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# System to represent objects in Mathematics

Hi.

Usually, Computer Programmers use Flow Charts, Algorithms, or UML diagrams to build a great software or system. In the same manner, in Mathematics, what do Mathematicians use to build a great system that they want to build.

Category Theory is at the highest level of abstraction; then comes Set Theory which can be used to talk about the system in little more detail?

Could you please demonstrate Category Theory and Set Theory with an example?

My attempt to know the System in Math that lets me communicate with other Mathematicians.

Example: There is a book. I can study the information that it has. To describe what the book is about I will use Category Theory and to see what words are on what pages I would use Set Theory?

In short, I want to know the System in Mathematics that lets me communicate my ideas with other mathematicians.

Thanks.

jedishrfu
Mentor
There is only the common notation and conventions agreed upon by mathematicians worldwide who want to communicate and collaborate.

There is no math system in use akin to what programmers might use to design and document their programming systems. Even these programmer tools are seldom used except when deconstructing and reverse engineering a software system.

@fresh_42 is better able to comment on this topic.

Mark44
Mentor
Example: There is a book. I can study the information that it has. To describe what the book is about I will use Category Theory and to see what words are on what pages I would use Set Theory?
I don't think this is a good metaphor to describe the differences between these two areas of mathematics..

Have you done any basic research? For category theory, you might start with this wiki page: https://en.wikipedia.org/wiki/Category_theory.

fresh_42
Mentor
Category Theory is at the highest level of abstraction; ...
According to which order?
... then comes Set Theory which can be used to talk about the system in little more detail?
There is no category theory before set theory. Math is somehow the formalism of our human characteristic of pattern recognition. So basic arithmetic and the concepts of sets is basically the foundation everything else is built upon, category theory included. There is no certain system of categorization or language to communicate in. Each area has its own code. Of course they all usually use first order logic, which in the end is the language you might be looking for.
Could you please demonstrate Category Theory and Set Theory with an example?
##\mathcal{F}\, : \,(\text{ vector spaces },\text{ linear functions }) \longrightarrow (\text{ sets },\text{ mappings })## is the forget functor in category theory and the Boolean algebra ##(\mathcal{P}(\mathbb{R}),\cap ,\cup ,\{\,\,\}^C)## is an example of set theoretic considerations.

The main difficulty with your question is, that it's not clear what you mean as mathematics doesn't work this way. Last time mathematicians have tried resulted in a mathematical revolution, cp. Hilbert's program, Russel, Gödel.

As soon as you will have properly and well defined what you're talking about, it can be called mathematics. The difference to the real world is, that in life nobody makes the effort to do so, and furthermore attaches undefined and implicit meanings to words and concepts.

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Assume that there is a Rocket. The Rocket lifts off from initial point 'C' and then it moves from point 'A' to point 'B' and then it returns from point 'B' to point 'A'. Now the Rocket is back at the point 'C' - returns to the same point from where it was first launched.

How do I say about the quantity, structure, space, change of the Rocket in Mathematics at the highest level of abstraction so that I can give a rough idea about the Rocket for the same to my Math friends?

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jbriggs444
Homework Helper
Assume that there is a Rocket. The Rocket lifts off from initial point 'C' and then it moves from point 'A' to point 'B' and then it returns from point 'B' to point 'A'. Now the Rocket is back at the point 'C' - returns to the same point from where it was first launched.
View attachment 225784
How do I say about the quantity, structure, space, change of the Rocket in Mathematics at the highest level of abstraction so that I can give a rough idea about the Rocket for the same to my Math friends?
It's a fools errand.

There is no fixed and objective mapping between mathematical statements and physical reality. Typically, one picks a correspondence between some mathematical model and some aspects of the real world. Like the correspondence between the number of sheep in a pasture and the natural numbers. One then proceeds to reason about sheep by reasoning about natural numbers.

What behavior in your rocket scenario do you wish to reason about? One possibility is that you want to know after 1,000,000 rocket rides starting at point A, will the rocket end up at A, B or C. There is a useful mapping for that.

Stephen Tashi
Hi.

Usually, Computer Programmers use Flow Charts, Algorithms, or UML diagrams to build a great software or system. In the same manner, in Mathematics, what do Mathematicians use to build a great system that they want to build.

In short, I want to know the System in Mathematics that lets me communicate my ideas with other mathematicians.

Your examples from computer programming are precisely defined methods of communication. There is no generally accepted formal system for constructing and communicating mathematics. Mathematics is communicated using a "natural language" (such as English) together with certain conventions of notation and conventions about terminology. Each special branch of mathematics has its own conventions of notation and terminology.

Common to all mathematics is the use of logic and the understanding of how the "game" of mathematics is played. Logic involves things like understanding how to reason with quantifiers "for each" and "there exists", how to do proof by induction etc. Understanding the "game" of mathematics involves things like understanding that definitions must be precise and that definitions are theoretically arbitrary, not something that is proven.

Some mathematicians are interested in developing concise sets of axioms that are sufficient to develop all of mathematics and this work may be conducted with formal symbolic languages. Some mathematicians are interested in formulating mathematical ideas so that computers can prove theorems. However, the majority of mathematics is not expressed in a formal language or a computer language.

Okay, but what I am saying is: assume that if I have a Physics experiment that I want to perform then I take a sheet of paper and write down on it the following things:
1. Aim
2. Hypothesis
3. Apparatus
4. Method
5. Observation
6. Conclusion

In the same manner, if I want to build a System by using Mathematics and I need Algebra, Trigonometry, Vector Space then Category Theory or Set Theory could be a system that would let me convey the requisites like Algebra, Trigonometry, Vector Space etc, on a piece of paper, just like what I write on a paper when I perform Physics experiment?

Thanks!

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fresh_42
Mentor
Okay, but what I am saying is: assume that if I have a Physics experiment that I want to perform then I take a sheet of paper and write down on it the following things:
1. Aim
2. Hypothesis
3. Apparatus
4. Method
5. Observation
6. Conclusion
Sounds good. However, usually there is also a part, often in the introduction, where the following can be found within the world of existing results in this area. For short: What's known and what's new?
In the same manner, if I want to build a System by using Mathematics and I need Algebra, Trigonometry, Vector Space then Category Theory or Set Theory could be a system that would let me convey the requisites like Algebra, Trigonometry, Vector Space etc, on a piece of paper, just like what I write on a paper when I perform Physics experiment?
What do you mean? You can assume that your readers will already know, what a ring is, what the cosine means, what linearity is, and even what a contravariant functor and certainly power sets are. So there is no need to repeat those. On the other hand, if you want to write down a result from scratch, you'll have to start by defining your language and symbols, in which case I would first read a book on the theory of formal languages.

