# All Questions

**5**

votes

**1**answer

318 views

### Why do $12$ and $120$ occur very often in the denominators of $\zeta(-n)$ for odd $n$?

$\zeta(-n) = - \dfrac{B_{n+1}}{n+1}$
$\zeta(-2n) = 0$
$\zeta(-1) = - \dfrac{1}{12}$
$\zeta(-3) = \dfrac{1}{120}$
$\zeta(-5) = - \dfrac{1}{252}$
$\zeta(-7) = \dfrac{1}{240}$
$\zeta(-9) = - ...

**6**

votes

**1**answer

158 views

### Extending compact operators

Let $X$ be a separable, infinite-dimensional complex Banach space and $Y\subseteq X$ an infinite-dimensional closed subspace. Suppose $K:Y\to X$ is an arbitrary compact operator. I would like to ...

**0**

votes

**0**answers

76 views

### What are some continued fractions for $2\pi$ and for $\dfrac{\pi}{2}$? [closed]

I've found on Wikipedia three simple and beautiful continued fractions for $\pi$ :
I would like to see some continued fractions for $2\pi$ and for $\dfrac{\pi}{2}$.

**-1**

votes

**1**answer

79 views

### terminology: “complex” and “sequence” in homological algebra

It appears that the terms "complex" and "sequence" are used synonymously in homological algebra.
But there seem to be collocations (in the linguistic sense) that prefer one of those words. For ...

**2**

votes

**1**answer

62 views

### Triangulations of translation surfaces whose edges are shorter than the diameter

Let $S$ be a compact translation surface (i.e. a surface endowed with a singular flat metric such that singular points are locally isometric to a cone of angle an integer multiple of $2\pi$, and that ...

**-1**

votes

**0**answers

24 views

### convex analysis [closed]

So I read this theorem in a convex analysis book saying Let f: R^N --> R(Bar) be convex with x(bar) in dom f. the following are equivalent. (i) f is continuous at x(bar) (ii) x(bar) in int(dom f) ...

**0**

votes

**0**answers

39 views

### Semicubical parabola homeomorphic to C^2 [closed]

a semicubical parabola $L$ in $\mathbb C^2$ is given by $y^2=x^3$.
I showed that a bijective function $f\colon\mathbb C \to L$ defined by $t \mapsto (t^2, t^3)$ becomes a homeomorphism regarding the ...

**-1**

votes

**0**answers

43 views

### Characterization of a family of interval graphs

Let $G=(V,E)$ be a graph, where $V$ is a set of integral intervals from $[1,n]$ and $\left \{i,j \right \} \in E$ if $i \cap j \neq \emptyset $. Is the family of these graphs a proper subset of the ...

**4**

votes

**2**answers

116 views

### Isotypic components of the action of the symmetric group on polynomials

The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ decomposes as a direct sum of isotypic components for the action of the symmetric group $S_n$. The isotypic component of the trivial representation is ...

**2**

votes

**0**answers

85 views

### Slice a compact C1 surface in R3 by a moving transverse plane. Does the length of the slice depend C1 on the plane?

To be more precise I am interested in questions similar to the one below
(I asked the question below on math.stackexchange last week but got not answer.)
I have a $C^1$ function $f:[0,1]^2 \to ...

**2**

votes

**2**answers

106 views

### $(LLP(Epi), Epi)$ is a WFS on any variety of algebras

This entry in the Joyal catlab claims without proof that in a category $\bf V$ which is a "variety of algebras" the two classes $(LLP(Epi), Epi)$ form a weak factorization system. I interpret this ...

**1**

vote

**0**answers

63 views

### Definition of primitive divisor of a Lucas sequence

If $\{a_n\}_{n=1}^\infty$ is a sequence of integers, then the definition of primitive divisor of one of its terms is quite natural:
Def. 1. A prime number $p$ is a primitive divisor of $a_n$ if $p ...

**1**

vote

**0**answers

54 views

### How is this transformation related to the Legendre transform?

I stumbled over the following transform in a statistical mechanics paper:
Unfortunately, no mathematical details were given there, which is why I wanted to ask here about this transform.
Let $s : ...

**2**

votes

**0**answers

87 views

### Applying Lemma 2.12 of Deligne's/ Milne's “Tannakian Categories” on an irreducible representation

since this is my first question here, I'm not very certain whether I state it properly. I'm thankful for any helpful remarks.
Currently I'm trying to understand the above-mentioned article which can ...

**-1**

votes

**0**answers

105 views

### Question about non trivial zeros of Riemann zeta function [closed]

I would like to know if is it true that $$-\frac{1}{2\pi i}\underset{n\geq1}{\sum}\frac{1}{n\rho^{n}}\in\mathbb{R}-\mathbb{Z}$$where $\rho$ is a non trivial zero of Riemann zeta function. How can I ...

**0**

votes

**0**answers

43 views

### Solution of parabolic PDE system [on hold]

For the following parabolic PDE system, $u(x,t)$ and $v(x,t)$ are functions of independent variables $x$ and $t$, $x\in[a,b]$.
\begin{equation}
\begin{cases}
\frac{\partial}{\partial ...

