text: Name Jianchun Wu Soochow University Session Jan 10 15:20~15:50 E11-101 Free degrees of homeomorphisms on compact surfaces For a compact surface $M$, the free degree $FRD(M)$ of homeomorphisms on $M$ is the minimum positive integer $n$ with the property that for any self homeomorphism $\xi$ of $M$, at least one of the iterates $\xi,\xi^2,\cdots,\xi^{n}$ has a fixed point. This is to say $FRD(M)$ is the maximum of least periods among all periodic points of self homeomorphisms on $M$. We show that $FRD(F_{g,b}) \leq 24g-24$ for $g \geq 2$ and $FRD(N_{g,b}) \leq 12g-24$ for $g \geq 3$. Joint work with Xuezhi Zhao. 101-1-10/draft.pdf


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Akida Models

# Akida models

This package contains a zoo of TensorFlow/Keras defined models that can be quantized and that are compatible for Akida conversion.

## Download files

Download the file for your platform. If you're not sure which to choose, learn more about installing packages.

### Source Distribution

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## Lutz Bungart

Ph.D. Princeton University 1962
Dissertation: Holomorphic Fuctions with Values in Locally Convex Spaces and Applications to Integral Formulas


text:
21765 articles – 15575 references  [version française]

hal-00455207, version 1

## Spectral theory of discontinuous functions of self-adjoint operators and scattering theory

Alexander Pushnitski, Dimitri Yafaev () 1

Journal of Functional Analysis 259, 8 (2010) 1950-1973

Abstract: In the smooth scattering theory framework, we consider a pair of self-adjoint operators $H_0$, $H$ and discuss the spectral projections of these operators corresponding to the interval $(-\infty,\lambda)$. The purpose of the paper is to study the spectral properties of the difference $D(\lambda)$ of these spectral projections. We completely describe the absolutely continuous spectrum of the operator $D(\lambda)$ in terms of the eigenvalues of the scattering matrix $S(\lambda)$ for the operators $H_{0}$ and $H$. We also prove that the singular continuous spectrum of the operator $D(\lambda)$ is empty.

• 1:  Institut de Recherche Mathématique de Rennes (IRMAR)
• CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
• Domain : Mathematics/Spectral Theory
Mathematics/Functional Analysis
• Keywords : scattering matrix – Carleman operator – absolutely continuous spectrum – spectral projections
• Comment : 21 page – Latex

• hal-00455207, version 1
• oai:hal.archives-ouvertes.fr:hal-00455207
• From:
• Submitted on: Tuesday, 9 February 2010 17:10:36
• Updated on: Tuesday, 12 June 2012 17:10:07


text:
# Mathematische Annalen (MATH ANN )

Publisher: Springer Verlag

## Description

Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein David Hilbert Otto Blumenthal Erich Hecke Heinrich Behnke Hans Grauert und Heinz Bauer.

• Impact factor
1.38
Show impact factor history

Impact factor
.
Year
• 5-year impact
1.30
• Cited half-life
0.00
• Immediacy index
0.49
• Eigenfactor
0.02
• Article influence
1.75
• Website
Mathematische Annalen website
• Other titles
Mathematische Annalen
• ISSN
0025-5831
• OCLC
1639684
• Material type
Periodical, Internet resource
• Document type
Journal / Magazine / Newspaper, Internet Resource

## Publisher details

• Pre-print
• Author can archive a pre-print version
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• Author can archive a post-print version
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• Authors own final version only can be archived
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• Must link to publisher version
• Set phrase to accompany link to published version (The original publication is available at www.springerlink.com)
• Articles in some journals can be made Open Access on payment of additional charge
• Classification
 green