And you can always simply go to arxiv.org, chose an arbitrary mathematical article and analyse how it is written. (I would only recommend to select a couple of them in order to reduce the risk of selecting a strangely written one.)

FactChecker
FactChecker
Gold Member
I would assume that Category theory is good for generalizing abstract concepts, but not a very good (efficient?/effective?) way to define specific concrete mathematical concepts. In other words, going from the specific to define the abstract is easier than going from the abstract to define the specific. By its very definition, the abstract approach has intentionally stripped away and avoided specifics.

fresh_42
Mentor
I would assume that Category theory is good for generalizing abstract concepts, but not a very good (efficient?/effective?) way to define specific concrete mathematical concepts. In other words, going from the specific to define the abstract is easier than going from the abstract to define the specific. By its very definition, the abstract approach has intentionally stripped away and avoided specifics.
I agree and it reflects how the common people among us understand complex subjects; an example in mind. I just want to add a statement which I incidentally came upon just yesterday. It is from van der Waerden's obituary for Emmy Noether (1935):

Franz Lemmermeyer und Peter Roquette: Helmut Hasse und Emmy Noether - Universitätsverlag Göttingen 2006

"The maxim from which Emmy Noether has always been guided can be formulated as follows: All relations between numbers, functions, and operations only become transparent, generalizable, and really fruitful when they are detached from their particular objects and reduced to general conceptual contexts. For her, this guiding principle was not the result of her experience of the scope of scientific methods, but an a priori principle of her thinking. She could not accept and process any theorem, proof in her mind until it was abstractly conceived and thereby made transparent to her spirit eye."

So for the gifted, abstraction seem to be very necessary at times.

FactChecker and jedishrfu
Stephen Tashi
Okay, but what I am saying is: assume that if I have a Physics experiment that I want to perform then I take a sheet of paper and write down on it the following things:

In the same manner, if I want to build a System by using Mathematics and I need Algebra, Trigonometry, Vector Space then Category Theory or Set Theory could be a system that would let me convey the requisites like Algebra, Trigonometry, Vector Space etc, on a piece of paper, just like what I write on a paper when I perform Physics experiment?

I don't know what you mean by "build a System".

There is such a thing as "experimental mathematics", but for the most part, describing experiments and doing mathematics are very different processes.

When we describe an experiment, we don't present a summary or diagram of all the knowledge necessary to understand the description. For example, we usually take it for granted that the reader is famililar with the system of physical units used and the mathematical notation used to express the physical laws mentioned in the hypothesis.

1. Aim
2. Hypothesis
3. Apparatus
4. Method
5. Observation
6. Conclusion

Are you seeking an outline for writing a mathematical paper?

jedishrfu
Mentor
There is one comparable system of notation that was developed for computers, the APL language which allowed one to describe mathematical formulas in a computable form. I think Iverson (inventor of APL) tried to systematize math with a common notation suitable for computation too.

http://cl-informatik.uibk.ac.at/teaching/ws12/bob/reports/FA.pdf

http://www.math.bas.bg/bantchev/place/apl.html

At one time, IBM's opcodes for System 370 were defined in terms of APL symbols.

Lastly, with respect to physics and math:

https://www.amazon.com/dp/0691154562/?tag=pfamazon01-20

The author shows how many math concepts can be derived from physical experiments.

fresh_42
Are you seeking an outline for writing a mathematical paper?
No.

What I am trying to know is that if I need a Circle then I could do "something" to get a Circle.
Do we have a name for this "something"? Yes, that something could be Arithmetic, Algebra, Space, et cetera.
So, if I want to tell others to "build" a Circle then I can guide them by saying: "Use Arithmetic, Algebra, Space.. to build your Circle, please".
Why I am doing this? I'm doing this because I don't want my Circle to be dependent on Arithmetic, Algebra, Space, et cetera.
The question is what else would I do besides Arithmetic, Algebra, Space et cetera to build a Circle.
So, if I abstract out Arithmetic, Algebra, Space, et cetera then I get more opportunities to "build" a Circle.
Arithmetic, Algebra, Space et cetera are theories.
Do we have something in Mathematics that gives me a flexibility in choosing a theory to build a Circle?
Category Theory? Set Theory? Can the elements of a Set be equations?
Can Category Theory and Set Theory tell me how I can swap these theories?

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fresh_42
Mentor
What I am trying to know is that if I need a Circle then I could do "something" to get a Circle. Do we have a name for this "something"?
Yes, set theory and arithmetic: ##\mathcal{C}=\{\,(x,y)\in \mathbb{R}^2\,|\,x^2+y^2-1=0\,\}##. Everything beyond this point, depends on what you want to have the circle for: differential geometry, algebraic geometry, geometry, topology, group theory, analysis, number theory and whatever else.

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FactChecker
Gold Member
I'm afraid that the very definition of a circle must be given in a way that only makes sense in a particular context. I don't think you can define a circle without using Arithmetic, Algebra, Space, et cetera. So trying to "abstract that away" will probably fail.

I'm afraid that the very definition of a circle must be given in a way that only makes sense in a particular context. I don't think you can define a circle without using Arithmetic, Algebra, Space, et cetera. So trying to "abstract that away" will probably fail.
I am not asking anyone to define a circle without using Arithmetic, Algebra, Space, et cetera.
I am just asking how to write "abstract that away" on a piece of paper in Mathematics. Category Theory? Set Theory?

fresh_42
Mentor
What does "abstract that away" mean? It would be clearer if you said it in Samoyedic and we used google translate. I have not the least idea what you are talking about. There is no abstraction beyond set theory and arithmetic. You could invent one, but what for?

FactChecker
Gold Member
I am not asking anyone to define a circle without using Arithmetic, Algebra, Space, et cetera.
I am just asking how to write "abstract that away" on a piece of paper in Mathematics. Category Theory? Set Theory?
There certainly are things more abstract than algebra, etc., but the question is whether they are concrete enough to define what a "circle" is. I think that most people would not be satisfied with anything which can not specify curvature, distance from a center point, etc. In topology, you would be able to define a simple closed curve, but that is not enough to be called a circle. In general, you can't be very abstract and still retain specific geometric and algebraic properties.

