# All Questions

**-2**

votes

**0**answers

12 views

### Induction Proof: 3^n > n^4

$3^n > n^4$ when $n>8$ and $n \in \mathbb{N}$
Attempt Solution:
Base Case: $n=8, 3^n > n^4$, which is $ 3^8=6561 > 8^4=4096$
Induction hypothesis: $3^n \geq n^4, n \in \mathbb{N}, n ...

**-2**

votes

**0**answers

8 views

### Discrete Math proof problem, unsure where to start

Let { m1, m2, ....., mk } be pairwise relatively prime positive integers. Prove that there cannot be more than one solution to the system of congruence's $$ \langle x ≡ ai (mod mi) \rangle $$ in ...

**0**

votes

**0**answers

21 views

### connectedness of semi algebraic sets

We know the inequalities $x_ix_j >\theta_{ij}$ or $x_ix_j<\theta_{ij}$ for some $\theta_{ij}$>0, some $i,j\in\{1,\cdots,n\}$, $i\neq j$ defines the easiest semi algebraic set in $R^n_{\geq 0}$, ...

**13**

votes

**1**answer

267 views

### Who proved that $l^1$ and $L^1[0,1]$ are not isomorphic?

$l^1$ has the Schur property (every weakly convergent sequence is norm convergent) and $L^1[0,1]$ does not, so the two spaces cannot be isomorphic.
Is this folklore, or is it credited to someone? ...

**0**

votes

**0**answers

19 views

### On Schrijver Lower bound

Shrijver lower bound gives number of perfect matchings on a $k$-regular bipartite graphs as $\Big(\frac{(k-1)^{k-1}}{k^{k-2}}\Big)^n$. What is the corresponding lower bound for min-degree $k$ and ...

**1**

vote

**1**answer

59 views

### Does $\omega_C\simeq N_{C/S}$ always happen on Enriques surfaces?

Let $S$ be an Enriques surface and $C\subset S$ a smooth irreducible curve of genus $g$.
Consider the condition $$\omega_C\simeq N_{C/S}$$
For example, when $g=1$ then $\omega_C=\mathcal{O}_C$ and ...

**0**

votes

**0**answers

27 views

### Moving a result from the unconditional to the conditional

I'm generally wary when lifting a result stated unconditionally to a situation where I'm conditioning on a random variable. Consider the following classical result in weak convergence:
Theorem. Let ...

**2**

votes

**0**answers

63 views

### Why do we care about simplicity of the spectrum in Oseledets' theorem?

Oseledets' theorem is a fundamental result in Ergodic theory (see for example here, or Chapter 4 of Lectures on Lyapunov Exponents by Marcelo Viana).
The simplicity of the spectrum has been studied ...

**-3**

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**0**answers

54 views

### Rational power Napier number [on hold]

Help me with the following question.
Prove that $2^e$ irrational, where $e$ is the Napier number.

**4**

votes

**1**answer

372 views

### Langlands program vs Shimura-Taniyama-Weil conjecture

Edward Frenkl said that "we can see Langlands program as a generalization of Shimura-Taniyama-Weil conjecture in the case of elliptic curves "
I hope I'm not distorting his phrase, can someone ...

**1**

vote

**0**answers

25 views

### Algebraic $K_1$ group for a $C^*$-algebra

Let $A$ be a $C^*$-algebra: then one defines topological $K_1$ group as $GL_{\infty}(A^+)/\Big(GL_{\infty}(A^+)\Big)_0$ where $A^+$ denotes $A$ with the unit adjointed (even if $A$ already had a unit: ...

**-6**

votes

**0**answers

115 views

### Great Mathematicians Without a PhD [on hold]

While listing to some music, I was wondering which great mathematicians did not have or do not have a PhD. This is a very subjective question, since "great" is not formally defined. But to describe it ...

**-5**

votes

**0**answers

28 views

### Proof about a measure zero set [on hold]

Let $A$ be a Lebesgue measurable subset of $\Bbb R$.
Show that there exists a Borel measurable subset $B$ of $\Bbb R$ such that $A\subseteq B$ and such that $l^*(B\setminus A)=0$. Also, show that ...

**-8**

votes

**0**answers

102 views

### I have no experience with math research [on hold]

Consider the polynomial expansion for $\frac {\sin x}{x} = p(x)= 1-\frac {x^2}{3!}+\frac {x^4}{5!}–\frac {x^6}{7!} + \cdots$
$p(x) = \prod(x-a_i)$ for $i = 1 →∞$ where the $a_i$ are the roots of ...

**0**

votes

**2**answers

134 views

### How to write $\mathbb{C}[G/U_-]=\oplus_{\lambda} V_{\lambda}$ explicitly?

Let $G=GL_n$ and $U_-$ the set of all lower unipotent triangular matrices. Then by Gauss Decomposition, we have $G = U_-B$, where $B$ is the set of all upper triangular matrices. The group $U_-$ acts ...