## Publications in this journal

• ##### Article: Classification of conformal minimal immersions of constant curvature from S^2 to { HP}^2
[Hide abstract]
ABSTRACT: In this paper, we study geometry of conformal minimal two-spheres immersed in quaternionic projective spaces. We firstly use Bahy-El-Dien and Wood’s results to obtain some characterizations of the harmonic sequences generated by conformal minimal immersions from $$S^2$$ to the quaternionic projective space $${ HP}^2$$ . Then we give a classification theorem of linearly full totally unramified conformal minimal immersions of constant curvature from $$S^2$$ to the quaternionic projective space $${ HP}^2$$ .
Mathematische Annalen 08/2014; 359(3-4).
• ##### Article: Characterization of Siegel cusp forms by the growth of their Fourier coefficients
Mathematische Annalen 06/2014; 359(1-2).
• ##### Article: Local maximal function and weights in a general setting
[Hide abstract]
ABSTRACT: For a proper open set [TEX equation: \Omega ] immersed in a metric space with the weak homogeneity property, and given a measure [TEX equation: \mu ] doubling on a certain family of balls lying “well inside” of [TEX equation: \Omega ] , we introduce a local maximal function and characterize the weights [TEX equation: w] for which it is bounded on [TEX equation: L^p(\Omega ,w d\mu )] when [TEX equation: 1 Document Type: Research Article DOI: http://dx.doi.org/10.1007/s00208-013-0973-7 Affiliations: 1: Instituto de Matemática Aplicada del Litoral, CONICET-UNL, Güemes 3450, 3000 , Santa Fe, Argentina, Email: harbour@santafe-conicet.gov.ar 2: Instituto de Matemática Aplicada del Litoral, CONICET-UNL, Güemes 3450, 3000 , Santa Fe, Argentina, Email: salinas@santafe-conicet.gov.ar 3: Instituto de Matemática Aplicada del Litoral, CONICET-UNL, Güemes 3450, 3000 , Santa Fe, Argentina, Email: viviani@santafe-conicet.gov.ar Publication date: April 1, 2014 $(document).ready(function() { var shortdescription =$(".originaldescription").text().replace(/\\&/g, '&').replace(/\\, '<').replace(/\\>/g, '>').replace(/\\t/g, ' ').replace(/\\n/g, ''); if (shortdescription.length > 350){ shortdescription = "" + shortdescription.substring(0,250) + "... more"; } $(".descriptionitem").prepend(shortdescription);$(".shortdescription a").click(function() { $(".shortdescription").hide();$(".originaldescription").slideDown(); return false; }); }); Related content In this: publication By this: publisher By this author: Harboure, Eleonor ; Salinas, Oscar ; Viviani, Beatriz GA_googleFillSlot("Horizontal_banner_bottom");
Mathematische Annalen 04/2014; 358(3-4).
• ##### Article: Global W^{2,p} estimates for solutions to the linearized Monge–Ampère equations
[Hide abstract]
ABSTRACT: In this paper, we establish global $W^{2,p}$ estimates for solutions to the linearized Monge–Ampère equations under natural assumptions on the domain, Monge–Ampère measures and boundary data. Our estimates are affine invariant analogues of the global $W^{2,p}$ estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin’s global $W^{2,p}$ estimates for the Monge–Ampère equations.
Mathematische Annalen 04/2014; 358(3-4).
• ##### Article: Selmer groups over {\mathbb {Z}}_p^d-extensions
[Hide abstract]
ABSTRACT: Consider an abelian variety $A$ defined over a global field $K$ and let $L/K$ be a ${\mathbb {Z}}_p^d$ -extension, unramified outside a finite set of places of $K$ , with ${{\mathrm{Gal}}}(L/K)=\Gamma$ . Let $\Lambda (\Gamma ):={\mathbb {Z}}_p[[\Gamma ]]$ denote the Iwasawa algebra. In this paper, we study how the characteristic ideal of the $\Lambda (\Gamma )$ -module $X_L$ , the dual $p$ -primary Selmer group, varies when $L/K$ is replaced by a strict intermediate ${\mathbb {Z}}_p^e$ -extension.
Mathematische Annalen 03/2014;
• ##### Article: Nuclear dimension, [TEX equation: \mathcal{Z }] -stability, and algebraic simplicity for stably projectionless [TEX equation: C^*] -algebras
[Hide abstract]
ABSTRACT: The main result here is that a simple separable [TEX equation: C^*] -algebra is [TEX equation: \mathcal{Z }] -stable (where [TEX equation: \mathcal{Z }] denotes the Jiang-Su algebra) if (i) it has finite nuclear dimension or (ii) it is approximately subhomogeneous with slow dimension growth. This generalizes the main results of Toms (Invent Math 183(2):225–244, 2011) and Winter (Invent Math 187(2):259–342, 2012) to the nonunital setting. As a consequence, finite nuclear dimension implies [TEX equation: \mathcal{Z }] -stability even in the case of a separable [TEX equation: C^*] -algebra with finitely many ideals. Algebraic simplicity is established as a fruitful weakening of being simple and unital, and the proof of the main result makes heavy use of this concept.
Mathematische Annalen 01/2014; 358.
• ##### Article: [TEX equation: p] -Adic interpolation of square roots of central [TEX equation: L] -values of modular forms
[Hide abstract]
ABSTRACT: We construct a meromorphic function on part of the eigencurve that interpolates a square root of a ratio of quadratic twists of the central modular [TEX equation: L] -value.
Mathematische Annalen 01/2014; 358.
• ##### Article: A note on “Quasihyperbolic boundary conditions and Poincaré domains”
[Hide abstract]
ABSTRACT: We establish optimal Poincaré inequalities under a logarithmic growth condition on the quasihyperbolic metric.
Mathematische Annalen 11/2013;
• ##### Article: Some mixed Hodge structures on l^{2}-cohomology groups of coverings of Kähler manifolds
[Hide abstract]
ABSTRACT: We give methods to compute $l^{2}$ -cohomology groups of a covering manifold obtained by removing the pullback of a normal crossing divisor to a covering of a compact Kähler manifold. We prove that in suitable quotient categories, these groups admit natural mixed Hodge structure whose graded pieces are given by the expected Gysin maps.
Mathematische Annalen 11/2013;
• ##### Article: A characterization of compact complex tori via automorphism groups
[Hide abstract]
ABSTRACT: We show that a compact Kähler manifold $X$ is a complex torus if both the continuous part and discrete part of some automorphism group $G$ of $X$ are infinite groups, unless $X$ is bimeromorphic to a non-trivial $G$ -equivariant fibration. Some applications to dynamics are given.
Mathematische Annalen 11/2013;
• Source
##### Article: Upper bounds for the Euclidean minima of abelian fields of odd prime power conductor
[Hide abstract]
ABSTRACT: The aim of this paper is to give upper bounds for the Euclidean minima of abelian fields of odd prime power conductor. In particular, these bounds imply Minkowski’s conjecture for totally real number fields of conductor $p^r$ , where $p$ is an odd prime number and $r \ge 2$ .
Mathematische Annalen 10/2013;
• Source
##### Article: Some surfaces with maximal Picard number
[Hide abstract]
ABSTRACT: For a smooth complex projective variety, the rank of the N\'eron-Severi group is bounded by the Hodge number h^{1,1}. Varieties with rk NS = h^{1,1} have interesting properties, but are rather sparse, particularly in dimension 2. We discuss in this note a number of examples, in particular those constructed from curves with special Jacobians.
Mathematische Annalen 10/2013; 259(3).
• Source
##### Article: Bounding S(t) and S_1(t) on the Riemann hypothesis
[Hide abstract]
ABSTRACT: Let $\pi S(t)$ denote the argument of the Riemann zeta-function, $\zeta (s)$ , at the point $s=\frac{1}{2}+it$ . Assuming the Riemann hypothesis, we present two proofs of the bound \begin{aligned} |S(t)| \le \left(\frac{1}{4} + o(1) \right)\frac{\log t}{\log \log t} \end{aligned} for large $t$ . This improves a result of Goldston and Gonek by a factor of 2. The first method consists of bounding the auxiliary function $S_1(t) = \int _0^{t} S(u) \> \text{ d}u$ using extremal functions constructed by Carneiro, Littmann and Vaaler. We then relate the size of $S(t)$ to the size of the functions $S_1(t\pm h)-S_1(t)$ when $h\asymp 1/\log \log t$ . The alternative approach bounds $S(t)$ directly, relying on the solution of the Beurling–Selberg extremal problem for the odd function $f(x) = \arctan \left(\frac{1}{x}\right) - \frac{x}{1 + x^2}$ . This draws upon recent work by Carneiro and Littmann.
Mathematische Annalen 09/2013; 356(3).

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text:
# What demonstrates the law of supply?

## What demonstrates the law of supply?

The law of supply says that a higher price will induce producers to supply a higher quantity to the market. Supply in a market can be depicted as an upward-sloping supply curve that shows how the quantity supplied will respond to various prices over a period of time.

## Which of the following is an example of the law of supply?

Which of the following is the best example of the law of supply? A sandwich shop increases the number of sandwiches they supply every day when the price is increased. When the selling price of a good goes up, what is the relationship to the quantity supplied? It becomes practical to produce more goods.

## What is an example of law of supply in economics?

The Law of Supply Definition In economics, supply is the number of goods an individual or business provides to the market which refers to the amount they produce at a specific point in time. For example, if Apple manufactures 100 iPhones, then this is the supply that is brought to the market.

## Which of the following statements best represents the law of supply?

Which of these statements best represents the law of supply? When the price of a good decreases, sellers produce less of the good. represents the sum of the quantities supplied by all the sellers at each price of the good.

## What is an example of supply in economics?

When the price of an orange is 65 cents the quantity supplied is 300 oranges a week. If the price of copper falls from $1.75/lb to$1.65/lb, the quantity supplied by a mining company will fall from 45 tons a day to 42 tons a day.

## What is an example of the law of supply and demand?

These are examples of how the law of supply and demand works in the real world. A company sets the price of its product at $10.00. No one wants the product, so the price is lowered to$9.00. Demand for the product increases at the new lower price point and the company begins to make money and a profit.

## What demonstrates the law of demand?

Key Takeaways. The law of demand is an economic principle that states that consumer demand for a good rises when prices fall and decline when prices rise. The law of demand comes into play during Black Friday saleswhen consumers rush to buy products at deep discounts.

## What is an example of the law of supply?

Examples of the Law of Supply The law of supply summarizes the effect price changes have on producer behavior. For example, a business will make more video game systems if the price of those systems increases. The opposite is true if the price of video game systems decreases.

