**7**

votes

**1**answer

275 views

### About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

I've asked this question http://math.stackexchange.com/questions/1407451/about-the-relation-between-the-categories-textsch-textlrs-and-text on math.stackexchange , however I don't think I will receive ...

**4**

votes

**1**answer

105 views

### Kan extensions of pseudofunctors

Can anyone suggest a reference for (left) Kan extensions of pseudofunctors?
In particular, say we are given bicategories $\mathscr{A,B,C}$ and pseudo functors $\mathscr A \xrightarrow{G} \mathscr ...

**1**

vote

**1**answer

97 views

### Need an explanation of a deduction

When I was reading the paper of Winfried Kohnen, Yuk-Kam Lau and Igore E. Shparlinski (ON THE NUMBER OF SIGN CHANGES OF HECKE EIGENVALUES OF NEWFORMS), I found this result (which is Theorem 2 of the ...

**0**

votes

**1**answer

82 views

### Are constructive characterisations of k-regular (simple) graphs known?

By a constructive characterisation I mean a theorem giving a list of base graphs and a list of operations such that every graph in a given class is generated from the base graphs by applying some ...

**2**

votes

**1**answer

78 views

### Second order estimates of Monge-Ampere equations

In order to prove existence of solutions of real and complex Monge-Ampere equations in various modifications (e.g. as in the Calabi problem) one often uses the method of a priori estimates. One of the ...

**2**

votes

**1**answer

119 views

### Homotopy fibers and stratified fibrations

Suppose I have a map $f:X \to Y$ of topological spaces and a nice stratification of $X$ ( say such that the inclusion of each stratum is a Hurewicz cofibration) such that the restriction of $f$ to ...

**6**

votes

**2**answers

188 views

### Zero divisors with support of size 3 in group algebras of finite groups

Are there a finite group $G$ and a field $\mathbb{F}$ such that $\gcd(3,|G|)=1$ and the group algebra $\mathbb{F}[G]$ contains a zero divisor whose support is of size $3$?
Recall that the support of ...

**5**

votes

**0**answers

101 views

### Has unconditional convergence ever been proved other than by deducing it from absolute convergence?

Nobody's answering this question so I'll try it here. This is really a reference request: Has a certain kind of proof ever been used?
A series $\displaystyle\sum_n a_n$ converges absolutely if ...

**13**

votes

**1**answer

326 views

### Integral cohomology ring of K(Z,3)

Computing the cohomology of Eilenberg Maclane spaces is a feasible but difficult problem in algebraic topology. The general answer is quite complicated (see the MO answer and the reference therein ...

**12**

votes

**3**answers

757 views

### May integration spoil real-analyticity?

Is there an example of a function $f:(a,b)\times(c,d)\to\mathbb{R}$, which is real analytic in its domain, integrable in the second variable, and such that the function
$$ g:(a,b)\to\mathbb{R},\qquad ...

**5**

votes

**3**answers

134 views

### Probability of random geodesics on the half-sphere intersecting

4 end points (a,b,c,d say) are chosen uniformly randomly and connected a to b and c to d by two geodesics on the 2-dim half-sphere. Here, uniform means that, probability that a point lies on a surface ...

**1**

vote

**1**answer

117 views

### Curves on $SU(4)$ whose adjoint action on $\mathfrak{su}(4)$ integrates to $0$

Given $\xi \in \mathfrak{su}(4)$ and positive $T \in \mathbb{R}$, is it possible to find all smooth curves $U_s \in SU(4)$ with $U_0 = I$ such that
$$\int_0^T U_s \xi U_s^{\dagger} ds =0\; ?$$

**-4**

votes

**0**answers

26 views

### What is the upper bound for training Linear separable set with Perceptron, Rosenblatt rule? [on hold]

I have the following neural networks problem and couldnt find any answer on the web. Any hints would help. I am not looking for a complete solution, just some pointing in the right direction.
...

**1**

vote

**1**answer

63 views

### The Laplacian of an expression involving the Ricci tensor

While doing some computations on a compact Riemannian manifold I have reached the following expression:
$$ \Delta_y \big( Ric_y (\exp_y ^{-1} x, \exp_y ^{-1} x) \big) (x)$$
where $\Delta_y$ is the ...

**7**

votes

**1**answer

261 views

### Irreducible cubics modulo primes

Is there a small finite (perhaps of cardinality two or three) collection of cubic polynomials $p_1, \dotsc, p_k \in \mathbb{Z}[x]$ such that for every prime $p$ at least one of these is irreducible?

**2**

votes

**0**answers

85 views

### Lie algebra of holomorphic vector fields

It's well known that the holomorphic vector fields on a complex manifold form a Lie algebra. In simplest situations, this Lie algebra can be described explicitly.
For example, take $X=\mathbb{P}^n$, ...

**1**

vote

**0**answers

79 views

### Category-theoretic characterization of zero-dimensional spaces

Some background: a zero-dimensional space is one admitting a basis of clopen sets, whereas an extremely disconnected space is one where the closures of open sets are open. In the category CHauss of ...