**-1**

votes

**1**answer

45 views

### Finding functional equations that a given function satisfies

Suppose we're given a function, for example a function $f:\mathbb{C}\rightarrow \mathbb{C}$ such that $f(x)=ax+b $ with $a,b \in \mathbb{C} $. I would like to know which functional equations are ...

**-7**

votes

**0**answers

51 views

### What are singular value of $A$? [on hold]

Let $
A = \left( {\begin{array}{*{20}{c}}
{x + (\frac{3}{4} + y)i}&1&1\\
0&{(x - \frac{5}{4}) + iy}&1\\
0&0&{(x + \frac{3}{4}) + iy}
\end{array}} \right)$, and $x,y\in ...

**-7**

votes

**0**answers

59 views

### I've read on the internet that 43 is congruent to 8 modulo 5+3w, can you explain? [on hold]

Initially, I wanted to determine whether 19 is a cube modulo 43.
Reading online, I've come across this computation that includes
a step that I don't understand. The step in question states that
(43 ...

**19**

votes

**1**answer

413 views

### Does $E_8$ know $Spin(7)$?

One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For ...

**5**

votes

**0**answers

12 views

### natural weak factorization systems

I am trying to understand the definition of natural weak factorization systems from this article by Tholen and Grandis, and these notes by Emily Riehl.
In Riehl's notes, the splitting $s,t$ are of ...

**36**

votes

**5**answers

1k views

### Undergraduate ODE textbook following Rota

I imagine many people are familiar with the extremely entertaining article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations" by Gian-Carlo Rota. (If you're not, do ...

**2**

votes

**1**answer

483 views

### A paper by Y. Morita

The corresponding bibliographical details are:
Yoshihito Morita, Elementary proofs of the commutativity of rings satisfying $x^{n}=x$. Mem. Defense Acad. 18 (1978), no. 1, 1–24.
Does anybody here ...

**3**

votes

**1**answer

88 views

+50

### Conserved quantities for the Cauchy momentum equation

I apologize if this question is too elementary for mathoverflow; I asked it (unsuccessfully) on MATH.SE first.
As a bit of background: one way to study the mechanics of deformation of a continuous ...

**228**

votes

**68**answers

107k views

### Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results?
(One could ask if this is of interest to mathematicians, and I ...

**12**

votes

**1**answer

139 views

### Applications of the Central Limit Theorem in dynamical systems

There are very many papers in the area of (possibly non-uniformly) hyperbolic dynamical systems whose aim is to prove the Central Limit Theorem. In a dynamical context, this means that one:
has a ...

**12**

votes

**1**answer

562 views

### What is the precise relationship between o-minimal theory and Grothendieck's “Esquisse d'un programme”?

I have seen various references in the literature to such a connection but they tend to assume that the reader is familiar with the connection, and limit themselves to providing additional detail. So ...

**1**

vote

**1**answer

72 views

### Elliptic regularity and inhomogeneous Neumann boundary condition

Consider a harmonic function $u$ defined on $D : = \{ (x, y) \in \mathbb{R}^2 | (x, y) \in \overline{B(0, 2)}, y \geq 0\}$, that is, the closed upper half ball centered at $0$ and radius $2$. Let $u$ ...

**117**

votes

**46**answers

39k views

### Ways to prove the fundamental theorem of algebra

This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?
Please give a new way in each answer, and if possible give reference. I start by giving two:
...

**1**

vote

**0**answers

92 views

### weak-* versus entropy growth

General question. Let $\eta_{n}$ be a sequence of invariant measures on $\{0,1,2,...,p-1\}^{\mathbb{N}}$ and $B$ the Bernoulli uniform measure. Knowing that $\eta_{n} \rightarrow B$ in the weak-* ...

**80**

votes

**17**answers

11k views

### Are there examples of non-orientable manifolds in nature?

Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence:
"The unorientable surfaces are never discussed ...

**82**

votes

**11**answers

9k views

### Checkmate in $\omega$ moves?

Is there a chess position with a finite number of pieces on the infinite chess board $\mathbb{Z}^2$ such that White to move has a forced win, but Black can stave off mate for at least $n$ moves for ...

**8**

votes

**2**answers

296 views

### The category of elements, enrichment, and weighted limits

This is a crosspost of this MSE question.
Every so often, when reading notes online or skimming through books, the category of elements and the Grothendieck construction pop up. I don't know ...

**41**

votes

**9**answers

4k views

### Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?
I've found some examples:
1) In MO-Q111339 ...

**4**

votes

**1**answer

439 views

### Comparing norms on tensor products of matrices

Given a Hilbert space $H$, let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is
$$ \Vert T \Vert_1 = \sum_{j\geq 1} |s_j| $$
where ...

