# All Questions

**14**

votes

**5**answers

2k views

### Proof that no differentiable space-filling curve exists

Could someone provide a reference or a sketch of a proof that no differentiable space-filling curve exists?
Or piecewise differentiable?
Must every continuous space-filling curve be nowhere ...

**1**

vote

**0**answers

93 views

### When does an algebraic space that is a torsor over a scheme have to be a scheme?

In Group actions on stacks and applications (Section 4 of part A), M.Romagny gives a definition of $G$-torsor over a scheme $S$ in which the total space need not be a scheme, just an algebraic space. ...

**2**

votes

**1**answer

110 views

### Curvature of a principal bundle and the exterior covariant derivative

I am sorry if this is too elementary; I had posted it on math.stack but no one answered.
Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a ...

**7**

votes

**2**answers

219 views

### Tauberian theorem with better error term

This is a fairly vague question.
Suppose we have a sequence of positive numbers $(c_n)_n$ and we want to find an asymptotic formula for $S(x) = \sum_{n \leq X} c_n$. In favorable circumstances, ...

**0**

votes

**0**answers

27 views

### Is the minimal solution of a Pell equation a positive integral power of the fundamental unit? [migrated]

Let $k=\mathbb{Q}(\sqrt{d})$ -- $d$ is a positive square-free integer --
be a real quadratic field, and let $\varepsilon_k$ be its fundamental unit.
Let $(x,y)$ be the minimal solution to the Pell ...

**9**

votes

**1**answer

284 views

+50

### Example of a ring $R$ such that $\dim(R[[X]])<\dim(R[X])$

Dimension refers to the Krull dimension of a commutative ring.
In the paper "Prime ideals in power series rings" J. Arnold gives an example of such a ring:
Let $k$ be a field and $K=k(t)$ a ...

**20**

votes

**0**answers

262 views

### Identities for power series like $\sum_n z^{n^3}$

Probably, one of the first power series that every mathematician encounter is the geometric series
$$\sum_{n=0}^\infty z^n = \frac1{1-z}, \quad z \in \mathbb{C},\; |z| < 1 .$$
Also, a particular ...

**0**

votes

**0**answers

72 views

### Are fibers at points of morphism of schemes closed subschemes? [on hold]

Is $X \rightarrow Y$ is a morphism of schemes and $y \in Y$, is the fiber of $y$ a closed subscheme of $X$? Is is true that the fibers of the projection $X \times_S Y \rightarrow Y$ (with $X,Y$ ...

**-7**

votes

**0**answers

75 views

### Conjecture about Prime Numbers [on hold]

Conjecture 1.
Being P the product of the multiplication of several different prime numbers, and being Np ˂ P any prime number not being prime factor of P; being NL ˂ │P^(1/2)│ any prime number not ...

**2**

votes

**1**answer

67 views

### Criteria for Compactness of a Closed in $L^2$ Spaces [on hold]

$(X, \mathcal{B}, \mu)$ is a measure space.
Is there any well-known criteria for compactness of a closed set in $L^2(X, \mu)$?
If the answer is negative what about $L^2(\mathbb{R}^n,\mu)$(in this ...

**4**

votes

**1**answer

148 views

### Hausdorff measure of the graph

Is there any example of a real valued function on the real line whose domain has Lebesgue measure zero but the graph (in the plane) has positive one dimensional Hausdorff measure?
Of course if such ...

**-5**

votes

**0**answers

173 views

### Are numbers fundamental mathematical entities? [on hold]

This question came to my mind after seeing Vi Hart's video on YouTube about the "number" Wau and the answer I gave there to the question "what is the number Wau?". As far as I know, numbers have ...

**0**

votes

**0**answers

25 views

### Is there a term for “ranked distance” matrices?

In a n by n "ranked distance matrix" each element has a rank $r_{ij}$ between 1 and n that indicates it is the $r_{ij}$th smallest element in column $i$ of a corresponding Euclidean distance matrix. ...

**5**

votes

**0**answers

136 views

### Is there a higher, “orientalish” version of geometric realisation?

Geometric realisation of simplicial sets can be roughly thought of like this:
In some category $\mathcal{C}$, we choose an object for every abstract $n$-simplex. In topological spaces, we would ...

**5**

votes

**1**answer

171 views

### Representability of morphism of stacks

A morphism of Artin stacks $f:X\to Y$ over $\mathbb Q$ is representable by algebraic spaces if and only if its geometric fibres are algebraic spaces. I would like to know if one can use this to prove ...

**1**

vote

**0**answers

122 views

### Generalization of Little Fermat Theorem for a particular $a$ and perfect shuffles

I'm looking for the smallest $n\in \mathbb{N}$ that solves the following equation:
$$2^n=1 \mod m$$
For an odd $m$. I know that Little Fermat Theorem and Euler Totient give me a solution but they ...