What does "abstract that away" mean?
Assume that I have ten distinct theories for each mathematical operation: Addition, Subtraction, Multiplication, Division.
How do I write on a piece of paper that I have opted for ninth theory to perform addition? That's it. This is what I want to know: I want to present my thoughts in standard ways for communication. So, how do I say that I have chosen ninth theory to perform addition in a standard way?

Little more explanation to my question:

If you look at the code below, you will notice that IWater is of two types: normal water and soda water. In the same manner, how do I say to someone that I have such flexibility of choosing theories on a piece of paper?
Code:
using System;

namespace ConsoleApp1
{
public interface IWater
{
string ThisWater();
}

public class Water
{
IWater _obj;
public Water(IWater _obj)
{

this._obj = _obj;
}
public void MyWater()
{
string myString = _obj.ThisWater();
Console.WriteLine(myString);
}
}

public class NormalWater : IWater
{
public string ThisWater()
{
return "Normal Water";
}
}

public class SodaWater : IWater
{
public string ThisWater()
{
return "Soda Water";
}
}
class Program
{
static void Main(string[] args)
{
Water nwater = new Water(new NormalWater());
nwater.MyWater();
Water swater = new Water(new SodaWater());
swater.MyWater();
}
}
}

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fresh_42
Mentor
These operations are binary operations on certain types of objects aka sets with certain properties aka arithmetic rules. Rules and sets can vary a lot, so the binary operation is all what they have in common: ##\circ \, : \,\mathcal{U} \times \mathcal{V} \longrightarrow \mathcal{W}##. Normally ##(*)\quad\mathcal{U}=\mathcal{V}=\mathcal{W}## but not necessarily, as e.g. ring modules or inner products. Of course, it makes not a lot of sense only to view such an operation without any specifications of the sets or the rules. Even if we chose categorial terms, we normally need to specify which category. If we only define categories which allow a binary operation of its objects, it will be necessary to at least define something about the allowed objects as e.g. ##(*)##. A binary operation between any of its objects is so arbitrary that few can be said in such a generality. It will be a collection of categories, rather than one on its own. But in principle, you can do this. However, it will be challenging even to define morphisms properly.

Mark44
Mentor
Assume that I have ten distinct theories for each mathematical operation: Addition, Subtraction, Multiplication, Division.
How do I write on a piece of paper that I have opted for ninth theory to perform addition? That's it. This is what I want to understand.
We have several different rules (these are NOT theories) for addition, including
2. Addition of signed (positive or negative) numbers
7. ... and possibly others that don't spring to mind immediately.
You don't just arbitrarily pick one of these as the "best" -- you use the one that works for the set of things you're considering. So the context in which you're working dictates which form of "addition" you should use.

If you look at the code below, you will notice that IWater is of two types: normal water and soda water. In the same manner, how do I say to someone that I have such flexibility of choosing theories on a piece of paper?
Going back to my example of addition (only), you don't have flexibility in choosing which operation to use. Whatever set of things you're working with will dictate which operation you have to use. If you need to write something on a piece of paper, it would be a big if ... else if ... else if ... decision structure, where the type of things you're adding determines which rule for addition you will use.

My question for you is, why are you doing this? What's your purpose?

fresh_42
fresh_42
Mentor
... and the addition doesn't even have to be an addition in the classical sense. It could as well be the union on sets or the OR in Boolean algebras, or simply the only operation which is just written with a ##+##.

My question for you is, why are you doing this? What's your purpose?
I'm curious and I want a PhD. in Mathematics.

fresh_42
Mentor
I'm curious and I want a PhD. in Mathematics.
Good luck!

By the way, many objects which define a category on their own can be realized as homomorphic images of tensor spaces. Here you have a natural construction ##\mathcal{U} \otimes \mathcal{V}## from which you can build a quotient for the defining rules.


text:
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text:
Report Open Access

D2.1 PIACERE DevSecOps Framework Requirements specification, architecture and integration strategy - v1

Emmanuelle Morganti

Dublin Core Export

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<dc:creator>Emmanuelle Morganti</dc:creator>
<dc:date>2022-07-06</dc:date>
<dc:description>This document will contain 1) all the functional,
non-functional and technical requirements of the PIACERE DevSecOps Framework and of all the components to be developed in the context of the technical WPs (WP3, WP4, WP5 and WP6). This deliverable is tightly related to the corresponding use cases requirements deliverable in WP7 (D7.1). 2) the architecture of the DevSecOps framework [KR13] and the workflow, 3) the requirements of the DevOps infrastructure to be used in the development of PIACERE as well as the definition of the strategy and steps to be followed for the continuous integration of the PIACERE solution. This deliverable is the result of Task 2.1 and Task 2.3.</dc:description>
<dc:identifier>https://zenodo.org/record/6801782</dc:identifier>
<dc:identifier>10.5281/zenodo.6801782</dc:identifier>
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<dc:language>ang</dc:language>
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<dc:relation>url:https://zenodo.org/communities/101000162</dc:relation>
<dc:rights>info:eu-repo/semantics/openAccess</dc:rights>
<dc:subject>Architecture, Integration Strategy</dc:subject>
<dc:title>D2.1 PIACERE DevSecOps Framework Requirements specification, architecture and integration strategy - v1</dc:title>
<dc:type>info:eu-repo/semantics/report</dc:type>
<dc:type>publication-report</dc:type>
</oai_dc:dc>

24
22
views


text:
Hardcover | $36.00 Short | £24.95 | ISBN: 9780262015257 | 296 pp. | 6 x 9 in | 60 figures, 3 tables| March 2011 Ebook |$25.95 Short | ISBN: 9780262296496 | 296 pp. | 6 x 9 in | 60 figures, 3 tables| March 2011

# Laws, Mind, and Free Will

## Overview

In Laws, Mind, and Free Will, Steven Horst addresses the apparent dissonance between the picture of the natural world that arises from the sciences and our understanding of ourselves as agents who think and act. If the mind and the world are entirely governed by natural laws, there seems to be no room left for free will to operate. Moreover, although the laws of physical science are clear and verifiable, the sciences of the mind seem to yield only rough generalizations rather than universal laws of nature. Horst argues that these two familiar problems in philosophy—the apparent tension between free will and natural law and the absence of "strict" laws in the sciences of the mind—are artifacts of a particular philosophical thesis about the nature of laws: that laws make claims about how objects actually behave.

