# All Questions

**1**

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**0**answers

3 views

### Help understanding the proof of a theorem about Cohomology of vector Bundles

I am trying to understand a paper called Betti tables of graded modules and cohomology of vector bundles, but i am stuck in Proposition 6.8 which states:
Let $\mathcal{E}$ be a vector bundle on ...

**0**

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**0**answers

13 views

### Computer Program for Calculations in Tensor Algebra Quotients

Let $V$ be a finite dimensional vector space, and $S$ a finite dimensional subspace of its tensor algebra ${\cal T}(V)$ for which $X := {\cal T}(V)/<S>$ is finite dimensional, where $S$ is the ...

**0**

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12 views

### Number of Dyck paths with k returns and b peaks

The number of Dyck paths from the origin to $(2n,0)$ which touch the $x$-axis $k+2$ times ($k$ internal touches) is given by
$$\frac{k}{2n-k}{2n-k \choose n}.$$
The number of Dyck paths from the ...

**-1**

votes

**0**answers

29 views

### a question about the branches of $\log z$ [on hold]

Let $U$ be a connected open set in the complex plane $C$ and there exists a coninuous mapping $f:U\longrightarrow C$ such that for every $z\in U$,$f(z)^2=z$.
I want to ask if there must exist a ...

**0**

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**0**answers

34 views

### Which homology classes from loop space?

Fix a closed connected manifold $Q$ and let $LQ$ denote its free loop space. We can get second homology classes on $Q$ by "doing things" to loops in $Q$.
For instance, if we have a loop of loops, it ...

**-1**

votes

**0**answers

16 views

### Chance of pulling (game) cards [on hold]

What is the chance of pulling 6 cards in the correct order?
The game I play gives you 3 cards at random to play out of my 6 card deck to start with.
From the 3 open cards given you pick one to ...

**0**

votes

**0**answers

24 views

### Automorphism group of a modular curve and its action on the set of cusps

Let $X$ be a modular curve, that is the compact Riemann surface obtained by adding cusps to a quotient of Poincaré half-plane $\mathbb H$ by a congruence subgroup $\Gamma$ of $SL_2(\mathbb Z)$. The ...

**-3**

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**0**answers

18 views

### how do binary logic extend to algebra and find an algebra for 3 valued logic [on hold]

how do binary logic extend to set theory and then extend to algebra?
and find an algebra for 3 valued logic? i guess no longer use add or multiplication
it will use from logic 1 to logic 16 and ...

**2**

votes

**0**answers

76 views

### Teaching stochastic calculus to students who know no measure theory (or PDE, or…)

I've got quite a challenge as my teaching assignment for the next Fall (not that I want to get rid of it, quite the contrary, but I still feel like asking for advice won't hurt :-)).
I'm to teach the ...

**0**

votes

**0**answers

23 views

### what is the first non-constant term in the Kronecker Limit formula?

The Kronecker Limit formula gives the constant term in the Laurent expansion about s=1 of the Eisenstein series E(s,\tau). What is the next term? I.e., the coefficient of the first power of (s-1)? I ...

**0**

votes

**0**answers

11 views

### Max flow with minimal requirements algo problem

While applying the algorithm to solve the max flow of the network with minimal requirements on edges, I have encountered a problem.
The algorithm states:
For graph G
create an edge from target to ...

**0**

votes

**0**answers

19 views

### is a network a sum of its subnetworks?

I was wondering if networks/graphs are the sum of their parts. Let's say you have a 15-node network. The spectral density of that network has X kurtosis and Y skewness. You also have a 20-node ...

**1**

vote

**0**answers

32 views

### Hausdorff dimension of wandering set

I am searching some papers about the dimension of wandering set. It seems that there are more papers considering the non-wandering sets. I will appreciate if any references are recommended.

**0**

votes

**1**answer

48 views

### Action of the pure braid group on the commutator subgroup of a free group

Let $P=P_n$ be the pure braid group on $n$ strands and $F=F_n$ the free group on $n$ generators. I'm interested in a nice description of the action of $P$ on the derived subgroup $F'$ which somehow ...

**0**

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**0**answers

18 views

### Optimization - Linear or Non Linear? [on hold]

I'm a total newbie so pardon me, here is my problem.
I want to find out the maximum value of a loan I can give to a customer so that the annual simple interest rate does not exceed say 25% and the ...

**3**

votes

**0**answers

13 views

### Analytical solution of diffusion PDE with Robin boundary condition

I need to find the analytical solution of the time-independent diffusion equation with constant coefficients on the unit disk with subject to Robin boundary conditions. The formulation is as follows:
...

**4**

votes

**0**answers

87 views

### Is it provable in $\mathsf{ZF}$ that there is a group structure on any set $X$? [duplicate]

Given a set $X$ is it provable in $\mathsf{ZF}$ that there is a binary operation $\ast: X\times X\to X$ such that $(X,\ast)$ is a group?