**-3**

votes

**0**answers

49 views

### Solution manual of real mathematical analysis PUGH [on hold]

Is there anybody know that where I can find the solution manual of real mathematical analysis PUGH ?
Thanks a lot.

**1**

vote

**1**answer

47 views

### approximate diameter of polytopes in high dimensions

I just came across the following problem:
Let us consider the unit corner of the n-cube
$$
\Delta^n = \left\{(t_1,\cdots,t_n)\in\mathbb{R}^n\mid\sum_{i = 1}^{n}{t_i} \leq 1 \mbox{ and } t_i \ge 0 ...

**9**

votes

**0**answers

223 views

### Intrinsic definition of the weight filtration

Let $X$ be a smooth quasiprojective complex variety. Then Deligne (Theorie de Hodge II) defined a weight filtration on the Betti cohomology of $X$. The general philosophy is quite simple: express the ...

**3**

votes

**1**answer

64 views

### Integral domains equal to intersection of their height one localizations

Which integral domains have the property that $R = \bigcap R_P$, the intersection being taken over all height one prime ideals of $R$?
It is a standard fact that Krull domains, and thus noetherian ...

**5**

votes

**1**answer

131 views

### Optimisation of betting strategy

Consider integers $r \geq 1$ and $k \geq 1$ and consider the following game:
We start with $r$ tokens and at each round we choose $i \in \{1,...,r\}$ tokens to bet (if we have $N<r$ tokens we ...

**2**

votes

**0**answers

54 views

### Explanation that Twistor Space of $S^4$ is $\mathbb{C}P^3$?

I am attempting to read Atiyah's paper on self-duality in four-dimensional Riemannian geometry, and I came across the following basic example:
Let $S_-$ be the $SU(2)$-bundle of anti-self dual ...

**-1**

votes

**0**answers

112 views

### Is $(X_G, d_G)$ , compact manifold?

Let compact topological group $G$ acts on $(X,d)$ . We define a relation $\sim$ on $(X, d)$ as follows: for $x,y\in X$:
$$x\sim y \Leftrightarrow x=gy \ \text{ for some } g\in G.$$
It is clear that ...

**29**

votes

**2**answers

862 views

### The coupon collector's earworm

[EDITED mostly to report on the answer by Kevin Costello
(and to improve the gp code at the end)]
I thank Nicolas Dupont for the following question
(and for permission to disseminate it further):
...

**1**

vote

**0**answers

57 views

### Integral homology of $S^{n-1}/\pi$, $H_*(S^{n-1}/\pi; \mathbb{Z}_p)$ [migrated]

Let $p$ be an odd prime number. Regard the cyclic group $\pi$ of order $p$ as the group of $p$th roots of unity contained in $S^1$. Regard $S^{2n-1}$ as the unit sphere in $\mathbb{C}^n$, $n \ge 2$. ...

**6**

votes

**0**answers

162 views

### On an unpublished result of Magidor

In 1970th, Magidor proved the following important results:
(1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$
is strong limit and ...

**0**

votes

**0**answers

46 views

### Conditions for Mellin inversion

Under which conditions is the function
$$
g(s)=a^{c(s-1)}\Gamma(s),\qquad a>0,c\in \mathbb{R}
$$
the Mellin transform of a probability density function $f$? If $c=-1$, then $f$ is the exponential ...

**5**

votes

**1**answer

217 views

### Constructing a simple $A$-module

Let $n \ge 2$, and let $A$ be the (unital and associative, but noncommutative) $\mathbb{C}$-algebra with generators $x_1, \dots, x_n$ and relations $x_ix_j + x_j x_i = 2\delta_{ij}$. What is a ...

**7**

votes

**2**answers

81 views

### Generate polyhedra by collapsing vertices of a polyhedron

I am looking for basic information about the following idea:
(I) Consider a square. By collapsing two adjacent vertices, we obtain a triangle.
(II) Consider a three-dimensional cube. By collapsing a ...

**4**

votes

**0**answers

56 views

### Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure.
Assume $\partial\Omega=\partial\Omega'$.
Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...

**7**

votes

**1**answer

116 views

### cohomology of iterated loop space on spheres

In the book The homology of iterated loop spaces, the homology Hopf algebra
(1)
$$
H_*(\Omega^n \Sigma^n X;\mathbb{Z}_p)
$$
for primes $p\geq 2$ is obtained on p. 226, Thm. 3.2. In particular, the ...

**-3**

votes

**1**answer

103 views

### Encyclopedia of Mathematics?(non-Alphabetical) [on hold]

Do you know any Encyclopedia of Mathematics which is in non-alphabetical order, like it starts from basic mathematics and then goes up to very advanced level.
And what's the difference between say, ...

**8**

votes

**0**answers

106 views

### Is tightness decidable?

Given a contact structure on a three-manifold, is there an algorithm to decide whether or not it tight?
For concreteness' sake, let's agree to represent the given contact three-manifold via an ...