## What is an example of supply?

The noun means an amount or stock of something that is available for use. That stock has been supplied. A mother, for example, may take a large supply of diapers (UK: nappies) with her when she goes on vacation with her baby. This means a large amount that is available for use.

## What are the types of law of supply?

The law of supply explains the reaction of the supplier when the prices in the market change. Market supply, short-term supply, long-term supply, joint supply, and composite supply are five types of supply.

## What is the law of supply quizlet?

law of supply. the principle that, other things equal, an increase in the price of a product will increase the quantity of it supplied, and conversely for a price decrease; directly related.

## What are some examples of supply?

When the price of an orange is 65 cents the quantity supplied is 300 oranges a week. If the price of copper falls from $1.75/lb to$1.65/lb, the quantity supplied by a mining company will fall from 45 tons a day to 42 tons a day.

## What is law of supply and demand cite an example?

Examples of the Law of Supply There is a drought and very few strawberries are available. More people want strawberries than there are berries available. The price of strawberries increases dramatically. A huge wave of new, unskilled workers come to a city and all of the workers are willing to take jobs at low wages.

## What is an example of law of demand?

For example, a company launching a new product might deliberately try to raise the price of its product by increasing consumer demand through advertising. At the same time, they might try to further increase their price by deliberately restricting the number of units they sell to decrease supply.

## What represents the law of supply?

The law of supply says that a higher price will induce producers to supply a higher quantity to the market. Supply in a market can be depicted as an upward-sloping supply curve that shows how the quantity supplied will respond to various prices over a period of time.

## Which of the following statements best represents the law of supply price and quantity?

Price and quantity supplied are directly related, ceteris paribus. This option is correct because the law of supply states that there exists one to one relationship between price and quantity supplied

## Which of the following correctly states the law of supply?

Which of the following is the best example of the law of supply? A sandwich shop increases the number of sandwiches they supply every day when the price is increased. When the selling price of a good goes up, what is the relationship to the quantity supplied? It becomes practical to produce more goods.

## What are some of the examples of supply and demand?

Examples of the Law of Supply There is a drought and very few strawberries are available. More people want strawberries than there are berries available. The price of strawberries increases dramatically. A huge wave of new, unskilled workers come to a city and all of the workers are willing to take jobs at low wages.

## What is meant by supply in economics?

Supply and Demand Examples

• Example #1: The Price of Oranges. In this case we will look at how a change in the supply of oranges changes the price The demand for oranges will stay the same.
• Example #2: Designer Jeans.
• Example #3: Finding the Right Price.
• Other Examples.

## What are the 3 types of supply in economics?

Supply is a fundamental economic concept that describes the total amount of a specific good or service that is available to consumers. Supply can relate to the amount available at a specific price or the amount available across a range of prices if displayed on a graph.

## What is an example of the law of demand?

What is law of demand with example? The law of demand dictates that when prices go up, demand goes down and when prices go down, demand goes up. For instance, a baker sells bread rolls for $1 each. They sell 50 each day at that price. However, when the baker decides to increase to price to$1.20 they only sell 40.

## What is an example that embodies the law of demand?

The law of demand says that everything being constant; as the price of the good increases, then there will be a decline in the quantity demanded of that good. An example of this can be; when the price of rice was $3, the quantity demand by Harry was 4kg; when the price increases to$4, the quantity demand falls to 3kg

## Which is an example of the law of demand at work?

Which is an example of the law of demand at work? Demand for pizza rises when the price of pizza falls. Demand for a good rises if its price is expected to rise. How might advertising lead to a shift in the demand curve?

## What reflects the law of demand?

The law of demand is one of the most fundamental concepts in economics. The law of demand states that quantity purchased varies inversely with price. In other words, the higher the price, the lower the quantity demanded. This occurs because of diminishing marginal utility.

## Which is an example of the law of supply quizlet?

Examples of the Law of Supply There is a drought and very few strawberries are available. More people want strawberries than there are berries available. The price of strawberries increases dramatically. A huge wave of new, unskilled workers come to a city and all of the workers are willing to take jobs at low wages.


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Select Page

• This is an assessment test.
• These tests focus on the basics of Maths and are meant to indicate your preparation level for the subject.
• Kindly take the tests in this series with a pre-defined schedule.

## Basic Maths: Test 27

Congratulations - you have completed Basic Maths: Test 27. You scored %%SCORE%% out of %%TOTAL%%. You correct answer percentage: %%PERCENTAGE%% . Your performance has been rated as %%RATING%%
 Question 1
$\frac{4\frac{9}{13}}{4+\frac{3}{3+\frac{1}{2}}}$
is equal to ?
 A 1.7 B 1.5 C 2.4 D 0.96
Question 1 Explanation:
We will solve this step by step
$\begin{array}{l}\frac{\frac{61}{13}}{4+\frac{3}{3+\frac{1}{2}}}\\=\frac{\frac{61}{13}}{4+\frac{3}{\frac{6+1}{2}}}\\=\frac{\frac{61}{13}}{4+\frac{6}{7}}\\=\frac{\frac{61}{13}}{\frac{28+6}{7}}\\=\frac{61}{13}\times \frac{7}{34}=\frac{427}{442}=0.96\end{array}$
So the right answer for the question is option (d).
 Question 2
$\frac{9}{26}$ is equal to
 A $\displaystyle \frac{1}{5+\frac{1}{7+\frac{1}{17}}}$ B $\displaystyle \frac{1}{2+\frac{1}{1+\frac{1}{8}}}$ C $\displaystyle \frac{1}{3+\frac{1}{1+\frac{1}{2+\frac{1}{4}}}}$ D $\displaystyle \frac{1}{16+\frac{1}{8+\frac{1}{2+\frac{1}{9}}}}$
Question 2 Explanation:
The fraction can be written by hit and trail method
$\begin{array}{l}\frac{1}{2+\frac{1}{1+\frac{1}{8}}}\\=\frac{1}{2+\frac{1}{\frac{8+1}{8}}}=\frac{1}{2+\frac{8}{9}}\\=\frac{1}{\frac{18+8}{9}}=\frac{9}{26}\end{array}$
 Question 3
Simplyfy
$1+\frac{5}{2+\frac{3}{5-\frac{1}{2}}}-\frac{1}{2}\,\left( 20\div 2 \right)$
 A -1/2 B 0 C -17/8 D 1/3
Question 3 Explanation:
$\begin{array}{l}1+\frac{5}{2+\frac{3}{5-\frac{1}{2}}}-\frac{1}{2}\,\times 10\\=1+\frac{5}{2+\frac{6}{9}}-5\\=1+\frac{5}{2+\frac{2}{3}}-5\\=1+\frac{5}{\frac{8}{3}}-5\\=1+\frac{5\times 3}{8}-5\\=\frac{15}{8}-4=\frac{15-32}{8}=\frac{-17}{8}\end{array}$
 Question 4
$\displaystyle \begin{array}{l}\frac{\left[ \left( 5+\frac{1}{9+\frac{1}{9}} \right)\times \left( 5+\frac{1}{9+\frac{1}{9}} \right)-\left( 5-\frac{1}{9+\frac{1}{9}} \right)\times \left( 5-\frac{1}{9+\frac{1}{9}} \right) \right]}{\left[ \left( 5+\frac{1}{9+\frac{1}{9}} \right)+\left( 5-\frac{1}{9+\frac{1}{9}} \right) \right]}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}$
 A $\displaystyle \frac{40}{101}$ B $\displaystyle \frac{9}{41}$ C $\displaystyle \frac{21}{101}$ D $\displaystyle \frac{20}{100}$
Question 4 Explanation:
We will solve this equation in sections
Let us consider first
Let us suppose
$\begin{array}{l}5+\frac{1}{9+\frac{1}{9}}=\frac{419}{82}=a\\5-\frac{1}{9+\frac{1}{9}}=\frac{401}{82}=b.\\if\,you\,see\,\,it\,\,is\,\,a\,\,formula\,\,of\,\,\\\frac{{{a}^{2}}-{{b}^{2}}}{\left( a+b \right)}=\frac{\left( a+b \right)\left( a-b \right)}{\left( a+b \right)}\\=\left( a-b \right)\\=\frac{419}{82}-\frac{401}{82}=\frac{18}{82}=\frac{9}{41}\end{array}$
So the right answer for this question is option b
 Question 5
$1+\frac{3}{2+\frac{1}{5}}=?$
 A $\frac{26}{11}$ B $\frac{13}{6}$ C $\frac{15}{6}$ D $\frac{11}{26}$
Question 5 Explanation:
The given fraction can be simplified by
Expression
$=1+\frac{3}{2+\frac{1}{5}}$
$=1+\frac{3}{2+\frac{1}{5}}=1+\frac{15}{11}=\frac{11+15}{11}=\frac{26}{11}$
Therefore the answer for the question is option a
Once you are finished, click the button below. Any items you have not completed will be marked incorrect.
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+0