**0**

votes

**0**answers

28 views

### Lie bialgebras cohomology

I am wondering if there exist a universal construction of (co)-chain complex associated to a given algebra to study deformations as Hochschild homology for associative algebras, or Chevalley-Eilenberg ...

**1**

vote

**0**answers

57 views

### How to test if the power of some algebraic number is the rational combination of two specific algebraic numbers?

Suppose we are given three algebraic numbers $\alpha,\beta,\gamma$ by presenting their minimal polynomial (degree less than $m$), the goal is to compute all positive integers $n$ such that $\alpha^n$ ...

**7**

votes

**2**answers

185 views

### Inclusion-preserving bijection between subsets of cardinality k and n-k

Let $n$ be a positive integer. A subset of $[n] := \{1,2,...,n\}$ having $k$ elements will be called a $k$-subset.
For $n,k \in \mathbb{N}$ with $k \leq \lfloor n/2 \rfloor$, it is clear that one can ...

**6**

votes

**1**answer

132 views

### Bounded operator on a normed space with empty spectrum

A bounded operator acting on a complex Banach space has non-empty spectrum, and the proof of this fact uses the completeness of the space.
Is there any example of bounded operator acting on a ...

**2**

votes

**1**answer

69 views

### How many pairwise non-homeomorphic compact, zero-dimensional topologies are there on $\mathbb{N}$?

To make the question more precise:
We call a topological space $(X,\tau)$ zero-dimensional if for $x\neq y \in X$ there is a clopen set $U\subseteq X$ with $x\in U, y\notin U$.
Let $\mathcal{C}$ be ...

**-2**

votes

**0**answers

64 views

### About Noncommutative Geometry [on hold]

I have some questions following:
1- what is the noncommutative geometry?
2- what are the prerequisites to study noncommutative geometry?
3- what are the branches of the noncommutative geometry?
...

**2**

votes

**0**answers

23 views

### Maps between spaces of non-empty compact subsets with the Hausdorff distance (reference request)

Let $X, Y$ be metric spaces, and let $PX$ (resp. $PY$) be the set of all non-empty compact subsets of $X$ (resp. $Y$) with the Hausdorff metric. A continuous map $f\colon X\to Y$ induces a continuous ...

**0**

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**0**answers

35 views

### abstract affine representations of semisimple Lie groups

in 1933 van der Waerden proved that any abstract unitary representation of a compact semisimple Lie group is necessarily continuous. Is any kind of similar result known for abstract affine ...

**0**

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**0**answers

23 views

### Mathematical simulation of viscous material behaviour

I have a non linear first order differential equation of the type:
$[y(t)]^n + a \frac{dy(t)}{dt} = b(t)$
where $y(t)$ is a real function, the exponent n is a real number greater than $2$, but not ...

**4**

votes

**1**answer

27 views

### Are epimorphisms (defined via an obvious action) of free Boolean algebras whose set of generators is a group automorphisms?

Let $G$ be a group. Consider $B$, the free Boolean algebra with generating set (I'll call them "variables") $G$. Let $F$ be some formula (that is, some fixed element of $B$). Define an endomorphism ...

**-2**

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**0**answers

22 views

### Question on inverse normal distribution [on hold]

We we're asked the following question among many, however I'm not quite sure how to start:
InvNormal(x) + InvNormal(1-x) = ?
Is this homework? Absolutely, ...

**0**

votes

**0**answers

26 views

### Central limit theorems for unequal probability sampling (weak but ill-defined dependence)

Suppose we are choosing samples of size $s$ from a finite population
$\{a_1, a_2, \dots , a_n\}$
where our sampling is with unequal probabilities. Construct
$$
S_n = \sum_{k=1}^{n} a_k
$$
Under what ...

**2**

votes

**0**answers

27 views

### How can the generators of subalgebra $\mathfrak g^{\sigma}$ of $\sigma$-stable elements be expressed through generators of Lie algebra $\mathfrak g$?

Let $\mathfrak g$ be the semisimple Lie algebra of type $D_{4}$. Let $\sigma$ be the 3-rd order automorphism of $\mathfrak g$ induced by the triality of $D_{4}$:
$$
...

**5**

votes

**0**answers

34 views

### Obstruction to the existence of global resolution of coherent sheaf

It is well known that any coherent sheaf on a complex manifold (or more generally a complex space) admits locally a resolution with locally free sheaves. It is also well known that for non-algebraic ...

**0**

votes

**0**answers

48 views

### Defining density of a random function using Radon-Nikodym Theorem

Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$.
Let $X$ be random function defined ...

**1**

vote

**1**answer

21 views

### Does $L^2$ progressive measurable processes form a Hilbert space?

Let $(\Omega, \mathcal F_1, {\mathbb P}, \mathbb F = \{\mathcal F_t\}_{0\le t \le 1})$ is a
filtered probability space. Let $L^2_{\mathbb F}$ be a collection of all $\mathbb F$ progressive measurable
...

**1**

vote

**1**answer

88 views

### What is the cubic casimir element of sl_3?

I have been thinking about this for some time but have had no luck. I have found some sources that say higher Casimir elements can be obtained by generalizing the second order Casimir, which is ...