**12**

votes

**5**answers

1k views

### How do you define the strict infinity groupoids in Homotopy Type Theory?

In the setting of Homotopy Type Theory, how would you construct $\mathrm{isStrict} : U \rightarrow U$ which is inhabited exactly when the first type is (equivalent to?) a strict $\infty$-groupoid?
...

**14**

votes

**6**answers

3k views

### Metrization of weak convergence of signed measures

Edit: Changed from "Hausdorff" to "metric" spaces.
Let $\mathcal{M}(\Omega)$ denote the space of signed regular Borel measures on a compact metric space $\Omega$. By Riesz-Markov, this is the dual ...

**27**

votes

**6**answers

3k views

### How do we recognize an integer inside the rationals?

My question is fairly simple, and may at first glance seem a bit silly, but stick with me. If we are given the rationals, and we pick an element, how do we recognize whether or not what we picked is ...

**34**

votes

**5**answers

8k views

### Theories of Noncommutative Geometry

[I have rewritten this post in a way which I hope will remain faithful to the questioner and make it seem more acceptable to the community. I have also voted to reopen it. -- PLC]
There are many ...

**8**

votes

**3**answers

1k views

### Is there the Whitehead theorem for cohomology theory?

We know that there are Whitehead theorem for homotopy and homology theory.
Is there the Whitehead theorem for cohomology theory for 1-connected CW complexes?

**0**

votes

**1**answer

77 views

### Relation between Harmonic vector field and Harmonic 1-form

Definition 1: A unit vector field $X$ side to be harmonic if it is critical point for the following energy function
$$E(X)=\int_M\|dX\|dvol_g=\frac{m}{2}vol(M,g)+\int_M\|\nabla X\|^2dvol_g.$$
...

**2**

votes

**2**answers

120 views

### Definable curves in definable sets

Suppose that I have an unbounded subset $X \subset \mathbb{R}^n$, definable in the $o$-minimal structure $\mathbb{R}_{an, exp}$. Is it possible to find an unbounded, analytic and definable curve (i.e. ...

**0**

votes

**0**answers

56 views

### Upper bound on the norm of the inverse of matrices with zero limit

Posted here too, with no answer yet:
http://math.stackexchange.com/questions/1766281/upper-bound-on-the-norm-of-the-inverse-of-matrices-with-zero-limit
Let $\{L(\sigma)\}_{\sigma}$ be a family of ...

**5**

votes

**1**answer

100 views

### Intuition for density comonad in relation to lifting problems

In Emily Riehl's Categorical Homotopy Theory, there is a section on Garner's Small Object Argument which I'm trying and failing to understand. Originally I followed most of Garner's paper, using the ...

**1**

vote

**1**answer

121 views

### List is a monad, but is it a comonad with these natural transformations?

List is known to be a monad. It takes a set and maps it to lists of elements of that set. The natural transformations are, singleton and flatten, whereby we map a set to a set of singleton lists ...

**5**

votes

**1**answer

177 views

### “Identity tensor transpose” as a map $M_n \hat{\otimes} M_n \to M_n \overline{\otimes} M_n$

Equipping $M_n$ with its usual operator space structure,
$\newcommand{\ptp}{\widehat{\otimes}}$
we can form the projective tensor product of operator spaces $M_n\ptp M_n$. In particular this puts a ...

**7**

votes

**1**answer

85 views

### Can a weak fibration category be non saturated?

A weak fibration category is a category $\mathcal{C}$ equipped with two subcategories
$$\mathcal{F}, \mathcal{W} \subseteq \mathcal{C}$$
containing all the isomorphisms, such that the following ...

**26**

votes

**1**answer

561 views

### A long-lasting conjecture of Pólya & Szegő

There is a conjecture by Pólya & Szegő (~1950) which is as follows:
"Of all $n$-gons of a fixed area, the regular $n$-gon minimizes the first Dirichlet eigenvalue."
Surprisingly, this is still ...

**14**

votes

**1**answer

322 views

### Generalizing the Mazur-Ulam theorem to convex sets with empty interior in Banach spaces

The Mazur-Ulam theorem (1932) states that any isometry of a normed linear space is affine. See Nica (Expo. Math. 30 (2012), 397-398; arXiv:1306.2380) for a very elegant proof.
Question: Let $M$ be a ...

**8**

votes

**1**answer

195 views

### Operads and the Stable Module Category

I am seeking references to places where operads and their algebras have been studied for the stable module category. Colored operads are fine too.
Let $k$ be a field and $R$ a $k$-algebra. The stable ...

**11**

votes

**2**answers

501 views

### The Simultaneous Conjugacy Problem in the symmetric group $S_N$

We are interested in the following notions in the case $G=S_N$, the symmetric group on
$\{1,\dots,N\}$.
Fix a group $G$ and a number $d$. For $(g_1,\dots,g_d)\in G^d$ and $x\in G$, define
...