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Contribute to SSAGES¶

The SSAGES project is built on an inclusive and welcoming group of physicists, chemists, and chemical engineers working on complex Molecular Dynamics simulations employing Metadynamic techniques. Metadynamics is an exciting and fast developing field and similarly this project is designed to facilitate the usage and implementation of a wide array of Metadynamics methods. And we welcome you heartily to join us and to embark with us on this great adventure.

There are many ways to contribute to SSAGES and you do not necessarily need programming skills to be part of this project (even though they surely help). But, if you decide to work on the code base, you will be happy to find that SSAGES is designed to be easy to use and is just as easy to extend. We put a high priority on maintaining a readable and clearly structured code base as well as an inclusive community welcoming new ideas and contributions.

Here is a short summary of ideas how you can become part of SSAGES:

Reporting, Triaging, and Fixing Bugs
No software is without errors, inconsistencies, and strange behaviors. Even with zero programming knowledge, you can help tremendously by reporting bugs or confirming issued bugs. Read more...
Improving the SSAGES documentation
SSAGES would like to have a detailed yet comprehensive documentation on what it does and how it does it. This should include concise introductions to the methods, quick to learn tutorials, complete coverage of the nooks and crannies of each method, and of course helpful pointers in case you run into errors. And while the documentation is already expansive, improvements on it never go unappreciated. Read more...
Including your Method and CV in SSAGES
You have developed a new Metadynamics scheme or a Collective Variable and want to make it available to the community via SSAGES? Great! Read more...
Working on the core SSAGES system
If you would like to climb into the heart of SSAGES and get your hands dirty, this task is for you. Read more...

Todo

Improving the Documentation¶

Great documentation and great code produces great software. -SSAGE advice

Improvements on the documentation are always highly appreciated. The SSAGES documentation is split into two parts: The User Manual (which you are reading right now), and the API documentation. While the Manual uses the Sphinx documentation and contains all information necessary to use the program, the API docs are bulit on Doxygen and describe the usage of the underlying classes and functions for everyone willing to extend and improve SSAGES.

Here are a few ideas on how you can help:

• Fix typos: Even though we have thoroughly checked, there are certainly still a few hidden somewhere.
• Check if all internal and external links are working.
• Make sure that the documentation is up to date, i.e. that it reflects the usage of the latest version.
• Add examples: An examples on how to use a method, avoid a common problem, etc. are more helpful than a hundred pages of dry descriptions.
• Write a tutorial.

Building the documentation¶

Before you can work on the documentation, you first have to build it. The documentation is part of the SSAGES source code. It is assumed that you have already downloaded and built the source code as described in the Getting Started section. You will find a collection of rst files comprising the User Manual under doc/source/ where the file ending rst stands for ReStructured Text. The API documentation on the other hand resides directly in the header files right next to the classes and functions they describe.

Assuming you have already built SSAGES, building the documentation is as easy as typing make doc in your build directory. In order to make correctly check that you have the following programs installed:

• Sphinx (with PyPI via pip install Sphinx for example)
• Doxygen
• dot (in Ubuntu this is part of the graphViz package)
• Sphinx “Read the docs” theme (via pip install sphinx_rtd_theme)

Once you have successfully built the documentation you will find the User Manual under doc/Manual/ and the API documentation under doc/API-doc/html/ (relative to your build directory - do not confuse it with the doc/ folder in the main directory of the project). To view it in your favorite web browser (using FireFox as an example) just type

firefox doc/Manual/index.html

for the User Manual or

firefox doc/API-doc/html/index.html

for the API documentation.

How to write documentation¶

Here are a few pointers on how to write helpful documentation, before we dive into the details of Sphinx and Doxygen for the User Manual and the API documentation:

• Write documentation “along the way”. Do not code first and write the documentation later.

• Use helpful error messages. These are considered part of the documentation and probably are the part that is read most frequently.