# help

0
44
1

In triangle ABC, BC = 6.  Point E is on AC so that angle CEB is 80 degrees.  Point D is on AB so that angle BDC is 100 degrees.  The bisectors of angles ECD and DBE intersect at F.  Find the numerical value of CF^2 + FB^2.

Dec 18, 2019

#1
+118
+1

If we show that the measure of angle f in the triangle BCF is 90 degrees, then the law of cosines

would tell us that

$$6^2=BF^2+CF^2-2(BF)(CF)cos(f)=BF^2+CF^2-2(BF)(CF)cos (90)$$

$$=BF^2+CF^2$$

since cos(90) = 0.

But f is indeed 90 degrees:

The three interior angles of BFC add up to 180, so

$$f+\alpha +c+\beta+b=180$$

The three interior angles of DBC add up to 180, so

$$\alpha+2b+\beta=80$$

and from the interior angles of EBC we get

$$\beta+2c+\alpha=100$$

Adding the second and third equation we get

$$2\alpha+2\beta+2b+2c=180$$, giving us

$$\alpha+\beta+b+c=90$$.

This fact added to the first equation results in

$$f=\alpha+\beta+b+c=90$$

.
Dec 18, 2019


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# zbMATH — the first resource for mathematics

Connectivity of the space of ending laminations. (English) Zbl 1190.57013
For a surface of genus at least four or a punctured surface of genus at least two, the main result of the present paper states that the space of ending laminations (or of arational measured foliations) on the surface is connected (this is also the space of ending laminations of geometrically infinite Kleinian groups, without accidental parabolics and isomorphic to the fundamental group of the surface). Since this space is known to be homeomophic to the Gromov boundary of the complex of curves of the surface, as a consequence also the Gromov boundary is connected. This in turn implies that quasi-isometry of the complexes of curves of two surfaces implies homeomorphism of the surfaces (rigidity of the complex of curves). As another consequence, also the space of doubly degenerate Kleinian surface groups is connected (by the ending lamination theorem, these Kleinian groups are determined by their two ending laminations).

