# Tagged Questions

This tag is used if a reference is needed in a paper or textbook on a specific result.

**4**

votes

**3**answers

90 views

### Does this property of a first-order structure imply categoricity?

Let $\mathfrak{A}$ be a first-order structure over a relational language and let $\kappa$ be an infinite cardinal. Lets say that $\mathfrak{A}$ has the $\kappa$-property if for every structure ...

**25**

votes

**9**answers

2k views

### Are there some other notions of “curvature” which measure how space curves?

I am learning differential geometry and have a few questions on curvature. -- Background:
Gauss invented "Gauss curvature" to measure how surface curves.
Riemann gives an ingenious generalization ...

**16**

votes

**5**answers

795 views

+50

### What arrangement of unit cubes minimizes surface area?

For each of these two questions, one can assume that the arrangements are polycubes (for which a definition can be found in the excerpt-image below).
Question A. How does one arrange $n$ unit cubes ...

**4**

votes

**1**answer

103 views

### In a Simplicial group the Degenerate elements don't intersect the kernel of the face maps.

Let $G_{\bullet}$ be a simplical group. I denote by $D_n \subset G_n$ the subgroup generated by all the degeneracy maps $s_i:G_{n-1} \to G_n$.
I also denote $$M_n = \bigcap_{i>0} \mathrm{Ker} d_i ...

**-2**

votes

**0**answers

88 views

### Mathematics of volleyball [on hold]

I'm working on a mathematical model that should calculate probabilities of various things in the game of volleyball and I thought it might not be a bad idea to see if there is already some research on ...

**0**

votes

**0**answers

34 views

### Repeatedly changing queue behavior

I'm not sure if this question is suited to MO. I will happily delete if not.
Situation
Consider a general queueing system $\mathscr{S}$, whose customer arrival times are independent, and whose ...

**6**

votes

**0**answers

156 views

### Geometric meaning of the black hole horizon

It is widely accepted that the singularity of the Schwarzschild metric at the event horizon is purely an artifact of the coordinates and no physical singularity exists at the horizon. However, as ...

**3**

votes

**1**answer

120 views

### Optimal lower bounds for the sum of digits in base $b$

Let $b \geq 2$ be an integer and let $s_b(n)$ be the sum of the digits of the base-$b$ representation of the nonnegative integer $n$
(e.g., $s_{10}(726)=7+2+6$). From the weak law of large numbers, it ...

**10**

votes

**3**answers

2k views

### Has the Fundamental Theorem of Algebra been proved using just fixed point theory?

Question:
Is there already in the literature a proof of the fundamental theorem of algebra as a consequence of Brouwer's fixed point theorem?
N.B. The original post contained superfluous ...

**2**

votes

**0**answers

135 views

### The topology of the classifying space of U(n)

In $\textit{Atiyah and Bott: The Yang-Mills equations over riemann surfaces}$, there is a sentence about the topology of BU(n) (i.e. the classifying space of U(n)):
Here $K(\mathbb{Z},n)$ means the ...

**3**

votes

**2**answers

268 views

### Aspheric functors and Grothendieck fibrations

Following Grothendieck, let us say that a category is aspheric if its nerve is weakly contractible and a functor $u : \mathcal{A} \to \mathcal{B}$ is aspheric if for every object $b$ in $\mathcal{B}$, ...

**9**

votes

**2**answers

267 views

### Behavior of the spectrum of the Laplacian under pointed smooth convergence

The Laplacian on a compact Riemannian manifold has a discrete spectrum. For example on a circle of perimeter $L$ the $n$-th eigenvalue starting at $0$ is $-\lambda_n = -(2\pi/L)^2 n^2$.
On the other ...

**4**

votes

**1**answer

127 views

### Is the absolute of a compact space the projective limit of the Stone-Čech compactifications of its open dense subsets?

Is the following statement true, and if it is, does someone have a reference?
Let $X$ be a compact (i.e., compact and Hausdorff) topological space. Then the Gleason space (=Iliadis absolute, ...

**1**

vote

**0**answers

92 views

### Exactness of the relative de Rham complex restricted to subschemes

I think that the statement below about relative de Rham complex is true (am I wrong?) If it is the case, a reference would be very helpful. (I admit that the statement sounds somewhat technical and ...

**5**

votes

**0**answers

95 views

### minimal energy of affine Lie algebra reps

Let $\mathfrak g$ be a simple Lie algebra.
Let $\widetilde{L\mathfrak g}$ be the universal central extension of $L\mathfrak g:=\mathfrak g[t,t^{-1}]$. Let $V_\lambda$ be a positive energy ...

**4**

votes

**1**answer

162 views

### Proof of the Dunford-Pettis theorem

I would like to know where to find a complete proof of the Dunford-Pettis theorem:
A sequence $(f_n)_{n\geq 0} \subset L^1$ is uniformly integrable if and only if it is relatively compact for the weak ...