• Do everything you can to structure the text. Let’s face it: Most people will just skim the documentation. Feel encouraged to use all techniques that help to spot the relevant information, for example:

• Format your text bold, italic, code, etc.
• Write in short paragraphs, use headers
• Use lists, code blocks, tables, etc.

Note

These Note blocks are extremely helpful for example.

Warning

Warnings work great, too!

Here you can find more examples for helpful Sphinx markup: http://www.sphinx-doc.org/en/stable/markup/para.html

• Use examples, a lot of them

• In the initial stages: Don’t be a perfectionist. Missing documentation is the worst kind of documentation. “It is better to have written and coded than to have never written at all.” -SSAGE advice

How to write Sphinx¶

The Sphinx documentation system uses ReStructured text which is loosely based on the markdown format. Examples for documentations written with Sphinx include:

The following tutorials are extremely helpful:

One of the great things of Sphinx is that most documentations have a “view page source” link where you can take a look at the Sphinx source code. Thus, the best way to learn Sphinx is to click on this link right now and look at the source code of this page. But here is a short summary of the most important commands:

• Markup: You can use *italic*, **bold**, and code for italic, bold and code.
• Headers. Underline your headers with at least three === for titles, --- for subtitles, ^^^ for subsubtitles and ~~~ for paragraphs.
• Bullet lists are indicated by lines beginning with *.

Note

These highlighted blocks can be created with .. note::. The content of this block needs to be indented. You can also use warning and seealso. Even more can be found here.

How to write Doxygen¶

Doxygen follows a very different philosophy compared to Sphinx and is more steered towards API documentation, exactly what we use it for in SSAGES. Instead of maintaining the documentation separate from the source code, the classes and functions are documented in the same place where they are declared: The header files. Doxygen then reads the source code and automatically builds the documentation. Examples for documentation created with Doxygen:

The mainpage of the Doxygen documentation is written in a separate header file, in our case doc/mainpage.h. A good introduction to the Doxygen syntax can be found at

The basic rule is that Doxygen comments start with //! or /*! and document the class, namespace or function that directly follows it. Let’s start with a short example:

//! Function taking the square of a value
/*!
* \param val Input value
* \returns Square of the input value
*
* This function calculates the square of a given value.
*/
double square(double val)
{
return val*val;
}

This example documents the function square() which simply calculates the square of a number. The first line, starting with //!, is the brief description and should not be longer than one line. The second comment block, starting with /*! is the full description. Here, two special commands are used:

\param
This command documents one parameter of the function
\returns
This command documents the return value of the function

There are many special Doxygen commands. They all start with a backslash and the most important, apart from the two mentioned above, are:

\tparam
Used to document a template parameter.
\ingroup
This class is part of a group, such as Methods or Core. The groups are defined in doc/mainpage.h.

Helpful are also boxes highlighting a given aspect of the function, such as:

\attention
Puts the following text in a raised box. A blank line ends the attention box.
\note
Starts a highlighted block. A blank line ends the note block.
\remark
Starts a paragraph where remarks may be entered.
\see
\deprecated
The documented class or function is deprecated and only kept for backwards compatibility.
\todo
Leave a ToDo note with this command.

You can also highlight your text:

\em
For italic word. To highlight more text use <em> Highlighted text </em>.
\b
For bold text. To highlight more text use <b> Bold text </b>.
\c
For typewriter font. To have more text in typewriter font, use <tt>Typewriter Font</tt>.
\code
Starts a code block. The block ends with \endcode.
\li
A line starting with \li is an entry in a bullet list.

Another big benefit of doxygen is that you can use a lot of LaTeX syntax. For example:

\f$Starts and ends an inline math equation, similar to$ in Latex.
\f[ and \f]
Start and end a display-style LaTeX equation.
\cite <label>
Cite a reference. The references are listed in doc/references.bib and follow the BibTex syntax.

Doxygen is very clever in producing automatic links. For example, there exists a class Method in SSAGES. Thus, Doxygen automatically creates a link to the documentation of this class where the word “Method” appears. This does, however, not work for the plural, “Methods”. Instead, you can write \link Method Methods \endlink. On the other hand, if you want to prevent Doxygen from creating an autolink, put a % in front of the word.