**0**

votes

**0**answers

137 views

### What is the status on questions related to Bhargava's factorial function?

In Manjul Bhargava's The Factorial Function and Generalizations he motivates a new type of factorial $n!_S$ using by generalizing a few theorems like:
For $k, l \in \mathbb{Z}$, we have $k! \times ...

**0**

votes

**0**answers

36 views

### Relation between Chebyshev Interpolation and Expansion

I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials.
Pointwise Lagrange ...

**1**

vote

**1**answer

83 views

### The automorphism groups of smallest grammars of a language string are isomorphic

Let $s \in \Sigma^*$ be a formal language string. Consider the automorphism group of $s$, defined to be the set of all permutations of positions of $s$ that leave $s$ fixed. For instance $G(abab) = ...

**0**

votes

**0**answers

75 views

### Can anyone comment on uniformizing parameters and uniformizing coordinates?

Let $V$ be an algebraic variety ($\dim V = r$) over an algebraically closed field $k$, $U \subseteq V$ an open subset (in Zariski topology), and W a prime divisor of V, that is, the closed subvariety ...

**0**

votes

**0**answers

61 views

### How local is the exponent in the definition of a function with analytic/algebraic singularities?

In Demailly's Analytic Methods in Algebraic Geometry (available on his web page), the definition of a (plurisubharmonic) "function with analytic singularities" is a (plurisubharmonic) function $ u: ...

**0**

votes

**1**answer

111 views

### A $C^{*}$ algebra associated to a group [on hold]

Let $G$ be a compact group which act on a Hilbert space $H$. We define a linear map $T$ on the dual space $H^{*}$ with $$T(\phi)(x)=\int_{G} \phi(g.x)$$ The integration is based on the Haar ...

**4**

votes

**1**answer

171 views

### Average probability that a random cosine polynomial with bernoulli coefficients is small

Let $P_{n}(t)=\sum_{k=0}^{n}\varepsilon_{k}\cos(kt)$ where $\varepsilon_{i}$ are independent random variables taking values in $\left\{-1,1\right\}$ with equal probability. Is is true that for any ...

**8**

votes

**4**answers

848 views

### number theory which is close to analysis

I have basic training in Fourier and Harmonic analysis. And wanting to enter and work in area of number theory(and which is of some interest for current researcher) which is close to analysis.
...

**2**

votes

**0**answers

184 views

### Functor of points of a tensor triangulated category

Is there is a functor of points approach to tensor triangulated categories parallel to Balmer's theory of prime spectra?
Given a tensor triangulated category $\mathcal{T}$ an $R$-point can be ...

**0**

votes

**0**answers

7 views

### dual basis of cohomology algebra [migrated]

Let $H^*(M)$ be the cohomology algebra of oriented manifold $M$ with rational coefficients.
Let $\{b_i\}$ be a basis of $H^*(M)$ as a vector space over $\mathbb{Q}$.
Let the dual basis be
...

**-3**

votes

**0**answers

37 views

### shortest line segment that equally divide a triangle? [on hold]

Given the coordinates of all three vertexes of a triangle, what's the length of the shortest line segment that divide a triangle into two equal-area halves?
This is a problem that a friend gave to ...

**22**

votes

**2**answers

580 views

### Interplay between Loop Quantum Gravity and Mathematics

It is known that there are many interesting connections between String Theory and modern Mathematics, with a rich feedback going on in both directions: there have been advances in mathematics thanks ...

**7**

votes

**0**answers

86 views

### Thales Style Level Sets

Encouraged by Joseph O'Rourke ( and inspired by the discussion at
Thales' semicircle theorem in higher dimensions ), I ask about level sets in three
dimensional space occuring from considering ...

**3**

votes

**3**answers

127 views

### Probable direction of deviations from the expected value in binomial and hypergeometric cases

Suppose I have an urn with N marbles, with frequencies p and q for red and black marbles, and with p > 0,5. I take a sample of r marbles.
It sounds intuitive to say that deviations from the mean ...

**3**

votes

**0**answers

73 views

### Is there any counterpart to Thales' semicircle theorem in higher dimensions?

It was established by TMA, @WillSawin, and @DouglasZare, in their responses to
the MO question,
"Thales' semicircle theorem in higher dimensions,"
that the natural generalization of Thales' semicircle ...

**3**

votes

**1**answer

208 views

### Classifying spaces of topological groups whose underlying spaces are homotopy equivalent

Let $G$, $H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism which happens to be a homotopy equivalence of the underlying topological spaces. Let us assume that $G$, $H$ ...

**5**

votes

**1**answer

64 views

### A decision problem for clones

E. Post proved that there are only countably many clones on a two-element set (classes of operations closed under superposition and containing all projections). All these clones are finitely ...