**3**

votes

**0**answers

129 views

### Homotopy type of a locally contractible compact

Does a locally contractible compact space have the homotopy type of a finite CW complex? (I think it probably does, but I need a reference anyway.)
EDIT: My intuition was wrong [to see why, read ...

**0**

votes

**0**answers

93 views

### Any result, example or conjecture about the computational complexity of transcendental number that can not be computed by linear time Turing Machine [on hold]

The Hartmanis-Stearns Conjecture is restated by Prof.Lipton as:
Suppose that a linear time Turing Machine computes the first $n$ digits of the real number $r$ in base ten. Then, the number is either a ...

**0**

votes

**0**answers

72 views

### Are there any relevance between coefficients of simple continued fraction of quadradic algebraic number and algebraic number with degree $2^n$ [on hold]

Let $\sqrt{c}$ be quadratic algebraic number, We know that $[a_0;a_1,a_2,\dots ]$ the coefficients of simple continued fraction of $\sqrt{c}$ the quadratic algebraic number is periodical. ...

**0**

votes

**0**answers

128 views

### Applications of infinite permutations [closed]

I was looking at approximation in the forlmula of Products of necklaces:
$n \to \infty$ we have $\prod_{p=1}^n N(p,a) \approx \frac {a^n} {n!}$ $\prod_{p=1}^n \frac {1-a^p} {1-a}$. (The number of ...

**1**

vote

**0**answers

108 views

### convergence of $e^{it\Delta}f$

I heard of a conjecture that
$e^{it\Delta}f\rightarrow f$ a.e. as $t\rightarrow 0$ for $f\in H^{1/4+\epsilon}$
but couldn't find a proper reference.

**1**

vote

**1**answer

210 views

### Deformations of holomorphic/algebraic vector bundles over $\mathbb{P}^3$

I would like to know what can be said about (global) deformations of holomorphic/algebraic rank two vector bundles on $\mathbb{P}^3$. I am particularly interested in the case of topologically trivial ...

**0**

votes

**0**answers

48 views

### Function Related to Jordan Curves

I am looking for a solution to the following problem:
given
a Jordan curve $c(s) = (x(s),y(s))$ with $\dot x(s)^2+\dot y(s)^2 = 1$ and $c(s+L)=c(s)\,$
an integrable function $g(s): c(s)\mapsto ...

**9**

votes

**4**answers

458 views

### A metric space of geometric shapes

My research involves geometric shapes in $R^2$, and I need a metric with several properties such as:
Families of similar shapes, such as squares, are closed in this metric. Also more general ...

**8**

votes

**2**answers

178 views

### Clifford algebras for quadratic modules over ringed spaces

What is the earliest possible reference for definition and basic properties of Clifford algebras associated to quadratic modules over a ringed space? The ringed space does not need to be locally ...

**6**

votes

**2**answers

310 views

### Twist in identification with singular cohomology

Let $X$ be a smooth projective variety over $\mathbb{Q}$ and $$V = H^m(X(\mathbb{C}), \mathbb{Q} \cdot (2\pi i)^r)$$ Then I've seen people write the comparison with complex cohomology (an isomorphism ...

**6**

votes

**2**answers

293 views

### A continuous notion of realizability

I have been interested in non-classical logics, off and on, for quite a while. This question is probably very basic, and I hope it is not too low-level for MO. My question stems from an attempt to ...

**2**

votes

**0**answers

82 views

### What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer
As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= ...

**0**

votes

**0**answers

203 views

### What does a Turing machine compute? [closed]

I suspect that it might be necessary to define for a Turing machine how its inputs and outputs are to be interpreted in order to be able to say e.g. that a Turing machine $T$ computes an arithmetical ...

**4**

votes

**1**answer

108 views

### Equations and random subgroups in compact groups

EDIT: Here is a more specific question.
Let $G$ be a compact group and let $w$ be a word in $d$ variables. Then the solution set $S$ of the equation of $w=1$ is a closed subset of the product $G^d$ ...

**2**

votes

**3**answers

161 views

### How to define the input of computable function or Turing machine over real numbers

Computation or computability over $\mathbb{N}$ can be extended to computation or computability over $\mathbb{R}$ or even computation or computability over $\mathbb{C}$.The following is a formal ...

**0**

votes

**1**answer

75 views

### Absolute irreducibility of a symmetric square?

This is a question I received today by email, which somebody more experienced with finite group representations can probably answer directly. Take $F:=\mathbb{F}_q$ for some prime power $q$, so ...

**2**

votes

**1**answer

111 views

### Lp estimate for resolvent of Laplace operator

Consider for $1<p<\infty$ operator $A_p:L_p(0,1)\to L_p(0,1), \ D(A_p)=\{u\in W^2_p(0,1): u'(0)=u'(1)=0\}, \ A_pu=u''$, i.e. $L_p$-realisation of the Laplace operator with Neumann bcd on the ...

