equation is unknown to have any further integer solutions other than $(x, y, z) = (1, 1, 1)$ and $(4, 4, -5)$.
I am a prospective undergraduate mathematics student in Zimbabwe ...