**9**

votes

**1**answer

304 views

### Relationship between the syntomic cohomology of Kato and of Fontaine-Messing

Fix a prime $p$ and let $X$ be a $\mathbb{Z}_{p}$-scheme. Write $X_{n}:=X\otimes\mathbb{Z}/p^{n}$ and $\phi:X_{1}\rightarrow X_{1}$ for the absolute Frobenius. Let $X\hookrightarrow Z$ be a (suitable) ...

**10**

votes

**3**answers

424 views

### Continued Fractions from Digit Streams

let $x=\sum_{i=1}^{\infty}\delta_i2^{-i},\ \delta_i\in\{0,1\}$.
Is there an algorithm that converts the sequence $(\delta_0,\ \delta_1,\ ...)$ of the binary digits of $x$ to the sequence ...

**9**

votes

**4**answers

515 views

### Is this an instance of any existing convex pentagonal tilings?

Inspired by Wikipedia's article on pentagonal tiling, I made my own attempt.
I believe this belongs to the 4-tile lattice category, because it's composed of pentagons pointing towards 4 different ...

**-3**

votes

**0**answers

17 views

### segment intersecting a tetrahedron coordinates [on hold]

I am trying to write C++ code to find the intersection points of a segment intersecting a tetrahedron. I reduced the problem like this:
For each face of the tetrahedron (a triangle), find the ...

**-1**

votes

**0**answers

37 views

### A question on spectrum [migrated]

Let $A,B \in {C^{n \times n}}$ and ${\sigma (A + B)}$ is spectrom of $(A+B)$.
Suppose
$M = \left\{ {\lambda \in C:\lambda \in \sigma (A + B),\left\| B \right\| \le \varepsilon } \right\}$
$F(A) = ...

**0**

votes

**0**answers

23 views

### Log-concave distributions: Weighted sum of pdfs

Assuming $f_n(\cdot)$ is a log concave function (e.g., pdf of Gaussian distribution) and $0\le q_n\le 1$ for all $n\le N$, I am trying to find conditions under which the following holds
...

**4**

votes

**3**answers

365 views

### Bateman-Horn, continued even further

As before, consider the "singular series", which shows up in the Bateman-Horn conjecture: for an irreducible polynomial $f,$ this is equal to
$$
s(f) = \prod_p \frac{1-\frac{n_f(p)}p}{1-\frac1p},
$$
...

**-3**

votes

**1**answer

36 views

### upper bound for a convex fractional function [on hold]

Consider the following convex fractional function
$$f\left( {\bf{x}} \right) = \frac{1}{{1- {\bf{x}}}}$$
where ${1- {\bf{x}}} > 0$. Is it possible to obtain a linear or quadratic upper bound ...

**2**

votes

**1**answer

96 views

### Odds of residue being small

Given $\mathsf{c\geq1}$, what is the probability that if you choose $\mathsf{A,B,\alpha\in\Bbb N}$ such that $\mathsf{A,B<\alpha<AB}$ holds we will have both ...

**2**

votes

**0**answers

81 views

### An equality of discriminant and resultant divisors

Let $\Phi$ be the root system of a split group $G$ over a field $k$. The differentials $d\alpha$ of the roots define a polynomial called the discriminant
$$\prod_{\alpha\in\Phi}d\alpha$$
on $\mathfrak ...

**4**

votes

**1**answer

58 views

### Is every pair of writable reals one-tape-ITTM-computable?

I've been reading this paper, in which authors prove that not all ITTM-computable functions $\Bbb R\rightarrow\Bbb R$ are 1-tape-computable, but if we put some restriction on the output of the ...

**5**

votes

**2**answers

227 views

### cup product and Steenrod operations in Serre spectral sequence

Let $F\to E\to B$ be a fibration with $B$ simply-connected. Suppose all differentials in the cohomology Serre spectral sequence (corresponding to the above fibration) are zero maps. Then as a graded ...

**10**

votes

**1**answer

429 views

### Is an irreducible ideal in $R$ also irreducible in $R[x]$?

Let $R$ be a commutative Noetherian ring and $I\subset R$ an ideal that is irreducible in the sense that if $I = J_1 \cap J_2$, then $I=J_1$ or $I=J_2$. Is (the ideal generated by) $I$ irreducible in ...

**1**

vote

**1**answer

78 views

### Projective dimension of a quotient ring

Assume $A$ and $B$ are commutative algebras with $1$, $B = A[z] = A[Z]/(h(Z))$,
$Z$ an indeterminate.
The first comment in this question says that, if $A$ is noetherian, then
$pd_{B\otimes_A B}(B) ...

**-1**

votes

**0**answers

31 views

### Why do there is a unique continuous homomorphism? [migrated]

Is this a right place to ask help for an exercise?
Let $n\geq 2$ be an integer and $D=\mathbb Z[1/n]$. Let $A$ be a complete commutative ring with unit for the $I$-adic topology, where $I$ is an ...

**14**

votes

**3**answers

358 views

### Brouwer's theorem for the Cauchy reals

Brouwer famously proved, using principles motivated by intuitionistic choice sequences, that every function $\mathbb{R}\to \mathbb{R}$ is continuous. In Sheaves in geometry and logic (section VI.9), ...