What to document¶

We are aiming for a comprehensive documentation of all the methods available in SSAGES as well as the core features. Thus, for each method the documentation should include

• An introduction into the method, what it does and how it does it.
• A short tutorial based on one of the working examples. The reader should be able to complete the tutorial in ~30min and should leave with a sense of accomplishment, e.g. a nice energy profile or a picture of a folded protein.
• A detailed description on how to use the method, the parameters, constraints, requirements, etc.

See here for an introduction to how to develop your own method.

So, you have developed a new Metadynamics method or a new collective variable (CV)? Great! SSAGES is about collaboration and integrating your new CV or method is a priority. But before we do that, make sure you check the following boxes:

• Your code needs to compile and run (obviously).
• If you have implemented a new method, this method should have been published in a peer reviewed journal and the publication should be cited in the documentation of the method (see next point). If you have implemented a CV, please give a small example of usage. In which case(s) does the new CV come in handy?
• Your method needs to come with the necessary documentation. For others to be able to use your method, you will have to explain how it works. You can take a look at the section “How to improve the documentation” for a starter on how to write good documentation.
• Please provide an example system. This could be the folding of an Alanine Dipeptide molecule, a NaCl system or just a toy model with a simple energy landscape. As long as the system is small and the method can easily complete within a few hours, it will be fine.

Once these boxes have been checked, our team of friendly code-reviewers will take a look at your source code and help you meet the high standard of the SSAGES code.

Working on the core classes¶

Todo

Describe SSAGES development


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» » » A cuboid has a diagonal of length 9 cm and a square base of side 4 cm. What is i...

# A cuboid has a diagonal of length 9 cm and a square base of side 4 cm. What is i...

### Question

A cuboid has a diagonal of length 9 cm and a square base of side 4 cm. What is its height?

### Options

A)
9 cm
B)
$$\sqrt{65}$$ cm
C)
4 $$\sqrt{2}$$ cm
D)
7 cm

format: plain


text:
# why there does not exist any homomorphism f : Q → Qpos.

If G is a group and b ∈ G, then a square root of b is an element a ∈ G such that a^2= b. For example 3 is a square root of 6 in 〈Z,+>

Let 〈Q,+> be the group of rational numbers with the operation of addition and let 〈Qpos, . >be the group of positive rational numbers with the operation of multiplication. Use the fact that you proved in part a to explain why there does not exist any homomorphism f : Q → Qpos.

SO FAR , I know let f be homomorphism from 〈Q,+> to　〈Qpos, . >defined by f(x+y) = x times y, let x = 0, and y = 1 we will get f(1)= 0 but 0 is not in Qpos. do you think my method is right? because I didn't use square root in this part of question...

the part a is about {a. Suppose that G and H are groups and that f : G → H is a homomorphism of G onto H. Prove that if every element of G has a square root then every element of H also has a square root.}

• Qpos is the second group? and do you mean no non-trivial hom? – GiantTortoise1729 Dec 4 '15 at 3:55
• What have you tried? Where did you get stuck? (Also, what did you prove in part a?) EDIT: All that you've shown is that one specific map is not a homomorphism; you need to show that there is no homomorphism, at all. – Noah Schweber Dec 4 '15 at 3:55
• @NoahSchweber that part a is (a. Suppose that G and H are groups and that f : G → H is a homomorphism of G onto H. Prove that if every element of G has a square root then every element of H also has a square root.) I have already solve it. – Helen Dec 4 '15 at 3:58
• @GiantTortoise1729 I edit my question, Thank you! – Helen Dec 4 '15 at 4:00