**26**

votes

**14**answers

4k views

### Explicit computations using the Haar measure

This question is somewhat related to my previous one on Grassmanians. The few times I've encountered the Haar measure in the course of my mathematical education, it's always been used in a very ...

**1**

vote

**1**answer

65 views

### Algorithm for Polynomial Reduction in a Quotient Ring

Any reference or suggestion for the following problem would be greatly appreciated.
I am working on the quotient ring $Q=R[X_1,\dots,X_n]/<f_1,\dots,f_k>$. Given polynomials $p$ and $q$ I want ...

**3**

votes

**0**answers

45 views

### Polynomials connected with Gale's condition and cyclic polytopes

I was reading about cyclic polytopes and (I think naturally) became interested in polynomials $f(X) \in \mathbb{R}[X]$ which have a factorization of the form $f(X)=g(X)g(X+1)$ for some $g(X) \in ...

**0**

votes

**0**answers

59 views

### reference help indecomposable representations of SL(2,R)

Let $\mathfrak{g}$ be the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, $K=SO(2)$ the maximal compact subgroup of $SL_2(\mathbb{R})$. Then the classification of irreducible admissible ...

**2**

votes

**0**answers

313 views

### Help with my research topic [closed]

I have a masters degree in mathematics and I'm currently a PHD student. Since the beginning of my studies (2 years ago) I haven't progressed and still don't have a research topic. I was a very good ...

**3**

votes

**3**answers

102 views

### Basic Questions about Radon Transforms

I am currently working on a problem that may be interpreted as recovering an unknown function from its Radon transform.
Unfortunately I don't have any background in Radon transform, but need to ...

**2**

votes

**0**answers

45 views

### How to prove that all smooth vector bundles on a given vector bundle are the pull back of a vector bundle on the base [migrated]

Recently, during a conversation, I heard about the result (previously mentioned also here on MO), whose statement is reported below. Not having the specific background necessary to reconstruct a proof ...

**5**

votes

**1**answer

277 views

### What is the current state of the crystalline analogue of the Weil conjectures?

In "F-isocrystals on open varieties results and conjectures" Faltings says:
"Finally, we extend the theory of weights and show as much as possible of the crystalline analogue of the Weil ...

**5**

votes

**1**answer

238 views

### Morava $K(n)$'s are not $E_{\infty}$

I am looking for a reference/proof that shows that the Morava $K$-theory spectra, $K(n)$ are not $E_{\infty}$ ring spectra. I suspect that this should be a calculation but I can't quite get it right.
...

**-1**

votes

**0**answers

23 views

### Proof that $G(3)\le 7$ [migrated]

Let $G(k)$ be the minimal $n$ s.t. every sufficiently large integer is the sum of $n$ nonnegative $k$th powers.
Does anybody know where I can find Vaughan's proof that $G(3)\le 7$? I can't find a ...

**1**

vote

**2**answers

139 views

### Plucker embedding and tautological/universal quotient bundle

Let $G$ be a Grassmannian and $Q$ the tautological/universal quotient bundle of $G$. As far as I understand, the associated tautological quotient line bundle for the Plucker embedding of the ...

**2**

votes

**1**answer

47 views

### A possible minimal aperiodic set of corner Wang Tile

From one of my previous question Aperiodic set of corner Wang Tile (although it is put on hold), I realize there is a systematic way to construct aperiodic corner type of Wang tile from edge type ...

**3**

votes

**1**answer

56 views

### Reference for Elliptic PDE on $\mathbb{R}^d$

Could anyone suggest a textbook, article, or lecture notes that covers elliptic PDE theory (existence, uniqueness, regularity) on all of $\mathbb{R}^d$, as opposed to the Dirichlet or Neumann problem ...

**0**

votes

**0**answers

75 views

### Superelliptic Curves [duplicate]

I'm trying to find information on superelliptic curves and how to solve them over the integers. The equation is $$y^k = f(x)$$ where $k=3$ and $f$ has degree $d=3$. Does anyone know any ...

**19**

votes

**0**answers

464 views

### Are there “chain complexes” and “homology groups” taking values in pairs of topological spaces?

Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A ...

**1**

vote

**0**answers

101 views

### Reference on calculation of 2nd cohomology group

Let $G$ be a finitely generated, infinite, countable discrete nonamenable group with zero first Betti number, I.e., $H^1(G, \ell^2(G))=0$, e.g., $G=F_2\times F_2$, the product of free groups of two ...

**3**

votes

**1**answer

253 views

### An identity for elementary symmetric functions

Trying to understand a result in a representation theoretical paper, I realized that it implies the following elementary identity for symmetric functions. My question is whether this identity is true, ...