# All Questions

**4**

votes

**2**answers

141 views

### The Hadwiger number of $L(K_n)$

For $n\in\mathbb{N}$ we consider the set $\{1,\ldots,n\}$ and define the line graph $L(K_n)$ of the complete graph $K_n$ as follows:
$V(L(K_n)) = \big\{\{a,b\}: a,b\in \{1,\ldots, n\}, a\neq b ...

**4**

votes

**1**answer

96 views

### Diameter of sum-graph over a meager set

We say that $S\subseteq \mathbb{N}$ is meager if $$\text{lim sup}\frac{S\cap\{1,\ldots, n\}}{n} = 0.$$
Given $S\subseteq \mathbb{N}$, we associate to $S$ the sum-graph $G_S = (\mathbb{N}, E)$ where ...

**138**

votes

**19**answers

19k views

### Geometric Interpretation of Trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandry;
Is there a geometric interpretation of the trace of a matrix?
This question ...

**50**

votes

**11**answers

5k views

### Why aren't representations of monoids studied so much?

It seems to me like every book on representation theory leaps into groups right away, even though the underlying ideas, such as representations, convolution algebras, etc. don't really make explicit ...

**44**

votes

**28**answers

25k views

### Applications of the Chinese remainder theorem

As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...

**5**

votes

**1**answer

154 views

### How many cospectral graphs available for a given number of nodes?

Two graphs are said to be cospectral if they have same eigenvalues wrt adjacency matrix, Normalised or Signless laplacian matrix. How many graphs has cospectral mates for a given number of nodes? We ...

**5**

votes

**1**answer

96 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

335 views

### When is a power of an indeterminate in an ideal with 2 generators?

If I have an ideal ${\frak a} \colon= (f(X,S), g(X,S))$ of height $2$ in ${\Bbb C}[X, S]$, is it easy to know what power of $S$ is contained in $\frak a$? For example, what is the minimal number $m$? ...

**24**

votes

**20**answers

2k views

### Why are so few operations with arity bigger than 2?

In usual algebraic structures, like groups, rings, monoids, etc, or in algebras coming from logics like Boolean algebras, Heyting algebras and the like the operations are usually of arity 0 ...

**16**

votes

**1**answer

657 views

### Goodstein's theorem without transfinite induction

Goodstein's theorem is an example of a theorem that is not provable from first order arithmetic. All proofs of the theorem seem to deploy transfinite induction and I've wondered if one could prove the ...

**15**

votes

**2**answers

1k views

### On mathematical aspects of the most recent Nobel prize in economics winners' work

Can somebody briefly introduce the mathematical aspects, in particular, those related to math finance, of the three economists who were just awarded this year's Nobel Memorial Prize in Economic ...

**9**

votes

**2**answers

406 views

### Model for the (infinity,1)-category of functors preserving certain homotopy limits

This question is a follow up to: Model for the (infinity,1)-category of (homotopy-)limit preserving functors.
Warm-up Question: Given a simplicial model category $M$, what model category models ...

**45**

votes

**2**answers

14k views

### Is the analysis as taught in universities in fact the analysis of definable numbers?

Ten years ago when I studied in the university I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows:
All numbers are divided into two classes: those ...

**22**

votes

**3**answers

525 views

### What sets of self-maps are the continuous self-maps under some topology?

An open question on MSE, http://math.stackexchange.com/questions/427634/a-topology-such-that-the-continuous-functions-are-exactly-the-polynomials , asks whether there is an infinite field and a ...

**34**

votes

**1**answer

2k views

### Optimization problem arising from the study of zeta zeros

Motivation: The following problem arose in [1] while studying the vertical distribution of the zeros of the Riemann zeta-function. At the time, my collaborators and I were unable to solve it and I ...

**5**

votes

**1**answer

145 views

### Left Properness of Simplicial Commutative Algebras

A bit of light googling turns up several sources asserting that the model structure on simplicial commutative algebras over a ring is left proper (for example, 2.9 in Charles Rezk's paper Every ...

**3**

votes

**1**answer

319 views

### Valuations on tensor products

Let $A$ be a commutative ring, $B$ (resp. $C$) be a commutative $A$-algebra endowed with a valuation $v$ (resp. $w$), not necessarily of rank 1. Assume that $v$ and $w$ induce equivalent valuations on ...

**13**

votes

**2**answers

534 views

### Categories where projective objects are flat

In many categories arising in the theory of algebras or modules, projective objects are flat. For example each projective module over a ring with identity is flat. However, there are categories where ...