There is such a homomorphism, namely the trivial homomorphism where everything is mapped to $1$. However this is the only possibility.
Proof. In $\def\Q{{\Bbb Q}}\Q$ you can take the square root of any element. Repeating, you can take the square root of the square root of the square root of... the square root of any element, with as many square roots as you like. If there is a homomorphism $\phi$ from $\Q$ to $\Q^+$, then the same thing holds in $R={\rm range}\,\phi$. But in $R$, square root has its normal meaning, and given any element $a$ of $\Q^+$, the only way you can take the square root of the square root... indefinitely many times and always get rational results is if $a=1$. Therefore ${\rm range}\,\phi=\{1\}$, and $\phi$ is the trivial homomorphism.


text:
Obtenir des infos en relation avec de votre demande, tous les résultats web dans une page unique
1940000 Résultats pour

# Normal Distribution Python

Page 1/1 (Temps écoulé: 8.7985)

1 Numpy.random.normal — Numpy V1.15 …
numpy.random.normal¶ numpy.random.normal (loc=0.0, scale=1.0, size=None) ¶ Draw random samples from a normal (Gaussian) distribution.

 https://docs.scipy.org/doc/numpy/reference/generated/numpy.random.normal.html
2
>calculate Probability In Normal …
How to calculate probability in normal distribution given mean, std in Python? I can always explicitly code my own function according to the definition like the OP in this question did: Calculating Probability of a Random Variable in a Distribution in Python. Just wondering if there is a library function call will allow you to do this.

 https://stackoverflow.com/questions/12412895/calculate-probability-in-normal-distribution-given-mean-std-in-python
3 Python - Normal Distribution - …
The normal distribution is a form presenting data by arranging the probability distribution of each value in the data.Most values remain around the mean value making the arrangement symmetric. We use various functions in numpy library to mathematically calculate the values for a normal distribution.

 https://www.tutorialspoint.com/python/python_normal_distribution.htm
4 Probability Distributions In Python
The frozen form creates an object with the distribution parameters set. For example, you could evaluate the PDF of a normal(3, 4) distribution at the value 5 by. …

 https://www.johndcook.com/blog/distributions_scipy/
5 How To Implement Probability …
1. Objective. After studying Python Descriptive Statistics, now we are going to explore 4 Major Probability Distributions: Normal, Binomial, Poisson, and Bernoulli ...

 https://data-flair.training/blogs/probability-distributions-with-python/
6 Python Code And Normal Distribution: …
02/04/2014 · Normal Distribution is one of 'the' most applicable distribution in statistics. For some time I have been trying to develop my own library code to get ...

 https://malishoaib.wordpress.com/2014/04/02/python-code-and-normal-distribution-writing-cdf-from-scratch/
7 Python - Calculating Percentile Of …
I just chose $8401$ as an example of the kinds of numbers you might expect. $\Phi(1) = 0.8413\ldots$ and so if you generate $10^4$ samples of a standard normal distribution, you should expect close to $8413$ of the $10000$ samples to have value $\leq 1$.

 https://stats.stackexchange.com/questions/3438/calculating-percentile-of-normal-distribution
8 Statistics/normal Distribution - Rosetta …
Statistics/Normal distribution You are encouraged to solve this task according to the task description, using any language you may know.

 https://rosettacode.org/wiki/Statistics/Normal_distribution
9 9.6. Random — Generate Pseudo-random …
This module implements pseudo-random number generators for various distributions. For integers, uniform selection from a range. For sequences, uniform selection of a random element, a function to generate a random permutation of a list in-place, and a function for random sampling without replacement.

 https://docs.python.org/2/library/random.html
10 Tf.distributions.normal | Tensorflow
Class Normal. Inherits From: Distribution. Aliases: Class tf.contrib.distributions.Normal; Class tf.distributions.Normal; Defined in tensorflow/python/ops ...

 https://www.tensorflow.org/api_docs/python/tf/distributions/Normal
11 Normal Distribution - Wikipedia
For the normal distribution, the values less than one standard deviation away from the mean account for 68.27% of the set; while two standard deviations ...

 https://en.wikipedia.org/wiki/Normal_distribution
12 8.2. Math — Mathematical Functions — …
Note that the Python expression x % y may not ... function can be used to compute traditional statistical functions such as the cumulative standard normal distribution:

 https://docs.python.org/3.2/library/math.html
13 Python Gaussian Distribution - Youtube
28/06/2016 · Python Gaussian Distribution. Category ... Python Tutorial | ... Creating a Graph of the Standard Normal Distribution in Excel - Duration: ...