**0**

votes

**0**answers

32 views

### Relation eccentricity/diameter in undirected tree [on hold]

I know in my mind that it's very obvious, but I just can't seem to prove the following statement:
Let $G$ be an undirected non-trivial tree with at least $3$ vertices. Let $u$ be an arbitrary vertex ...

**2**

votes

**0**answers

100 views

### Intersecting balls with convex regions and a bisector thereof

This question is related to my previous posting
Angle subtended by the shortest segment that bisects the area of a convex polygon
Let $C$ be a convex region in the plane and let $s$ be the shortest ...

**1**

vote

**0**answers

49 views

### freedom in choosing a smooth function of compact support [migrated]

Suppose $\Omega$ represents a bounded domain in $\mathbb{R}^n$. For ease, let $\Omega$ be a ball around the origin. Let $\varphi$ be a smooth radial function defined on $\Omega$. I am trying to ...

**0**

votes

**1**answer

84 views

### Exterior product

I have asked this question in the Mathematics forum but I received no answer.
Let $E$ be an algebraic vector bundle of rank $r$ and degree $d$,
Then $\Lambda^2 E$ is of rank $r'=r(r-1)/2$, but is of ...

**0**

votes

**1**answer

113 views

### homological invariant of the “universal elliptic curve” over the punctured $j$-line

My question considers the curve $E$ over the affine $j$-line $S$ given by
$$Y^2 - (j-1728)XY = X^3 - 36(j-1728)^3X - (j-1728)^5$$
This curve has the property that it's $j$-invariant is $j$ (see ...

**-4**

votes

**0**answers

111 views

### How does a mathematician sees the numbers? [on hold]

When I look at a tree - I see the tree. I feel the tree in my head, it's shape, it's presence - everything about it. When I look at a number or a math expression I see only the expression and the ink ...

**1**

vote

**0**answers

48 views

### Obtaining graphics of functions in non-standard analysis

In the context of $R(\varepsilon)$ or more broad fields, Levi-Civita field or $No(\omega_1)$, how can we obtain the graphics of functions on the infinitesimal range?
For instance, it is alleged that ...

**3**

votes

**1**answer

160 views

### Glueing a property via homotopy colimits

I have a problem concerning a fact which is stated without proof in this Rezk's draft: http://www.math.uiuc.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf .
In the proof of Lemma 2.11, we are given a ...

**-4**

votes

**0**answers

61 views

### Math mobile Apps [on hold]

Are there any interactive apps, which you can use your phone camera and point at things in your environment and the app regognizes the shapes of objects etc?

**10**

votes

**1**answer

173 views

### Relative Picard functor for the Zariski topology

I'm trying to understand better the relative Picard functor, as defined, for example, in Kleiman's article.
Let $X \to S$ be a smooth projective morphism of schemes whose geometric fibres are ...

**2**

votes

**1**answer

147 views

### When is finding an explicit inverse of an isomorphism not possible

My question is about Shapiro's lemma. Consider the isomorphism $\phi: H^n(G, Hom_{ZH}(ZG, A))\cong H^n(H,A)$ of shapiro's lemma. I would like to describe this via cochains.
So the obvious map is ...

**-2**

votes

**0**answers

53 views

### the special value in a L-funtion [on hold]

how to calculate the value of series
The Gamma function satisfying $\Gamma(n) = (n-1)!\quad\forall
n\in\mathbb N$ is via through the Euler integral
$$
\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt\,.
$$
...

**2**

votes

**0**answers

43 views

### About the space of function used in the harmonic extensions for elliptic problems involving the fractional laplacian

recently i am studying the following paper:
A concave–convex elliptic problem involving the fractional Laplacian -
C. Brandle, E. Colorado, A. de Pablo and U. Sánchez.
At the Pgs 41, 42, the ...

**0**

votes

**0**answers

17 views

### How to character the norm of elemental units in a quadratic number field [on hold]

if a prime number P≡1 mod 4,why the equation x^2-py^2=-4 has solutions in integer

**5**

votes

**1**answer

135 views

### Coherent sheaves on $\mathbb C^2$ and commuting matrices

Let $V$ be an $n$-dimensional complex vector space. The stack $Coh^n(\mathbb C^2)$ of coherent sheaves on $\mathbb C^2$ supported on $n$ points (not necessarily distinct) is equivalent to the stack ...

**4**

votes

**2**answers

150 views

### How can dimension depend on the point?

Let $M$ be a metric space.
For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension.
For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim_{r\to0}\dim(B(x,r))$; this limit ...

**3**

votes

**2**answers

78 views

### Polynomial invariants for unoriented links

I have seen that usually one finds polynomial invariants for oriented links (for example the Jones polynomial, the Hompfly polynomial). Does anyone know what polynomial invariants exist for ...