Horst argues against this Empiricist orthodoxy and proposes an alternative account of laws—an account rooted in a cognitivist approach to philosophy of science. Horst argues that once we abandon the Empiricist misunderstandings of the nature of laws there is no contrast between "strict" laws and generalizations about the mind ("ceteris paribus" laws, laws hedged by the caveat "other things being equal"), and that a commitment to laws is compatible with a commitment to the existence of free will. Horst’s alternative account, which he calls "cognitive Pluralism," vindicates the truth of psychological laws and resolves the tension between human freedom and the sciences.


text:
## An inverse $K$-theory functor

### Summary

Summary: Thomason showed that the $K$-theory of symmetric monoidal categories models all connective spectra. This paper describes a new construction of a permutative category from a $\Gamma$-space, which is then used to re-prove Thomason's theorem and a non-completed variant.

### Mathematics Subject Classification

19D23, 55P47, 18D10, 55P42

### Keywords/Phrases

gamma space, permutative category, connective spectrum


text:
Divisor
Overview

In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a divisor of an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

, also called a factor of , is an integer which divides without leaving a remainder
Remainder
In arithmetic, the remainder is the amount "left over" after the division of two integers which cannot be expressed with an integer quotient....

.
The name "divisor" comes from the arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

operation of division
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...

: if then is the dividend
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...

, the divisor, and the quotient
Quotient
In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...

.

In general, for non-zero integers and , divides , written:
if there exists an integer such that .


text:
# Why is the determinant $1$ not $-2$

I am trying to find the determinant of this matrix. Let $$A$$ be the matrix :

$$\begin{bmatrix}2&2&1\\1&0&5\\1&1&0\end{bmatrix}$$

Using row operations, I can change $$R_3$$ to ( $$-2R_3+R_1\rightarrowR_3$$). So, $$B$$ is the matrix :

$$\begin{bmatrix}2&2&1\\1&0&5\\0&0&1\end{bmatrix}$$

Then, using Laplace Expansion and expanding along the 2nd column,

$$det(A) = (2)(-1)^3 det(\begin{bmatrix}1&5\\0&1\end{bmatrix})= (2)(-1)(1) = -2$$

However, $$det(A)=det(B)$$ if a multiple of one row is added to another, but the determinant of $$A$$ is $$1$$. What am I doing wrong here?

Because of that $$-2R_3$$ that you wrote. When you do $$-2R_3+R_1\to R_3$$, part of you are doing is to multipliply the third row by $$-2$$, which changes the determinant of the matrix.
• But my textbook says adding a multiple of one row to another doesn't change the determinant, I'm just adding a multiple of $R_3$ to $R_1$. I apologize if it's an obvious question. Oct 15, 2019 at 11:38
• @MohamedTlili If you had done $-2R_3+R_1\to R_1$, then the determinant would've been unchanged. That would be "adding a multiple of $R_3$ to $R_1$". However, $-2R_3+R_1\to R_3$ doesn't just add a multiple of $R_1$ to $R_3$, it also multiplies $R_3$ by $-2$ first. Oct 15, 2019 at 11:38
• @MohamedTlili First you replaced $R_3$ with $-2R_3$, multiplying the determinant by $-2$. Then you added $R_1$ to the third row, which had no effect on the determinant. Writing down the intermediate matrix may help you follow this argument.
You're not adding a multiple of $$R_3$$ to $$R_1$$. You're replacing $$R_3$$ by a multiple of $$R_3$$ plus $$R_1$$.
I warn my students at the start that there's a place coming up later where it will be important to realize that (if $$j\ne k$$) $$R_j+cR_k\to R_j$$is a row operation, while $$R_j+cR_k\to R_k$$is not an official elementary row operation. This it that place. As long as $$c\ne0$$ both versions work fine for finding an echelon form, but the second changes the determinant, while the first does not.


text:
# The Informative Vector Machine: A Practical Probabilistic Alternative to the Support Vector Machine

Neil D. Lawrence, Matthias Seeger, Ralf Herbrich
, 2004.

#### Abstract

We present a practical probabilistic alternative to the popular support vector machine (SVM). The algorithm is an approximation to a Gaussian process, and is probabilistic in the sense that it maintains the process variance that is implied by the use of a kernel function, which the SVM discards. We show that these variances may be tracked and made use of selection of an active set which gives a sparse representation for the model. For an active set size of $d$ our algorithm exhibits $O(d^{2}N)$ computational complexity and $O(dN)$ storage requirements. It has already been shown that the approach is comptetive with the SVM in terms of performance and running time, here we give more details of the approach and demonstrate that kernel parameters may also be learned in a practical and effective manner.


text:
# IMABS

Returns the absolute value of a complex number.

### Sample Usage

`IMABS("3+4i")`

`IMABS("6j")`

### Syntax

`IMABS(number)`

• `number` - The complex number to calculate the absolute value of.

### Notes

• The absolute value (or modulus) is defined as follows:

### Examples

1 A B
2 Formula Result
3 `=IMABS("3+4i")` 5
4 `=IMABS(COMPLEX(3, 4, "j"))` 5


text:
#### Real-Time Locomotion on Soft Grounds with Dynamic Footprints

Frontiers in Virtual Reality, 2022

Eduardo Alvarado [1], Chloé Paliard [2], Damien Rohmer [1], Marie-Paule Cani [1]
[1] LIX, Ecole Polytechnique/CNRS, Institut Polytechnique de Paris, Palaiseau, France
[2] LTCI, Télécom Paris, Institut Polytechnique de Paris, Palaiseau, France

## Abstract

When we move on snow, sand, or mud, the ground deforms under our feet, immediately affecting our gait. We propose a physically based model for computing such interactions in real time, from only the kinematic motion of a virtual character. The force applied by each foot on the ground during contact is estimated from the weight of the character, its current balance, the foot speed at the time of contact, and the nature of the ground.

We rely on a standard stress-strain relationship to compute the dynamic deformation of the soil under this force, where the amount of compression and lateral displacement of material are, respectively, parameterized by the soil’s Young modulus and Poisson ratio. The resulting footprint is efficiently applied to the terrain through procedural deformations of refined terrain patches, while the addition of a simple controller on top of a kinematic character enables capturing the effect of ground deformation on the character’s gait. As our results show, the resulting footprints greatly improve visual realism, while ground compression results in consistent changes in the character’s motion. Readily applicable to any locomotion gait and soft soil material, our real-time model is ideal for enhancing the visual realism of outdoor scenes in video games and virtual reality applications.