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
Full Text:
##### References:
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Bonahon, Bouts des variétés hyperboliques de dimension $$3$$ , Ann. of Math. (2) 124 (1986), 71–158. JSTOR: · Zbl 0671.57008 · doi:10.2307/1971388 · links.jstor.org [9] B. H. Bowditch, Intersection numbers and the hyperbolicity of the curve complex , J. Reine Angew. Math. 598 (2006), 105–129. · Zbl 1119.32006 · doi:10.1515/CRELLE.2006.070 [10] J. F. Brock, Continuity of Thurston’s length function , Geom. Funct. Anal. 10 (2000), 741–797. · Zbl 0968.57011 · doi:10.1007/PL00001637 [11] J. F. Brock and K. W. Bromberg, On the density of geometrically finite Kleinian groups , Acta Math. 192 (2004), 33–93. · Zbl 1055.57020 · doi:10.1007/BF02441085 [12] J. F. Brock, R. D. Canary, and Y. N. Minsky, The classification of Kleinian surface groups, II : The ending lamination conjecture, · Zbl 1253.57009 · arxiv.org [13] R. D. Canary, A covering theorem for hyperbolic $$3$$ -manifolds and its applications, Topology 35 (1996), 751–778. · Zbl 0863.57010 · doi:10.1016/0040-9383(94)00055-7 [14] R. D. Canary, D. B. A. Epstein, and P. Green, “Notes on notes of Thurston” in Analytical and Geometric Aspects of Hyperbolic Space (Coventry/Durham, England, 1984) , London Math. Soc. Lecture Note Ser. 111 , Cambridge Univ. Press, Cambridge, 1987, 3–92. · Zbl 0612.57009 [15] C. J. Earle and J. Eells, A fibre bundle description of Teichmüller theory , J. Differential Geometry 3 (1969), 19–43. · Zbl 0185.32901 [16] A. Eskin, H. Masur, and A. Zorich, Moduli spaces of abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants , Publ. Math. Inst. Hautes Études Sci. 97 (2003), 61–179. · Zbl 1037.32013 · doi:10.1007/s10240-003-0015-1 · numdam:PMIHES_2003__97__61_0 · eudml:104192 [17] B. Farb and L. Mosher, Convex cocompact subgroups of mapping class groups , Geom. Topol. 6 (2002), 91–152. · Zbl 1021.20034 · doi:10.2140/gt.2002.6.91 · emis:journals/UW/gt/GTVol6/paper5.abs.html · eudml:122881 [18] A. Fathi, F. Laudenbach, and V. PoéNaru, eds., Travaux de Thurston sur les surfaces , reprint of the 1979 ed., Astérisque 66–67, Séminaire Orsay, Soc. Math. France, Montrouge, 1991. [19] D. Gabai, Almost filling laminations and the connectivity of ending lamination space , Geom. Topol. 13 (2009), 1017–1041. · Zbl 1165.57015 · doi:10.2140/gt.2009.13.1017 [20] F. P. Gardiner, Teichmüller Theory and Quadratic Differentials , Pure Appl. Math. (N.Y.), Wiley-Interscience, New York, 1987. · Zbl 0629.30002 [21] U. HamenstäDt, “Train tracks and the Gromov boundary of the complex of curves” in Spaces of Kleinian Groups , London Math. Soc. Lecture Note Ser. 329 , Cambridge Univ. Press, Cambridge, 2006, 187–207. · Zbl 1117.30036 [22] J. Hubbard and H. Masur, Quadratic differentials and foliations , Acta Math. 142 (1979), 221–274. · Zbl 0415.30038 · doi:10.1007/BF02395062 [23] S. P. Humphries, “Generators for the mapping class group” in Topology of Low-Dimensional Manifolds (Chelwood Gate, England, 1977) , Lecture Notes in Math. 722 , Springer, Berlin, 1979, 44–47. · Zbl 0732.57004 · doi:10.1007/BFb0063188 [24] N. V. Ivanov, Subgroups of Teichmüller Modular Groups , revised by the author, trans. by E. J. F. Primrose, Trans. Math. Monogr. 115 , Amer. Math. Soc., Providence, 1992. · Zbl 0776.57001 [25] R. P. Kent Iv and C. J. Leininger, Shadows of mapping class groups: Capturing convex cocompactness , Geom. Funct. Anal. 18 (2008), 1270–1325. · Zbl 1282.20046 · doi:10.1007/s00039-008-0680-9 [26] R. P. Kent Iv, c. j. leininger, and S. Schleimer, Trees and mapping class groups , · arxiv.org [27] S. Kerckhoff, H. Masur, and J. Smillie, Ergodicity of billiard flows and quadratic differentials , Ann. of Math. (2) 124 (1986), 293–311. JSTOR: · Zbl 0637.58010 · doi:10.2307/1971280 · links.jstor.org [28] E. Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller space , preprint, 1999. · Zbl 1011.30035 [29] R. S. Kulkarni and P. B. Shalen, On Ahlfors’ finiteness theorem , Adv. Math. 76 (1989), 155–169. · Zbl 0684.57019 · doi:10.1016/0001-8708(89)90046-7 [30] C. J. Leininger, m. mj, and S. Schleimer, The universal Cannon-Thurston map and the boundary of the curve complex , · Zbl 1248.57003 · arxiv.org [31] A. Marden and K. Strebel, The heights theorem for quadratic differentials on Riemann surfaces , Acta Math. 153 (1984), 153–211. · Zbl 0577.30038 · doi:10.1007/BF02392377 [32] H. A. Masur and Y. N. Minsky, Geometry of the complex of curves, I: Hyperbolicity , Invent. Math. 138 (1999), 103–149. · Zbl 0941.32012 · doi:10.1007/s002220050343 [33] D. Mccullough, Compact submanifolds of $$3$$ -manifolds with boundary, Quart. J. Math. Oxford Ser. (2) 37 (1986), 299–307. · Zbl 0628.57008 · doi:10.1093/qmath/37.3.299 [34] Y. N. Minsky, “Curve complexes, surfaces and 3-manifolds” in International Congress of Mathematicians, Vol. II , Eur. Math. Soc., Zürich, 2006, 1001–1033. · Zbl 1099.57012 [35] -, The classification of Kleinian surface groups, I: Models and bounds , · Zbl 1193.30063 · doi:10.4007/annals.2010.171.1 · annals.princeton.edu [36] M. Mitra, Cannon-Thurston maps for hyperbolic group extensions , Topology 37 (1998), 527–538. · Zbl 0907.20038 · doi:10.1016/S0040-9383(97)00036-0 [37] L. Mosher, Train track expansions of measured foliations , preprint, 2003. [38] J.-P. Otal, The Hyperbolization Theorem for Fibered $$3$$-Manifolds , translated from the 1996 French original by Leslie D. Kay, SMF/AMS Texts Monogr. 7 , Amer. Math. Soc., Providence, 2001. · Zbl 0986.57001 [39] K. Rafi and S. Schleimer, Curve complexes with connected boundary are rigid , · Zbl 1227.57024 · arxiv.org [40] G. P. Scott, Compact submanifolds of $$3$$ -manifolds, J. London Math. Soc. (2) 7 (1973), 246–250. · Zbl 0266.57001 · doi:10.1112/jlms/s2-7.2.246 [41] W. P. Thurston, Three-Dimensional Geometry and Topology, Vol. 1 , Princeton Math. Ser. 35 , Princeton Univ. Press, Princeton, 1997. · Zbl 0873.57001 [42] -, The geometry and topology of, $$3$$-manifolds , Princeton lecture notes, 1979. [43] -, Hyperbolic structures on $$3$$-manifolds, II: Surface groups and $$3$$-manifolds which fiber over the circle , · arxiv.org
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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 7th 2012
• (edited Jan 7th 2012)

I have edited fibration

• promted by an email question I have added more information on when the pullback of a fibration is a homotopy pullback;

• in the discussion of “transport” in topological spaces I added a pointer to Flat ∞-parallel transport in Top which gives details;

• I fixed a mistake where quasifibration was mentioned and pointed to, but, fibration in the Joyal model structure was meant (despite the previous warning of exactly this trap…)

• I added a subsection “Related concepts”

FInally, I noticed that the following old discussion was sitting there, which hereby I move fromthere to here

begin forwarded discussion

+–{.query} Tim: I do not quite agree with ’transport’ as being the main point of fibrations. Rather ’lifting’ is the main point, in particular lifting of homotopies, at least in topological situations. For transport, one needs connections of some sort to get things working well, but in many cases there is only a very weak notion of action, so perhaps that should be derived as a property rather than taken as a ’defining property’ in some sense.

Perhaps a reference to Stasheff and Wirth

James Wirth & Jim Stasheff

Homotopy Transition Cocycles

math.AT/0609220.

and the discussion

http://golem.ph.utexas.edu/category/2006/09/wirth_and_stasheff_on_homotopy.html

on the cafe would be a good idea to add.

Urs: In situations where one wants to talk of transport, the fibration usually arises as the pullback of some “universal fibration”, a generalized universal bundle. For instance (split op-)fibrations of categories are precisely the pullbacks of the universal $Cat$-bundle $Cat_* \to Cat$ along a functor $F : C \to Cat$.

If one looks at this kind of situation where we do have an established notion of (parallel) transport one sees:

• it is the classifying functor $F : C \to Cat$ which should be addressed as the “(parallel) transport”, while the corresponding fibration is its “action object” as in action groupoid, i.e. the thing whose objects are all possible things that the parallel transport can transport and whose morphisms take these things to the image of that transport. So it’s a subtle difference, but an important one.

For instance, to make this more concrete, consider the category of smooth groupoids (which is a category of fibrant objects), let for any manifold $X$ the groupoid $P_1(X)$ be the groupoid of smooth thin-homotopy classes of paths in $X$, let $G$ be any Lie group, $\mathbf{B} G$ the corresponding one-object Lie groupoid and consider the _universal fibration _ $\mathbf{E} G \to \mathbf{B}G$ – the groupoid incarnation of the universal $G$-bundle as described at generalized universal bundle. Then

Theorem: $G$-bundles with connection on $X$ are equivalent to functors $tra : \widehat{P_1(X)} \to \mathbf{B}G$ out of acyclic fibrations $\widehat{P_1(X)} \to P_1(X)$ over $P_1(X)$ (i.e. smooth anafunctors $P_1(X) \to \mathbf{B}G$). These functors are literally the corresponding “parallel transport”: indeed, evaluated on a path $\gamma$ in $X$ there is locally a 1-form $A \in \Omega^1(X, Lie(G))$ such that the group element $tra(\gamma)$ is the traditional parallel transport of that 1-form, $tra(\gamma) = P exp(\int_\gamma A)$.