14 Basic Data Plotting With Matplotlib Part …
11/07/2011 · Basic Data Plotting with Matplotlib Part 3: ... This is the common “normal” distribution, ... Basic Data Plotting with Matplotlib Part 3: Histograms

 https://bespokeblog.wordpress.com/2011/07/11/basic-data-plotting-with-matplotlib-part-3-histograms/
15 Python Code Normal Distribution - …
Self-contained Python code for computing Phi(x), the distribution function (CDF) of a standard normal probability

 https://www.johndcook.com/blog/python_phi/

Suggestions Relatives :

### Python Chr

Recherches Associées :


text: One promising technique is to use programs such as the open-source Passpal to reduce cracking time by identifying patterns exhibited in a statistically significant percentage of intercepted passwords. For example, as noted above, many website users have a propensity to append years to proper names, words, or other strings of text that contain a single capital letter at the beginning. Using brute-force techniques to crack the password Julia1984 would require $6^{29}$ possible combinations, a “keyspace” that’s calculated by the number of possible letters (52) plus the number of numbers (10) and raising the sum to the power of nine (which in this example is the maximum number of password characters a cracker is targeting). Using an AMD Radeon HD7970, it would still take about 19 days to cycle through all the possibilities.


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## Configuration Spaces of Graphs

### Configuration spaces

Definition: Let $X$ be a topological space and $n\in\mathbb{N}$, then the configuration space of $n$ ordered particles in $X$ is defined as the set $$\mathrm{Conf}_n(X) = \{ (x_1, \ldots, x_n) \,|\, \text{x_i \neq x_j for i\neq j} \}$$ endowed with the subspace topology of $X^n$.
The symmetric group $\Sigma_n$ acts by permuting indices.

### Graphs

Definition: A graph $G$ is a 1-dimensional CW-complex.
It consists of sets
• $V(G)$ of 0-dimensional cells called vertices and
• $E(G)$ of 1-dimensional cells called edges.

### Main result

Theorem (Chettih-L)
If $T$ is a finite tree, then $H_\bullet (\mathrm{Conf}_n(T); \mathbb{Z})$ is free and generated by products of disjoint basic classes.
Definition: A basic class is a class in the image of the map $$H_1(\mathrm{Conf}_n(\iota))\colon H_1(\mathrm{Conf}_n(G)) \to H_1(\mathrm{Conf}_n(T))$$ induced by a piecewise linear embedding $\iota\colon G\hookrightarrow T$, where $G$ is a star graph or the H-shaped graph.

### Main result

Theorem (Chettih-L)
If $T$ is a finite tree, then $H_\bullet (\mathrm{Conf}_n(T); \mathbb{Z})$ is free and generated by products of disjoint basic classes.

### A combinatorial model

We will work with a combinatorial cube complex model of $\mathrm{Conf}_n(G)$, analogous to Świątkowski's model in the unordered case.
Its zero dimensional cubes are given by configurations where the particles on each edge are equidistant.

### A combinatorial model

One dimensional cubes move one particle from a vertex to an edge.
Move mouse along the interval

### A combinatorial model

A two dimensional cube moves two particles from two distinct vertices onto edges.
Move mouse along the square

### A combinatorial model

Theorem (L)
If $G$ is a finite graph, then $\mathrm{Conf}_n(G)$ deformation retracts to a cube complex of dimension $\mathrm{min}\{n, |V|\}$, where $V$ is the set of essential vertices.
Therefore, for graphs with exactly one essential vertex the combinatorial model is itself a graph and it is easy to determine its homology.
Example: $\mathrm{Conf}_2(Y)\simeq S^1$
Move mouse along the combinatorial model

### A combinatorial model

If the number of particles is at least $2|V|$, then the dimension of this combinatorial model is equal to the homological dimension of the space.