Code: [Github]

Paper: [HAL] [Frontiers]

## Bibtex

title = {Real-Time Locomotion on Soft Grounds With Dynamic Footprints},
author = {Alvarado, Eduardo and Paliard, Chloé and Rohmer, Damien and Cani, Marie-Paule},
doi = {10.3389/FRVIR.2022.801856},
journal = {Frontiers in Virtual Reality},
volume = {3},
pages = {3},
year = {2022},
month = {mar},


text:
# Studies of Single-Particle Structure in the N=16 Region Using Transfer Reactions

Abstract : We have developed a new experimental setup based at the GANIL/SPIRAL facility in Caen, France to measure one-nucleon transfer reactions in inverse kinematics in order to study the evolution of the single particle structure of exotic nuclei. The setup couples together three state-of-the-art detection systems: the TIARA Si array, the large-acceptance magnetic spectrometer VAMOS and the high-efficiency segmented Ge gamma-ray array EXOGAM. In a first experiment, the $^{24}$Ne(d,p)$^{25}$Ne reaction has been studied to probe the N=16 shell closure. Details of the setup, data analysis and preliminary results are presented.
Keywords :
Document type :
Conference papers

http://hal.in2p3.fr/in2p3-00116972
Contributor : Michel Lion <>
Submitted on : Wednesday, November 29, 2006 - 10:35:06 AM
Last modification on : Saturday, December 26, 2020 - 1:46:04 PM

### Citation

R. C. Lemmon, B. Fernandez-Dominguez, C. Timis, M. Labiche, W. N. Catford, et al.. Studies of Single-Particle Structure in the N=16 Region Using Transfer Reactions. FUSION06: Reaction Mechanisms and Nuclear Structure at the Coulomb Barrier, Mar 2006, Venice, Italy. pp.285-290, ⟨10.1063/1.2338391⟩. ⟨in2p3-00116972⟩

Record views


text:
# Complexity, drivers and modulators in life and the law

This video is fascinating for a host of reasons. In particular, it illustrates a concept that is critical in my continuing series of posts about the legal ecosystem.

#### Patterns

The video shows an aggregation of anchovies — a shoal of fish obeying simple rules, but creating a constantly changing unpredictable pattern in the sea (just as a murmuration of starlings does in the air). What I find especially interesting is the way the fish react to humans in their midst.

Studies of fish show that they observe very simple rules when they shoal (as starlings do when they flock):

1. Move in the same direction as your neighbour;
2. Remain close to your neighbours;
3. Avoid collisions with your neighbours.

Armed with these rules, it is possible to simulate the behaviour of shoals. It is even possible to predict large-scale changes in behaviour (such as migration). What is impossible is predicting the precise pattern traced by the shoal itself as it moves through the water. There are simply too many variables to allow this — for all intents and purposes each pattern is unique for the fleeting moment that it exists. This can be demonstrated mathematically.

For any number of items (N), the number of links (L) between pairs of items can be expressed thus:

$\L=N(N-1)/2\$

Likewise, the number of patterns (P) that can be generated by connection items can be calculated thus:

$P=2^{L}$

A simple table shows how the numbers of patterns grows exponentially as items are added:

N=4 L=6 P=64
N=10 L=45 P=35 trillion
N=12 L=66 P=73.8 quintillion

The number of possible patterns generated by hundreds of fish is inconceivable.

#### Drivers and modulators

We often refer to actions driving change in life and work. The word ‘driver’ suggests a direct and predictable link between cause and effect: if I do this, the outcome will be that — every time. Drivers of this type do exist, even in some quite complicated systems. When I turn the steering wheel in my car, I need the result to be predictable. It usually is, unless the system is broken or some other factor (ice or gravel, perhaps) has been introduced without my knowledge.

As the number of components in a system increases and the connections between them are loosened, the behaviour of the system become less predictable. However, when we see the outcome it often looks inevitable; we are persuaded by hindsight that it should have been predictable.

This can be seen in the video when people swim towards the anchovies. Sometimes the shoal just moves away from the swimmer. Sometimes it parts and rejoins beyond the swimmer. Sometimes it forms a ring around the swimmer. None of these outcomes was predictable, but they all appear to follow the same rules — the fish maintain a constant distance from each other and follow the course of their neighbours, but they keep a greater distance from the alien body (whilst not fleeing from it altogether).

Similar things happen in organisations. Patterns of behaviour might look fairly constant and predictable, but can be disturbed by significant events (a change of leadership, for example, or some external pressure). The consequence of that disturbance is unpredictable before it happens, but may look obvious afterwards — hindsight makes us think that it was inevitable.

It is at this point that the ‘driver’ fallacy comes into being. If we see a number of events that appear to form an inevitable sequence of events, it is natural to think that repeating the initial cause will drive the same outcome. If the outcome looks good, then we are likely to take the same initial action expecting it to result in the same outcome. This is the thought process that underpins so-called ‘best practice’ and books like Good to Great. These hold out a promise that success will follow emulation of others.

Dave Snowden has suggested that it is more appropriate to think of change in complex systems being effected by modulators rather than drivers.

Imagine that you have a round flat table and around that table are a series of electro-magnets. They can vary in strength and also polarity. Some you control, some are controlled by people you know and some appear to change at random. In the middle of the table are a lot of iron filings. Now as long as the magnets don’t change, the iron filings will form a complex stable pattern. However as the magnets fluctuate in strength the pattern changes. if some of them change polarity then change is sudden and drastic before a new stability emerges. At the same time some of the iron filings get magnetised in turn as they pass through electric currents, making the situation even more complex. I may not even be aware of some modulators until they suddenly come into play and their impact is seen.

The magnets in this case modulate the system. They interact with each other and with the system as a whole, they make it inherently unpredictable. Understanding what modulators are in play will help us understand emergent behaviour of the system, but not to predict its future state. Attributing cause to a limited number of dominant modulators (that is what I think people mean by drivers) is a mistake as the level of interaction is too much. I can build models to simulate the behaviour of the system, however simulation does not lead to prediction.

#### Modulating change in law

A common type of ‘driver error’ arises when two systems appear to have the same objectives and basic structure. If one of the systems appears to have a good way of achieving a particular outcome, it is natural to consider transposing it into the other system. This sometimes occurs in legal and political systems, when adoption of different approaches to similar problems from foreign jurisdictions might be proposed. It also happens in law firms and other organisations that are apparently similar in scope and purpose. Otto Kahn-Freund skewered the notion of transplanting between legal cultures in his 1973 Chorley Lecture, “On the Uses and Misuses of Comparative Law”. His view was that there is a continuum of actions ranging from the organic (rooted in unique cultural, social and political soil) to the mechanical. The closer a particular process is to the mechanical end of the continuum the more likely it is that a transplant will be successful.