Now, we can form the fibration which is associated with this parallel transport, namely the pullback

$\array{ tra^* \mathbf{E} G &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ \widehat{P_2(X)} &\stackrel{tra}{\to}& \mathbf{B}G \\ \downarrow \\ P_2(X) } \,.$

This fibration $tra^* \mathbf{E}G \to \widehat{P_2(X)}$ is what is properly speaking the action groupoid of $tra$ acting on the fibers of the principal $G$-bundle.

Mike: Can you clarify the distinction between “lifting” and “transport”? In what way does the lifting of a path $f$ starting at a point $e$ not transport $e$ along $f$? Certainly in geometric situations to get a parallel notion of transport, you need a connection, but I see that as a stronger requirement.

forwarded discussion continued in next entry

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJan 7th 2012

continuation of forwarded discussion

Mike adds: In fact, any topological fibration over $X$ (not just a bundle) induces a “transport” map from (the fundamental $\infty$-groupoid of) $X$ to the $(\infty,1)$-category $Top$, taking each point to its fiber. Putting extra structure on the fibration, such as making it a $G$-bundle, corresponds to restricting the codomain of the transport functor to land in some smaller subcategory, such as $\mathbf{B}G$. A reference in classical homotopy theory is May’s “Classifying spaces and fibrations.”

I agree with what Urs said that the “transport” properly refers to the corresponding functor into $Top$, $Cat$, or $\mathbf{B}G$, with the fibration being the “action object.” Then the property of “being a fibration” means “can be equipped with a transport” or “can be obtained from a transport.” These are all important points to be added, but I think that what I wrote is still correct.

-Tim The transport aspect of course is there, but as the lifting is not unique the idea of the transport being ’up to coherent homotopy’ or ’$\infty$-homotopy is much more sophisticated than the simple point of the existence of a ’lifting’. My point is that although correct your definition may be accused of being ’mathematics made difficult’. To have a definition of fibration that somehow requires one to have understood $(\infty,1)$-categories and the fundamental $\infty$-groupoid of a space is, to me, the wrong way around. The simply stated lifting property leads eventually naturally to the beautiful transport description (and is more general, more classically based, and probably more approachable). That process is only recently understood and is still only understood by a handful of people who work with fibrations.

If you look at the classical treatment of fibre spaces (and I found a copy of Grothendieck’s 1950s notes on them in a pile of stuff today), there is no thought of transport, just of lifting. Serre, who revolutionised the algebraic topology of fibre spaces and fibrations is using lifting, not transport. The link between the transport functor and the notion of fibration emerged via the ’Grothendieck construction’ (due to Ehresmann as well.. and there transport was part of the story that he wanted) but was part of the image of fibre spaces\fibrations as being like semi-direct products and that took a long time to get to the present state.

Mike: Well, I think a lot of what we do here could be accused of being “mathematics made difficult.” Cf. the first sentence of group. But I believe that it is useful and important to have a correct conceptual understanding, in addition to any technically simpler definition that one works with in practice.

For instance, I believe that the “real” definition of a monoidal category requires that “all diagrams of constraints commute,” while the fact that it’s enough to check two particular diagrams is a happy accident. Presenting the pentagon and unit axioms as the God-given definition of a monoidal category, without mentioning that the only reason this is an okay definition is because you can prove a coherence theorem from it, leads the student quite naturally to wonder why God likes pentagons.

Likewise, I believe that the only reason the definition of a fibration via lifting properties is a correct definition is that it does lead to the complete functoriality up to higher homotopies. (One doesn’t need complicated notions of $\infty$-categories to get an intuitive feel for how this works: it’s enough to observe that lifting of 2-cells gives you homotopy-uniqueness for lifts of paths, lifting of 3-cells gives you homotopy-uniqueness for lifts of 2-cells, etc.) The fact that it took many years for this to be understood is not, to me, an argument against its truth, or against its expository helpfulness.

Also, I wrote this page not to be just about the topological notion of fibration, but to include categorical (Grothendieck) fibrations as well and show the commonality between them. And for that I think the notion of transport is indispensable, since categorical fibrations can’t be defined by a simple lifting property.

fowarded discussion to be continued in the next comment

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJan 7th 2012

continuation of forwarded discussion

Mike, again: Okay, I basically rewrote the whole page, trying to take your comments into account. What do you think of it now?

Tim : It looks good. My objection was, sort of, pedagogical as I did not want to put off the reader who might be wanting to understand fibrations from the infinity cat point of view. You nicely ease the way in, suggesting how the understanding of the basic classical notion evolves. I like it. Thanks.

PS. Wikipedia entries for fibration and fibred category exist and could be used if you think they are consistent enough with you wishes for this.

Urs: thanks, nice entry. Eventually it would be nice if we could say more about which fibrations are fibrations: which notion of fibred ($n$-)categories/groupoids are fibrations with respect to which model structure. I once tried to collect a bit of data on this, but I need to dig out my old notes, and they were likely very incomplete anyway.

Ronnie: I made a few minor changes, but I also find it odd to define a fibration in terms of a hypothetical fundamental $\infty$-groupoid of $B$. What is this animal?

With regard to transport I like the old papers of Philip R. Heath on operations and fibrations-

Heath, Philip R. Groupoid operations and fibre-homotopy equivalence. I. Math. Z. 130 (1973), 207–233. (the first of 3 papers)

If $p:E \to B$ is a fibration then there is an operation of the paths on the base on the fibres the laws of an operation holding up to homotopy.

At Hull, Phil and I were amused by the fact that the unit interval $I=[0,1]$ is a cogroupoid up to homotopy where homotopy is defined using $I$!. This led to his thesis work on fibrations, getting away from the base point approach of the loops on the base operates on the fibre. Maybe effort should be put in to defining and constructing an $\infty$-cogroupoid (up to homotopy)!

=–

end of forwarded discussion

• CommentRowNumber4.
• CommentAuthorjim_stasheff
• CommentTimeJan 7th 2012
I beg to differ with some of this.

@Urs
In situations where one wants
to talk of transport, the fibration usually arises as the
pullback of some "universal fibration", a
generalized universal bundle.

Yes, but that is usually a theorem not a definition. A fibration is (almost always) defined as a map such that...

@Urs it is the classifying functor
F:C→Cat which should be addressed as the "(parallel) transport",

why confuse related ideas by naming them all with the same address'
cf the damage created by multiple equivalent defintiions of connection'

while the corresponding fibration is its "action object" as in action groupoid, i.e. the thing whose objects are all possible things that the parallel transport can transport and whose morphisms take these things to the image of that transport. So it's a subtle difference, but an important one.

which is why to be careful about names

These functors are literally

NOT literally

the corresponding "parallel transport": indeed, evaluated on a path
γ in
X there is locally a 1-form
A∈Ω 1(X,Lie(G))

Notice you have now abandoned topolgoical fibrations by switching to 1-forms such that the group element
tra(γ) is the traditional parallel transport of that 1-form,
tra(γ)=Pexp(∫ γA).

Now, we can form the fibration which is associated with this parallel transport, namely the pullback
tra *EG → EG ↓ ↓ P 2(X)̂ →tra BG ↓ P 2(X).