**0**

votes

**0**answers

78 views

### Must group homomorphisms on the infinite symmetric group be identical if they agree on transpositions? [on hold]

Suppose we have two group homomorphisms $f, g: S_A \to S_B$, where $A$ and $B$ are infinite sets, and that $f(x) = g(x)$ for all transpositions $x \in S_A$. Does it follow that $f = g$?

**2**

votes

**2**answers

76 views

### When do zero-simplices of a simplicial diagram determine its homotopy colimit?

Suppose that I have a diagram of simplicial sets $X_\bullet:\mathscr{C} \to Set^{\Delta^{op}},$ with $\mathscr{C}$ a small category such that for each $C \in \mathscr{C},$ $X_\bullet(C)$ is a Kan ...

**0**

votes

**0**answers

25 views

### Why is the bisector in the complex hyperbolic ball not totally geodesic [on hold]

Why is the bisector, i.e. the real hyperplane defined by the equation real part of the first variable is zero, not a totally geodesic submanifold in complex hyperbolic space?

**1**

vote

**2**answers

98 views

### Handle body of 3-manifold with boundary

We know from Morse theory that smooth manifold(with or without boundary) is a handlebody.
However, I found a paper "Three-dimensional manifolds with boundary of nonnegative Ricci curvature" by Ananov, ...

**1**

vote

**0**answers

66 views

### Lebesgue point and regularity of functions

A known theorem says that for $f \in L_{loc}^1(\mathbb{R}^d)$, almost every point is a Lebesgue point.
I know too a theorem saying that for $f \in W_{loc}^{1,p}(\mathbb{R}^d)$ , every point is a ...

**2**

votes

**1**answer

72 views

### Bounds on imaginary parts of partial Kloosterman sums?

For a prime $p$ and integers $a,m$, $0<a,m<p$ define the (partial Kloosterman) sum
$$ S_p(a,m) = \sum_{0<k<m} \exp\left(\frac{2\pi\mathrm{i}}{p}(x + a x^{-1})\right), $$
where $x^{-1}$ is ...

**2**

votes

**0**answers

42 views

### counting irreducible factors

In How hard is it to compute the number of prime factors of a given integer? a question was asked on computing number of prime factors of an integer.
Suppose we have a polynomial $f(X)\in \Bbb Z[X]$ ...

**14**

votes

**4**answers

854 views

### What problem would you base your mathcoin on?

Recently, a variant of electronic currency, based on prime sextuplets,
broke the record in generating the largest known set of six primes, packed as closely as possible, that is, a sextuple ...

**4**

votes

**1**answer

117 views

### References to proofs of upper and lower bounds on the number of coprimes in an interval?

On the first page of the article "When the sieve works", the authors present upper and lower bounds for $S(T,T+x;\mathcal{E})$; the number of integers in the interval $(T,T+x]$ that are coprime to all ...

**10**

votes

**1**answer

228 views

### What's the volume of $\{x\in[0,1]^n|\sum x_i\le t\}$ for real $t$?

EDIT: Thanks for your answers and comments. There is indeed a classical easy formula, given by Pietro Majer (with a simple nice proof) in his answer below.
Given $x\in\mathbb{R}^n$, $x_i$ denotes ...

**0**

votes

**1**answer

55 views

### Frobenius Condition for a specific first order pde

I would appreciate it if Someone would be kind enough to share some insights about the following question:
Suppose $(M,g)$ is a 3 dimensional Riemannian manifold. Consider the following system of ...

**3**

votes

**0**answers

71 views

### Newvectors in tensor product representations

Let $\pi_p$ and $\pi_p^\prime$ two smooth admissible irreducible complex representations of ${\rm GL}_2(F)$ where $F$ is a non archimedean local field of residual characteristic $p$ of central ...

**9**

votes

**0**answers

184 views

### Do we know that 'most' finite groups are Galois groups of number fields?

The inverse Galois problem is a classical problem in mathematics and asks whether every finite group can be realized as the Galois group of a finite field extension of the rational numbers. The ...

**4**

votes

**1**answer

110 views

### Intersections of hypersurfaces of degree $d$ in $\mathbb CP^n$

Is it true that every projective sub-variety of degree $d$ in $\mathbb CP^n$ is an intersection of some number of hypersurfaces of degree $d$? Is there some simple proof of this fact? (I believe this ...

**8**

votes

**2**answers

105 views

### Elementary embeddings with the same critical point

Question: Is it consistent (relative to the existence of large cardinals) that there is an elementary embedding $j\colon V\to M$ (where $M$ is transitive model) that factors as $j = j_n \circ k_n$ for ...

**-1**

votes

**0**answers

41 views

### A general Formula for calculating Bezout's identity? [on hold]

I know how to calculate Bezout's identity by using the extended Euclidian algorithm (running the regular algorithm "backwards" where each step is calculated before I proceed to the next one, like so ...