### Configuration spaces with sinks

The map collapsing a subgraph $$G \to G/H$$ does not induce a map on configuration spaces.

### Configuration spaces with sinks

To solve this, we introduce sinks to our graphs. Sinks are vertices where particles are allowed to collide.
Definition: For a graph $G$ and a subset $Z$ of the vertices, define $$\mathrm{Conf}_n(G,Z) := \{ (x_1,\ldots,x_n) | \text{x_i\neq x_j or x_i\in Z} \}.$$

### Configuration spaces with sinks

Collapsing a subgraph $H$ of $G$ now induces a map $$\mathrm{Conf}_n(G) \to \mathrm{Conf}_n(G/H, H/H).$$
Collapsing and computing the homology of small graphs with sinks allows us to compute the homology of configurations in trees.

### Configuration spaces with sinks

Theorem (Chettih-L)
If $G$ is a finite graph and $Z$ a subset of the vertices, then $\mathrm{Conf}_n(G,Z)$ deformation retracts to a cube complex of dimension $\mathrm{min}\{n, |V| + |E_\mathrm{sink}| \}$, where $V$ is the set of essential vertices and $E_\mathrm{sink}$ is the set of edges incident to two sinks.
Example: $\mathrm{Conf}_2([0,1], \{0,1\})\simeq S^1$
Move mouse along the combinatorial model

### Explicit calculations

For star graphs, the combinatorial model is good enough to compute the homology:
Theorem (Ghrist)
For $n\ge 1$ and $k\ge 3$ we have $$\tilde{H}_q(\mathrm{Conf}_n(\mathrm{Star}_k)) \cong \begin{cases} \mathbb{Z}^{r(k,n)} & \text{q = 1,}\\ 0 & \text{else,} \end{cases}$$ where $r(k,n) = 1 + \frac{(n+k-2)!}{(k-1)!} (n(k-2) - k + 1)$.

### Explicit calculations

For the interval with sinks, we get:
Proposition (Chettih-L)
For $n\ge 1$ we have $$\tilde{H}_q\left(\mathrm{Conf}_n([0,1], \{0\})\right) =0,$$ and $$\tilde{H}_q\left(\mathrm{Conf}_n([0,1], \{0,1\})\right) \cong \begin{cases} \mathbb{Z}^{r(n)} & \text{q = 1,}\\ 0 & \text{else,} \end{cases}$$ where $r(n) = 1 + (n-2) 2^{n-1}$.

### Explicit calculations

It turns out that these are the only building blocks we need to describe the homology $H_*(\mathrm{Conf}_n(T))$ for any finite tree $T$.
To illustrate the approach of our proof, we will now compute $H_*(\mathrm{Conf}_n(\mathrm{H}))$.

### The Mayer-Vietoris spectral sequence

Let $n\ge 1$, $G$ a finite graph, $Z$ a subset of the vertices and $\{V_i\}_{i\in I}$ an open cover of $G$.
For each map $$\phi\colon \{1,\ldots,n\}\to I$$ define the open subset $U_\phi$ of $\mathrm{Conf}_n(G,Z)$ as $$U_\phi = \bigcap_{1\le j\le n} \pi_j^{-1}( V_{\phi(j)}),$$ where $\pi_j\colon\mathrm{Conf}_n(G,Z)\to G$ projects to the $j$-th particle.
This defines an open cover $\{ U_\phi \}$ of $\mathrm{Conf}_n(G,Z)$.

### The Mayer-Vietoris spectral sequence

Associated to this open cover $\{ U_\phi \}$ of $\mathrm{Conf}_n(G,Z)$ there is the Mayer-Vietoris spectral sequence given by $$E^1_{p,q} = \bigoplus_{\phi_0,\ldots,\phi_p} H_q( U_{\phi_0} \cap\cdots\cap U_{\phi_p})$$ converging to the homology $H_*(\mathrm{Conf}_n(G,Z))$.
Instead of computing the pages directly, we compare the spectral sequences for different graphs and different sets of sinks.