**6**

votes

**1**answer

163 views

### Can the Cohen forcing collapse cardinals?

Let $\kappa$ be a regular cardinal, and let $\mathbb{P} = Add(\kappa,1)$ be the standard forcing notion for adding a new subset of $\kappa$ using partial function from $\kappa$ to $2$ with domain of ...

**1**

vote

**0**answers

118 views

### Towards an enhanced version of homological mirror symmetry for affine varieties

Let $X$ and $X^\vee$ be a mirror pair, homological mirror symmetry relates the symplectic geometry of $X$ to the complex geometry of $X^\vee$ via the equivalence of triangulated categories
...

**0**

votes

**0**answers

59 views

### Integral representation of the function

I try find integral representation to following function ($d_1<d_2$ and $d_2-d_1=1(\mod 2)$
$f(d_1,d_2)=\frac{(-1)^{\frac{d_2-d_1-1}{2}}}{2^{d_1-1} d_1!}\sum_{k=[\frac{d_1}{2}]+1}^{ d_1} { d_1 ...

**-3**

votes

**0**answers

44 views

### Binomial theorem [closed]

I have a problem with binomial theorom.
What is the result of solving of inequality:
(n 1) + (n 2) + (n 3) + ... (n n) > 32
Sorry for this notation.
Thanks for answer.

**10**

votes

**2**answers

367 views

### Is every order type of a PA model the \omega of some ZFC model?

Let $N$ be a model of first-order Peano arithmetic, and let $\sigma$ be its order-type. Does it follow that there is a (non-transitive, expect when $M$ is the standard model) $ZFC$-model $M$ such that ...

**-1**

votes

**1**answer

65 views

### CAT spaces and Metric Measure Spaces [on hold]

This is very vague question but here it goes: are there any results of analysis on metric spaces in CAT spaces?, for instance as in the paper of Sturm (On the geometry of metric measure spaces) or as ...

**3**

votes

**1**answer

307 views

### what would be the consequences on the distribution of primes of $\Lambda=\infty$?

It is widely believed that the quantity $\Lambda:=\lim\sup\dfrac{t_{n+1}-t_{n}}{2\pi/\log t_{n}}$, where $t_{n}$ is the imaginary part of the $n$-th non-trivial zero on the critical line of the ...

**0**

votes

**0**answers

17 views

### Some Galois theory [migrated]

I have a question on field extensions, and I can't seem to find precise
answers when browsing through online notes etc.
Here it is: suppose $K$ and $k$ are fields with $k \leq K$ and $[K : k] = m$
...

**0**

votes

**0**answers

24 views

### Question about Skorokhod embedding problem

Let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion on some probability space. Now for every centered probability distribution $\mu$ on $R$, i.e. $\int_{R}|x|d\mu(x)<+\infty$ and ...

**3**

votes

**0**answers

120 views

### Moduli interpretation of Eisenstein series

Let $N \geq 11$ be an integer and consider the basis of Eisenstein series for $M_2(\Gamma_0(N))$ described in Theorem $4.6.2$ of Diamond--Shurman's book. Pick and Eisenstein series $F$ in this basis. ...

**3**

votes

**1**answer

198 views

### Is $L(\ell_2,\ell_2)$ dense in $L(\ell_2,c_0)$?

Let $\ell_2:=\{ x:\mathbb{N} \to \mathbb{R}: \sum_{j=1}^\infty x_j^2<\infty\}$ and
$c_0:=\{ x:\mathbb{N} \to \mathbb{R}: \lim_{j\to\infty}x_j=0,\, \sup_{j\in\mathbb{N}}|x_j|<\infty\}$ denote the ...

**0**

votes

**1**answer

38 views

### A hyperbolic partial differential equation (wave-like) with variable-dependent coefficient and possibly singular in one variable

First, I beg your pardon since the title of the question is a bit confusing I guess. I'm working on a physical equation of the wave-like form. Explicitly, it reads
...

**2**

votes

**1**answer

71 views

### Weight polytopes of the fundamental representations of simple Lie groups

Where can I find a description of the weight polytopes of the fundamental representations of the classical complex simple Lie groups?
Thanks in advance

**0**

votes

**0**answers

138 views

### Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1)) $, where $ p > 1 $. Let
$$
(D_{n})_{n \in \mathbb{N}_{0}} =
\left( \left\{
I^{n}_{j} \stackrel{\text{df}}{=} \left[ ...

**4**

votes

**2**answers

249 views

### Are Banach space norms (up to equivalence) unique?

Here is a naive question: is a "completing" norm of a vector space unique (up to equivalence) or can one find a vector space and two non-equivalent norms $\|.\|$ and $|||.|||$ that both induce a ...

**3**

votes

**0**answers

60 views

### Integral group rings on which stably free modules are free

Let $G$ be a torsion-free group and $ZG$ the integral group rings. Recall that a projective module $P$ over $ZG$ is stably free if there is an isomorphism $P \oplus ZG^n \cong ZG^m$. Are there known ...