I prefer the Cynefin framework to Kahn-Freund’s continuum. It is better rooted in theory, as well as being more subtle — the linearity of a continuum is an immanent flaw. I intend to explore that further in a future blog post.

Whatever explanatory tool one uses, it should be clear that some practices translate better between organisations than others. At the moment, there are a few practical changes that are fairly universally recommended to law firms as panaceas to help them ride out changes in the legal market. These include legal project management, process mapping, fixed-fees, and so on. Some of these will work for some firms, but how can we know at the outset which and why? The answer is that we cannot do so reliably. Instead, it is important to test things out. There should be clarity about how the experiment can be evaluated — what does success or failure look like? — and there should be a safe fall-back position in the case of failure. Anything else is wishful thinking.

An experiment might be fairly small-scale, but it can also be quite audacious, as shown in this video explaining what happened when wolves were reintroduced to Yellowstone National Park.

The sequence of events described here — leading eventually to changes in the physical environment — is unique. It is not possible to say that reintroducing wolves in other places would have the same effect. A number of other factors also played their part:

• The patterns of grazing behaviour of deer
• The topography of the park itself
• The availability of other species (beaver, coyotes, bears, birds, etc)
• The time taken for plant species to regenerate
• and so on…

So, for a law firm, introducing new ways of working or doing business might be a really good idea. The success of such changes depends on a host of components responding in particular ways. Beneficial outcomes are neither inevitable nor predictable.

Better outcomes arise from a process like this:

• An understanding of why particular changes might work (and knowing what ‘working’ means for your firm);
• Testing the change;
• Evaluating if it is working or not (by reference to the first step);
• If it works, continue;
• If it doesn’t, revert to the previous safe state.

## 1 thought on “Complexity, drivers and modulators in life and the law”

1. […] This necessary fallibility is akin to, if not the same as, the complexity that I described in an earlier blog post here. […]

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text:
# REQ Solution to differential equation

1. Mar 21, 2008

### Kurret

Find the solution to $$\frac{c}{y^2}+k=y''$$

I have not started with differential equations yet, so i hav eno idea if its possible to solve, but i stumbled upon it when investigating a physics idea i got. Thanks for any help!

Last edited: Mar 21, 2008
2. Mar 22, 2008

### Kurret

3. Mar 22, 2008

### MathematicalPhysicist

well we can first sketch a solution for y''=k which is y=kx^2/2+Ax+B
but for: y''y^2=c I don't think there's an analytical answer.

4. Mar 22, 2008

### dextercioby

Multiply by y' and you'll be able to integrate both terms of the new equation.

5. Mar 23, 2008

### Kurret

How?? Cant make it work...


text:
# ➥命题等价式

$p$ $\lnot p$ $p\lor \lnot p$ $p\land \lnot p$
$T$ $F$ $T$ $F$
$F$ $T$ $T$ $F$

## ➥逻辑等价式

• $\lnot(p\land q) = \lnot{p}\lor\lnot$
• $\lnot(p\lor q) = \lnot{p}\land\lnot$

$p\land T\equiv p$ $p\lor F \equiv p$ 恒等律
$p\lor T\equiv T$ $p\land F\equiv F$ 支配律
$p\lor p\equiv p$ $p\land p \equiv p$ 幂等律
$\lnot(\lnot p)\equiv p$ 双重否定律
$p\lor q\equiv q\lor p$ $p\land q\equiv q\land p$ 交换律
$(p\lor{q})\lor r\equiv q\lor(p\lor r)$ $(p\land q)\land r\equiv q\land(p\land r)$ 结合律
$p\lor(q\land{r})\equiv(p\lor{q})\land(p\lor{r})$ $p\land(q\lor r)\equiv(p\land q)\lor(p\land r)$ 分配律
$\lnot(p\land q)\equiv\lnot p\lor\lnot q$ $\lnot(p\lor q)\equiv\lnot p\land\lnot q$ 德·摩根律
$p\lor(p\land q)\equiv p$ $p\land(p\lor q)\equiv p$ 吸收律
$p\lor\lnot p\equiv T$ $p\land\lnot p\equiv F$ 否定律

• $p\to q \equiv \lnot p \lor q$
• $p\to q \equiv \lnot q \to \lnot p$
• $p\lor{q}\equiv\lnot{p}\to q$
• $p\land{q}\equiv\lnot({p}\to\lnot{q})$
• $\lnot(p \to q)\equiv p \land\lnot q$
• $(p\to{q})\land(p\to{r})\equiv{p\to(q\land{r})}$
• $(p\to{q})\lor(p\to{r})\equiv{p\to(q\lor{r})}$
• $(p\to{r})\land(q\to{r})\equiv{p\lor{q}}\to r$
• $(p\to{r})\lor(q\to{r})\equiv{p\land{q}}\to r$

• $p\harr{q}\equiv(p\to{q})\land(q\to{p})$
• $p\harr{q}\equiv\lnot{p}\harr\lnot q$
• $p\harr{q}\equiv(p\land{q})\lor(\lnot{p}\land\lnot{q})$
• $\lnot(p\harr{q})\equiv p\harr\lnot q$

$\lnot(p_1\lor p_2\lor \dots \lor p_n)=\lnot{p_1} \land \lnot{p_2}\land\dots\land\lnot{p_n}\\ \lnot(p_1\land p_2\land \dots \land p_n)=\lnot{p_1} \lor \lnot{p_2}\lor\dots\lor\lnot{p_n}\\$

## ➥逻辑等价式证明

$p$ $q$ $r$ $q\land r$ $p\lor(q\land{r})$ $(p\lor{q})\land(p\lor{r})$ $q\lor$ $p\land(q\lor{r})$ $(p\land{q})\lor(p\land{r})$
$T$ $T$ $T$ $T$ $T$ $T\land{T}=T$ $T$ $T$ $T\lor{T}=T$
$T$ $T$ $F$ $F$ $T$ $T\land{T}=T$ $T$ $T$ $T\lor{F}=T$
$T$ $F$ $T$ $F$ $T$ $T\land{T}=T$ $T$ $T$ $F\lor{T}=T$
$T$ $F$ $F$ $F$ $T$ $T\land{T}=T$ $F$ $F$ $F\lor{F}=F$
$F$ $T$ $T$ $T$ $T$ $T\land{T}=T$ $T$ $F$ $F\lor{F}=F$
$F$ $T$ $F$ $F$ $F$ $T\land{F}=F$ $T$ $F$ $F\lor{F}=F$
$F$ $F$ $T$ $F$ $F$ $F\land{T}=F$ $T$ $F$ $F\lor{F}=F$
$F$ $F$ $F$ $F$ $F$ $F\land{F}=F$ $F$ $F$ $F\lor{F}=F$