### The H-shaped graph

As open cover for the graph $\mathrm{H}$ we take $V_1$ and $V_2$ defined as follows:
This defines a spectral sequence $E^*_{\bullet,\bullet}$.

### The H-shaped graph

Each intersection of sets $U_\phi$ is a disjoint union of spaces of the form $$\mathrm{Conf}_{S_1}(V_1) \times \mathrm{Conf}_{S_I}(V_1\cap V_2) \times \mathrm{Conf}_{S_2}(V_2)$$ for some $S_1\sqcup S_I \sqcup S_2 = \{1,\ldots,n\}$.

### The H-shaped graph

The connected components of $\mathrm{Conf}_{S_I}(V_1\cap V_2)$ are contractible and determined by the order of the particles $S_I$. By the Künneth theorem, the $q$-th homology of those products is given by $$\bigoplus_{q_1+q_2=q} H_{q_1}( \mathrm{Conf}_{S_1}(V_1) )\otimes H_{q_2}( \mathrm{Conf}_{S_2}(V_2)).$$

### The H-shaped graph

This implies that only the three bottom rows of $E^1_{\bullet,\bullet}$ are non-trivial.

### The H-shaped graph

We now compute the bottom row $E^2_{\bullet,0}$.
Entries are sums of modules $$H_0(\mathrm{Conf}_{S^j_1}(V_1))\otimes H_0(\mathrm{Conf}_{S^j_2}(V_2))$$ These groups don't change if we turn the vertices into sinks.
$\cong H_\bullet(\mathrm{Conf}(\mathrm{H},V(\mathrm{H}))$
$\cong H_\bullet(\mathrm{Conf}([0,1],\{0,1\}))$

### The H-shaped graph

We now compute the first row $E^2_{\bullet,1}$.
Entries are sums of modules $$H_1(\mathrm{Conf}_{S^j_1}(V_1))\otimes H_0(\mathrm{Conf}_{S^j_2}(V_2))$$ and $$H_0(\mathrm{Conf}_{S^j_1}(V_1))\otimes H_1(\mathrm{Conf}_{S^j_2}(V_2))$$ These groups don't change if we turn one of the vertices into a sink.

### The H-shaped graph

This describes the infinity page.

### The H-shaped graph

This gives a generating set of $H_*(\mathrm{Conf}_n(\mathrm{H}))$ in terms of classes in the configuration spaces of the $\mathrm{Y}$-graph and the interval with two sinks.
The general proof proceeds in the same way by glueing star graphs to trees.


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Hardcover | $40.00 Short | £32.95 | 256 pp. | 6 x 9 in | June 2017 | ISBN: 9780262036283 eBook |$28.00 Short | June 2017 | ISBN: 9780262340434
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## Truth in Husserl, Heidegger, and the Frankfurt School

Critical Retrieval

## Overview

The idea of truth is a guiding theme for German continental philosophers from Husserl through Habermas. In this book, Lambert Zuidervaart examines debates surrounding the idea of truth in twentieth-century German continental philosophy. He argues that the Heideggerian and critical theory traditions have much in common—despite the miscommunication, opposition, and even outright hostility that have prevailed between them—including significant roots in the phenomenology of Edmund Husserl. Zuidervaart sees the tensions between Heideggerian thought and critical theory as potentially generative sources for a new approach to the idea of truth. He argues further that the “critical retrieval” of insights from German continental philosophy can shed light on current debates in analytic truth theory.

Zuidervaart structures his account around three issues: the distinction between propositional truth and truth that is more than propositional (which he calls existential truth); the relationship between propositional truth and the discursive justification of propositional truth claims, framed in analytic philosophy by debates between epistemic and nonepistemic conceptions of truth; and the relationship between propositional truth and the objectivity of knowledge, often presented in analytic philosophy as a conflict between realists and antirealists over the relation between “truth bearers” and “truth makers.” In an innovative and ambitious argument, drawing on the work of Husserl, Heidegger, Horkheimer, Adorno, and Habermas, Zuidervaart proposes a new and transformative conception of truth.

