# All Questions

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### generalization of developable helicoids

Please indicate references to parametrization of a constant negative Gauss curvature $K = -1/a^2$ helicoid. How to modify the developable helicoid parametrization to get it?
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### Computing the bourbaki ideals

By virtue the Griffith's paper and subsequently e.g. Goto's paper several examples of several desired class of Noetherian normal domains with specific finite length local cohomologies are constructed ...
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### Electrodynamics modelled by U(1) gauge theory

As the article 'Electrodynamics in general spacetime' greatly explains, the U(1)-gauge theory is a good base for working in non-simply connected spaces. But I wonder whether there is a deep reason to ...
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### How can I interpolate between these sets of algebraic integers?

Consider the set $S_d(m)$ of algebraic integers whose minimal polynomials are of degree $\leq d$ and have constant and leading coefficients $+1$, and all other coefficients chosen from the set ...
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### What are the results about quotient of two finite Blaschke products?

We all know that product of any two finite Blaschke product is again a finite Blaschke product. I am interested about what are results concerning about the Quotient of two finite Blaschke products? ...
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### Simple $Z^{*}$ algebra

What is an example of a simple $C^{*}$ algebra which all elements are (two sided or equivalently one sided) zero divisor?
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### faithful action of Hecke algebra

Let $G$ be a connected reductive group split over a number field $F$, $\mathbb{A}$ the adeles. Let $v$ be a finite place and $\mathcal{H}_{v}$ the spherical hecke algebra at palce $v$. ...
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### Parabola: Relation between arc length and span length

I am analysing a bridge cable. The self weight of the cable is distributed along the arc-length of the cable whereas the external vertical uniform length is distributed along the span length of the ...
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### Simplest non-constructible set of integers compatible with the nonexistence of $0^\sharp$?

What is the simplest non-constructible set of integers (say, in the analytical hierarchy) that is compatible with the nonexistence of $0^\sharp$? In particular, can there still be a non-constructible ...
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### Exp of an Infinitesimal Value? [on hold]

I'm reading a book and found $e^{iA(x)\delta}=1+iA(x)\delta$ if $\delta$ is some infinitesimal value. Why is this?
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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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.
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### 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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### 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 ...
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### 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: ...
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### 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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### 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 ...
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### 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 $\alpha^n$ ...
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### 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 ...
172 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
### 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 $D_{4}$:  ...