**4**

votes

**2**answers

315 views

### Cyclically symmetric functions

Where can I learn about the invariant theory associated with actions of cyclic groups (as opposed to symmetric groups)?
E.g., do the functions $x+y+z$, $xy+yz+zx$, and $x^2y+y^2z+z^2x$ generate the ...

**1**

vote

**3**answers

382 views

### Metric properties for $d:X\times X\times…X\rightarrow\mathbb R$ [closed]

Let us define $d:X^n\rightarrow\mathbb R$. How can we define metric properties such as symmetry, triangle inequality equivalent property etc for such a function?

**4**

votes

**7**answers

3k views

### Are real numbers countable in constructive mathematics?

We are talking about ordinary reals in constructive mathematics.
Let represent each real number by infinite converging series:
$$r = [\;(a_0,b_0),(a_1,b_1),...,(a_i,b_i),...\;]$$
$$where\quad a_i ...

**11**

votes

**1**answer

2k views

### Software for symbolic matrix calculus?

Is it possible to get widely available math software (Maple/Matlab/Mathematica, etc) to symbolically differentiate vector and scalar functions of matrices, returning the result in terms of the ...

**4**

votes

**6**answers

2k views

### How do researchers carry out computational experiments in Graph Theory?

There are packages like, Combinatorica in Mathematica, GA, SA, PSO can be comparatively easily done in MATLAB, C++ has boost, Java has JGraphT and so on and on. My question is,
How do Graph ...

**0**

votes

**1**answer

126 views

### Smooth morphism to homogeneous spaces and fibers

Let $f:X \to Y$ be a smooth morphism between projective varieties. Suppose $Y$ is a homogeneous space. Under what additional condition on $f$, can we conclude that every fibers of $f$ are isomorphic?

**3**

votes

**0**answers

49 views

### Kullback Leibler “variance”: does that divergence have a name?

If you consider two probability distributions $p$ and $q$, one way to measure the distance between the two is the Kullback-Leibler divergence:
$$KL(p,q)=\int p \log (p/q) = E_p(\log p/q)$$
and this ...

**3**

votes

**2**answers

119 views

### Determinant of block tridiagonal matrices

Is there a formula to compute the determinant of block tridiagonal matrices, when the determinants of the involved matrices are known? In particular, I am interested in the case
$A = \begin{pmatrix} ...

**-1**

votes

**0**answers

29 views

### Can I calculate another model? [on hold]

Let's assume that I have following linear model:
y = b + c*x
can I just do following to get linear regression model for x?
It will be a good model?
y - b = c*x
x = -b/c + 1/c*y
What if I had more ...

**0**

votes

**0**answers

9 views

### equivalence of Lie group and Lie algebra intertwiner [migrated]

I encountered this problem while working on my research. Let $G$ be a Lie group, and consider an intertwiner of the complex representations (possibly infinite-dimensional)
$$
\pi:G\rightarrow ...

**471**

votes

**193**answers

121k views

### Examples of common false beliefs in mathematics

The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...

**78**

votes

**3**answers

6k views

### Has the Lie group E8 really been detected experimentally?

A few months ago there were several math talks about how the Lie group E8 had been detected in some physics experiment. I recently looked up the original paper where this was announced,
"Quantum ...

**21**

votes

**7**answers

1k views

### What's that shape? Inferring a 3D shape from random shadows

Let $P$ be a bounded, simply connected region of $\mathbb{R}^3$.
$P$ could be a polyhedron, or a smooth shape, or an arbitrary shape;
I'll assume below that $P$ is a (non-degenerate, perhaps ...

**1**

vote

**1**answer

238 views

### optimization of inverse matrix with constraint on matrix elements

everyone! I have this optimization problem with constraint.
$D$ and $T$ are symmetric matrices, where T is known and D is the unknown parameter.
$x$ and $v$ are two known p-dimensional vectors.
The ...

**71**

votes

**14**answers

6k views

### Why do we care about L^p spaces besides p = 1, p = 2, and p = infinity?

I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ...

**0**

votes

**1**answer

282 views

### Supremum in a Markov chain model

A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...

**35**

votes

**12**answers

7k views

### Examples of undergraduate mathematics separation from what mathematicians should know

I'm looking for examples of four kinds of things:
Material that is usually covered in standard undergraduate mathematics courses and/or in first-year graduate work (or tested in qualifying ...