**1**

vote

**0**answers

12 views

### How to restructure adjacency matrix $A$ from shortest distance matrix $B$ in Network topology inference

An undirected graph with $n$ nodes could be referred to as an adjacency matrix $A$. $A=[a_{ij}]_{n×n}$ with $a_{ij}=a_{ji}=1$ standing for there being an edge between node $i$ and node $j$, and no ...

**-4**

votes

**0**answers

43 views

### Statistics, probability [closed]

A statistician-gone-mad has concocted the following multi-part experiment. For the first part of the experiment, a fair, seven sided die is rolled and the upper-most facing number is noted. If the ...

**3**

votes

**0**answers

114 views

### Total variation and Hellinger distance inequality between truncated Gaussians

We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, ...

**2**

votes

**0**answers

70 views

### motivic integration and jacobian ideal

When we consider the change of variables in motivic integration, we have a birational map $f:Y\rightarrow X$ with Y smooth and we have to consider two invariants the order of the Jacobian ideal of $X$ ...

**1**

vote

**1**answer

134 views

### What is the ring $A_{\Gamma}$ in the Cohen-Lenstra Heuristics?

I understand the work in Cohen and Lenstra's paper that leads up to the heuristics themselves, where they count weighted averages of functions defined over isomorphism classes of $A$-modules, where ...

**6**

votes

**0**answers

119 views

### “extended TQFT” versus “TQFT with defects”

There are two ways in which higher categories appear in topological field theory: in extended TFTs and in TFTs with defects. How are these appearances related?
According to the Atiyah-Segal axioms, a ...

**2**

votes

**0**answers

58 views

### Geodesics on a perturbed submanifold of $\mathbb{R}^m$ [closed]

Let us consider $M$, a Riemannian manifold of dimension $n$, isometrically embedded in $R^m$. Let us consider a geodesic $\gamma$ on $M$. Now, let us "perturb" (in other words, change slightly the ...

**-1**

votes

**0**answers

10 views

### 1 dimensional flows and phase portraits [migrated]

I have a flow defined by $\dot{x} = x-x^4+1 :=f(x)$. I need to sketch its phase portrait. Firstly, I have to find its fixed points, these occur at $f(x)=x$. So, $x^4=1 \Rightarrow x=
\pm1$. Next, I ...

**-2**

votes

**0**answers

48 views

### Theorem on the algebraic manipulation of divergent series [closed]

There has been much debate over the values of divergente series. Applying the normal rules of algebraic manipulation to series such as 1+2+4+8+... can produce seemingly legit results such as -1, in ...

**1**

vote

**1**answer

148 views

### A generalisation of Narayana-like numbers (walks on the 2D lattice)

I apologize in advance if this question has a trivial answer. I am pretty sure this kind of problem was already studied and I am mostly asking for good references.
Given integers $0 < k \le n+1,$ ...

**-1**

votes

**1**answer

92 views

### Oddify an even function and vice versa: need a Fourier transform-based formula [closed]

I have constructed an operator that applied to an odd function will give its even counterpart and also an inverse operator that would transform an even function into an odd one. I need a ...

**3**

votes

**2**answers

131 views

### Is every Montel locally convex vector space compactly generated?

Let $X$ be a Hausdorff locally convex vector space. Recall (my reference is the book of H. Jarchow, Locally Convex Spaces. B.G. Teubner, 1981) that we say that $X$ is a semi-Montel space if every ...

**-2**

votes

**0**answers

36 views

### Bounding and continuity in Banach spaces [closed]

Prove that if mapping in Banach space is bounded, it is continuous.

**2**

votes

**2**answers

91 views

### proof that “small” sets in an extension by iterated forcing already appear in an earlier stage

In Kunen's book (introduction to independence proofs, ) the following lemma is proved (chapter 8, lemma 5.14):
Assume that in M, $\alpha$ is a limit ordinal,
$( ( \mathbb{P}_\xi : \xi \leq \alpha) , ...

**-5**

votes

**0**answers

32 views

### Proving that Riesz map is bijection [closed]

1) Prove that Riesz map is bijection
2) Prove that Riesz map is monomorphism

**11**

votes

**4**answers

1k views

### Is it possible to formulate the axiom of choice as the existence of a survival strategy?

Consider the following situation:
There is an infinite set $G$ of giraffes.
A lion comes and announces a set $C$ of all possible colours and an infinite cardinal $\kappa$.
The hungry lion ...

**6**

votes

**1**answer

189 views

### non(Meager) in Random times Random extension

Suppose the least size of a non meager set of reals is $\kappa$. Is it still $\kappa$ after forcing with Random $\times$ Random?

**1**

vote

**0**answers

103 views

### Gauss's Cirlce Problem #lattice points in circle

Can someone link me the proof that $E(r)/r^{1/2}$ -> infinity when $r$->infinity?
Where #lattice points in circle = $Pi*r^2+E(r)$