$p$ $q$ $\lnot p$ $\lnot q$ $p\land q$ $p\lor q$ $\lnot(p\land{q})$ $\lnot{p}\lor\lnot q$ $\lnot(p\lor{q})$ $\lnot{p}\land\lnot q$
$T$ $T$ $F$ $F$ $T$ $T$ $F$ $F$ $F$ $F$
$T$ $F$ $F$ $T$ $F$ $T$ $T$ $T$ $F$ $F$
$F$ $T$ $T$ $F$ $F$ $T$ $T$ $T$ $F$ $F$
$F$ $F$ $T$ $T$ $F$ $F$ $T$ $T$ $T$ $T$

• (a) 证明：$p\lor(p\land{q})\equiv{p}\:$
$p=T$时，无论$p\land{q}$为何真值，$p\lor(p\land{q})$都为$T$
$p=F$时，无论$q$为何真值，$p\land{q}$总为$F$$F\lor{F} = F$
所以，$p\lor(p\land{q})$$q$的真值无关，所以$p\lor(p\land{q})\equiv$
• (b)证明：$p\land(p\lor{q})\equiv{p}$
$p=T$时，无论$q$为何真值，$p\lor{q}$总为$T$$T\land{T} = T$
$p=F$时，无论$p\land{q}$为何真值，$p\land(p\lor{q})$都为$F$
所以，$p\land(p\lor{q})$$q$的真值无关，所以$\land(p\lor{q})\equiv p$

## ➥构造新的逻辑等价式

$\begin{array}{ll} p\lor{q} \to r &\equiv \lnot(p\lor{q})\lor{r} \\ &\equiv(\lnot{p}\land\lnot{q})\lor{r}\\ &\equiv(\lnot{p}\lor{r})\land(\lnot{q}\lor{r})\\ &\equiv(p\to{r})\land(q\to{r}) \end{array}$

$\begin{array}{ll} \lnot(p\to{q}) &\equiv \lnot(\lnot{p}\lor{q}) \\ &\equiv \lnot\lnot{p}\land\lnot{q}\\ &\equiv p\land\lnot{q} \end{array}$

$\begin{array}{ll} (p\land{q})\to(p\lor{q}) &\equiv \lnot(p\land{q})\lor(p\lor{q})\\ &\equiv\lnot{p}\lor\lnot{q}\lor(p\lor{q})\\ &\equiv(\lnot{p}\lor{p})\lor(\lnot{q}\lor{q})\\ &\equiv T\lor{T}\\ &\equiv T \end{array}$

## ➥其他

$NAND$（与非）和$NOR$（或非）

$p$ $q$ $p\:\mid\:q$ $(p\land q)$ $\lnot(p\land q)$ $p\:\darr\:q$ $p\lor q$ $\lnot(p\lor{q})$
$T$ $T$ $F$ $T$ $F$ $F$ $T$ $F$
$T$ $F$ $T$ $F$ $T$ $F$ $T$ $F$
$F$ $T$ $T$ $F$ $T$ $F$ $T$ $F$
$F$ $F$ $T$ $F$ $T$ $T$ $F$ $T$
• $p\:\mid\:q\equiv\lnot(p\land{q})$
• $p\:\darr\:q\equiv\lnot(p\lor{q})$

Power by 不蒜子


text:
# How to align piecewise functions?

I have two piecewise functions defined like this:

\begin{align*}
\left\{
\begin{array}{ll}
i = x^2 + y^2 \\
j = a^2 + b^2 \\
\end{array}
\right.
\\
\left\{
\begin{array}{ll}
i = 1 \\
j = 2 \\
\end{array}
\right.
\end{align*}

The problem is that they're not aligned. How do I align them?

-
Place an & before the \left\{ of each. –  qubyte Dec 10 '11 at 15:37
It would be better to provide a complete MWE, from \begin{document} to \end{document} with all needed packages. This way it's easier to test your code and provide a solution. –  Ignasi Dec 10 '11 at 15:47

The quick fix is a simple addition of two ampersands before opening the large curly brackets. In the amsmath align environment, along with the starred version, the ampersand tells LaTeX what elements should be aligned.

\begin{align*}
&\left\{
\begin{array}{ll}
i = x^2 + y^2 \\
j = a^2 + b^2 \\
\end{array}
\right.
\\
&\left\{
\begin{array}{ll}
i = 1 \\
j = 2 \\
\end{array}
\right.
\end{align*}

Perhaps the following code using split is a little better to align the = signs, although it's a matter of taste. The ampersand once again allows you to align the brackets on one level, and the equality signs on the level down. This is done inside the starred equation environment to allow split to be used, which is provided by the amsmath package.

\begin{equation*}
\begin{split}
&\left\{
\begin{split}
i &= x^2 + y^2 \\
j &= a^2 + b^2 \\
\end{split}
\right.
\\
&\left\{
\begin{split}
i &= 1 \\
j &= 2 \\
\end{split}
\right.
\end{split}
\end{equation*}

If the reduced space between the bracket and the equations bothers you in this version, you can insert horizontal space, for example with \, or with \hspace{2pt} just after the \left\{ of each group. Using \; spacing in this solution gives the final version of this code:

\begin{equation*}
\begin{split}
&\left\{\;
\begin{split}
i &= x^2 + y^2 \\
j &= a^2 + b^2 \\
\end{split}
\right.
\\
&\left\{\;
\begin{split}
i &= 1 \\
j &= 2 \\
\end{split}
\right.
\end{split}
\end{equation*}

-
Would you also explain why? Perhaps suggesting cases instead of split. –  egreg Dec 10 '11 at 15:45
@egreg: Good point. I added some more detail to explain what the ampersands are for. –  qubyte Dec 10 '11 at 15:51
@egreg: Cases has been suggested at the time of this comment in Gonzalo's answer. –  qubyte Dec 10 '11 at 15:58
@egreg: In addition, cases behave badly when you try to align them on the equalities. –  qubyte Dec 10 '11 at 16:19

To simplify, in this case, you can use two cases environments inside a align* using & to align both structures:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
&\begin{cases}
\rlap{$i$}\phantom{j} = x^2 + y^2 \\
j = a^2 + b^2
\end{cases}\\
&\begin{cases}
\rlap{$i$}\phantom{j} = 1 \\
j = 2
\end{cases}
\end{align*}
\end{document}