**0**

votes

**1**answer

323 views

### Size of KL-divergence neighbourhoods

I am new here. I was reading another
post
here and this got me wondering what can be said about the size of the following kl divergence neighborhoods.
Consider these two kl-divergence neighbourhood ...

**6**

votes

**1**answer

248 views

### Cellular model structures on continuous functors

The category of enriched functors from finite based CW complexes to based topological spaces
has a projective model structure. The fibrations
are the objectwise Serre fibrations and the weak ...

**6**

votes

**2**answers

530 views

### All mapping space between CW complexes is a CW complex?

Let $\mathrm{Map}(X,Y)$ denote the (unbased) cellular mapping space from $X$ to $Y$.
If $X$ and $Y$ are finite CW complexes, is $\mathrm{Map}(X,Y)$ a CW complex?
Can we know the cell structure of ...

**12**

votes

**5**answers

3k views

### Why do I need densities in order to integrate on a non-orientable manifold?

Integration on an orientable differentiable n-manifold is defined using a partition of unity and a global nowhere vanishing n-form called volume form. If the manifold is not orientable, no such form ...

**16**

votes

**2**answers

1k views

### Prospects for reverse mathematics in Homotopy Type Theory

Reverse mathematics, as I mean here, is the study of which theorems/axioms can be used to prove other theorems/axioms over a weak base theory. Examples include
Subsystems of Second Order Arithmetic ...

**13**

votes

**2**answers

657 views

### $\infty$-categorical interpretation of type theory

One can read at several places that Martin-löf type theory should be the internal language of a locally Cartesian closed infinity category, and that the univalence axiom should distinguished infinity ...

**0**

votes

**0**answers

79 views

### golden number and stability of 3 body problem [closed]

In Cedric Villani talk "Of Particles, Stars, and Eternity" this video around 45':00" the speaker says that the golden number $\phi=\frac{1+\sqrt{5}}{2}$ is the worst approximated by rational numbers ...

**2**

votes

**1**answer

132 views

### Is every weight of an integrable highest weight module in the Tits cone?

Let $\mathfrak{g}$ be a Kac-Moody algebra with Cartan subalgebra $\mathfrak{h}$, Weyl group $W$, and simple roots and coroots $\alpha_i, \check{\alpha_i}, i \in I$, respectively. Let $L$ be an ...

**0**

votes

**1**answer

122 views

### Chinese remainder theorem for cyclic subfactor planar algebras

This post was inspired by an exchange with the indian woman mathematician Ajit Iqbal Singh.
The chinese remainder theorem can be stated as follows:
Let $n_1, \dots, n_r \ge 2$ be positive integers ...

**7**

votes

**1**answer

627 views

### Uniform boundedness of an $L^2[0,1]$-ONB in $C[0,1]$

Assume that we have an orthonormal basis of smooth functions in $L^2[0,1]$. Are there useful practical criteria to determine whether the sup-norm of the basis functions has a uniform bound? I am sure ...

**0**

votes

**0**answers

162 views

### Non-university jobs suited for pure mathematician turned computational neuroscientists, with coding experience [on hold]

I also asked this question on academia stack exchange,
http://academia.stackexchange.com/questions/48057/type-of-non-university-research-jobs-suitable-for-a-mathematician-turned-comput
but asking ...

**9**

votes

**0**answers

372 views

### One-to-one correspondance between zeta zeros and the prime powers? [on hold]

This question is highly speculative, but I would really appreciate some insight into the problem. Previously asked on MSE without answer here.
I have noticed an interesting property related to the ...

**12**

votes

**4**answers

332 views

### Maximum of the Vandermonde determinant / minimum of the logarithmic energy

The problem is to find the asymptotics (as $n\to\infty$) of the maximum (say $M_n$) of the Vandermonde determinant
$$V_n:=\prod_{0\le i<j\le n-1}(a_j-a_i)
$$
over all $a_0,\dots,a_{n-1}$ such ...

**2**

votes

**1**answer

76 views

### Shared maximum eigenvector

Let us consider two arbitrary Hermitian square matrices $\mathbf{A,B}$ with the same dimension. Given $\mathbf{v}$ the eigenvector associated to the maximum eigenvalue of $\mathbf{A}$:
Are there ...

**0**

votes

**0**answers

53 views

### tessellation of an arbitrary shape

Is there any shapes that we can tessellate any plane shapes (with arbitrary shapes) by them? i.e. if I generate a random shape, how can I tessellate by some shapes?