Can do, but why invoke the universal as oppose to working directly over X?

@Mike: Can you clarify the distinction between "lifting" and "transport"? In what way does the lifting of a path
f starting at a point e not transport e along f? Certainly in geometric situations to get a parallel notion of transport,
you need a connection, but I see that as a stronger requirement.

Transport as a verb - you are right but as a noun it (usually) refers to a lifting with certain properties.
cf. "can be equipped with a transport" or "can be obtained from a transport."
Yes, geometry needs more than topology.

@Tim: My point is that although correct your definition may be accused of being 'mathematics made difficult'. To have a definition of fibration that somehow requires one to have understood
(∞,1)-categories and the fundamental
∞-groupoid of a space is, to me, the wrong way around. The simply stated lifting property leads eventually naturally to the beautiful transport description (and is more general, more classically based, and probably more approachable).

Amen!

@Mike But I believe that it is useful and important to have a correct conceptual understanding, in addition to any technically simpler definition that one works with in practice.

Yes, we need to walk on 2 legs
• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJan 7th 2012

Jim., this is some very old discussion that I moved over here only because we agree’d to do so, instead of deleting it.

Is there anything in the current entry fibration that you feel needs to be changed? Let’s talk about that.

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeSep 20th 2019

Looking at this page I was surprised to see there’s no mention of type theory, so I added a section to connect to dependent types.

There’s no doubt loads more tying in one could do.

• CommentRowNumber7.
• CommentAuthorjesuslop
• CommentTimeJan 11th 2020

In section 4, “Fibrations in Category Theory”, Grothendiek and Cartesian fibrations are treated on the same footing saying that the fibration functor $p:E \to B$ produces a pseudofunctor $B \to Cat$, but I think that in the Cartesian case the functors should be $(\infty, 1)$.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeJan 12th 2020

I think that whoever wrote this paragraph was using all four terms to mean the same thing. It’s somewhat of an accident of history that our page Grothendieck fibration is about the 2-categorical version and our page Cartesian fibration is about the $(\infty,1)$-categorical one; there’s nothing intrinsically more “Grothendieck” about the 2-categorical version or “Cartesian” about the $(\infty,1)$-categorical one. It might be better if the page currently called “Cartesian fibration” were renamed something like “Cartesian fibration of quasi-categories”. But I agree, because of that accident, the current phrasing on the page fibration is potentially confusing.


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# Conservapedia:Godspeed

(Redirected from Godspeed)
 This Conservapedia-related article is of largely historical interest and is no longer the focus of RationalWiki today.Conservapedia (and religious fundamentalism to an extent) was a major focal point in the early history of RationalWiki, but long ago ceased coming up with new ways to appall and amuse.Our energies are now spent debunking other, fresher examples of pseudoscientific claims, authoritarianism, and deceit.For RationalWiki's less ancient content, try the Best of RationalWiki.
For those living in an alternate reality, Conservapedia has an "article" about Godspeed
 “”Whosoever transgresseth, and abideth not in the doctrine of Christ, hath not God ... If there come any unto you, and bring not this doctrine, receive him not into your house, neither bid him Godspeed: For he that biddeth him Godspeed is partaker of his evil deeds. —2 John 1:9-11, King James Version

Godspeed is a form of salutation, originating in the Middle English words for "May God prosper you" and nowadays meaning "Good Luck" or "Safe Journey." However, when a conservative says Godspeed to a liberal, it is generally believed to translate, roughly, as "Go Fuck Yourself." In some cases, the transmitted meaning seems to be more along the lines of "I acknowledge your existence, peon. Now go contribute more." There is a 98.7459964% chance that the above statements are true, 3.14159% chance of falsehood and an approximately 1% surplus that will be contributed to the national debt.

## Examples

Andy Schlafly - Official Master of Godspeed

Example of the "good luck" meaning

Example of the "neutral" meaning

Example of the "Go fuck yourself" meaning

• Aschlafly to JohnSmith: "Thanks, but I'll be debating others here who don't embrace falsehoods. Godspeed in your efforts to find someone else to debate you."[3]
• Aschlafly to RSchlafly (!!): "Would any mathematician embrace a flawed proof because it is better than other flawed attempts to prove the same theorem? I don't think so. Godspeed."[4]
• TK to Baruch: "That won't be necessary. Godspeed, Baruch, Godspeed!" [5]
• TK to ClementB: "As the owner of this site that is indeed his right, his prerogative, and we have indeed differed before, and will undoubtedly do so again. In the meantime, I wish you Godspeed." [6]

## Origins

The term derives from an ancient Icelandic saga explaining the source of rain and rivers. It was originally titled "When the Gods Peed".[citation NOT needed]

## As an actual speed

Since the emergence of the theory of relativity[7], the term may be referring to one of the ways[8] God used to dilate time. The purpose of this usage includes delaying the Second Coming, and fitting the age of the universe into a 6000 years timeframe.

Depending the assumptions used it gives 299792457.998875 m/s to 299792457.999987 m/s, compared to the speed of light at exactly 299792458 m/s.

Currently, only one thing is known to move at Godspeed, and that's bullshit.

### Calculating Godspeed

Godspeed can be obtained using special relativity, using the following method[9].

God and the Earth are travelling at Godspeed relative to the rest of the universe[10], and therefore the differences in age between the actual age of the universe and God's ideas on the subject can be corrected by special relativity. The effect of gravitational time dilation (general relativity) is assumed to be negligible; See our article on white hole cosmology for an explanation of this phenomenon. To determine Godspeed we have several approaches:

1. Fit the age of the universe (13.7 billion years, ± 0.12 billion years) to a framework starting 4004 BC (6020 years, give or take 100 years)
2. Adjusted version of 1.: We have to assume that the last ~2000 years (Up until birth of Jesus Christ) are already in slowed down state in order for the scientists not to find too much discrepancies about it, so the framework to fit in is only 4004 years
3. From passages of the bible, particularly the following phrase:
2 Peter 3:8
But, beloved, be not ignorant of this one thing, that one day is with the Lord as a thousand years, and a thousand years as one day.

### Results

Godspeed as a correction factor is very close to the speed of light (299792458 m/s[14]), and the difference is 1 part in 1012(< 1.2 mm/s) to 1014 (< 13 μm/s) depending on the adjustment used.

Apparently Peter is only one order of magnitude off[15] (Astonishing when you don't even have telescopes, redshifts, etc.) when determining how long a day for God should be, in order to adjust for the young earth. (Expected response from Peter: Dang it, if only I have written "ten thousand years" instead of "a thousand years" I would have saved more people with the Bible code)

### Technically speaking

To the atheist/agnostic, godspeed is the equivalent of making no change in velocity relative to the surroundings[16]. When arguing with a creationist (on Conservapedia or elsewhere), the discussion is likely to go absolutely nowhere as well.

## Other uses

Andy enjoys a good Godspeed in any medium

The term Godspeed appears in the name of the Anarchist Canadian band Godspeed You! Black Emperor. However, the band's name is from the Japanese documentary of the same name.

Godspeed is a graphic novelisation of the life of Kurt Cobain.

Said by Mission Control both times John Glenn lifted off into space.