Lambert Zuidervaart is Professor of Philosophy Emeritus at the Institute for Christian Studies in Toronto. He is the author of Adorno’s Aesthetic Theory (MIT Press), Artistic Truth, Social Philosophy after Adorno, Religion, Truth, and Social Transformation, and other books.

## Endorsements

“Charting a new path toward a comprehensive conception of truth by examining strengths and weaknesses of twentieth-century German philosophical approaches, Zudervaart’s ‘critical retrieval’ adroitly moves beyond the sterile isolation of these approaches from their Anglophone brethren and from each other.”
Dan Dahlstrom, Silber Professor of Philosophy, Boston University; author of Heidegger’s Concept of Truth

“Lucidly and succinctly Lambert Zuidervaart critically retrieves crucial insights into truth from seminal writings in twentieth-century German philosophy. The dynamic dialogues he constructs speak to the main concerns of contemporary theories, while showing the limitations of their focus on propositional truth. His book takes important steps toward the development of a more comprehensive conception, in which truth, intersecting with goodness, calls for societal transformation.”
Maeve Cooke, Professor of Philosophy, University College Dublin

“In his meticulously researched and elegantly written new book, Lambert Zuidervaart offers an account of the adventures of the concept of truth in Husserl, Heidegger, and the Frankfurt School. This informative and insightful new work has done scholars and students of critical theory and modern German philosophy a huge service.”
James Gordon Finlayson, Director of the Center for Social and Political Thought, University of Sussex

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# Borel's theorem for Banach's space valued functions

Let $(a_n)_{n=0}^\infty$ be an arbitrary sequence in a real Banach space $X$. Does there exist a smooth function $f: \mathbb {R} \rightarrow X$ such that $f^{(n)}(0)=a_n$ for $n=0,1,2,\ldots$?

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According to The Convenient Setting of Global Analysis (Kriegl and Michor), this is due to Wells (1973). The statement given is:

15.4. Borel's Theorem. [Wells, 1973]. Suppose a Banach space $E$ has $C^\infty_b$-bump functions. Then every formal power series with coefficients in $L_{\text{sym}}^n(E;F)$ for another Banach space $F$ is the Taylor-series of a smooth mapping $E \to F$.

In this case, $E = \mathbb{R}$ so you have $C^\infty_b$-bump functions.

### References:

• The Convenient Setting of Global Analysis, Kriegl and Michor. MR1471480
• Differentiable functions on Banach spaces with Lipschitz derivatives, Wells. MR370640
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I strongly believe that the original question has an affirmative answer for ALL Banach spaces. I would try to use the closed range theorem for Frechet spaces as e.g. in the Functional Analysis book of Meise and Vogt. –  Jochen Wengenroth Sep 3 '12 at 14:15

The result you are aiming at should follow from the scalar case by using tensor products since $\mathscr C^\infty(\mathbb R,X)= \mathscr C^\infty (\mathbb R) \tilde{\otimes} X$ and $X^{\mathbb N_0} = \mathbb R^{\mathbb N_0} \tilde{\otimes} X$. Because of the nuclearity of $\mathscr C^\infty (\mathbb R)$ and $\mathbb R^{\mathbb N_0}$ the tensor norm does not matter and tensorizing a surjective (hence open) continuous linear operator with the identity leads again to a surjection.

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(Replying to your comment on my answer here so you get notified of it) By "the original question", do you mean "the question asked here" or the general question of applying Borel's theorem to all pairs of Banach spaces, so dropping the requirement (in Wells' theorem) that the source space has sufficiently nice bump functions? –  Loop Space Sep 3 '12 at 17:50
@Andrew: Sorry, I just misread the theorem you quote in your answer (confusing the roles of $E$ and $F$). I don't know whether the generalized Borel theorem (which Kriegl/Michor attribute to Wells although I did not see the result in the quoted article) holds for all Banach spaces. –  Jochen Wengenroth Sep 4 '12 at 6:46