-
Cases behave badly when you try to align them on the = symbol. –  qubyte Dec 10 '11 at 16:20
@MarkS.Everitt: I know; that's why I said in my answer in this case. In this concrete example there's no need to use & for the equal symbol. –  Gonzalo Medina Dec 10 '11 at 16:27
To be fair, in this case is a little ambiguous. –  qubyte Dec 10 '11 at 16:29
@MarkS.Everitt: in fact; I've just noticed a little misalignment issue at the = symbols; I've updated my answer to correct this. –  Gonzalo Medina Dec 10 '11 at 16:34

Here is one way of aligning these two components:

\documentclass{article}
\usepackage{amsmath}% http://ctan.org/pkg/amsmath
\begin{document}
\begin{align*}
\left\{
\begin{array}{r@{\;}l}
i &= x^2 + y^2 \\
j &= a^2 + b^2
\end{array}
\right.
\\
\left\{
\begin{array}{r@{\;}l}
i &= \rlap{1}\phantom{x^2 + y^2} \\
j &= \rlap{2}\phantom{a^2 + b^2}
\end{array}
\right.
\end{align*}
\end{document}

You didn't use any column separation & in your array. I've inserted that at the alignment point (before =). Otherwise you would have noticed the difference in the unaligned equality signs due to different widths of i and j. I also changed your ll column specification into r@{\;}l to right-align the left-hand side, and left-align the right-hand side, plus add the appropriate spacing \; between the relational operator.

The use of the \rlap (short for right overlap) and \phantom (leaves space without typesetting) combination is aimed at making the lower array have the same width as the upper one, thereby establishing the appropriate horizontal spacing.

-
Quite complicated a solution, isn't it? A & before \left\{ is all that's needed. –  egreg Dec 10 '11 at 15:44
@egreg: Nope. With only that, the equal signs won't line up. Also, the left braces spacing is slightly modified in split. –  Werner Dec 10 '11 at 15:56
The & would align the braces so the \phantom commands become unnecessary, taking also into account that the two left hand side terms are the same in both arrays. Whether to align these equals signs is just a matter of personal taste. –  egreg Dec 10 '11 at 16:24


text:
Ask Your Question

Revision history [back]

Could you describe what the problem is that you are having?

One thing to check is the path to your mesh: cwd for a node is not where there source file is located, but by default is ~/.ros. So relative paths tend to not work so well.

Just to test, I'd replace it with an absolute path.

For a proper solution, you could use rospkg or catkin_pkg to first get the path to the pkg where the resource is located, then concatenate that with the relative path you have now.

Alternatively, you could set the cwd of your node to the pkg dir that contains the resource. See roslaunch/XML/node - Attributes for that.


text:
### July 2007

I have uploaded a new design to my capitalist shop at CaféPress. It features the slogan: Category theorists go Hom for the holidays. You can check it out at www.cafepress.com/cctops.

I ran across this awesome video on Zero Divides. I think it must be the best math-inspired song I’ve ever heard! The song is written and performed by a group of graduate students at Northwestern University, calling themselves the Klein Four. They run a web site at www.kleinfour.com.

I stumbled upon this self referential aptitude test at The Unapologetic Mathematician. It reminded my of the first problem set I was given in my first year linear algebra course as an undergrad. I think the problems there where from a book by Lewis Carroll, the author of Alice in Wonderland. Anyway, the aptitude test looks like a lot of fun, I can’t wait to sit down and try to figure it out!

I added my favorite math blogs to the blog roll. Some of them updated almost daily while others seem to hibernate for long periods of time. All of them, however, feature great posts on math and related subjects. Check them out!

I have succumbed to capitalism and opened a store at CafePress.com. The direct link is www.cafepress.com/cctops and if you are wondering the “cc” in “cctops” stands for crazy cool, (I know that’s really lame, by the way). So far I’ve only uploaded one design featuring your favorite math mistresses, $\alpha, \beta, \gamma$. I hope to upload some more elaborate designs soon.

I experimented with a new format for the midterm of my single variable calculus class (which is running during the summer session here at Brown). The midterm consisted of 10 problems and at the beginning of the class the students were allowed to work on all the problems but at the end they each had to choose 5 problems to turn in. The remaining problems then turned into a take-home midterm which was due the next day.

The students liked this format but found the midterm itself a bit hard. I’m in the midst of grading it right now and it seems to have come out just fine.

Here is why I like this format:

• Students don’t need to know and worry about every single type of problem that I might put on the midterm.
• It takes most of the time pressure away.
• It is less likely that students collaborate on the take-home part since they might not even have the same problems chosen.

I think that next time I give a midterm I’ll experiment with different ratios, only allowing students to take 3-4 problems to work on at home.


text:
## Scalable algebraic multilevel preconditioners with application to CFD.(English)Zbl 1398.76068

Tromeur-Dervout, Damien (ed.) et al., Parallel computational fluid dynamics 2008. Parallel numerical methods, software development and applications. Proceedings of the 20th international conference, Lyon, France, May 19–22, 2008. Berlin: Springer (ISBN 978-3-642-14437-0/hbk; 978-3-642-26515-0/pbk978-3-642-14438-7/ebook). Lecture Notes in Computational Science and Engineering 74, 15-27 (2010).
Summary: The solution of large and sparse linear systems is one of the main computational kernels in CFD applications and is often a very time-consuming task, thus requiring the use of effective algorithms on high-performance computers. Preconditioned Krylov solvers are the methods of choice for these systems, but the availability of “good” preconditioners is crucial to achieve efficiency and robustness. In this paper we discuss some issues concerning the design and the implementation of scalable algebraic multilevel preconditioners, that have shown to be able to enhance the performance of Krylov solvers in parallel settings. In this context, we outline the main objectives and the related design choices of MLD2P4, a package of multilevel preconditioners based on Schwarz methods and on the smoothed aggregation technique, that has been developed to provide scalable and easy-to-use preconditioners in the Parallel Sparse BLAS computing framework. Results concerning the application of various MLD2P4 preconditioners within a large eddy simulation of a turbulent channel flow are discussed.
For the entire collection see [Zbl 1201.76012].

### MSC:

 76F65 Direct numerical and large eddy simulation of turbulence 65F08 Preconditioners for iterative methods 65Y05 Parallel numerical computation

### Software:

MLD2P4; BLAS; MPI; UMFPACK; ML; SuperLU; BoomerAMG; pARMS; PSBLAS; 2LEV-D2P4
Full Text:

### References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.