Godspeed also refers to the rush one gets when, instead of simply snorting one's communion powder, one melts it and shoots... it... into... one's... Aaaaaaaaah... v e  i   n    s

"God's peed" is an unfalsifiable statement. Its truth depends on how similar to his image you believe man is.

The less said about God's Speedo™, the better, especially in regards to Exodus 33:23.

7. Julian year (365.25 days; used in astronomy) is used in this calculation; Using Gregorian calendar gives 365242.5 on average (97 extra days from leap years every 400 years) and ${\displaystyle 1-{\frac {v}{c}}\approx 3.748113\times 10^{-12}}$ (v in m/s is the same for the first 15 significant figures)
9. For those who is interested in coming up with extra bible code, it's approximately ${\displaystyle e^{\sqrt {5}}}$


text:
# A flatness result of Fiedorwicz for amalgamated free products of monoids in connection with classifying spaces of monoids

In Lemma 5.2(a) of Z. Fiedorowicz, Classifying Spaces of Topological Monoids and Categories American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 301-350 the author proves the following.

Lemma 5.2(a) Suppose that $\{M_i\}_{i\in I}$ is a collection of monoids with a common submonoid $W$ such that the monoid ring $\mathbb ZM_i$ is flat as a left $\mathbb ZW$-module for each $i\in I$. Let $M$ be the amalgamated free product (or pushout) of the $M_i$ over $W$. Then $\mathbb ZM$ is flat as a left $\mathbb ZM_j$-module for each $j\in I$.

His proof is one line, but I cannot understand it. I would very much love to use this result. I'm a bit nervous about it because the proof he sketches would seem to work in the category of rings, or more generally $k$-algebras over a commutative ring with unit $k$, but papers I saw of Cohn on amalgamated free products of rings seem to involve much more complicated colimits diagrams than Fiedorwicz is using. Moreover, Warren Dicks pointed out to me an example of an amalgamated free product of $k$-algebras, with $k$ a field, where the factors are flat over the subring but the amalgamated free product is not flat over the factors. He derives it from the paper https://arxiv.org/abs/math/0205034.

I would be happy for a proof I can understand in the case of two factors. I did try to contact the author, but he is retired and didn't respond. I now reproduce his proof.

Proof. This is immediate because $\mathbb ZM$ is the direct limit of

$$\mathbb ZM_j\otimes_{\mathbb ZW} \mathbb ZM_{i_1}\otimes_{\mathbb ZW} \mathbb ZM_{i_2}\otimes_{\mathbb ZW}\cdots \otimes_{\mathbb ZW}\mathbb ZM_{i_k}$$ $i_1,\ldots, i_k\in I$ as a left $\mathbb ZM_j$-module and hence is flat over $\mathbb ZM_j$.

What I don't understand is exactly what is the directed (or filtered) diagram that this direct limit is indexed over (i.e., what are the maps). Each of the terms is flat, so if $\mathbb ZM$ is such a direct limit, then I am fine with the proof. There is a map from each of these iterated tensors to $\mathbb ZM$ induced by multiplication (and the inclusions'' into the pushout) but I don't see how to organize this into a direct limit (=filtered colimit).

• I notice one of the tensor symbols is missing a ZW. Is this indeed a faithful reproduction, and does the omission have significance? Gerhard "Something Does Not Look Right" Paseman, 2018.04.11. – Gerhard Paseman Apr 11 '18 at 15:22
• Also (not that I understand it that well) M is described as a kind of pushout and ZM is a monoid ring over M. Is that not enough to pin down what ZM is? Gerhard "Not Feeling Very Decorative Presently" Paseman, 2018.04.11. – Gerhard Paseman Apr 11 '18 at 15:27
• @GerhardPaseman, I fixed the typo of the missing ZW. ZM is the ring pushout. This is why I think if the argument is correct it should apply to rings. Explicitly describing ring pushouts seems complicated. For example Mark Sapir showed amalgamated free products of finite monoids can have undecidable word problem so there is no nice normal form in general. Of course flatness is a strong condition. – Benjamin Steinberg Apr 11 '18 at 16:00

This appears to be false to me (unless maybe there are commutativity hypotheses), unless I've made a mistake below. (This is based on an example that I saw Andrew Ranicki give, a number of years ago, about non-exactness of Cohn localization.)

Consider the diagram of monoids $$\Bbb Z \leftarrow \Bbb N \rightarrow \Bbb N \ast \Bbb N$$ where the right-hand arrow is the inclusion of the first factor into the free product. On taking monoid algebras, this is a diagram of rings $$\Bbb Z[x^{\pm 1}] \leftarrow \Bbb Z[x] \rightarrow \Bbb Z\langle x,y\rangle.$$ The left factor is clearly flat over $\Bbb Z[x]$ and the right factor is free on the basis $$\{1\} \cup\{y\cdot w \mid w \text{ is a monomial in }x,y\}.$$

Thus both factors are flat left modules. The monoid algebra on the pushout $M$ is the ring $\Bbb Z\langle x^{\pm 1}, y\rangle$ by a universal property argument.

However, consider the exact sequence of left $\Bbb Z\langle x,y\rangle$-modules $$0 \to \Bbb Z\langle x,y\rangle \oplus \Bbb Z\langle x,y\rangle \xrightarrow{[x,y]} \Bbb Z\langle x,y\rangle \to \Bbb Z \to 0,$$ where the first map sends $(a,b)$ to $ax+by$ and the second sends $x$ and $y$ to zero. Tensoring on the left with $\Bbb Z\langle x^{\pm 1}, y\rangle$ gives the sequence $$0 \to \Bbb Z\langle x^{\pm 1},y\rangle \oplus \Bbb Z\langle x^{\pm 1},y\rangle \xrightarrow{[x,y]} \Bbb Z\langle x^{\pm 1}, y\rangle \to 0 \to 0.$$ This is not left exact, because the first factor of the direct sum is taken isomorphically to the middle term by invertibility of $x$.

• Are you making $\mathbb Z$ a module over $\mathbb Z\langle x,y\rangle$ by having $x,y$ annihilate it? – Benjamin Steinberg Apr 12 '18 at 2:42
• It looks ok to me. Are there stronger hypotheses that would yield flatness besides freeness assumptions? – Benjamin Steinberg Apr 12 '18 at 2:52
• Yes, $x$ and $y$ annihilate $\Bbb Z$. I'm not sure what additional assumptions to give: the example I give here is of a localization, and if one can show that it is an Øre localization instead then you should get flatness. Ideally there would be a hypothesis that covers both the free case and the Øre localization case. – Tyler Lawson Apr 12 '18 at 3:00
• It is really quite shocking that the failure is so extreme. I never tried the case where one inclusion is a free factor and the other is a group completion of a cancellative commutative monoid. I would have thought such a nice case would work. – Benjamin Steinberg Apr 12 '18 at 13:08
• @BenjaminSteinberg Indeed, I find this example shocking not only because it is so simple but because it is so severe. In fact, this argument shows that if $f\colon \Bbb Z\langle x,y\rangle \to R \neq 0$ is a ring homomorphism and $x$ is a unit in $R$, then $f$ is not flat. – Tyler Lawson Apr 12 '18 at 13:37