# All Questions

**0**

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6 views

### Minimal surfaces + Semi-Geodesic Coordinates

Let $(M,g)$ be a three dimensional smooth Riemannian manifold and suppose that $\Gamma$ is an embedded minimal surface in $M$. Define the Fermi or semigeodesic coordinates around this surface through ...

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13 views

### Are these three different notions of a graph Laplacian?

I seem to see three different things that are being called the Laplacian of a graph,
One is the matrix $L_1 = D - A$ where $D$ is a diagonal matrix consisting of degrees of all the vertices and $A$ ...

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13 views

### Proving equivalence of two sums involving Krawtchuk polynomials and binomial co-efficients

I seem to have chanced upon a new characterization for Kravchuk polynomials.
[http://en.wikipedia.org/wiki/Kravchuk_polynomials].
To begin with, let us define the function $\omega(n,p)$ as [Assuming ...

**1**

vote

**0**answers

7 views

### Reconstructing density from integrals along specific manifolds

Let $\Phi_t : \mathbb R^n \to \mathbb R^n$ be the time-$t$-map associated to an ODE $\dot{x}=F(x)$ and let $H: \mathbb R^n \to \mathbb R$. Let $F$ and $H$ be sufficiently smooth (e.g. $C^k$ or ...

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27 views

### Automorphism of simple lie type groups

I will be so thankful for any comment or answer.
Suppose $S$ is a simple Lie type group of characteristic $p$ and $S\subseteq G \subseteq Aut(S)$ and $G_0$ is a subgroup of $G$ generated by all inner ...

**-1**

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46 views

### Affine communication lemma and finite limits in the category of rings

Let $S$ be a scheme and $\mathrm{Spec}(B) = V \subseteq S$ be an open affine. When using the affine communication lemma (c.f. Theorem 6.3.2, Vakil's notes, Foundations of Algebraic Geometry), one ...

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31 views

### Isothermal-related functions in higher dimensions

I am interested in getting some geometrical or analytical perspective in studing the following complex pde. I would appreciate any help.
Consider $ (M,g)$ to be a 3 dimensional Riemannian manifold ...

**1**

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24 views

### Obstructions for existence of a Riemannian metric such that a given function is harmonic

Let $f:\mathbb{R}^{n}\to \mathbb{R}$ be a smooth function. What type of obstructions exist for existence of a Riemannian metric $g$ on $\mathbb{R}^{n}$ such that $f$ is a harmonic ...

**5**

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**0**answers

106 views

### Failure of “equivariant triangulation” for finite complexes equipped with a $G$-action

Let $\mathcal{S}$ be the $\infty$-category of spaces, and let $G$ be a finite group, and let $BG$ be the groupoid with one object and automorphisms given by $G$.
Consider the $\infty$-category ...

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19 views

### A question on extension of $Z^{*}$ algebras

A $Z^{*}$ algebra is a $C^{*}$ algebra which all elements are(two sided or equivalently one sided) zero divisor.
Are there two $Z^{*}$ algebras $A,B$ such that for every short exact sequence ...

**2**

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**0**answers

45 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 ...

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36 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 ...

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27 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

39 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 ...

**2**

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**0**answers

86 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

22 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 ...

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**0**answers

39 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 ...

**-4**

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**0**answers

34 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 ...

**5**

votes

**1**answer

215 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 ...

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**0**answers

32 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 ...

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**0**answers

15 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 ...

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**0**answers

23 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 ...

**2**

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**0**answers

44 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

57 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 ...

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**0**answers

21 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

98 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?

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32 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

86 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

**3**answers

288 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

168 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

85 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**

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**0**answers

70 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

25 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

38 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

30 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

24 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**

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**0**answers

29 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

29 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

36 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**

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**0**answers

52 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

23 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

96 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 ...

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**0**answers

84 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

85 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

26 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

101 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

71 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

82 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 ...